Canonical Ramsey Theory on Polish Spacesgasarch/COURSES/858/S13/canrampol.pdf · Canonical Ramsey Theory on Polish Spaces Vladimir Kanovei Institute for the Information Transmission

Canonical Ramsey Theoryon Polish Spaces

Vladimir KanoveiInstitute for the Information Transmission Problems

Moscow Institute for Transport Engineering

Marcin SabokAcademy of Sciences, Poland

Jindrich ZapletalUniversity of Florida

Academy of Sciences, Czech Republic

Marcin Sabok was supported through AIP project MEB051006.Jindrich Zapletal was partially supported through the following grants: AIP

project MEB051006, AIP project MEB060909, Institutional Research Plan No.AV0Z10190503 and grant IAA100190902 of Grant Agency of the Academy ofSciences of the Czech Republic, NSF grant DMS 0801114, and INFTY, the

ESF research networking programme.

vii

Contents

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Outline of results . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Outline of concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Background facts 112.1 Descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Invariant descriptive set theory . . . . . . . . . . . . . . . . . . . 122.3 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Shoenfield absoluteness . . . . . . . . . . . . . . . . . . . . . . . 172.5 Generic ultrapowers . . . . . . . . . . . . . . . . . . . . . . . . . 212.6 Idealized forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.7 Other tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Tools 293.1 Integer games connected with σ-ideals . . . . . . . . . . . . . . . 293.2 The selection property . . . . . . . . . . . . . . . . . . . . . . . . 363.3 A Silver-type dichotomy for a σ-ideal . . . . . . . . . . . . . . . . 413.4 Equivalence relations and models of set theory . . . . . . . . . . 47

3.4a The model V [x]E . . . . . . . . . . . . . . . . . . . . . . . 473.4b The model V [[x]]E . . . . . . . . . . . . . . . . . . . . . . 543.4c Comparison with the symmetric models of ZF . . . . . . . 60

3.5 Associated forcings . . . . . . . . . . . . . . . . . . . . . . . . . . 653.5a The poset PEI . . . . . . . . . . . . . . . . . . . . . . . . . 653.5b The reduced product posets . . . . . . . . . . . . . . . . . 69

4 Classes of equivalence relations 734.1 Smooth equivalence relations . . . . . . . . . . . . . . . . . . . . 734.2 Countable equivalence relations . . . . . . . . . . . . . . . . . . . 744.3 Equivalences classifiable by countable structures . . . . . . . . . 794.4 Orbit equivalence relations . . . . . . . . . . . . . . . . . . . . . . 854.5 Hypersmooth equivalence relations . . . . . . . . . . . . . . . . . 86

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vi CONTENTS

5 Classes of σ-ideals 895.1 σ-ideals σ-generated by closed sets . . . . . . . . . . . . . . . . . 89

5.1a Calibrated ideals . . . . . . . . . . . . . . . . . . . . . . . 915.1b The σ-compact ideal and generalizations . . . . . . . . . . 965.1c The meager ideal . . . . . . . . . . . . . . . . . . . . . . . 100

5.2 Porosity ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.3 Halpern-Lauchli forcing . . . . . . . . . . . . . . . . . . . . . . . 1135.4 The E0-ideal and its products . . . . . . . . . . . . . . . . . . . . 1155.5 The E1 ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.5a The σ-ideals and their dichotomies . . . . . . . . . . . . . 1245.5b Canonization results . . . . . . . . . . . . . . . . . . . . . 1295.5c Borel functions on pairs . . . . . . . . . . . . . . . . . . . 134

5.6 The E2 ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.7 Products of perfect sets . . . . . . . . . . . . . . . . . . . . . . . 1415.8 Silver cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.9 Ramsey cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545.10 Weak Milliken cubes . . . . . . . . . . . . . . . . . . . . . . . . . 1605.11 Strong Milliken cubes . . . . . . . . . . . . . . . . . . . . . . . . 1695.12 Products of superperfect sets . . . . . . . . . . . . . . . . . . . . 1725.13 Laver forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1765.14 Fat tree forcings . . . . . . . . . . . . . . . . . . . . . . . . . . . 1795.15 The Lebesgue null ideal . . . . . . . . . . . . . . . . . . . . . . . 1825.16 Finite product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1865.17 The countable support iteration ideals . . . . . . . . . . . . . . . 189

5.17a Iteration preliminaries . . . . . . . . . . . . . . . . . . . . 1905.17b Weak Sacks property . . . . . . . . . . . . . . . . . . . . . 1915.17c The preservation result . . . . . . . . . . . . . . . . . . . 1945.17d A few special cases . . . . . . . . . . . . . . . . . . . . . . 199

Chapter 1

Introduction

1.1 Motivation

The Ramsey theory starts with a classical result:

Fact 1.1.1. For every partition of pairs of natural numbers into two classesthere is a homogeneous infinite set: a set a ⊂ ω such that all pairs of naturalnumbers from a belong to the same class.

It is not difficult to generalize this result for partitions into any finite numberof classes. An attempt to generalize further, for partitions into infinitely manyclasses, hits an obvious snag: every pair of natural numbers could fall into itsown class, and then certainly no infinite homogeneous set can exist for such apartition. Still, there seems to be a certain measure of regularity in partitionsof pairs even into infinitely many classes. This is the beginning of canonicalRamsey theory.

Fact 1.1.2 ([11]). For every equivalence relation E on pairs of natural numbersthere is an infinite homogeneous set: a set a ⊂ ω on which one of the followinghappens:

1. pEq ↔ p = q for all pairs p, q ∈ [a]2;

2. pEq ↔ min(p) = min(q) for all pairs p, q ∈ [a]2;

3. pEq ↔ max(p) = max(q) for all pairs p, q ∈ [a]2;

4. pEq for all pairs p, q ∈ [a]2.

In other words, there are four equivalence relations on pairs of natural numberssuch that any other equivalence can be canonized : made equal to one of thefour equivalences on the set [a]2, where a ⊂ ω is judiciously chosen infinite set.It is not difficult to see that the list of the four primal equivalence relations isirredundant: it cannot be shortened for the purposes of this theorem. It is also

1

2 CHAPTER 1. INTRODUCTION

not difficult to see that the usual Ramsey theorems follow from the canonicalversion.

Further generalizations of these results can be sought in several directions.An exceptionally fruitful direction considers partitions and equivalences of sub-structures of a given finite or countable structure, such as in [38]. Anotherdirection seeks to find homogeneous sets of larger cardinalities. In set theorywith the axiom of choice, the search for uncountable homogeneous sets of arbi-trary partitions leads to large cardinal axioms [25], and this is one of the centralconcerns of modern set theory. A different approach will seek homogeneoussets for partitions that have a certain measure of regularity, typically expressedin terms of their descriptive set theoretic complexity in the context of Polishspaces. This is the path this book takes. Consider the following classical result:

Fact 1.1.3 ([51]). For every partition [ω]ℵ0 = B0 ∪ B1 into two Borel pieces,one of the pieces contains a set of the form [a]ℵ0 , where a ⊂ ω is some infiniteset.

Here, the space [ω]ℵ0 of all infinite subsets of natural numbers is consideredwith the usual Polish topology which makes it homeomorphic to the space ofirrational numbers. This is the most influential example of a Ramsey theoremon a Polish space. It deals with Borel partitions only as the Axiom of Choice canbe easily used to construct a partition with no homogeneous set of the requestedkind.

Are there any canonical Ramsey theorems on Polish spaces concerning setson which Borel equivalence relations can be canonized? A classical example ofsuch a theorem starts with an identification of Borel equivalence relations Eγon the space [ω]ℵ0 for every function γ : [ω]<ℵ0 → 2 (the exact statement anddefinitions are stated in Section 5.9) and then proves

Fact 1.1.4 ([41, 36]). If f : [ω]ℵ0 → 2ω is a Borel function then there is γ andan infinite set a ⊂ ω such that for all infinite sets b, c ⊂ a, f(b) = f(c)↔ b Eγ c.

Thus, this theorem deals with smooth equivalence relations on the space [ω]ℵ0 ,i.e. those equivalences E for which there is a Borel function f : [ω]ℵ0 → 2ω suchthat b E c ↔ f(b) = f(c), and shows that such equivalence relations can becanonized to a prescribed form on a Ramsey cube. Other similar results can befound in the work of Otmar Spinas [57, 56, 34].

1.2 Outline of results

This book extends this line of research in two directions. Firstly, we considercanonization properties of equivalence relations more complicated than smoothin the sense of Borel reducibility complexity rating of equivalence relations [27,14]. It turns out that distinct complexity classes of equivalence relations possessvarious canonization properties, and most of the interest and difficulty lies in thenon-smooth cases. Secondly, we consider the task of canonizing the equivalence

1.2. OUTLINE OF RESULTS 3

relations on Borel sets which are large from the point of view of various σ-ideals. Again, it turns out that σ-ideals commonly used in mathematical analysisgreatly differ in their canonization properties, and a close relationship with theirforcing properties, as described in [67], appears.

Given a σ-ideal I on a Polish space X, the strongest canonization resultone can obtain is the total canonization for analytic equivalence relations: forevery I-positive Borel set B ⊂ X and every analytic equivalence relation E onB, there is an I-positive Borel set C ⊂ B consisting either only of pairwiseinequivalent elements, or only of pairwise equivalent elements.

Theorem 1.2.1. The following σ-ideals have the total canonization for analyticequivalence relations:

1. (Theorem 5.1.6) all Π11 on Σ1

1 σ-ideals on compact metrizable spaces whichare σ-generated by compact sets and are calibrated;

2. (Corollary 5.2.14) all metric porosity σ-ideals on compact metric spaces;

3. (Solecki-Spinas [55], Theorem 5.1.15) the σ-ideal on ωω σ-generated bycompact sets.

The class considered in the first item contains many σ-ideals commonly studiedin abstract analysis; the σ-ideal σ-generated by closed sets of measure zero,the σ-ideal σ-generated by closed sets of uniqueness, or the σ-ideal of sets ofextended uniqueness are just three of an army of examples.

In all cases named above, we get an even stronger result modeled afterthe classical Silver dichotomy for coanalytic equivalence relations: if B is anI-positive Borel set and E is a Borel equivalence relation on B, either B decom-poses into countably many equivalence classes and an I-small set, or it containsa Borel I-positive subset consisting of pairwise inequivalent elements.

In most cases, such a strong canonization result is either not available or wedo not know how to prove it. One can then attempt to canonize only equivalencerelations of limited Borel reducibility complexity:

Theorem 1.2.2. The following σ-ideals have the total canonization for equiv-alence relations classifiable by countable structures:

1. (Corollary 4.3.10) the σ-ideal σ-generated by sets of continuity of a Borelfunction;

2. (Corollary ??) Laver σ-ideal on ωω;

3. (Corollary 4.3.10) the σ-ideal σ-generated by sets of finite Hausdorff mea-sure of a fixed dimension, on a compact metric space;

4. (Corollary 5.6.4) the σ-ideal of Borel sets on which the equivalence relationE2 is classifiable by countable structures.

Theorem 1.2.3. The following σ-ideals have the total canonization for equiv-alence relations Borel reducible to equality modulo an analytic P -ideal on ω:


1. the σ-ideal σ-generated by sets of continuity of a Borel function;

2. (Michal Doucha) the σ-ideal of nondominating sets on ωω.

In other cases still, the total canonization may fail because there are clearlyidentifiable obstacles to it. There, we strive to canonize up to the known obsta-cles:

Theorem 1.2.4 (Theorem 5.4.2). Let I be the σ-ideal on 2ω generated by Borelsets on which the equivalence relation E0 is smooth. If B is a Borel I-positiveset and E is an analytic equivalence relation, then there is a Borel I-positiveset C ⊂ B such that either C consists of pairwise inequivalent elements, or Cconsists of pairwise equivalent elements, or E � C = E0.

Theorem 1.2.5 (Corollary 5.5.11). Let I be the σ-ideal on (2ω)ω generated byBorel sets on which the equivalence relation E1 is Borel reducible to E0. If B is aBorel I-positive set and E is an a hypersmooth equivalence relation, then there isa Borel I-positive set C ⊂ B such that either C consists of pairwise inequivalentelements, or C consists of pairwise equivalent elements, or E � C = E1.

Theorem 1.2.6 (Theorem 5.16.2). Let n ∈ ω, let {Ii : i ∈ n} be σ-ideals onrespective compact spaces {Xi : i ∈ n}, each Π1

1 on Σ11, σ-generated by compact

sets, and with covering property. If {Bi : i ∈ n} are Borel Ii-positive sets andE is an analytic equivalence relation on

∏iBi, there is a set a ⊂ n and Borel

I-positive sets {Ci ⊂ Bi : i ∈ n} such that E �∏i Ci =equality on indices in

the set a.

In a similar vein, we achieve a number of ergodicity results. There, an obvi-ous obstacle to canonization is identified and all equivalence relations containingthis obstacle and simpler than it are shown to have a large equivalence class.

Theorem 1.2.7 (Hjorth, Kechris ??? and Theorem 5.15.3). Let E be a Borelequivalence relation on 2ω; the space 2ω is equipped with the usual topologyand a Borel probability measure. If E2 ⊆ E and E is classifiable by countablestructures then E has a comeager class as well as a conull class.

Theorem 1.2.8 (Theorems 5.1.27 and 5.15.9). Let E be a Borel equivalencerelation on (2ω)ω; the space (2ω)ω is equipped with the usual product topologyand the product Borel probability measure. If F2 ⊆ E and F2 is not Borelreducible to E, then E has a comeager class as well as a conull class.

The weakest canonization results we obtain reduce the Borel reducibilitycomplexity of equivalence relations to a specified class. It should be notedthat no parallel to results of this kind exists in the realm of finite or countablecanonization results.

Theorem 1.2.9 (Theorems 5.7.5 and 5.7.7). Let E be an equivalence relation on(2ω)ω which is classifiable by countable structures or Borel reducible to equalitymodulo an analytic P-ideal on ω. Then there are nonempty perfect sets 〈Pn :n ∈ ω〉 such that E �

∏n Pn is smooth.

1.3. OUTLINE OF CONCEPTS 5

Theorem 1.2.10 (Mathias, Theorem 5.9.5). Let E be an essentially countableBorel equivalence relation on [ω]ℵ0 . Then there is an infinite set a ⊂ ω suchthat E � [a]ℵ0 is Borel reducible to E0.

There is a number of anticanonization results. Again, the classification ofanalytic equivalence relations via their Borel reducibility complexity allows usto prove theorems that have no immediate counterpart in the finite or countablerealm:

Theorem 1.2.11. There is a Borel equivalence relation on ωω bireducible withEKσ such that for every Borel dominating set B ⊂ ωω, E � B is still bireduciblewith EKσ .

Many results in the book either connect canonization properties of σ-idealswith the forcing properties of the associated quotient forcing PI of Borel I-positive sets ordered by inclusion as studied in [67], or use that connection intheir proofs. This allows us to tap a wealth of information amassed about suchforcings by people otherwise not interested in descriptive set theory. A samplerof results of this kind:

Theorem 1.2.12. Let I be a σ-ideal on a Polish space X such that the quotientposet PI is proper.

1. (Proposition 4.1.2) PI adds a minimal real degree if and only if I has totalcanonization for smooth equivalence relations;

2. (Corollary 4.2.6) if PI is nowhere c.c.c. and adds a minimal forcing ex-tension then it has total canonization for equivalence relations classifiableby countable structures;

3. (Theorem 4.3.8) if PI adds only finitely many real degrees then every equiv-alence relation classifiable by countable structures simplifies to an essen-tially countable equivalence relation on a Borel I-positive set.

1.3 Outline of concepts

The central canonization notion of the book is the following.

Definition 1.3.1. A σ-ideal I on a Polish space X has total canonization fora class of equivalence relations if for every Borel I-positive set B ⊂ X andevery equivalence relation E on B in this class there is a Borel I-positive setC ⊂ B consisting only of pairwise E-inequivalent elements or only of pairwiseE-equivalent elements.

The classes of equivalence relations considered in this book are nearly alwaysclosed under Borel reducibility; the broadest class would be that of all analyticequivalence relations. The total canonization is closely related to an ostensiblystronger notion:


Definition 1.3.2. A σ-ideal I on a Polish space X has the Silver property fora class of equivalence relations if for every Borel I-positive set B ⊂ X and everyequivalence relation E on B in this class, either there is a Borel I-positive setC ⊂ B consisting of pairwise E-inequivalent elements, or B decomposes into aunion of countably many E-equivalence classes and an I-small set.

Thus, the usual Silver theorem can be restated as the Silver property for coan-alytic equivalence relations for the σ-ideal of countable sets. Unlike the totalcanonization, the Silver property introduces a true dichotomy, as the two op-tions presented cannot coexist for σ-ideals containing all singletons: the BorelI-positive set C from the first option would have to have a positive intersectionwith one of the countably many equivalence classes from the second option,which is of course impossible. The Silver dichotomy also has consequences forundefinable sets. If there is any I-positive set C ⊂ B consisting of pairwiseE-inequivalent elements, then there must be a Borel such set, simply becausethe second option of the dichotomy is excluded by the same argument as above.

The Silver property certainly implies total canonization within the sameclass of equivalence relations: either there is a Borel I-positive set B ⊂ Cconsisting of pairwise inequivalent elements, or one of the equivalence classesform the second option of the dichotomy must be I-positive and provides thesecond option of the total canonization. On the other hand, total canonizationdoes not imply the Silver property as shown in Section 5.3. In many commoncases though, various uniformization or game theoretic properties of the σ-idealI in question can be used to crank up the total canonization to Silver property.This procedure is described in Section 3.3 and it is the only way used to arguefor the Silver property in this book.

There are a number of tricks used to obtain total canonization for variousrestricted classes of equivalence relations used in this book. However, the totalcanonization for all analytic equivalence relations is invariably proved via thefollowing notion originating in Ramsey theory:

Definition 1.3.3. A σ-ideal I on a Polish space X has the free set property iffor every I-positive analytic set B ⊂ X and every analytic set D ⊂ B ×B withall vertical sections in the ideal I there is a Borel I-positive set C ⊂ B suchthat (C × C) ∩D ⊂ id.

The free set property immediately implies the total canonization for analyticequivalence relations. If E is an analytic equivalence relation on an I-positiveBorel set B ⊂ X, then either there is an I-positive E-equivalence class, whichimmediately yields the second option of total canonization. Or, the relationE ⊂ B×B has I-small vertical sections, and the E-free I-positive set postulatedby the free set property yields the first option of total canonization. It is not atall clear how one would argue for the opposite implication though, and we arecoming to the first open question of this book.

Question 1.3.4. Is there a σ-ideal on a Polish space that has total canonizationfor analytic equivalence relations, but not the free set property?

1.3. OUTLINE OF CONCEPTS 7

The free set property is typically verified through fusion arguments as inTheorem 5.1.6, or through some version of the mutual generics property.

Definition 1.3.5. A σ-ideal I on a Polish space X has the mutual genericsproperty if for every Borel I-positive set B and every countable elementarysubmodel M of a large enough structure containing I and B there is a BorelI-positive subset C ⊂ B such that its points are pairwise generic for the productforcing PI × PI .

This is the first place in this book where forcing makes explicit appearance. Themutual generics property is somewhat ad hoc in that in many cases, the argu-ments demand that the product forcing is replaced by various reduced productforcings, see Section 5.3. The resulting tools are very powerful and flexible, andwhere we cannot find them, we spend some effort proving that no version ofmutual generics property can hold. The concept used to rule out all its versionsis of independent interest:

Definition 1.3.6. A σ-ideal I on a Polish space X has a coding function ifthere are a Borel I-positive set B ⊂ X and a Borel function f : B × B → 2ω

such that for every Borel I-positive set C ⊂ B, f ′′(C × C) = 2ω.

Clearly, if a σ-ideal has a coding function then no I-positive Borel set below thecritical set B (which is always equal to the whole space in this book) can consistof points pairwise generic over a given countable model M , since some pairs inthis set code via the function f for example a transitive structure isomorphicto M . We will show that many σ-ideals have a coding function (Theorem 5.1.9or 5.1.19) and that coding functions can be abstractly obtained from the failureof canonization of equivalence relations in many cases (Corollaries 5.5.18, 5.6.12,5.17.21).

In the cases where total canonization is unavailable, we can still providea number of good canonization results. In the spirit of traditional Ramsey-theoretic notation, we introduce an arrow to record them:

Definition 1.3.7. Let E, F be two classes of equivalence relations, and let Ibe a σ-ideal on a Polish space X. E→I F denotes the statement that for everyI-positive Borel set B ⊂ X and an equivalence E ∈ E on B there is a BorelI-positive set C ⊂ B such that E � C ∈ F.

The strongest anti-canonization results we can achieve will be cast in termsof a spectrum of a σ-ideal. Note that it has no obvious counterpart in thecanonical Ramsey theory on finite structures.

Definition 1.3.8. An analytic equivalence relation E is in the spectrum of aσ-ideal I on a Polish space X if there is a Borel set B ⊂ X and an equivalencerelation F on B which is Borel bireducible with E, and also for every I-positiveBorel set C ⊂ B, F � C remains bireducible with E.

For example, the equivalence relations such as E0 and F2 are in the spectrumof the meager ideal, and EKσ is in the spectrum of the ideals associated with


the Silver forcing and Laver forcing. From the point of view of Ramsey theory,finding a nontrivial equivalence relation in the spectrum amounts to a strongnegative result. However, the results of this sort may be quite precious inthemselves and have further applications.

1.4 Navigation

Chapter 2 contains background material which has on its face nothing to dowith canonization of equivalence relations. Most results are standard and statedwithout proofs, even though there are exceptions such as the proof of canon-ical interpretation of Π1

1 on Σ11 classes of analytic sets in generic extensions

(Theorem 2.4.7). Section 2.6 describes the basic treatment of quotient posetsas explored in [67]. For a σ-ideal I on a Polish space X we write PI for theposet of Borel I-positive subsets of X ordered by inclusion, and outline its basicforcing properties.

Chapter 3 introduces a number of general concepts useful in the treatmentof the particular canonization theorems later. Section 3.4 explains the main toolunderlying much of the current of thought contained in this book: the corre-spondence between analytic equivalence relations and models of set theory. Itenables proofs of canonization theorems essentially by a discussion of the pos-sibilities for intermediate forcing extensions of the PI models; such possibilitieshave been in many cases explored independently by mathematicians interestedpurely in forcing. Section 3.1 axiomatizes the treatment of integer games as-sociated with certain σ-ideals on Polish spaces. Many such games have beenpresented in [67]; however, the treatment here points out the common featuresas well as the connection with uniformization theorems for Borel sets. The in-teger games are ultimately used for two purposes in this book: first, to limitthe possibilities for intermediate forcing extensions as in Theorem 3.2.3, andsecond, in the proofs that improve the total canonization of Borel equivalencerelations to the much more informative Silver property. This latter connectionis explained in Section 3.3.

Chapter 4 discusses several classes of equivalence relations and their impacton the structure of forcing extensions. Thus, the smooth equivalence relations(Section 4.1) correspond to models of the form V [x] where x is a real. Thecountable Borel equivalence relations (Section 4.2) are associated with a veryspecific intermediate extension containing no new reals. The equivalence rela-tions classifiable with countable structures (Section 4.3) correspond to a (oftenchoiceless) model of the form V (a) where a is a hereditarily countable sets.The hypersmooth equivalence relations (Section 4.5) are mirrored in infinitedescending sequences of real degrees. The precise evaluation of these relation-ships helps us to obtain several remarkable general theorems; for example, if thequotient poset PI is proper and adds a minimal forcing extension then the σ-ideal I has total canonization for equivalence relations classifiable by countablestructures–Theorem 4.3.8(3).

Chapter 5 contains the main contributions of this book. It goes from one

1.5. NOTATION 9

class of σ-ideals to another, with the intention of proving the strongest canon-ization theorem possible. The task at hand is enormous and a great number ofnatural σ-ideals remains untouched.

1.5 Notation

We employ the set theoretic standard notation as used in [22]. If E is anequivalence relation on a set X and x ∈ X then [x]E denotes the equivalenceclass of x. If B ⊂ X is a set then [B]E denotes the saturation of the set B, theset {x ∈ X : ∃y ∈ B x E y}. If E,F are equivalence relations on respectivePolish spaces X,Y , we write E ≤B F if E is Borel reducible to F , in other wordsif there is a Borel function f : X → Y such that x0 F x1 ↔ f(x0) E f(x1). Iff : X → Y is a Borel function between two Polish spaces and F is an equivalencerelation on the space Y , then f−1F is the pullback of F , the equivalence relationE on the space X defined by x0 E x1 ↔ f(x0) F f(x1); so E ≤B F . If T isa tree then [T ] denotes the set of all of its infinite branches. id is the identityequivalence relation on any underlying set, ee is the equivalence relation makingevery two points of the underlying set equivalent. If x ∈ 2ω and m ∈ ω thenx flip m is the element of 2ω that differs from x exactly and only at its m-thentry. If h ∈ 2<ω is a finite binary sequence then x rew h is the sequencein 2ω obtained from x by rewriting its initial segment of corresponding lengthwith h. If σ is a winning strategy for one of the players in an integer game andy ∈ ωω is a sequence of moves of the other player, the symbol σ ∗ y denotes thesequence that the strategy σ produces in response to y. If P is a forcing, τ is aP -name, and G ⊂ P is a generic filter, then τ/G is the valuation of the nameτ according to the generic filter G. If P has an obvious “canonical” genericobject x (typically, an element of a Polish space), we will abuse this conventionto write also τ/x for the valuation. If M is a model of set theory and x is a set,we say that x is M -generic if x belongs to a (set) generic extension of M . If κis a cardinal then Hκ denotes the collection of sets whose transitive closure hascardinality < κ. Several theorems in the book use a rather weak large cardinalassumption stated as “ω1 is inaccessible to the reals”, which is to say that forevery x ∈ 2ω, the ordinal ω1 as computed in the model L[x], the class of setsconstructible from x, is countable.

Theorem is a self-standing statement ready for applications outside of thisbook. Claim is an intermediate result within a proof of a theorem. Fact is aresult that has been obtained elsewhere and is not going to be proved in thisbook; this is not in any way to intimate that it is an unimportant or easy orperipheral result.


Chapter 2

Background facts

2.1 Descriptive set theory

Familiarity with basic concepts of descriptive set theory is assumed throughout.[33] serves as a standard reference.

Definition 2.1.1. A collection I of subsets of a Polish space X is Π11 on Σ1

1 iffor every analytic set A ⊂ 2ω ×X the set {y ∈ 2ω : {x ∈ X : 〈y, x〉 ∈ A} ∈ I} iscoanalytic.

Fact 2.1.2. [33, Theorem 28.10] If I is Π11 on Σ1

1 then every analytic set in Ihas a Borel superset in I.

Fact 2.1.3. (Luzin–Novikov, [33, Theorem 18.10]) If A ⊂ X × Y is a Borelset with countable vertical sections, then A is the countable union of graphs ofBorel functions from X to Y .

Fact 2.1.4. [33, Theorem 18.6] Let X,Y be Polish spaces. If A ⊂ X × Y isa Borel set then the set B = {x : Ax is not meager} is Borel, and the setA ∩ (B × Y ) has Borel uniformization.

Fact 2.1.5. (Arsenin–Kunugui, [33, Theorem 18.18]) Let X,Y be Polish spaces.If A ⊂ X × Y is a Borel set with Kσ vertical sections, then A has a Boreluniformization.

Fact 2.1.6. (Jankov–von Neumann, [33, Theorem 18.1]) Let X,Y be Polishspaces. If A ⊂ X × Y is an analytic set then it has a σ(Σ1

1)-measurable uni-formization. In particular, the uniformization is Baire measurable.

Fact 2.1.7. (Luzin–Suslin, [33, Theorem 15.1]) A Borel one-to-one image of aBorel set is Borel.

In several instances, we will need the tools of effective descriptive set theoryas described in [27, Chapter 2].

11

12 CHAPTER 2. BACKGROUND FACTS

Fact 2.1.8. 1. The set {(x, y) ∈ 2ω × 2ω : y ∈ ∆11(x)} is Π1

1;

2. every Σ11(x) subset of 2ω with a non-∆1

1(x)-element is uncountable;

3. (effective boundedness) If A is a Σ11(x) collection of wellfounded trees on

ω then there is a wellordering of ω recursive in x whose rank is larger thanranks of all trees in A.

Let X be a recursively presented Polish space. The Gandy-Harrington forc-ing is the poset of nonempty Σ1

1 subsets of X, ordered by inclusion. The posethas obvious relativizations for every real parameter.

Fact 2.1.9. If G is a Gandy-Harrington generic filter, then⋂G = {xgen} for

some point xgen ∈ X, and G = {A ∈ Σ11 : xgen ∈ A}.

As an interesting application of the Gandy-Harrington forcing, we will provethe following simple

Proposition 2.1.10. Let A ⊂ (2ω)ω be a Σ11 set containing a sequence x such

that x(n) /∈ ∆11 for every number n ∈ ω. Then for every countable set a ⊂ 2ω

there is a sequence x′ ∈ A such that rng(x′) ∩ a = 0.

Proof. Let A′ = {x ∈ A : ∀n x(n) /∈ ∆11}; this is a nonempty Σ1

1 set. Let M bea countable elementary submodel of a large enough structure containing the seta, and let G be a Gandy-Harrington-generic filter over M containing A′, withthe attendant generic point x ∈ A′. We claim that rng(x) ∩ a = 0.

And indeed, for every point y ∈ 2ω and every number n ∈ ω, the set D ={B ⊂ A′ : for some basic open neighborhood O ⊂ 2ω of y, for every z ∈ B,z(n) /∈ Oy} is open dense in the Gandy-Harrington forcing below A′. To see this,let B′ ⊂ A′ be a nonempty Σ1

1 set. Consider its projection into n-th coordinate;this is a nonempty Σ1

1 subset of 2ω containing only non-∆11-elements; therefore,

it is uncountable and it must contain some element distinct from y, separatedfrom y by some basic open neighborhood O. Then B = {x ∈ B′ : x(n) /∈ O}is the desired nonempty Σ1

1 set in D below B′. The rest follows from standarddensity arguments.

2.2 Invariant descriptive set theory

If E,F are Borel or analytic equivalence relations on Polish spaces X,Y thenE ≤B F (E is Borel reducible to F ) denotes the fact that there is a Borelfunction f : X → Y such that x0 E x1 ↔ f(x0) F f(x1). Bireducibility is anequivalence relation on the class of all Borel equivalence relations. ≤B turnsinto a complicated ordering on the bireducibility classes. There are severalequivalences occupying an important position in this ordering:

Definition 2.2.1. id is the identity equivalence relation; ee, or the full equiva-lence relation, is the equivalence relation that makes all elements of its domain

2.2. INVARIANT DESCRIPTIVE SET THEORY 13

equivalent. E0 is the equivalence on 2ω defined by x E0 y ↔ the symmetricdifference x∆y is finite. E1 is the equivalence relation on (2ω)ω defined by~x E1 ~y ↔ {n ∈ ω : ~x(n) 6= ~y(n)} is finite. E2 is the equivalence relation on 2ω

defined by x E2 y ↔ Σ{1/(n + 1) : x(n) 6= y(n)} < ∞. F2 is the equivalencerelation on (2ω)ω defined by ~x F2 ~y ↔ rng(~x) = rng(~y). EKσ is the equivalenceon Πn(n + 1) defined by x EKσ y ↔ ∃m∀n |x(n) − y(n)| < m. Ec0 is theequivalence on [0, 1]ω defined by x Ec0 y ↔ lim(x(n)− y(n)) = 0.

Definition 2.2.2. An equivalence E on a Polish space X is classifiable bycountable structures if there is a countable relational language L such that E isBorel reducible to the equivalence relation of isomorphism of models of L withuniverse ω.

Definition 2.2.3. An equivalence is smooth if it is Borel reducible to the iden-tity. An equivalence is hypersmooth if it is an increasing union of a countablesequence of smooth equivalence relations.

Definition 2.2.4. An equivalence E on a Polish space X is essentially countableif it is Borel reducible to a countable Borel equivalence relation: one all of whoseclasses are countable.

The countable Borel equivalences are governed by the Feldman–Moore the-orem:

Fact 2.2.5. ([13] [14, Theorem 7.1.4]) For every countable Borel equivalencerelation E on a Polish space X there is a countable group G and a Borel actionof G on X whose orbit equivalence relation is equal to E.

Fact 2.2.6. (Silver dichotomy, [52], [27, Section 10.1]) For every coanalyticequivalence relation E on a Polish space X, either X is covered by countablymany equivalence classes or there is a perfect set of mutually E-inequivalentpoints.

Corollary 2.2.7. For every coanalytic equivalence relation E on a Polish spaceX and every analytic set A ⊂ X, either A intersects only countably manyE-equivalence classes, or it contains a perfect set of pairwise E-inequivalentelements.

Proof. Let f : ωω → X be a continuous function such that rng(f) = A, and letF = f−1E; thus, F is a coanalytic equivalence relation on ωω and the Silverdichotomy can be applied to it. The corollary follows immediately.

Fact 2.2.8. (Glimm-Effros dichotomy, [18], [27, Section 10.4]) For every Borelequivalence relation E on a Polish space, either E ≤B id or E0 ≤B E.

Fact 2.2.9. [19] If E ≤B E2 is a Borel equivalence relation then either E isessentially countable, or E2 ≤B E.

Fact 2.2.10. [29], [27, Section 11.3] For every Borel equivalence relation E ≤ E1

on a Polish space, either E ≤B E0 or E1 ≤B E.


Fact 2.2.11. [44], [27, Section 6.6] Every Kσ equivalence relation on a Polishspace is Borel reducible to EKσ .

Fact 2.2.12. [44], [27, Chapter 18] For every Borel equivalence relation E thereis a Borel ideal I on ω such that E ≤B=I .

Here, =I is the equivalence on 2ω defined by x =I y ↔ {n ∈ ω : x(n) 6= y(n)} ∈I. It is not difficult to construct an Fσ-ideal I on a countable set such that =I

is bireducible with EKσ , and such an ideal will be useful in several arguments inthe book. Just let dom(I) be the countable set of all pairs 〈n,m〉 such that n ∈ ωand m ≤ n and let a ∈ I if there is a number k ∈ ω such that for every numbern ∈ ω, there are at most k many numbers m with 〈n,m〉 ∈ a. This ideal isclearly Fσ and so =I≤B EKσ by the above fact. On the other hand, EKσ ≤=I ,as the reduction f : Πn(n+1)→ dom(I) defined by f(x) = {〈n,m〉 : m ≤ x(n)}shows.

2.3 Forcing

Familiarity with basic forcing concepts is assumed throughout the book. If〈P,≤〉 is a partial ordering and Q ⊂ P then Q is said to be regular in P if everymaximal antichain of Q is also a maximal antichain in P ; restated, for everyelement p ∈ P there is a pseudoprojection of p into Q, a condition q ∈ Q suchthat every strengthening of q in Q is still compatible with p.

Regarding extensions of models of set theory, we will use the standardnomenclature of [22, Theorem 13.22 ff]. If V is a transitive model of ZF settheory containing all ordinals and a (perhaps a /∈ V ) is a set, then V (a) is thesmallest transitive model of ZF containing V as a subset and a as an element,and V [a] is the smallest model W of ZF containing V as a subset and such thata ∩W ∈ W . The two models are formed by transfinite induction in parallel tothe L(a) and L[a] constructions, except at successor stages of the induction, allsubsets of the constructed part which belong to V are added. If a is in addi-tion a generic filter over V , the models V [a] and V (a) coincide and there is analternative description of them as a forcing extension: V (a) = V [a] = {τ/a : τis a name in V }. The basic development of forcing up to the forcing theoremand product forcing theorem works in the context of ZF set theory withoutchoice, and there are several opportunities for considering forcing extensions ofchoiceless models.

Whenever κ is a cardinal, Coll(ω, κ) is the poset of finite partial functionsfrom ω to κ ordered by reverse extensions. If κ is an inaccessible cardinal,then Coll(ω,< κ) is the finite support product of the posets Coll(ω, λ) for allcardinals λ ∈ κ. The following fact sums up the homogeneity properties of thesepartial orders.

Fact 2.3.1. [22, Corollary 26.10] Let κ be a cardinal, P a poset of size < κ,and G ⊂ Coll(ω, κ) be a generic filter. In V [G], for every V -generic filterH ⊂ P there is a V [H]-generic filter K ⊂ Coll(ω, κ) such that V [G] = V [H][K].

2.3. FORCING 15

Identical statement holds true if κ is an inaccessible cardinal and Coll(ω, κ) isreplaced by the poset Coll(ω,< κ).

In several places, we need to invoke a forcing factorization theorem, and thefollowing is the exact statement that we will use:

Fact 2.3.2. (Forcing factorization theorem) If P,Q are posets and τ is a P -name for a Q-generic filter, then there is a condition q ∈ Q such that in alarge collapse extension, for every filter H ⊂ Q generic over V containing thecondition q there is a filter G ⊂ P generic over V such that H = τ/G.

Proof. Let κ be a cardinal larger than |P(P )|. Consider the statement φ of theQ-forcing language saying “Coll(ω, κ) there is a V -generic filter G ⊂ P suchthat H = τ /G”; here, H is the Q-name for the generic filter on Q. Note thatthe above Fact 2.3.1 implies that the largest condition in Coll(ω, κ) decides thestatement on the right of the forcing relation, as all its parameters are in themodel V [H].

First argue that there must be a condition q ∈ Q that forces φ. If this failedthen Q ¬φ. Let G ⊂ P be a filter generic over V , and let K ⊂ Coll(ω, κ) bea filter generic over V [G]. Let H = τ/G. Now, Fact 2.3.1 shows that V [G][K]is a Coll(ω, κ)-extension of the model V [H]. By the forcing theorem and thefailure of φ, there should be no V -generic filter on P in the model V [G][K] thatgives H. However, there is such a filter there, namely G; a contradiction.

Thus, fix a condition q ∈ Q forcing φ; we claim that q works as required. LetK ⊂ Coll(ω, κ) be a filter generic over V , and let H ⊂ Q be a V -generic filter inV [K], containing the condition q. By Fact 2.3.2, V [K] is a Coll(ω, κ) extensionof V [H], and the forcing theorem applied to φ and H shows that there is a filterG ⊂ P generic over V in the model V [K] such that H = τ/G.

The following definition sums up the common forcing properties used in thisbook:

Definition 2.3.3. A forcing P is proper if for every condition p ∈ P andevery countable elementary submodel M of large enough structure containingP, p there is a master condition q ≤ p which forces the generic filter to meetall dense subsets in M in conditions in M . The forcing is bounding if everyfunction in ωω in its extension there is a ground model function with largervalue on every entry. The forcing preserves outer Lebesgue measure if everyground model set of reals has the same outer measure whether evaluated in theground model or in the extension. The forcing preserves Baire category if everyground model set of reals is meager in the ground model if and only if it ismeager in the extension. The forcing does not add independent reals if everysequence in 2ω in the extension has an infinite ground model subsequence.

The phrase “countable elementary submodel of a large enough structure” ap-pears in essentially every paper dealing with proper forcing. The large enoughstructure should be interpreted as the structure with universe Hκ, the collection


of sets of hereditary cardinality < κ for some cardinal κ such that Hκ containsevery object named in the proof before the phrase occurs and its powerset. Thestructure has a binary relation ∈ and a designated unary predicate for everyobject named in the proof before the phrase occurs.

If M is a model of some portion of set theory and P ∈ M is a poset, wewill say that a filter G ⊂ P is P -generic over M if for every dense set D ⊂ Pin M there is a condition p ∈ D ∩M ∩ G. If the poset P adds a “canonical”object nominally different from a generic filter (such as some point xgen in aPolish space from which the generic filter is recoverable), we will extend thisusage to say that x is P -generic over M if the filter on P recovered from x is.In contradistinction, the phrase “a is M -generic” will mean that a belongs tosome generic extension of the transitive isomorph of M ; it will be used only ifM is wellfounded.

We will need several facts about the interplay between elementary submodelsand generic extensions. If M is an elementary submodel of a large structure,P ∈ M is a poset and G ⊂ P is a filter generic over V , let M [G] = {τ/G : τ ∈M}.

Proposition 2.3.4. Let P be a partial order, let κ > |P |, let M ≺ Hκ, and letG ⊂ P be a generic filter over V .

1. 〈M [G],∈〉 ≺ 〈Hκ[G],∈〉 = 〈HV [G]κ ,∈〉;

2. if G is moreover generic over M then 〈M [G],∈,M〉 ≺ 〈Hκ[G] ∈〉;

3. if, in addition, Q ∈M is another poset and K ∈M [G] is a filter Q-genericover both M and V , then M [K] = M [G] ∩Hκ[K].

Proof. The first item is proved by a straighforward application of the Tarskicriterion. Let φ be a formula of set theory and let τi : i ∈ n be Q-names inthe model M . Piecing together various names, there is a name σ in Hκ suchthat P if Hκ[G] |= ∃x φ(x, τi : i ∈ n) then Hκ[G] |= φ(σ, τi : i ∈ n). By theelementarity of the model M , there must be such a name σ in the model M .Thus, Hκ[G] |= ∃x φ(x, τi/G : i ∈ n) → φ(σ/G, τi/G : i ∈ n) and the Tarskicriterion follows.

For the second item observe that M [G] ∩ HVκ = M . To see this, if τ ∈ M

is a P -name, then the set D = {p ∈ P : either p τ /∈ HVκ or there exists

a ∈ HVκ such that p τ = a} is open dense subset in P , in the model M by

the elementarity of M . Since the filter G is generic over M , there must be somecondition p in the intersection G∩M ∩D. If p τ /∈ HV

κ then τ/G /∈ V . If, onthe other hand, there is a ∈ Hκ such that p τ = a, then by the elementarityof the model M it is the case that there must be such an a in the model M ,and then τ/G = a ∈M .

The third item follows from the second. If τ ∈ M is a Q-name then τ/K ∈M [G] since M [G] is an elementary submodel of Hκ[G] containing both τ andK. On the other hand, if a ∈ M [G] ∩ Hκ[K] then there is a Q-name τ ∈ Hκ

such that a = τ/K, and by the elementarity in (2) such a name must exist inM .

2.4. SHOENFIELD ABSOLUTENESS 17

2.4 Shoenfield absoluteness

Nearly every argument in this book uses the following classical fact:

Fact 2.4.1. (Shoenfield, [22, Theorem 25.20]) Let x ∈ X be an element of aPolish space, and φ a Π1

2 formula with one variable. Then the truth value ofφ(x) is the same in all forcing extensions.

Shoenfield absoluteness is necessary to establish the existence of canonicalinterpretations of many notions of descriptive set theory in forcing extensions,a fact consistently underappreciated in existing literature, such as [67]. Here weprovide the necessary proofs.

Theorem 2.4.2. Every Polish space has a canonical interpretation in everyforcing extension.

The precise definition of the canonical interpretation is to a great extent amatter of taste, and as such is likely to generate endless scholastic discussions onthe imaginary virtues and advantages of one approach over another. We pick thecompletion approach, since it was the first reaction of all set theorists contactedin an ad hoc survey and since it shows the role of Shoenfield absoluteness mostclearly. We will also show that another perfectly good approach using Gδ subsetsof the Hilbert cube is equivalent.

Proof. Let X be a Polish metric space, with a complete compatible metric d.Let V [G] be a generic extension of V , and consider the completion Xd in V [G]of the metric space 〈X, d〉 there. This will be our canonical interpretation ofX in V [G]. We must show that this interpretation does not depend on thechoice of the complete metric d in the sense that if d, e are two compatiblecomplete metrics on X in the ground model, then the two spaces Xd and Xe arehomeomorphic in V [G]. What is more, if id : X → Xd and ie : X → Xe are thetwo canonical metric embeddings, then there is a homeomorphism h : Xd → Xe

such that ie = h ◦ id. Such a homeomorphism must necessarily be unique.Return to the ground model. Let A be a countable dense subset of X and

consider the usual equivalence relation Ed on d-Cauchy sequences and Ee thesimilar equivalence on e-Cauchy sequences.

Claim 2.4.3. Every Ed-equivalence class intersects exactly one Ee-equivalenceclass. Moreover, this is a Π1

2 statement.

Proof. First observe that for every point x ∈ X there is a sequence of points inA converging to x that is simultaneously e-Cauchy and d-Cauchy. To find sucha sequence, note that the 2−n-balls around x in the sense of the metric e and dare both open in the topology of the space X, and pick the n-th point on thesequence in the open intersection of these two balls.


The first sentence of the claim now immediately follows: every Ed-equivalenceclass consists of d-Cauchy sequences converging to some fixed point x, and assuch may intersect only the unique Ee-class consisting of e-Cauchy sequencesconverging to the same point. It does intersect this unique Ee-class in a nonemptyset by the previous paragraph. The complexity evaluation is also nearly trivial.The first sentence of the claim says that for every d-Cauchy sequence there isan Ed-equivalent e-Cauchy sequence, and moreover, if two sequences are bothe-Cauchy and d-Cauchy then they are Ed-equivalent if and only if they areEe-equivalent.

In the generic extension V [G], the first sentence of the claim remains inforce by Shoenfield’s absoluteness. The d-completion Xd of X is equal to thed-completion of A and similarly on the e-side. The map h that sends everyd-equivalence class to the unique e-equivalence class that it intersects has therequired properties.

Another approach to a canonical interpretation of Polish spaces would usethe basic fact that every Polish space is homeomorphic to a Gδ subset of theHilbert cube, and that there is a good candidate for a canonical interpretationof any Gδ subset of the Hilbert cube in a forcing extension. The Shoenfieldabsoluteness will again appear in this approach at the point of showing that Gδsets homeomorphic in the ground model are still homeomorphic in the exten-sion, and the homeomorphism in the extension can be chosen so that it extendsthe homeomorphism in the ground model. We will only show that this approachis equivalent in that the interpretation of the Gδ set in the extension is home-omorphic to the completion interpretation obtained above. Let B =

⋂nOn be

a Gδ subset of [0, 1]ω, with a countable dense subset A ⊂ B, and a compatiblecomplete metric d. The following claim is nearly trivial.

Claim 2.4.4. For every d-Cauchy sequence of elements of A converges to somepoint in B. Vice versa, every point in B is a limit of some d-Cauchy sequenceof elements of A. Moreover, this is a Π1

2 statement.

Now move to any generic extension. The first two sentences of the claimremain in force by Shoenfield absoluteness. The map h from the d-completionof the set B (as interpreted in V ) to B as interpreted in V [G] mapping eachEd-class of elements of A to the point of B to which it converges has the requiredproperties.

We will use the above theorem in the following way. For any Polish spaceX and any forcing P , the symbol X will denote the P -name for the canonicalinterpretation of the space X in the generic extension. We may drop the dot ifthe symbol stands on the right hand side of the forcing relation; in such a case,the ground model version of X will be denoted by X∩V or XV . We will alwaysabuse the notation by considering the ground model version of X as a subset ofits canonical interpretation in the extension, and the canonical interpretationsof such simply definable spaces as 2ω or ωω will be the spaces with the samedefinition in the extension.

2.4. SHOENFIELD ABSOLUTENESS 19

Theorem 2.4.5. Analytic subsets of Polish spaces have canonical interpretationin any generic extension.

Proof. First, the canonical interpretation of closed sets is rather obvious. IfC ⊂ X is a closed subset of a Polish space, then the closure of C in the canonicalinterpretation of X in the generic extension is the canonical interpretation of C.If A ⊂ X is an analytic set, then it is a projection of a closed set C ⊂ X × ωω.The canonical interpretation of A in the generic extension will be the projectionof the interpretation of C into the interpretation of X. We must show that thisdefinition does not depend on the choice of the closed set C. Let D be anotherclosed subset of X × ωω projecting into A. Let U be a countable dense subsetof X × ωω. The following nearly trivial claim is left to the reader.

Claim 2.4.6. For every sequence of elements of U converging to an element ofC there is a sequence of elements of U converging to an element of D with thesame first coordinate, and vice versa. Moreover, this is a Π1

2 statement.

By Shoenfield absoluteness, the first sentence of the claim remains in forcein any generic extension, and so the projection of the interpretation of C is thesame as the projection of the interpretation of D.

Notationally, this feature is exploited in the following way. If A ⊂ X is ananalytic (or coanalytic) subset of a Polish space, the same letter A is used todenote the canonical interpretation of A in all forcing extensions.

The reader as well as the authors may equally wish to leave this seeminglybarren subject already. However, there is a much finer canonical interpretationproblem that must be resolved–the question of interpretation of Π1

1 on Σ11 classes

of analytic sets in generic extensions. It turns out that the verification of thevalidity of the obvious candidate definition is a highly nontrivial matter thatuses the methods of effective descriptive set theory; it is certainly missing in[67].

Theorem 2.4.7. Every Π11 on Σ1

1 class of analytic sets has a canonical Π11

on Σ11 interpretation in every generic extension. If the class is a σ-ideal of

analytic sets in the ground model, then its canonical interpretation in the genericextension is again a σ-ideal of analytic sets.

Proof. Let I be a Π11 on Σ1

1 collection of analytic sets on a Polish space X.For simplicity we assume that the space X is recursively presented; otherwise,replace it with its homeomorph, a Gδ-subset of the recursively presented space[0, 1]ω, and insert the code for the Gδ subset in appropriate places in the argu-ment.

Let A ⊂ 2ω × X be a good universal Σ11 set in the sense of ???. The

set C = {y ∈ 2ω : Ay ∈ I} is coanalytic; therefore, both A and C have acanonical interpretation in any forcing extension. The collection {Ay : y ∈ C},as interpreted in the generic extension, is a natural candidate for the canonicalinterpretation of I in the generic extension. It is necessary to check that inthe generic extension, this is a Π1

1 on Σ11 collection of analytic sets, and the


interpretation does not depend on the choice of the good universal analytic setA. These features will both follow very quickly as soon as we check that inthe generic extension, the set C still defines a collection of analytic sets, in thesense that the sentence φ = ∀y, z ∈ 2ω (Ay = Az)→ (y ∈ C ↔ z ∈ C) holds inthe generic extension. The sentence φ holds in the ground model; the challengeresides in the fact that φ appears to be more complex than Π1

2, and therefore it isimpossible to use Shoenfield absoluteness to transfer it to the generic extension.We need to resort to a patch that includes the use of effective descriptive settheory. Assume for simplicity that the set C is lightface Π1

1; otherwise insertthe parameter used in its definition in the appropriate spots in the argumentbelow.

Fix a standard parametrization of Borel subsets of X such as in ???. Thatis, there are Π1

1 sets D ⊂ 2ω and F ⊂ 2ω × X such that for every y ∈ D theequality Az = Fz holds, and for every parameter z ∈ 2ω and every ∆1

1(z) setG ⊂ X there is y ∈ ∆1

1(z) such that Fz = G. Moreover, this property of thesets A, D, and F is provable in ZFC.

Claim 2.4.8. For every z ∈ C there is y ∈ ∆11(z) such that y ∈ D ∩ C and

Az ⊆ Ay. Moreover, this is a Π12 statement.

Proof. The first sentence is just a restatement of the usual effective version ofthe first reflection theorem. For the evaluation of the complexity, note thaty ∈ D ∧ Az ⊆ Ay is a Π1

1 statement about z and y: for y ∈ D, Az ⊆ Ay isequivalent to ∀x x /∈ Az ∨x ∈ Gy. The complexity evaluation then follows fromthe well-known fact that the quantifier ∃y ∈ ∆1

1(z) translates into a universalquantifier ???

Claim 2.4.9. For every z ∈ 2ω and every y ∈ D ∩ C, if Az ⊆ Ay then z ∈ C.Moreover, this is a Π1

2 statement.

Proof. The first sentence just says that the collection I is closed under takingan (analytic) subset. This statement has an easy folklore proof: assume thatP ⊂ R ⊂ X are analytic sets and R ∈ I. Choose an analytic non-Borel setS ⊂ 2ω and consider the analytic set T = (S × R) ∪ (2ω × P ) ⊂ 2ω ×X. Theset U = {y ∈ 2ω : Ty ∈ I} is coanalytic as I is Π1

1 on Σ11. The vertical sections

Ty for y ∈ S are equal to R, so S ⊆ U . Since the set S is not coanalytic, theremust be a point y ∈ U \ S. For such a point y, Ty = P and we conclude thatP ∈ I as well.

For the complexity computation, recall from the previous proof that y ∈D ∧Az ⊆ Ay is a Π1

1 statement about y and z, and rewrite the first sentence ofthe claim as ∀z∀y (y /∈ D ∩ C) ∨ ¬(y ∈ D ∧ Az ⊆ Ay) ∨ (z ∈ C). The formulato the right of the universal quantifiers is a disjunction of Π1

1 and Σ11 formulas,

therefore Π12, and the complexity estimate follows.

2.5. GENERIC ULTRAPOWERS 21

The sentence φ formally follows from the conjunction of the two statementsin the claims: If z0, z1 ∈ 2ω are points such that Az0 ⊆ Az1 and z1 ∈ C, thenthe first claim provides a point y ∈ D ∩ C such that Az1 ⊆ Ay, and the secondclaim applied to Y0, z then shows that z0 ∈ C. The statements in the claims areΠ1

2 and therefore will hold in any forcing extension by Shoenfield absoluteness;so even φ will hold in every forcing extension.

It follows immediately that the canonical interpretation does not depend onthe choice of the analytic set A ⊂ 2ω × X. Let A∗ ⊂ 2ω × X be an arbitraryanalytic set and C∗ = {y ∈ 2ω : A∗y ∈ I}. We will show that in every forcingextension, for every y ∈ 2ω, y belongs to (the interpretation of) C∗ if and onlyif A∗y belongs to the interpretation of I described above. Since the set A is gooduniversal, there is a continuous function f : 2ω → 2ω such that for every y ∈ 2ω,A∗y = Af(y); moreover, for every y ∈ 2ω, y ∈ C∗ ↔ f(y) ∈ C by the definitionof the set C∗. This feature of the function f is Π1

2, and therefore persists to thegeneric extension. Thus, even in the generic extension C∗ = {y ∈ 2ω : A∗y ∈ I}.

The (canonical interpretation of the) collection I remains Π11 on Σ1

1 in everygeneric extension, again due to the choice of the good universal set A ⊂ 2ω×X.In the extension, if A∗ ⊂ 2ω × X is an analytic set, then there must be acontinuous function f : 2ω → 2ω such that for every y ∈ 2ω, A∗y = Af(y). Itfollows that the set C∗ = {y ∈ 2ω : A∗y ∈ I} is coanalytic, since it is simply thef -preimage of the (interpretation of) the set C.

Finally, if I is a σ-ideal of analytic sets in the ground model, then this iseasily restated as a Π1

2 statement using the good universal set A, and thereforeit persists to all forcing extensions.

We exploit the previous theorem in the following way. If I is a Π11 on Σ1

1 σ-ideal,we use the same letter I to denote its canonical interpretation in every genericextension.

As a final remark, in this book we are interested in many properties of Π11

on Σ11 σ-ideals, most of which seem to be impossible to restate as Π1

2 formulas,and their absoluteness into forcing extension remains open to question. Sabok[47] proved that the statement “I is σ-generated by closed sets” is absolute forsuch ideals. The following remains open.

Question 2.4.10. Let I be a Π11 on Σ1

1 σ-ideal on a Polish space. If I is c.c.c.,is (the canonical interpretation of) I c.c.c. in every forcing extension? If thequotient forcing PI is proper, is it again proper in every forcing extension?

2.5 Generic ultrapowers

In several arguments throughout the book, we will exploit the technique of (ill-founded) generic ultrapower outlined in [22, Section 35]. Let X be an arbitraryset and I be an ideal on it, and consider the quotient poset RI = P(X) moduloI. Whenever G ⊂ RI is a generic filter, one can form in the extension V [G] thegeneric ultrapower, the model N of equivalence classes of ground model func-tions with domain X, where f is equivalent to g if {x ∈ X : f(x) = g(x)} ∈ G,


and [f ] ∈N [g] if {x ∈ X : f(x) ∈ g(x)} in G. A modified proof of Los’s theoremshows that j : V → N defined by j(a) = [fa], where fa is the constant functionwith the single value a, is an elementary embedding. The model N may not bewellfounded. However, for every N -ordinal α which is wellfounded in V [G], wewill always identify the wellfounded relation 〈V Nα ,∈N 〉 with its transitive iso-morph. If I is a σ-ideal, then ωN is wellfounded, and (2ω)N , (ωω)N etc. becomesubsets of (2ω)V [G], (ωω)V [G] through this identification.

We will use the ultrapower technique exclusively through the following fact.

Fact 2.5.1. For every uncountable cardinal λ, in some generic extension thereis a generic ultrapower j : V → N such that

1. j′′Hλ exists in N and it is countable there;

2. for every analytic set A coded in the ground model, j(A) ⊂ A, and forevery Borel set B coded in the ground model, j(B) = B ∩N .

Note that both properties of the embedding j will survive into any furthergeneric extension of the ground model. It may happen that there is an analyticset A such that j(A) 6= A ∩N—for example if there is an illfounded countableordinal of N and A is the set of illfounded linear orders on ω. In such a case,the model N may be incorrect about such items as codes for Borel sets etc.

Proof. The argument is entirely standard. Let I be the σ-ideal on nonstationarysubsets of [Hλ]ℵ0 , and let RI be the quotient poset. Let G ⊂ RI be a genericfilter over V and form the generic ultrapower j : V → N . We claim that thisembedding works.

The first item follows from the normality of the nonstationary ideal on[Hλ]ℵ0 . Consider the equivalence class [id] ∈ N , and observe that [id] = j′′Hλ.First of all, if a ∈ Hλ then for a club subset of b ∈ [Hλ]ℵ0 it is the case thata ∈ b; in other words, the values of function fa constantly equal to a belong tothe values of the identity function everywhere but on a nonstationary set. By Los’s theorem, N |= j(a) = [fa] ∈ [id]. Second, if S N |= [f ] ∈ [id], then forall but nonstationarily many b ∈ S it is the case that f(b) ∈ b, and the normalityof the nonstationary ideal yields a stationary subset T ⊂ S and a ∈ Hλ suchthat ∀b ∈ T f(b) = a. Clearly, T [f ] = j(a). Finally, N |= M = j′′Hλ byLos’s theorem again, since the values of the id function are all countable sets.

Since M ∈ N has an initial segment of its ordinals isomorphic to λ, itfollows that λ is isomorphic to an initial segment of ordinals of the model N . Inparticular, the wellfounded (transitive) part of the model N includes (Vω+3)N

and for any Polish space X ∈ (Vω+3)V it is the case that j(X) ⊂ XV [G] .Suppose that A ⊂ X is an analytic set coded in the ground model, say A =rng(f) for some continuous function f : ωω → X. Then, as the model N hasstandard ω, j(f) ⊂ f , and so j(A) = rng(j(f)) ⊂ rng(f) = A. If B ⊂ X isa Borel set, then both B,¬B are analytic and respective ranges of continuousfunction f, g : ωω → X. By elementarity of the embedding j, rng(j(f)) andrng(j(g)) are complementary subsets of j(X) in N , and as in the previous

2.6. IDEALIZED FORCING 23

sentence j(B) = rng(j(f)) ⊂ rng(f) = B and j(¬B) = rng(j(g)) ⊂ rng(g) =¬B. The only way that this can happen is that j(B) = B ∩N .

2.6 Idealized forcing

The techniques of the book depend heavily on the theory of idealized forcing asdeveloped in [67]. Let X be a Polish space and I be a σ-ideal on it. The symbolPI stands for the partial ordering of all Borel I-positive subsets of X ordered byinclusion; we will alternately refer to it as the quotient forcing of the σ-ideal I.Its separative quotient is the σ-complete Boolean algebra of Borel sets moduloI. The forcing extension is given by a unique point xgen ∈ X such that thegeneric filter consists of exactly those Borel sets in the ground model containingxgen as an element. One can also reverse this operation and for every forcing Padding a generic point x ∈ X, one may form the σ-ideal J associated with theforcing P , generated by those Borel sets B ⊂ X such that P x /∈ B.

If I is a σ-ideal on a Polish space X, x ∈ X is a point and M is an elementarysubmodel of a large structure such as Hκ, the collection of all sets of hereditarycardinality < κ for some cardinal κ, and I ∈ M , we will say that the pointx is PI -generic over M if for every open dense set D ⊂ PI , D ∈ M , there isB ∈ D∩M with x ∈ B. The set g ⊂ PI ∩M consisting of those B ∈ PI ∩M forwhich x ∈ B is an M -generic filter on PI ∩M . We then consider M [x] as thetransitive model obtained from the transitive isomorph of M by adjoining thefilter g. If τ ∈M is a PI -name then τ/x denotes the evaluation of the transitivecollapse image of τ according to the filter g. Similar terminology will be appliedin case that M is instead some transitive model of set theory and M |= I is aσ-ideal on a Polish space X.

Fact 2.6.1. [67, Proposition 2.2.2] Let I be a σ-ideal on a Polish space X. Thefollowing are equivalent:

1. PI is proper;

2. for every Borel I-positive set B ⊂ X and every countable elementarysubmodel M of a large structure with I ∈ M , the set C ⊂ B of all pointsin B that are P -generic over the model M is I-positive.

There are many consequences of this characterization:

Fact 2.6.2. [67, Section 2.3] Let I be a σ-ideal on a Polish space X such thatthe quotient PI is proper. Then

1. V [xgen]∩2ω = {f(xgen) : f is a Borel function coded in the ground model};

2. for every Borel set B ∈ PI and every analytic set A ⊂ B × ωω withnonempty vertical sections, there is a Borel I-positive set C and a Borelfunction f : C → ωω such that f ⊂ A;


3. for every Borel set B ∈ PI and every Borel function f : B → ωω there isa Borel I-positive set C ⊂ B such that f ′′C is Borel.

4. whenever y ∈ V [xgen] is a real then V [y] ∩ 2ω = {f(y) : f is a groundmodel Borel function with y ∈ dom(f) and rng(f) ⊂ 2ω}.

The following simple fact is used throughout the book:

Fact 2.6.3. [67, Theorem 3.3.2] Suppose that I is a σ-ideal on a Polish spaceX such that the quotient poset PI is proper. The following are equivalent:

1. PI is bounding;

2. every Borel I-positive set has a compact I-positive subset, and every Borelfunction on an I-positive domain is continuous on a compact I-positiveset.

The following two properties deal with σ-ideals without any reference toforcing; nevertheless, they are particularly useful when idealized forcing is con-sidered.

Definition 2.6.4. A pair of σ-ideals I, J on respective Polish spaces X,Y hasthe Fubini property if whenever B0 ⊂ X and B1 ⊂ Y are I- and J-positive Borelsets respectively, then the product B0 × B1 cannot be covered by the union oftwo Borel subsets of X × Y , one with all vertical sections in J and the otherwith all horizontal sections in I.

Definition 2.6.5. A pair of σ-ideals I, J on respective Polish spaces X,Y hasthe rectangular Ramsey property if whenever B0 ⊂ X and B1 ⊂ Y are I- andJ-positive Borel sets respectively and the product B0 × B1 is covered by theunion

⋃nDn of countably many Borel subsets of X × Y , then there are Borel

respectively I-and J-positive sets C0 ⊂ B0 and C1 ⊂ B1 such that the productC0×C1 is a subset of one of the sets forming the union. The definition naturallyextends to finite or countable collections of σ-ideals.

The main purpose of the rectangular Ramsey property in [67] is the computationof the σ-ideal associated with the product PI × PJ . If the property holds, thenthe collection K of all Borel subsets of X×Y containing no Borel rectangle withI- and J-positive sides respectively is a σ-ideal of Borel sets, and the productis naturally isomorphic to a dense subset of the quotient PK . In this book, therectangular property will be also used to rule out various ergodicity phenomena.

Among the quotient forcings, the definable c.c.c. ones form a very specialclass that has been investigated repeatedly. In several places in this book, wewill need to discuss the global properties of this class. The following fact recordsthe features important here. If one is willing to make large cardinal assumptions,it will hold true of much larger class of σ-ideals than just Π1

1 on Σ11.

Fact 2.6.6. Let I be a c.c.c. Π11 on Σ1

1 σ-ideal on a Polish space. Then

1. if PI is bounding, then it is a Maharam algebra;

2.6. IDEALIZED FORCING 25

2. if PI is not bounding, then it adds a Cohen real;

3. if I contains all singletons and not the whole space, and PI preserves Bairecategory, then it is equivalent to adding one Cohen real.

Proof. This remarkable statement is in fact just a concatenation of various the-orems scattered throughout the literature. If PI is bounding, then Player II hasa winning strategy in the bounding game [67, Theorem 3.10.7]. Fremlin [23,Theorem 7.5] proved that the winning strategy yields a continuous submeasureφ on the Polish space such that I = {A : φ(A) = 0}. If PI adds an unboundedreal, then it in fact adds a Cohen real by a result of Shelah [3, Theorem 3.6.47].One has to adjust the final considerations of that proof very slightly to concludethat it works for all Π1

1 on Σ11 c.c.c. σ-ideals. The third item is in essence [67,

Corollary 3.5.6] or [?].

Corollary 2.6.7. Let I be a c.c.c. Π11 on Σ1

1 σ-ideal on a Polish space. Thequotient forcing adds an independent real.

Proof. Maharam algebras add independent reals by [1, Lemma 6.1], and a Cohenreal is an independent real in its usual presentation. The rest is handled by thefirst two items of the above Fact.

Corollary 2.6.8. Let I, J be c.c.c. Π11 on Σ1

1 σ-ideals on Polish spaces. Thepair I, J does not have the rectangular Ramsey property.

Proof. The first two items of the Fact again show that we only have to considerthe meager ideal, and the null ideals for the various continuous submeasures.As shown in [12], pairs of these σ-ideals even fail to have the Fubini propertyunless they are two null ideals obtained from Borel probability measures or twomeager ideals obtained from some Polish topologies. In these two cases, therectangular property fails again; the simplest way to see it is to consider theequivalence E0 on 2ω and the usual Borel probability measure or topology on2ω, and apply the Steinhaus theorem to see that there cannot be a product oftwo Borel positive sets which is either disjoint from E0 or a subset of E0.

The following humble theorem will be useful as well.

Theorem 2.6.9. Let I be a Π11 on Σ1

1 σ-ideal on a Polish space X such thatthe quotient forcing PI is proper. Every c.c.c. intermediate extension of the PIextension either is equal to the ground model or contains a new real.

In other words, the quotient does not add new branches through Suslin treesand Suslin algebras. Note that the intermediate c.c.c. extension may not begiven by a single real; posets such as adding ℵ1 Cohen reals with finite supportmay embed to PI under suitable circumstances.

Proof. Suppose for contradiction that the intermediate extension is effected bya c.c.c. forcing Q, a regular subforcing of PI which does not add reals. Let M bea countable elementary submodel of a large enough structure containing Q and


I. Since no real is added by Q, it forces that the intersection of its generic filterG with the model M belongs to the ground model. Fix a maximal antichainof conditions in the poset Q deciding the value of M ∩ G. The antichain iscountable, and so we can list all possible values for the intersection as {gn : n ∈ω}. Note that each of these filters in fact Q-generic over M : every antichainof Q is countable, therefore if it is in M then it is a subset of M and so Qforces the filter G to contain an element of such an antichain which is also inM . Let κ = |P(PI)|, let h ⊂ Coll(ω, κ) ∩ M be a filter in V , generic overall the models M [gn]; such a filter exists as the models contain together onlycountably many antichains of the poset Coll(ω, κ) ∩ M . The Borel set B ofPI -generic reals over M is equal to

⋂{⋃D ∩ M : D ∈ M is an open dense

subset of PI}; therefore, it has a Borel code in M [h] as the model M [h] cansee that the unions and intersections forming it are countable. The set B isI-positive in V by properness of PI and Fact 2.6.1, this positivity is an analyticstatement by the complexity assumption on I, and therefore it transfers to thewellfounded model M [h]. Let f : B → P(Q) ∩M be the function defined byf(x) = {q ∈ Q∩M : ∃C ∈ PI ∩M x ∈ C∧C forces q to belong to the Q-genericfilter}. This is a Borel function with code in the model M [h]. In that model,there are obviously two cases: either the f -preimage of every subset of Q ∩Mis in the ideal I, or there is a subset of Q ∩M with Borel I-positive preimageC ⊂ B. In the first case, the property of B and f is a coanalytic statementsince the ideal I is Π1

1 on Σ11, therefore transfers to the model V , and there B

forces that the intersection of the Q-generic filter with M is not in the groundmodel, contradicting the assumptions. In the latter case, C the intersectionof the Q-generic filter with M is in the model M [h] and therefore not on thelist {gn : n ∈ ω}, reaching a contradiction again.

2.7 Other tools

In several places, we will make good use of the concentration of measure phe-nomenon on simple finite metric spaces. The following is the simplest context:

Definition 2.7.1. A Hamming cube of dimension n is the set 2n equipped withthe normalized counting measure and the normalized Hamming distance, i.e.d(x, y) = |{m ∈ n : x(m) 6= y(m)}|/n.

Fact 2.7.2. For every ε, δ > 0 there is n ∈ ω such that in the Hamming cubeof dimension n, the ε-neighborhood of any set of mass ≥ 1/2 has mass ≥ 1− δ.

Note that it immediately follows that 2ε neighborhood of a set of mass ≥ ε musthave a mass 1− ε. This yields an attractive corollary:

Corollary 2.7.3. For every ε > 0 and every m > n there is n ∈ ω such thatwhenever ai : i ∈ m are subsets of the Hamming cube of dimension n of mass≥ ε then there is an ε-ball intersecting them all.

Observe that if c ⊂ n is a set and πc : 2n → 2n is the map flipping the bitsof its entries on the set c, then πc preserves both the Hamming measure and

2.7. OTHER TOOLS 27

the Hamming metric. Thus, replacing some of the sets ai above with their πcimages, one can realize all various metric configurations between the elementsof the sets ai with ε-precision.

We will also need the following more precise calculation:

Fact 2.7.4. [40, Theorem 4.2.5] Let X =∏i∈nMi be a product of finite metric

spaces, each with metric di of diameter ai and a probability measure µi. EquipX with the product measure µ and the l1-sum d of the metrics di. For everyε > 0 and every set of µ-mass at least 1/2, its d − ε-neighborhood has µ-mass

at least 1− 2e−ε2/16Σia

2i .

Let (ω)ω be the space of all infinite sequences a of finite sets of naturalnumbers with the property that min(an+1) > max(an) for every number n ∈ ω.Define a partial ordering ≤ on this space by b ≤ a if every set on b is a union ofseveral sets on a.

Fact 2.7.5. (Bergelson-Blass-Hindman [61, Corollary 4.49]) For every partition(ω)ω = B0 ∪ B1 into Borel sets there is a ∈ (ω)ω such that the set {b ∈ (ω)ω :b ≤ a} is wholly included in one of the pieces.

Suppose that 〈an, φn : n ∈ ω〉 is a sequence of finite sets and submeasures onthem such that the numbers φn(an) increase to infinity very fast. The necessaryrate of increase is irrelevant for the applications in this book; suffice it to saythat it is primitive recursive in n and the sizes of the sets {am : m ∈ n}.

Fact 2.7.6. [50] For every partition Πnan×ω = B0 ∪B1 into two Borel pieces,one of the pieces contains a product Πnbn× c where each bn ⊂ an is a set of φnmass at least 1 and c ⊂ ω is infinite.


Chapter 3

Tools

3.1 Integer games connected with σ-ideals

Many of the σ-ideals considered in this book have integer games associated withthem. As a result, they satisfy several interconnected properties, among themthe selection property that will be instrumental in upgrading the canonizationresults to Silver-style dichotomies for these σ-ideals.

The subject of integer games and σ-ideals was treated in [67] rather exten-sively, but on a case-by-case basis. In this section, we provide a general frame-work, show that it is closely connected with uniformization theorems, and provea couple of dichotomies under the assumption of the Axiom of Determinacy.The following definition will be central:

Definition 3.1.1. Let I be a σ-ideal on a Polish space X. We say that I is agame ideal if there is

1. a Borel basis B ⊂ ωω ×X for I–a Borel set all of whose vertical sectionsare in I and such that every set in I is a subset of one of vertical sectionsin B;

2. a scrambling tool–a Borel function f : ωω → X × ωω

such that for every closed set C ⊂ X × ωω, Player I has a winning strategy inthe game G(C) if and only if the projection of the set C to the space X is inI. Here, in the infinite game G(C), Players I and II alternate to play pointsy, z ∈ ωω respectively and Player II wins if f(z) ∈ C and the first coordinate ofthe point f(z) does not belong to the vertical section By.

Note that the games G(C) as in the definition above are all Borel and thereforedetermined by [35]. The notion is going to be used only for σ-ideals in thisbook, but it appears to be useful even in the more general context of hereditarycollections of subsets of Polish spaces.

The meaning of the definition must be illustrated on examples. The existenceof a Borel basis for a σ-ideal is typically easy to verify. Thus, the σ-ideal of

29

30 CHAPTER 3. TOOLS

countable subsets of X = 2ω has a Borel basis B naturally indexed by the space(2ω)ω), and 〈y, x〉 ∈ B ↔ ∃n x = y(n). To produce a Borel basis B for the σ-ideal of meager subsets of X = 2ω, note that the collection Z of closed nowheredense subsets of 2ω is Gδ in K(2ω) and therefore Polish, index the basis by thespace Zω, and let 〈y, x〉 ∈ B ↔ ∃n x ∈ y(n). There are many σ-ideals of Borelsets without a Borel basis. One way how that can happen is when the σ-idealin question is not σ-generated by Borel sets of fixed Borel rank. Such is thecase of the E0-ideal of Section 5.4. A more subtle failure may occur in the casewhen I is σ-generated by Borel sets of a fixed Borel rank (e.g. closed sets), butthe generators cannot be organized into a Borel collection. Such is the case ofthe σ-ideal generated by closed sets of uniqueness by a deep theorem of [8].

The scrambling tool is a more sophisticated concept. To motivate it, supposethat A ⊂ X is a set, and attempt to design an integer game which detects themembership of A in the σ-ideal I: Player I produces a set in the basis, andPlayer II produces a point in X. Player II wins if his point is in the set Aand outside of the set Player I played. We would like to set the game upso that Player I has a winning strategy if and only if A ∈ I. For that, wemust make the game as easy for Player II as possible; this is accomplished byapplying a scrambling tool to his moves so that during the play, Player I cannotinfer virtually anything about Player II’s resulting point from his moves. Thescrambling tool is typically obtained after a suitable basis is identified, and itwill have higher Borel rank than the basis.

For many σ-ideals–such as those investigated in [67, Chapter 4]–an integergame has been manually produced independently of the concerns of the presentbook, and these games can be trivially restated in the language of a basis and ascrambling tool. We will show how this is done in the typical case of the Laverideal of Section 5.13, and leave other similar examples to the reader.

Theorem 3.1.2. The Laver ideal I on ωω is a game ideal.

Proof. The σ-ideal I is generated by sets Ag = {x ∈ ωω : ∃∞n x(n) /∈ g(x � n)}as g varies over all functions from ωω

<ω

. This obviously yields a Borel basis forI: just find a bijection between ω and ω<ω, extend it to a continuous bijectionh between ωω and ωω

<ω

, and let B ⊂ ωω × ωω be described by the formulaBy = Ah(y).

Now revisit the Laver game exhibited for example in [67, Section 4.5.2].Suppose that C ⊂ ωω × ωω is a closed set, and consider the game G ∗ (C) inwhich Player II first indicates a finite sequence t ∈ ωω, and then in each roundi Player I plays ni ∈ ω, Player II responds with mi > ni, and at some roundsof his choice Player II also produces another natural number ki. Player II winsif 〈(tam0,m1, . . . ), (k0, k1, . . . )〉 ∈ C. [65] shows that Player I has a winningstrategy in the game G∗(C) if and only if the projection of C is I-positive.

To construct the scrambling tool, just ???

However, there are several general theorems that produce the scramblingtools for various ideals just from various abstract descriptive properties of the

3.1. INTEGER GAMES CONNECTED WITH σ-IDEALS 31

ideals. Surprisingly enough, the construction of games is closely connected withBorel uniformization theorems. The first example we give is associated withArsenin-Kunugui theorem [33, Theorem 18.18].

Theorem 3.1.3. Suppose that I be a σ-ideal on a Polish space X which hasa Borel basis consisting of Gδ-sets and moreover, if C ⊂ X × ωω is a closedset with I-positive projection then C has a compact subset with an I-positiveprojection. Then I is a game ideal.

For example, the σ-ideal of φ-null sets for any subadditive outer regular capacityφ on a Polish space is a game ideal by the Choquet’s theorem [33, Theorem 30.13]and the above theorem.

Proof. Let B ⊂ ωω × X be a Borel set with I-small Gδ vertical sections suchthat every I-small sets is a subset of a vertical section of B. Let W be the spaceof all compact subsets of X×ωω. Use Arsenin-Kunugui uniformization theorem[33, Theorem 18.18] to find a Borel function g : ωω ×W → X × ωω such thatwhenever 〈y, w〉 ∈ ωω ×W is a pair such that By does not cover the projectionof w then g(y) ∈ w is a point whose first coordinate is not in By. Let S be thespace of all strategies for Player I in integer games, and let h : ωω → S ×W bea Borel bijection. Finally, let f(z) = g(σ ∗ z, w) for every z ∈ ωω, where w issuch that h(z) = 〈σ,w〉. We claim that f works as required.

Indeed, if C ⊂ X × ωω is a closed set, then either its projection is in theσ-ideal I, in which case Player I can win G(C) just by producing y ∈ ωω suchthat p(C) ⊂ By. Or, the projection of C is I-positive, and by the assumptions Chas a compact subset w with I-positive projection. In such a case, no strategyσ can be winning for Player I in the game G(C): if z ∈ ωω is a point such thath(z) = 〈σ,w〉, then the definition of the function f shows that z is a winningcounterplay for Player II against the strategy σ.

The uniformization theorem for sets with nonmeager sections [33, Theorem18.6] yields the following:

Theorem 3.1.4. Let I be a σ-ideal on a Polish space X with a Borel basis.If I is the intersection of meager ideals for some collection of Polish topologiesyielding the same Borel structure, then I is a game ideal. Similarly, if I is theintersection of null ideals for some collection of Borel probability measures, thenI is a game ideal.

Proof. Both the meager and null case are similar. We start with the meagercase.Let B ⊂ ωω × X be the Borel basis for the σ-ideal I. Let W be thespace of all partial continuous maps from ωω to X ×ωω with Gδ domain with astandard Borel structure on it as in [33, Definition 2.5, Proposition 2.6]. Use thenonmeager uniformization theorem [33, Theorem 18.6] to find a Borel functiong : ωω×W → ωω such that whenever 〈y, w〉 ∈ ωω×W is a pair such that the set{u ∈ ωω: the first coordinate of w(u) is not in By} is nonmeager in ωω, then thefunctional value g(y, w) picks an element from this nonmeager set. Let S be the

32 CHAPTER 3. TOOLS

space of all strategies for Player I in integer games, and let h : ωω → S ×W bea Borel bijection. Let f(z) = g(σ ∗ z, w) for every z ∈ ωω, where h(z) = 〈σ,w〉.The function f is the scrambling tool we need.

To see this, suppose that C ⊂ X × ω is a closed set whose projection isnot in I, and suppose that σ ∈ S is a strategy for Player I; we must producea winning counterplay for Player II in the game G(C). The assumption on theσ-ideal I yields a topology t on the space X with the same Borel structure suchthat the σ-ideal It of t-meager sets is a superset of I, and proj(C) /∈ It. Theset proj(C) is analytic, it has the Baire property in the topology t, and so itis comeager in some nonempty t-open set O. There is a Borel isomorphismk : ωω → O which carries the meager ideal on ωω to It. The Jankov-vonNeumann uniformization theorem yields a partial Baire-measurable functionl : ωω → ωω such that whenever u ∈ ωω is a point with k(u) ∈ proj(C) thenl(u) is defined and 〈k(u), l(u)〉 ∈ C. There is a dense Gδ set P on which bothk, l are continuous maps (the continuity of k is considered with respect to theoriginal topology on X). Let w ∈W be the function u 7→ 〈k(u), l(u)〉 restrictedto P . Find z ∈ ωω such that h(z) = 〈σ,w〉 and show that z is the desiredwinning counterplay for Player II. Indeed, the set Bσ∗z is in the σ-ideal I, so inIt, the set {u ∈ ωω : g(u) /∈ Bσ∗z} is comeager in P , and therefore the functionalvalue f(z) is defined and picks a winning point in C.

The null case is similar, using the isomorphism theorem for measures ???

The class of ideals that fit into the scheme of Theorem 3.1.4 is much widerthan the class satisfying the assumption of Theorem 3.1.3. Remember thoughthat σ-ideals which are intersection of meager ideals or intersections of nullideals may not have a Borel basis and so can fail to be game ideals on thataccount.

Theorem 3.1.5. The following σ-ideals are intersections of meager ideals:

1. every Π11 on Σ1

1 σ-ideal I such that the quotient poset PI is proper andpreserves Baire category;

2. every hypergraph ideal;

3. the σ-ideal σ-generated by the classes of a given Borel equivalence relation.

Checking whether a given σ-ideal is an intersection of null ideals is much moredemanding, and also more informative. Such ideals are commonly referred to aspolar ideals in the literature. Choquet [9] proved that every strongly subadditivecapacity is the supremum of some collection of Borel probability measures, andtherefore its null ideal is a polar ideal. [67, Example 3.6.4] shows that for everydimension h > 0, the σ-ideal generated by sets of finite h-dimensional Hausdorffmeasure in a Euclidean space is polar. On the other hand, [20] works hard toprove that the σ-ideal of σ-porous sets on the real line is not a polar σ-ideal.

Theorem 3.1.6. The following σ-ideals are intersections of null ideals:

1. the σ-ideal of sets of zero mass for a strongly subadditive capacity;


2. the σ-ideal σ-generated by sets of finite Hausdorff measure of any fixeddimension, on any fixed Euclidean metric space.

Faced with such a powerful machinery to produce integer games, it appearshard to identify a σ-ideal with a Borel basis that is not a game ideal, and wehave no such an example. An obvious complexity computation shows that σ-ideals with Borel basis are in general Π1

2 on Σ11, while the game ideals are a tad

simpler:

Definition 3.1.7. A σ-ideal I on a Polish space X is ∆12 on Σ1

1 if for everyanalytic set A ⊂ 2ω ×X the collection {y ∈ 2ω : Ay ∈ I} is ∆1

2.

Theorem 3.1.8. All game ideals are ∆12 on Σ1

1.

Proof. Let A ⊂ 2ω ×X be an analytic set, with C ⊂ 2ω ×X × ωω a closed setprojecting to it. Then, for every z ∈ 2ω, the vertical section Ay is in the idealI if and only if Player I has a winning strategy in the game G(Cz) if and onlyif Player II has no winning strategy in the game G(Cz). These are a Σ1

2 and aΠ1

2 formulas respectively.

Question 3.1.9. Is every σ-ideal with a Borel basis a game ideal? Is every ∆12

on Σ11 σ-ideal with Borel basis a game ideal? Is there a simple uniformization-

type property of a σ-ideal that is equivalent to an existence of a game for I?

In the rest of this section, we will use the games to derive a couple of inter-esting corollaries in the context of the Axiom of Determinacy.

Theorem 3.1.10. (ZF+Axiom of Determinacy) Let I be a game ideal on aPolish space X. If every coanalytic set is either in I or contains an analyticsubset not in I, then every set is either in I or contains an analytic subset notin I.

The theorem has a substantial weakness in that for all game ideals we know, theinteger games can be readjusted to prove the assumption of the theorem underAD. In the special but frequent case of σ-ideals such that the quotient forcing isproper and preserves Baire category, the assumption on coanalytic sets can beproved to hold from AD as shown in [7]. However, we do not know for examplewhether for every outer regular capacity φ, every coanalytic subset of φ-positivecapacity contains an analytic subset of positive capacity.

Proof. Let X be the Polish domain of the σ-ideal I, let B ⊂ ωω ×X be a Borelbasis of I and f a scrambling tool as in Definition 3.1.1. Let A ⊂ X be anarbitrary set, consider the set C = A× ωω, and the associated game G(C). Bythe determinacy assumptions, either Player I has a winning strategy or PlayerII does.

If Player II has a winning strategy σ, let A′ = {x ∈ X : ∃y, z, v ∈ ωω z = σ∗yand f(z) = (x, v)}. This is clearly an analytic set, and since the strategy σis winning, it must be a subset of the set A. It also must be I-positive, sincewhenever By is a generator of the σ-ideal I for some y ∈ ωω, the first coordinate

34 CHAPTER 3. TOOLS

of the point f(σ ∗ y) is in the set A′ and not in the set By. Thus, the set Acontains an I-positive analytic subset.

If Player I has a winning strategy σ, let A′′ = {x ∈ X : ∀y, z, v ∈ ωω y = σ∗zand f(z) = (x, v) implies x ∈ By}. This is a coanalytic set, and certainlyA ⊂ A′′ since σ was a winning strategy for Player I. We claim that that A′′ ∈I, and so A ∈ I. Otherwise, by the assumption of the theorem, A′′ wouldcontain an analytic I-positive subset, which would be a projection of a closedset D ⊂ X ×ωω. Player II would have a winning strategy τ in the game G(D).The struggle between the strategies σ and τ would result in a point x ∈ A′′

contradicting the definition of the set A′′.In both cases, we verified the conclusion of the theorem.

Corollary 3.1.11. (ZF+DC+Axiom of Determinacy) Let E be a Borel equiv-alence relation on a Polish space X. Whenever A ⊂ X is a set, then either A iscovered by countably many E-equivalence classes or A contains a perfect subsetof pairwise E-inequivalent elements.

Note that some assumptions beyond ZF+DC are necessary to obtain thisalready for coanalytic sets, since in the constructible universe L, there is anuncountable coanalytic set without a perfect subset, and so the statement willfail already for E = id. Unlike the Silver dichotomy, the Corollary fails forcoanalytic equivalence relations. Consider the space X of all trees on ω, andlet E be the coanalytic relation defined by x E y if and only if x = y or(there is no level preserving homomorphic embedding of x into any of the treesy � t where t ∈ y is a node distinct from the root, and vice versa, there isno such embedding of y into any of the trees x � t). It is not difficult to seethat this is an equivalence relation that connects wellfounded trees with thesame rank, and illfounded trees form singleton equivalence classes. The set A ofwellfounded trees is coanalytic and it contains uncountably many equivalenceclasses. Moreover, every perfect subset of it meets only countably many ofthe classes by the boundedness theorem, and so it cannot consist of pairwiseinequivalent elements.

To derive the corollary, we have to deal with the case of analytic and co-analytic sets separately. This is in itself a quite interesting case that deservesseparate label as a theorem. It is a special case of forthcoming work of [7].

Theorem 3.1.12. Suppose that ω1 is inaccessible to the reals. Then everyΣ1

2 set either intersects only countably many equivalence classes of E, or elsecontains a nonempty perfect subset of pairwise inequivalent elements.

Proof. We will start with the case of a Π11 set A ⊂ X. Fix a real parameter

z ∈ 2ω such that A ∈ Π11(z) and let ≤ be a Π1

1(z) rank on A. For everycountable ordinal α consider the Coll(ω, α)-extension of the model L[z] givenby a function y ∈ αω; such a function exists by theassumption of inaccessibilityof ω1. The model L[z][y] contains a Borel code for the Borel set Aα of all pointsof A of ≤-rank < α. By the Silver dichotomy applied in the model L[x][y] to theequivalence relation E on Aα, we see that either L[x][y] |= Aα intersects only


countably many E-equivalence classes, or L[z][y] |= Aα contains a perfect setof pairwise inequivalent elements. If the former case prevails, then the perfectset of pairwise inequivalent elements retains its salient properties in V by thecoanalytic absoluteness between V and L[z][y], and we are done. Thus, we mayassume that for every α, the set Aα is forced by Coll(ω, α) to intersect onlycountably many equivalence classes, for all α ∈ ω1.

For every α ∈ ω1 fix names 〈τn,α : n ∈ ω in the model L[z] such that

Coll(ω, α) forces the set Aα to be covered by the set Bα =⋃n[τn,α]E . Note

that if E is a Borel equivalence relation of rank some β, then Bα is a name for aBorel set of rank at most β+1 (independently of α). Consider the set Cα = {x ∈X : L[z][x] |= Coll(ω, α) x ∈ Bα}. By the work of Jacques Stern [60], this is aBorel set and moreover, a code for it exists in every extension L[x][g] that makes

iL[x]β+1 countable. Such an extension exists by the inaccessibility assumption of

ω1, and in it there are only countably many codes for Borel sets, while there areuncountably many countable ordinals. Thus, there is a cofinal set R ⊂ ω1 anda fixed Borel set C such that for all ordinals α ∈ R, it is the case that Cα = C.We will show that A ⊂ C and C intersects only countably many E-classes. Thiswill conclude the proof for the case of a coanalytic set A.

First of all, whenever x ∈ A, then there is an ordinal α ∈ R such thatthe ≤-rank of x is smaller than α. Whenever y is a Coll(ω, α)-generic filterover L[z][x], then L[z][y] |= every element of Aα is equivalent to one of τn,α/y,by the coanalytic absoluteness this also holds in L[z][x][y], and therefore x isequivalent to one of τn,α/y. As y was arbitrary generic, this means that x ∈ Cαand so x ∈ C.

To prove that the set C intersects only countably many equivalence classes,assume for contradiction that it does not, and use the Silver dichotomy toproduce a perfect subset P ⊂ C of its pairwise inequivalent elements. Fix αsuch that C = Cα, consider the Polish space Y = αω where α has the discretetopology and Y has the product topology, and the maps f : Y → X givenby f(y) = τn,α/y. Thus, the maps fn are defined and continuous on a denseGδ subset of the space Y . Consider also the set D ⊂ Y × X of those points〈y, x〉 such that ∃n fn(y) E x. A brief review of the definitions will revealthat Cα = {x : Dx is comeager}, while for every y ∈ Y , the vertical sectionDx ∩ P can be at most countable. Therefore the set D ∩ Y × P contradicts theKuratowski-Ulam theorem for the product Y × P .

This completes the proof for a coanalytic set A. If A is instead Σ12, then it

is the image of some coanalytic set A′ under a continuous function f : ωω → X.The conclusion then follows from the application of the coanalytic case to thepullback equivalence relation E′ = f−1E and the coanalytic set A′.

Now we are ready for the proof of the corollary. By the Silver dichotomy,it is enough to show that every IE-positive set contains an IE-positive analyticsubset. Since the σ-ideal IE is a game ideal by Theorem 3.1.4, it is enough toverify this for coanalytic sets A ⊂ X. However, this isexactly the contents ofthe previous theorem.

36 CHAPTER 3. TOOLS

Corollary 3.1.13. (ZF+Axiom of Determinacy) Let φ be a subadditive outerregular capacity on a Polish space X. If every coanalytic φ-positive set con-tains an analytic φ-positive set, then every φ-positive set contains an analyticφ-positive set.

Proof. This is just an application of Theorem 3.1.3 and the previous theorem.Note that the collection of φ-null sets is a σ-ideal and in fact a game ideal, sinceit satisfies the assumptions of Theorem 3.1.3 by Choquet’s theorem [33, Section30.C].

3.2 The selection property

The most important application of game ideals and integer games in this bookis the following concept. It will lead towards many Silver-type dichotomies forvarious σ-ideals.

Definition 3.2.1. A σ-ideal I on a Polish space X has the selection propertyif for every analytic set A ⊂ 2ω ×X with I-positive sections there is a partialBorel map g : 2ω → X such that g ⊂ A and rng(g) /∈ I.

Theorem 3.2.2. All game ideals have the selection property.

Proof. Let A ⊂ 2ω ×X be an analytic set with all vertical sections I-positive,A equal to the projection of some closed set C ⊂ 2ω×X×ωω. We will first finda perfect set Z ⊂ Y and a continuous function τ with domain Z, assigning eachelement of Z a winning strategy τ(z) for Player II in the game G(Cz), whereCz ⊂ X × ωω is the vertical section of C associated with z.

To do this, consider the Sacks forcing P adding the generic point zgen to2ω. Since the statement “Player I has no winning strategy in neither game Czfor z ∈ 2ω” is Π1

2 and true in the ground model by the assumptions, it is alsotrue in the Sacks extension by the Shoenfield absoluteness 2.4.1. In particular,P Player I has no winning strategy in the game Czgen . Since the game isdetermined, there is a name τ for a winning strategy for Player II in this game.By the continuous reading of names for Sacks forcing, this name below somecondition reduces to a continuous function. Let M be a countable elementarysubmodel of a large enough structure, and let Z ⊂ 2ω be a perfect set consistingof M -generic Sacks reals below this condition only; thus, the name τ can beviewed as a continuous function e on the set Z. To see that e, Z work asrequired, let z ∈ Z and look in the model M [z]. There, by the forcing theorem,e(z) is a winning strategy for Player II in the game G(Cz). This is a coanalyticstatement about z, e(z) and therefore is absolute between the wellfounded modelM [z] and V . Thus, e(z) indeed is a winning strategy for Player II in the gameG(Cz).

Now, find any Borel bijection h : Z → ωω and let g : Z → X be thefunction defined in the following way: g(z) is the first coordinate of the pointf(e(z)∗h(z)). We claim that the function g is as required. It is certainly Borel.For every z ∈ Z, the strategy e(z) is winning for Player II and therefore it

3.2. THE SELECTION PROPERTY 37

produces a point in Az, so g ⊂ A. Finally, rng(g) /∈ I, since whenever By issome generator of the σ-ideal I and y = h(z) for some z ∈ Z, the point g(z) ∈ Xbelongs to rng(g) and not to By.

The main application of the selection property in this book–cranking up thetotal canonization to the Silver property for various σ-ideals–is postponed toSection 3.3. Here, we will show that the selection property can be used to provetotal canonization itself in a certain context. The argument is very abstract anduses a number of forcing tricks. First, in a rather surprising turn of events, weshow that the selection property rules out most nontrivial intermediate exten-sions of the PI extension. Then, in another unexpected twist, the trichotomytheorem 3.4.4 shows that equivalence relations are directly connected to such in-termediate extensions, and we conclude that the lack of such extensions impliesthe total canonization.

In the early versions of this book, this was the path through which totalcanonization was proved for a great number of σ-ideals. Later, in many caseswe found direct combinatorial arguments that avoid this path and shed morelight on the whole context. In the present version, this path is used to provetotal canonization only for the metric porosity σ-ideals of Section 5.2, reflectingthe lack of understanding of the combinatorial structure of these σ-ideals.

Theorem 3.2.3. Suppose that I is a σ-ideal on a Polish space X such thatthe quotient forcing PI is < ω1-proper in all σ-closed extensions. If I has theselection property, then every intermediate forcing extension of the PI extensionis either a c.c.c. extension of the ground model, or it is equal to the whole genericextension.

The wording of this theorem must be parsed carefully. First of all, σ-closedforcing extensions add no new reals, and therefore the space X, σ-ideal I andthe poset PI remain the same in such extension. Thus, we do not need anydefinability of the σ-ideal I to interpret it in σ-closed forcing extensions. Second,a forcing P is < ω1-proper if for every countable ordinal α and every ε-tower〈Mβ : β ∈ α〉 of countable elementary submodels of a large structure such as〈Vκ, ε〉 for a sufficiently large ordinal κ, with P ∈M0, for every condition p ∈ P∩M0 there is a strengthening q ≤ p which is simultaneously Mβ-master for eachordinal β ∈ α. This is a technical strengthening of properness due to Shelah; it issatisfied for all σ-ideals in this book as proved by a standard transfinite inductionon α in each case. Moreover, in each case the proofs just use the definition ofthe σ-ideals which remains the same in any σ-closed forcing extension, andtherefore yield < ω1-properness in all σ-closed extensions. Ishiu [21] provedthat < ω1-properness of a poset is equivalent to the Axiom A property of apartial order in forcing sense equivalent to the original poset. For the purposesof the theorem, this property does not seem to be replaceable by anythingdescriptive set theoretic in nature. The conclusion cannot be strengthened torule out intermediate c.c.c. extensions as such examples as the Cohen forcing(associated with the meager ideal, which is a game ideal and therefore does havethe selection porperty) will show.

38 CHAPTER 3. TOOLS

Proof. Let λ = c. Let A be a nowhere c.c.c. complete subalgebra of thecompletion of PI ; we must show that the generic filter on PI can be recov-ered from its intersection with A. Note that A contains a term for a functionf : λ → λ such that for every condition p ∈ PI there is δ ∈ λ such that theset {γ ∈ λ : ∃q ≤ p f(δ) = γ} is uncountable; here λ = cV . To see this, let〈pδ : δ ∈ κ〉 be an enumeration of PI , let Aδ be an uncountable antichain ofconditions in A still compatible with pδ with some fixed enumeration, and letf(δ) = γ if the generic filter contains the γ-th element of Aδ, and f(δ) =trash ifthe generic filter contains no elements of Aδ. We will show that PI forces thatxgen can be recovered from f ; this will be enough. The rest of the argumentdoes not use the complete algebra A anymore.

Move to any σ-closed generic extension V [G] in which the continuum hy-

pothesis holds; so |λ| = ℵ1. The poset PVI = PV [G]I is < ω1-proper and the

σ-ideal I still has the selection property. The rest of the proof takes place inthis extension.

Let M be a countable elementary submodel of a large structure containingI, f . Write BM for the Borel set of all points in X PI -generic over M , and writeFM : BM → (λ ∩M)λ∩M for the function defined by FM (x) = {〈δ, γ〉 : δ ∈λ∩M,γ ∈ λ∩M, and for some condition B ∈ PI ∩M such that B f(δ) = γ),x ∈ B}. Note that FM (x) really is a function from λ ∩M to itself since x is aPI -generic point over M . Topologizing λ ∩M with the discrete topology and(λ ∩M)λ∩M with the product topology, it is clear that FM is a Borel function.Note that M ∩ V ∈ V by the σ-closedness of the extension V [G], and thereforeBM , FM ∈ V and the definition of FM above is a definition from parametersin V . Also, if N ∈ M are two elementary submodels and x ∈ X is a pointPI -generic over both N and M , it is the case that FN (x) = FM (x) � N .

We will produce a countable elementary submodel M of a large structureand an I-positive Borel set B ⊂ BM such that FM � B is one-to-one. Thiswill conclude the proof. For suppose that x ∈ B is a PI -generic point overV [G]. In the model V [x] the Borel function FM (meaning the function withthe same definition as FM from the parameter M) is still one-to-one by coana-lytic absoluteness between V [G] and V [x], FM (x) = f/x � M . The coanalyticabsoluteness between the models V [f/x] and V [x] will yield a point y ∈ B inV [f/x] such that FM (y) = f/x � M . Since the function FM � B is one-to-one,we conclude that y = x and so V [f/x] = V [x] as desired.

The set B is obtained by an application of the selection property. We willfind a Borel set C ⊂ 2ω×X such that the vertical sections of C are I-positive andconsist of M -generic points for PI only, such that for distinct vertical sectionsof C have disjoint FM -images. Once this is done, the selection property yieldsa partial Borel function g : 2ω → X, whose range is I-positive. Since thevertical sections of the set C are pairwise disjoint, the function g is injective,and therefore its range B is a Borel I-positive set. The set B obviously has thedesired property.

Note that (regardless of the selection property) the vertical sections of Cform a perfect collection of pairwise disjoint conditions in PI . This should

3.2. THE SELECTION PROPERTY 39

be compared with the work of [?, Theorem 6.1], where a non-c.c.c. suitablydefinable poset is produced which has no perfect antichain, and the constructioncan produce a δ-proper poset for any ordinal δ ∈ ω1 fixed beforehand. The restof our argument uses only the < ω1-properness of the poset PI in the σ-closedforcing extension V [G]. The following claim is key:

Claim 3.2.4. For every a ∈ Hc+ , there are countable towers ~M0 and ~M1 ofcountable elementary submodels of Hc+ with respective largest models max( ~M0)

and max( ~M1) such that

1. a ∈ ~M0(0), ~M1(0), λ ∩max( ~M0) = λ ∩max( ~M1);

2. for every x ∈ X which is PI-generic over each model on ~M0 and ev-ery y ∈ X which is generic over each model on ~M1, it is the case thatFmax( ~M0)(x) 6= Fmax( ~M1).

Note that the functions Fmax( ~M0)(x), Fmax( ~M1) have the same domain, so the

claim really ascertains that they have a disagreement in the values.The construction of the required model M and the Borel set C follows im-

mediately from the claim. By induction on n ∈ ω build countable towers ~Mn,0

and ~Mn,1 such that the pair has the property from the claim and the tow-

ers ~Mn−1,0, ~Mn−1,1 belong to the first models on the n-indexed towers. LetM be the union of all models on the towers; this is an increasing union ofcountable elementary submodels of Hc+ , and so is elementary as well. De-fine C ⊂ 2ω × X by 〈y, x〉 ∈ C if x is PI -generic over every model on all

the towers ~Mn,y(n) : n ∈ ω. This is a Borel set. By the < ω1-properness,the vertical sections of the set C are I-positive. The claim shows that verti-cal sections of C have disjoint FM -images as desired. To see this, let y0 6= y1

and x0, x1 ∈ X be such that 〈y0, x0〉, 〈y1, x1〉 ∈ C and n ∈ ω is such thaty0(n) = 0 6= 1 = y1(n), then Fmax( ~M0

n)(x0) ⊂ FM (x0), Fmax( ~M1n)(x1) ⊂ FM (x1),

Fmax( ~M0n)(x0) 6= Fmax( ~M1

n)(x1), and so FM (x0) 6= FM (x1).

For the proof of the claim, let ~M ′0 be any ω1 tower of countable elementarysubmodels of some large structure such that the first model contains the set a,and let ~M1 be a tower of length 1, containing just one model, which in turncontains the tower ~M ′0. Let δ ∈ ω1 be the intersection of this single model with

ω1 and let ~M0 = ~M ′0 � δ. We claim that the two towers ~M0, ~M1 work as requiredin the claim.

Indeed, let D = { ~M ′0(α) ∩ λ : α ∈ ω1}. This is a closed unbounded set ofcountable subsets of λ. The choice of the name f shows that there must be anelement of D which is not closed under f . The single model on the sequence~M1 contains the set D as an element. Thus, if x ∈ X is a point generic for PI

for the single model on the sequence ~M1, there is α ∈ δ such that the set ~M ′0(α)is not closed under the function f/x. On the other hand, if x is a point generic

for PI for all the models on the sequence ~M1, all the sets ~M0(α) : α ∈ δ areclosed under the function f/x. This concludes the proof of the claim and of thetheorem.

40 CHAPTER 3. TOOLS

Theorem 3.2.5. Suppose that I is a Π11 on Σ1

1 σ-ideal on a Polish space Xsuch that the quotient forcing PI is < ω1-proper in all σ-closed extensions. IfI has the rectangular and selection properties, then I has total canonization foranalytic equivalence relations.

The Π11 on Σ1

1 part of the assumptions is used only to weed out possiblec.c.c. intermediate extensions of the PI -extension in the application of thetrichotomy theorem. The selection property assumption allows for cranking upthe conclusion to the much stronger Silver property of the σ-ideal I, as spelledout in Section 3.3.

Proof. First note that the assumptions imply that there is no model of ZFCstrictly between the ground model and the PI -extension of it. First, Theo-rem 3.2.3 shows that such an intermediate model would have to be c.c.c. andthis possibility is ruled out essentially by the dicussion of all the (very unlikely)cases. A nontrivial c.c.c. intermediate extension must contain a new real byTheorem 2.6.9. Thus, let B ∈ PI be a condition forcing that f(xgen) is such areal for some fixed Borel function f : X → Y . Let J be a σ-ideal on the Polishspace Y defined by A ∈ J ↔ f−1A ∈ I. Since f(xgen) is forced to be a c.c.c.real, falling out of all J-small sets, it must be the case that it is PJ generic. Ifthe ideal I is Π1

1 on Σ11, then so is J . However, the rectangular property rules

out all nontrivial possibilities here, as Fact 2.6.6 shows.Now, let B ⊂ X be a Borel I-positive set and E an analytic equivalence

relation on it. Let x ∈ B be a V -generic point for the poset PI and look atthe trichotomy theorem 3.4.4. The intermediate model case is ruled out by theprevious paragraph. If V [x] = V [x]E then there is an I-positive Borel set C ⊂ Bconsisting of pairwise inequivalent points. On the other hand, if V [x]E = V ,then there is a Borel I-positive set C ⊂ B on which E is ergodic. If every classof E � C was in the σ-ideal I, the rectangular property would have to produceI-positive Borel sets C0, C1 ⊂ C such that C0×C1∩E = 0; this would contradictthe ergodicity. Thus, in this case there has to be a Borel I-positive subset of Cconsisting of pairwise E-equivalent elements, and the theorem follows.

Reviewing the assumptions of the previous theorem, it becomes clear thatvery many interesting ideals are Π1

1 on Σ11, < ω1-proper, and have the selection

property by virtue of being game ideals. Such ideals appear in all the mainsections of [67, Chapter 4], and include, among others, the σ-ideals generatedby Borel collection of closed sets, σ-ideals generated by the sets of finite massfor a fixed Hausdorff measure, or the σ-ideals of sets of zero mass for a fixedpavement submeasure. The main problem on the path to total canonization forthese σ-ideals appears to be the verification of the rectangular property. Alas,this is a difficult problem which in all the interesting borderline cases is open,without any clear guidelines as to how it should be resolved.

Question 3.2.6. What is the status of the rectangular Ramsey property forthe following σ-ideals?

1. the σ-ideal σ-generated by sets of finite mass for a fixed Hausdorff measure;

3.3. A SILVER-TYPE DICHOTOMY FOR A σ-IDEAL 41

2. the σ-ideal σ-generated by sets of continuity of the Pawlikowski function;

3. the σ-ideal of sets of Newtonian capacity zero.

However, the connection between the rectangular Ramsey property and can-onization is not as tight as the previous theorem may suggest. Total canoniza-tion may hold even where the rectangular property fails–such is the case of theσ-ideal σ-generated by closed sets of zero Lebesgue mass, see Theorem 5.1.9.Rectangular property and total canonization may fail and still there may beuseful canonization features–such is the case for the Laver ideal of Section 5.13.

3.3 A Silver-type dichotomy for a σ-ideal

The main purpose of the selection property and the integer games in this bookis that they can be used to upgrade total canonization results to Silver-styledichotomies, which are significantly more striking. Recall from the introduction:

Definition 3.3.1. A σ-ideal I on a Polish space X has the Silver property fora class E of equivalence relations if for every I-positive analytic set B ⊂ X andevery equivalence relation E ∈ E on the set B, either B can be decomposedinto countably many equivalence classes and an I-small set or there is a BorelI-positive set C ⊂ B such that E � C = id. If E is equal to the class of all Borelequivalence relations, it is dropped from the terminology and we speak of theSilver property of I instead.

This should be compared with the classical Silver dichotomy 2.2.6, whichestablishes the Silver property for the ideal of countable sets. Observe thatunlike total canonization, the Silver property introduces a true dichotomy: thetwo options cannot coexist. It also has consequences for undefinable sets: if anequivalence relation E ∈ E has an I-positive set A ⊂ X consisting of pairwiseE-inequivalent points, then it has an I-positive Borel set B ⊂ X consisting ofpairwise E-inequivalent points, simply because the first clause of the dichotomycannot hold in such circumstances.

Let us offer a perhaps artificial reading of the dichotomy which neverthelessfits well with the techniques developed in this book or [67]. Given a σ-ideal Ion a Polish space X and a Borel equivalence E on X, consider the ideal I∗ ⊃ Iσ-generated by E-equivalence classes and sets in I. Then either I∗ is trivial,containing the whole space, or else the quotient forcing PI∗ is equal to PI belowsome condition.

Proposition 3.3.2. Suppose that I is a Π11 on Σ1

1 σ-ideal on a Polish spaceX such that every analytic I-positive set contains a Borel I-positive subset andthe quotient poset PI is proper. The following are equivalent:

1. I has total canonization and the selection property;

2. I has the Silver property;

42 CHAPTER 3. TOOLS

3. I has the Silver property, where I is the σ-ideal on X × ωω consisting ofthe sets with I-small projection.

If the σ-ideal is ∆12 on Σ1

1, then the conclusion of the proposition remains inforce under the additional assumption that ω1 is inaccessible to the reals.

Proof. First, the (1)→(3) direction. Let E be a Borel equivalence relation onX × ωω, let A ⊂ X × ωω be an analytic set and consider the set C = {x ∈ A :[x]E ∩A /∈ I}; this set is again analytic by the definability assumption on I, orif the σ-ideal I is merely ∆1

2 on Σ11, then the set C is ∆1

2.

Case 1. Either there are only countably many equivalence classes in the setC. In this case, the set C is even relatively Borel in A and A \C is analytic. IfA \ C ∈ I then we are in the first clause of the Silver property. On the otherhand, if A \ C /∈ I then the properness assumption shows that there is a BorelI-positive set B ⊂ p(A \ C) and a Borel function f : B → A \ C such that forevery x ∈ X, x is the first coordinate of f(x). Let F be the equivalence relationon B given by x F y iff f(x) E f(y). The equivalence F has I-small classesby the definition of the set C; the total canonization yields a Borel I-positivesubset B′ ⊂ B consisting of pairwise F -inequivalent elements. We may arrangethat f ′′B′ is Borel by thinning out B′ if necessary; the set f ′′B′ ⊂ A witnessesthe second clause of the Silver property.

Case 2. If Case 1 fails, then the classical Silver dichotomy [52] in the caseof a Π1

1 on Σ11 σ-ideal, or Theorem 3.1.12 in the case of a ∆1

2 on Σ11 σ-ideal,

provides a perfect set P ⊂ C of pairwise inequivalent points. Let D ⊂ P ×X bethe set defined by 〈y, x〉 ∈ D ↔ Ax∩ [y]E 6= 0; this is a Borel set with I-positivevertical sections. The selection property of the σ-ideal I yields a partial Borelfunction f : P → X such that ∀y ∈ dom(f) 〈y, f(y)〉 ∈ D and rng(f) /∈ I. Theproperness assumption applied through Fact 2.6.2 yields a Borel I-positive setB ⊂ rng(f) and a Borel function g : B → A such that for every x ∈ dom(g),the point g(x) ∈ Ax belongs to some [y]E such that x = f(y). Thinning out Bif necessary, we may assume that the set g′′B is Borel. It witnesses the secondclause of the Silver property!

The implication (3)→(2) is immediate from the definitions; thus, we areleft with the (2)→(1) implication. The Silver property clearly implies totalcanonization. If E is a Borel equivalence relation on X and A ⊂ X is an I-positive analytic set, then either A decomposes into countably many equivalenceclasses and a set in I, in which case one of the equivalence classes in A must beI-positive, or A contains a Borel I-positive subset consisting of pairwise inequiv-alent elements. In both cases, the total canonization follows. The verification ofthe selection property is significantly harder. Let A ⊂ 2ω×X be an analytic setwith I-positive vertical sections. Consider the set B = {x ∈ X :the horizontalsection Ax is uncountable}; this is an analytic set. There are two cases.

Case 1. Either, B ∈ I. By the first reflection theorem 2.1.2, there is aBorel set C ⊂ X such that B ⊂ X \ C ∈ I. By the first reflection theoremagain, there is a Borel set A′ ⊃ A ∩ (2ω × C) with nonempty and countablehorizontal sections. By the Novikov uniformization theorem 2.1.3, there are


Borel functions fn : C → 2ω for n ∈ ω such that the inverses of the graphsof fn cover the set A′. Let Fn be the equivalence relation on C ⊂ X definedby x Fn y if fn(x) = fn(y). Let Cn = {x ∈ C : 〈fn(x), x〉 ∈ A}. The Silverdichotomy for the σ-ideal I yields a split into two subcases.

Case 1a. Either there is a number n ∈ ω and an I-positive Borel setD ⊂ Cn such that D consists of fn-inequivalent elements. In that subcase, theproperness of the quotient forcing PI can be used to thin out D if necessary toget rng(fn � D) to be a Borel set as in Fact 2.6.2(3), and the inverse of fn � Dwill be the function required in the selection property.

Case 1b. Or, for every number n ∈ ω, the set Cn decomposes into countablymany Fn-equivalence classes (sets {x ∈ Cn : fn(x) = ymn } for some pointsymn ∈ 2ω) and an I-small set Dn ⊂ ω. This subcase is impossible, though: letz ∈ 2ω \ {ymn : n,m ∈ ω} be an arbitrary point, and let x ∈ Az ∩ C \

⋃nDn

be another arbitrary point. The ordered pair 〈z, x〉 must then both belong toA and not belong to it, which yields a contradiction.

Case 2. Or, B /∈ I. Thinning out the set B, we may assume that it isBorel. By a Shoenfield absoluteness argument, B forces in PI the set Axgen

to be uncountable. By a result of Velickovic and Woodin, [62, Theorem 3], inproper extensions adding new reals the set of ground model points does notcontain a nonempty perfect subset. Therefore, the proper forcing PI forces thatthe analytic set Axgen contains a point x which is not in the ground model.Thinning out the set B further if necessary, we may assume that there is aBorel function f : B → X such that B x = f(xgen). Since x is forced notto belong to the ground model, preimages of singletons under the function fmust be I-small, and another application of the Silver dichotomy for I showsthat we may thin out B such that f � B is one-to-one. Thinning out further ifnecessary, we may assume that for every x ∈ B, 〈f(x), x〉 ∈ A and rng(f � B)is Borel. It is clear that the inverse of f � B yields a function as required in thedefinition of the selection property.

A quick perusal of the argument shows that the proposition in fact holds evenif restricted to any class of Borel equivalence relations closed under reduction.Thus for example the ideal I(Fin × Fin) introduced in Section 5.1b is Π1

1

on Σ11, has the selection property, and has canonization for equivalences below

EKσ , even though it perhaps does not have total canonization. The proof showsthat in fact I(Fin × Fin) has the Silver property for the class of equivalencesreducible to EKσ .

The Silver property of a σ-ideal can be abstractly cranked up to an ostensiblystronger statement recently introduced by Ben Miller:

Definition 3.3.3. Let I be a σ-ideal on a Polish space X. We say that I hasthe finite multiple Silver property if for every analytic set A ⊂ X, every n ∈ ω,and every collection {Fi : i ∈ n} of Borel equivalence relations on X, eitherA can be covered by contably many equivalence classes of the various equiva-lences and an I-small set, or it contains a Borel I-positive subset consisting ofelements pairwise simultaneously Fi-inequivalent for all i ∈ n. The σ-ideal I

44 CHAPTER 3. TOOLS

has the infinite multiple Silver property if this holds even for countably infinitecollections of Borel equivalence relations.

Ben Miller proved what amounts to the infinite multiple Silver property ofthe σ-ideal of countable sets. Here, we will show that the simple Silver propertyautomatically yields the finite multiple Silver property for many σ-ideals I.

Theorem 3.3.4. Suppose that ω1 is inaccessible to the reals. If I is ∆12 on Σ1

1

σ-ideal on a Polish space X with the Silver property, then it also has the finitemultiple Silver property.

The proof relativizes to yield the same result restricted to any class of Borelequivalence relations closed under reducibility. The complexity computations donot seem to yield a general ZFC result even for the class of Π1

1 on Σ11 σ-ideals.

Note that all game σ-ideals are ∆12 on Σ1

1.

Proof. Let I be a ∆12 on Σ1

1 σ-ideal on a Polish space X with the Silver property.Let E be a Borel equivalence relation on X and consider the σ-ideal IE σ-generated by I and all equivalence classes of E. We will show that IE is againa ∆1

2 on Σ11 σ-ideal with the Silver property.

The theorem immediately follows. Suppose that {Fi : i ∈ n} is a finite listof Borel equivalence relations and A ⊂ X is an analytic set. Let Ji be theσ-ideal σ-generated by I and the equivalence classes of the various equivalences{Fj : j ∈ i}, for all i ≤ n. A repetitive use of the previous paragraph shows thatall of these σ-ideals have the Silver property. Now, there are two cases. Either,A ∈ Jn; then set An = A and by downward induction on i ∈ n find countablecollections Ci of equivalence classes of Fi such that the set Ai = Ai+1 \

⋃Ci is

in the σ-ideal Ji. In the end, A ⊂ A0∪⋃i<n

⋃Ci is covered by countably many

classes of the various equivalence relations and an I-small set. Or, A /∈ Jn; thenset An = A and by downward induction on i ∈ n use the Silver property of Jito find a Ji-positive Borel set Ai ⊂ Ai+1 consisting of pairwise Fi-inequivalentelements. In the end, the set A0 is a Borel I-positive set consisting of elementspairwise simultaneously inequivalent for all of the equivalence relations on thefinite list.

Now back to the σ-ideal IE . First observe that it is indeed a ∆12 on Σ1

1

σ-ideal. If A ⊂ X is an analytic set, the following are equivalent:

• A ∈ IE ;

• there is a countable set D ⊂ X such that A \ [D]E ∈ I;

• every analytic I-positive subset of A contains two distinct E-equivalentpoints.

The equivalence of the first two items is just the definition of IE , and theequivalence of the other two is exactly the Silver property of I. The seconditem is obviously Σ1

2 by the complexity assumption on I. The third item is Π12:

it asserts that for every analytic set B ⊂ X, either A∩B ∈ I or there are points


x0 6= x1 in A ∩ B with x0 E x1, and the complexity again follows from thecomplexity assumption on I. (It is exactly this complexity computation thatseems to fail if we attempt to prove a ZFC theorem for Π1

1 on Σ11 σ-ideals only.)

The main point of the proof is the verification of the Silver property of theσ-ideal IE . So let F be a Borel equivalence relation on the space X and letA ⊂ X be an analytic set. Consider the set C = {x ∈ A : [x]F ∩ A /∈ IE}.This is a ∆1

2 set by the previous paragraph. By Theorem 3.1.12, either C iscovered by countably many F -equivalence classes, or E contains a perfect setof F -inequivalent elements.

Case 1. If C is covered by countably many F -equivalence classes, then it isrelatively Borel in A and the set A\C is still analytic. By the Silver property ofI, there are two subcases. Either A \ C ∈ IE , and the verification of the Silverproperty for IE , A, and F is complete. Or, A \ C contains a Borel I-positivesubset D consisting of pairwise E-inequivalent elements. Now, the definition ofthe set C implies that every F -equivalence class in the set D is in I, and wecan apply the Silver property of the σ-ideal I to find a Borel I-positive subsetD′ ⊂ D consisting of F -inequivalent elements. This set must be IE-positive, andso the verification of the Silver property of IE in this subcase is again complete.

Case 2. There is a perfect set D0 ⊂ C consisting of pairwise F -inequivalentelements. We will now work towards a Borel I-positive set D′ ⊂ C consistingof pairwise simultanously E- and F -inequivalent elements. This will be the IE-positive Borel set witnessing the Silver property of IE in this case. There areagain two (not mutually exclusive) subcases.

Case 2a. There is a perfect subset D1 ⊂ D0 such that the set [D1]F ∩ Acontains only countably many I-positive E-classes. In this subcase, the setZ ⊂ D1 ×X given by 〈y, x〉 ∈ Z if x ∈ A, y F x, and [x]E ∩ [D1]F ∩ A ∈ I isanalytic with pairwise disjoint I-positive vertical sections: each vertical sectionDy is obtained from the IE-positive set [y]F ∩A by removing at most countablymany classes, and therefore it must be I-positive. The Silver property (moreprecisely, the selection property, which is implied by the Silver property byProposition 3.3.2) of the σ-ideal I shows that there must be a Borel I-positiveset D2 which meets each vertical section of the set Z in at most one point, andevery point of D2 belongs to some vertical section of the set Z. Now, such a setD2 consists of pairwise F -inequivalent elements, and the E-classes in it mustbelong to I. The Silver property of I applied again provides an I-positive Borelsubset of D2 consisting of pairwise E-inequivalent elements, and we are done inthis subcase.

Case 2b. In the case that no such perfect subset D1 ⊂ D0 exists, we willuse two claims of independent interest.

Claim 3.3.5. Suppose that ω1 is inaccessible to the reals. Suppose that G isa Borel equivalence relation on a Polish space Y and U ⊂ 2ω × Y is a ∆1

2 setwhose vertical sections cannot be covered by countably many G-classes. Thenthere is a perfect set P ⊂ 2ω and an analytic set U ′ ⊂ U such that none ofits vertical sections indexed by elements of P can be covered by countably manyG-classes.

46 CHAPTER 3. TOOLS

Proof. We will repeatedly use the following simple fact: if Z is Polish andC ⊂ 2ω × Z is a coanalytic set with nonempty vertical sections, then there is anonempty perfect set P ⊂ 2ω and a continuous map g : P → Z such that g ⊂ C.To see this, first use the Kondo uniformization to find a coanalytic total maph : 2ω → Z such that h ⊂ C. Such a map is Baire measurable, since preimagesof open sets are ∆1

2 and therefore have the Baire property by the assumptionabout ω1. Therefore h will be continuous on a dense Gδ set. Let P be anyperfect subset of this dense Gδ set, and let g = h � P .

Towards the proof of the claim, choose a coanalytic set C ⊂ 2ω × Y × ωωwhich projects to U . By the determinacy assumption and Theorem 3.1.12,for every y ∈ 2ω there is a nonempty compact perfect set K ⊂ Y which isa subset of Uy and consists of pairwise G-inequivalent points. Applying thefirst paragraph, thinning out the set K if necessary, we may also produce acontinuous function h : K → ωω such that the formula φ(y,K, h) =“for everyx ∈ K, 〈y, x, h(x)〉 ∈ C (and K consists of pairwise G-unrelated points)” holds.Note that the formula φ(y,K, h) is coanalytic. Applying the first paragraphagain, produce a nonempty perfect set P ⊂ 2ω and a continuous map g : P →K(X × ωω) such that for every y ∈ P , g(y) is a graph of some function h withdomain K such that φ(y,K, h) holds. Let U ′ = {〈y, x〉 : x is in the domain ofg(y)}.

Claim 3.3.6. Suppose that G is a Borel equivalence relation on a Polish spaceY and U ⊂ 2ω × Y is an analytic set such that vertical sections of U cannotbe covered by countably many G-classes. Then there is a partial continuousinjection f : 2ω → Y with perfect domain such that f ⊂ U and rng(f) consistsof pairwise G-unrelated points.

Proof. Note that the σ-ideal J σ-generated by E-equivalence classes is a gameideal by the remarks after Theorem 3.1.4. Therefore, there is a partial Borelselection g ⊂ U whose range is J-positive as in Theorem 3.2.2. The set rng(g)is analytic and cannot be covered by countably many equivalence classes. BySilver dichotomy or Corollary 2.2.7, there is a perfect set P ⊂ rng(g) consistingof pairwise inequivalent elements. The Jankov–von Neumann uniformizationtheorem 2.1.6 yields a Baire measurable inverse g−1 of g on P , which is thencontinuous on a dense Gδ subset P ′ ⊂ P . Let Q be the image of P ′ under g−1,let f = g � Q and if necessary thin out Q to a nonempty perfect subset on whichf is continuous.

Now, choose any Borel bijection π : 2ω × 2ω → D0, and define the set U ⊂2ω × X by 〈y, x〉 ∈ U if there is z ∈ 2ω such that x F π(y, z) and the set[x]E ∩ [π′′({y} × 2ω)]F is I-positive. This is a ∆1

2 set, its vertical sections arepairwise F -unrelated, if 〈y, x〉 ∈ U then [x]E ∩Uy is I-positive and moreover, novertical section of U is covered by countably many E-equivalence classes sinceSubcase 2a fails. Use the first claim above to find a perfect set P ⊂ 2ω and ananalytic subset U ′ ⊂ U such that for every y ∈ P , the vertical section U ′y cannotbe covered by countably many E-classes. The second claim yields a perfect setQ ⊂ P and a continuous function f : Q → X such that f ⊂ U and the range

3.4. EQUIVALENCE RELATIONS AND MODELS OF SET THEORY 47

of f consists of pairwise E-inequivalent elements. Let D ⊂ Q × X be theanalytic set defined by 〈y, x〉 ∈ D if 〈y, x〉 ∈ U ′ and x E f(y). By the selectionproperty of the σ-ideal I again, there is a partial Borel uniformization of theset D whose range is I-positive. The range is the desired analytic I-positive setwhich consists of pairwise E- and F -unrelated elements.

3.4 Equivalence relations and models of set the-ory

Let I be a σ-ideal on a Polish space X. The key idea underlying much of thecurrent of thought in this book is the correspondence between Borel equivalencerelations on X and intermediate forcing extensions of the generic extension givenby the quotient poset PI of Borel I-positive sets ordered by inclusion. Thecorrespondence is easiest to illustrate on smooth equivalence relations. If E is asmooth equivalence on X and f : X → 2ω is a Borel function reducing it to theidentity, then the model V [f(xgen)] depends only on E and not on the choiceof the reduction. However, we need to find a sufficiently general definition thatcovers the nonsmooth case, where most of interest and difficulty lies.

3.4a The model V [x]E

If x ∈ X is a V -generic point, we consider a ZFC model V [x]E that depends onthe E-equivalence class of x, not on x itself. The correct formalization:

Definition 3.4.1. Let E be an analytic equivalence relation on a Polish spaceX. Let x ∈ X be a V -generic point. Let κ be a cardinal greater than (2|P |)V

where P ∈ V is a poset from which the generic extension is derived, and letH ⊂ Coll(ω, κ) be a V [x]-generic filter. The model V [x]E is the collection of allsets hereditarily definable from parameters in V and from the equivalence class[x]E in the model V [x][H].

It is clear that V [x]E is a model of ZFC by basic facts about HOD type models[22, Theorem 13.26]. The basic observation:

Theorem 3.4.2. Let X be a Polish space, E an analytic equivalence relationon it, and let x ∈ X be a V -generic point.

1. V [x]E ⊂ V [x];

2. the definition of V [x]E does not depend on the choice of the cardinal κ orthe filter H;

3. if y ∈ X is a V -generic point such that x E y, then V [x]E = V [y]E.

The correct way to interpret (3) is that V is an inner model of some largeruniverse containing the points x, y. We only need V to be a transitive modelof some large fragment of ZFC; in particular, the case where V is perhaps

48 CHAPTER 3. TOOLS

countable in the larger universe is allowed. A typical context may arise whenV is the transitive collapse of some countable elementary submodel of a largestructure.

Proof. Fix P , a P -name τ ∈ V for the real x and a cardinal κ > 2|P |. Letφ(~v, [x]E , α) be a formula with parameters ~v ∈ V defining a set of ordinals in theColl(ω, κ) extension: a = {α ∈ β : φ(~v, [x]E , α)}. Homogeneity arguments suchas Fact 2.3.1 show that a = {α ∈ β : V [x] |= Coll(ω, κ) φ(~v, [x]E , α)} ∈ V [x].As φ was arbitrary, the first item follows.

For the second item, we will find a different formula θ( ~D, κ, [x]E , α) withparameters in the ground model and another parameter [x]E , which defines a inany other transitive model extending V containing x. This will show that theColl(ω, κ) extension of V [x] has the smallest possible HOD among all transitivemodels extending V containing x. As κ > 2|P | was arbitrary, this will prove thesecond item.

Let ~D = 〈Dα : α ∈ β〉 be the sequence of subsets of P in V defined by p ∈ Dα

if V |= p Coll(ω, κ) φ(~v, [τ ]E , α). Consider the formula ψ( ~D, [x]E , α) saying“there exists a V -generic filter g ⊂ P such that τ/g ∈ [x]E and g ∩ Dα 6= 0”.

The formula θ( ~D, κ, [x]E , α) says “Coll(ω, κ) ψ( ~D, [x]E , α)”; we claim that θworks as desired.

First, observe that in any transitive model extending V , containing x, andcontaining an enumeration of P(P )∩V in ordertype ω, the formula ψ( ~D, [x]E , α)is a statement about illfoundedness of a certain tree for any fixed ordinal α.Namely, let {Cn : n ∈ ω} enumerate all open dense subsets of P in V , letA ⊂ X × X × ωω be a closed set coded in V that projects to the equivalenceE, and let T be the tree of all finite sequences 〈pi, Oi : i ∈ j〉 such that theconditions pi ∈ P form a decreasing sequence such that p0 ∈ Dα, pi ∈ Ci, andOi ⊂ X×X×ωω form a decreasing sequence of basic open neighborhoods codedin V of radius < 2−i with nonempty intersection with A, such that x belongs tothe projection of Oi to the first coordinate, and pi τ belongs to the projectionof Oi in the second coordinate. It is immediate that ψ( ~D, [x]E , α) is equivalentto the statement that T contains an infinite branch.

Second, observe that the validity of the illfoundedness statement does notdepend on the particular enumeration of open dense subsets of P chosen–if onegets an infinite branch with one enumeration, then a subsequence of the condi-tions in P used in this branch will yield a branch for a different enumeration.Therefore, the formula ψ( ~D, [x]E , α) is absolute between all transitive models ofset theory containing V , x satisfying ”P(P )∩V is countable”. Ergo, the formula

θ( ~D, [x]E , α) is absolute between all transitive models containing V and x.Lastly, the formula θ does define the set a in all such models. It is enough

to check that this is so in V [x]. If α ∈ a then θ( ~D, [x]E , α) is satisfied: ifH ⊂ Coll(ω, κ) is a V [x]-generic filter, then ψ will be witnessed by the V -generic filter on the poset P obtained from the point x. On the other hand, ifθ( ~D, [x]E , α) holds, then for every V [x]-generic filter H ⊂ Coll(ω, κ) there is aV -generic filter g ⊂ P in V [x][H] such that τ/g E x and g∩Dα 6= 0, by Fact 2.3.1there is a V [g]-generic filter K ⊂ Coll(ω, κ) such that V [g][K] = V [x][H], and


by the forcing theorem applied in V [g], the model V [g][K] = V [x][H] satisfiesφ(~v, [τ/g]E , α), which is the same as φ(~v, [x]E , α). Thus α ∈ a as required!

For the third item, start with the case when the pair 〈x, y〉 is V -generic. Letκ be a cardinal larger than the density of the poset in the ground model thatyields the pair 〈x, y〉, let H ⊂ Coll(ω, κ) be a V [x, y]-generic filter, and let Wbe the class consisting of all sets hereditarily definable from parameters in Vand the additional parameter [x]E = [y]E in the model V [x, y][H]. Standardhomogeneity arguments using Fact 2.3.1 show that the model V [x, y][H] is aColl(ω, κ)-extension of both V [x] and V [y]. By the second item above, W =V [x]E = V [y]E .

Now look at the situation where x, y are each V -generic but perhaps thepair 〈x, y〉 is not. Let α ∈ V be an ordinal. We will show that Vα ∩ V [x]E =Vα ∩ V [y]E ; as α ∈ V is arbitrary, this will yield the desired conclusion. Let κbe a cardinal in V much larger than α and the posets P,Q ∈ V that are used toobtain the points x, y respectively; the necessary size of κ is determined below.Let G be a Coll(ω, κ)-generic filter over V such that x ∈ V [G]. Let ygen ∈ Vbe the name for the Q-generic point. The model V [G] satisfies the followingsentence φ: for every Q-generic filter H ⊂ Q over V such that ygen/H E x, it isthe case that Vα ∩ V [y]E = Vα ∩ V [x]E . We will show that this is a coanalyticstatement taking as parameters x, a real z coding a large part of V countable inV [G], and a code for the equivalence relation E. By the coanalytic absolutenessbetween the transitive model V [G] and the outer universe containing x, y,G, φwill hold in the outer universe as well. In particular, Vα ∩ V [x]E = Vα ∩ V [y]Eas desired.

To find the complexity of φ, recall the usual rank on Q-names: the canonicalname for the empty set has rank 0, and a Q-name σ has rank ≤ β if all namesmentioned in σ have rank < β. It is a standard fact of forcing theory that Qforces every set in Vα to be a realization of a name of rank < α. The collectionof all Q-names of rank < α forms a set Nα ∈ V , the cardinal κ must be chosenso large that Nα is countable in V [G], and the real z must code all of Nα, Q,and P(Q)∩V . Now in the model V [G], the sentence φ says that for every filterH ⊂ Q meeting all dense subsets of Q in V such that ygen/H E x there is amap f : Nα → (Vα ∩ V [x]E) ∪ {trash} such that

• f(τ) =trash iff there is a condition q ∈ H forcing τ /∈ V [ygen]E ;

• if f(τ), f(σ) ∈ V [x]E then f(τ) = f(σ) iff there is a condition q ∈ Hforcing τ = σ, and similarly for f(τ) ∈ f(σ);

• Vα ∩ V [x]E ⊆ rng(f).

Such a map is obtained as a consequence of the equality V [ygen/H]E = V [x]Eproved previously. Note that given a filter H ⊂ Q such that ygen/H E x, themap f is unique corresponding to the uniqueness of ∈-isomorphism between the(equal) transitive sets Vα∩V [x]E and Vα∩V [ygen/H]E . The coanalyticity of thesentence φ is now an immediate consequence of Lusin theorem: the quantifier”there exists exactly one” yields coanalytic sets [33, Theorem 18.11].

50 CHAPTER 3. TOOLS

The definition and properties of the model V [x]E may be somewhat myste-rious. Several more or less trivial examples will serve as a good illustration ofwhat can be expected. Let E = E0 and consider the cases of the Sacks real, theCohen real, and the Silver real. In the case of Sacks real x ∈ 2ω, a simple den-sity argument will show that there will be a perfect tree T in the Sacks genericfilter whose branches are pairwise E0-inequivalent. Then, x can be defined from[x]E0

as the only point of [T ]∩ [x]E0and therefore V [x] = V [x]E0

. In the case ofCohen generic real x ∈ 2ω, the rational translations of 2ω that generate E0 cantransport any Cohen condition to any other one, and the equivalence class [x]E0

is invariant under all of them, leading to the conclusion that V [x]E0 = V . In thecase of Silver generic real x, the model V [x]E0

is a nontrivial σ-closed genericextension of V generated by the quotient of Silver forcing in which E0-equivalentconditions are identified, roughly speaking–Theorem 5.8.4. On general grounds[22, Lemma 15.43], V [x]E is a generic extension of V , but it is not so easy tocompute the forcing that induces it. In a typical case, it is the poset of BorelE-invariant sets positive with respect to some σ-ideal as in Theorem 3.5.9.

The importance of the model V [x]E is clarified in the main result of thissection, a trichotomy theorem. Its wording requires a definition:

Definition 3.4.3. Let E be an equivalence relation on a Polish space X, let Ibe a σ-ideal on X, and let B ⊂ X be a set. We say that E � B is I-ergodic (orjust ergodic if the σ-ideal is clear from the context) if for all Borel I-positivesets C,D ⊂ B there are E-equivalent points x ∈ C, y ∈ D. The relation E � Bis nontrivially I-ergodic if it is I-ergodic and its equivalence classes are in theσ-ideal I.

Theorem 3.4.4. Suppose that X is a Polish space, I is a σ-ideal on it suchthat the quotient forcing PI is proper, and E is an analytic equivalence relationon X. Let V [G] be a PI-generic extension of V and xgen ∈ X its associatedgeneric point. One of the following holds:

1. (ergodicity) either V = V [xgen]E, and then there is a Borel I-positive setB ⊂ X such that E � B is ergodic;

2. (intermediate extension) or the model V [xgen]E is strictly between V andV [G];

3. (canonization) or V [xgen]E = V [G] and then there is a Borel I-positiveset B ⊂ X such that E � B = id.

The ergodicity is fairly close to saying that the E-saturation of every Borelset either has an I-small intersection with B, or contains all of B up to an I-small set. This is indeed equivalent to (1) if every coanalytic set is either in I orcontains a Borel I-positive set, or if every positive Borel set has a positive Borelsubset whose E-saturation is Borel. Both of these assumptions frequently hold:the former perhaps on the basis of the first dichotomy of the ideal I [67] or a


redefinition of the ideal I, and the latter perhaps on the basis of the quotientPI being bounding and the equivalence E being Kσ.

A trivial way to satisfy the first item is to find an I-positive equivalence classof E. A typical special case in which the ergodicity case holds nontrivially is thatof Laver forcing and the EKσ equivalence relation, see Section 5.13. The Laverforcing generates a minimal forcing extension, prohibiting the second item fromever occurring, but still there is an equivalence relation that cannot be simplifiedto id or ee on any Borel I-positive set. Another interesting special case is theCohen forcing and the F2 equivalence relation, see Theorem 5.1.29. There,the ergodicity is not at all surprising, but we get more information looking atchoiceless intermediate models of ZF as in Section 4.3.

In the second case we get an intermediate forcing extension strictly betweenV and the PI -extension V [G]. For many σ-ideals, the existence of such anintermediate model is excluded purely on forcing grounds, such as the ideal ofcountable sets or σ-compact subsets of ωω, or the more general Theorem 3.2.5.In such cases, we get a neat dichotomy. It is not true that every intermediateextension is necessarily obtained from a Borel equivalence relation in this way,as Section 5.3 shows. In special cases though (such as the countable lengthα ∈ ω1 iteration of Sacks forcing), the structure of intermediate models is well-known (each of them is equal to V [xγ : γ ∈ β] for some ordinal β ≤ α) andthey exactly correspond to certain critical equivalence relations (such as therelations Eβ on (2ω)α connecting two sequences if they are equal on their first βentries). We cannot compute the forcing responsible for the extension V [xgen]Ein a fully general situation, but in all specific cases we can compute, it is theposet of E-invariant I-positive Borel sets ordered by inclusion, as describedin Theorem 3.5.9 and its section. In the common situation that the modelV [xgen]E does not contain any new reals, we can compute the forcing leadingfrom V [xgen]E to V [G]–it is a quotient poset of the form PI∗ for a suitable σ-ideal I∗ ⊃ I which has the ergodicity property of the first item of the theorem.As Asger Tornquist remarked, in this context the theorem can be viewed asa general counterpart to Farrell-Varadarajan ergodic decomposition theorem[31, Theorem 3.3] for actions of countable groups by measure preserving Borelautomorphisms.

Proof ofTheorem 3.4.4. Consider the first case, V [xgen]E = V . In this case,consider the collection K of those Borel sets C ⊂ X coded in V such thatC ∩ [xgen]E 6= 0. This collection is certainly in the model V [xgen]E as we havejust defined it from the class [xgen]E , and therefore it is in V . There must be a

condition B ∈ PI and an element J ∈ V such that B K = J . If there weretwo Borel I-positive sets C0, C1 ⊂ B with [C0]E ∩ [C1]E = 0, then only one,say C0 of these sets can belong to the set L. But then C1 C1 ∈ K \ L! Thiscontradiction verifies the conclusion of the first case of the theorem.

The second possibility is that V ⊂ V [xgen]E ⊂ V [G] and these inclusions areproper. In this case, we are content to fall into the second item of the theorem.

Lastly, assume that V [xgen]E = V [G]; in particular, xgen ∈ V [xgen]E . Then,return to the ground model V and pick large enough cardinals λ ∈ κ. There

52 CHAPTER 3. TOOLS

must be a condition B ∈ PI forcing xgen ∈ V [xgen]E ; so, in the productPI × Coll(ω, κ), the condition 〈B, 1〉 forces xgen to be the only element x ∈ Xsatisfying φ(x, [xgen]E , ~v) for some sequence of parameters ~v ∈ V and a for-mula φ. We must find an I-positive Borel set C ⊂ B consisting of pairwiseE-inequivalent points.

Let H be a Coll(ω, κ)-generic filter over V ; the model V [H] contains a genericextension used in Fact 2.5.1 to produce an elementary embedding j : V → Nto a possibly illfounded model N such that M = j′′Hλ is a countable set in N .Recall that the wellfounded part of the model N is identified with its transitiveisomorph and with this identification, for every analytic set A coded in theground model, j(A) ⊆ AV [H].

Working in N , let C ′ = {x ∈ j(B) : x is j(PI) = Pj(I)-generic over M}.This is a j(I)-positive Borel set by Fact 2.6.1 applied in the model N : theset M = j′′Hλ is a countable elementary submodel of j(Hλ) by the Tarskicriterion, and the poset j(PI) = Pj(I) is proper in N . We shall show thatN |= C ′ consists of pairwise j(E)-inequivalent points. The desired set C ∈ Vwill then be obtained through elementarity of the embedding j.

Suppose N |= x 6= y ∈ C ′. Both points x, y are PI -generic over themodel V : for every maximal antichain A ⊂ PI in V , j(A) is a maximal an-tichain of j(PI), and since the points x, y are PjI -generic over M , there mustbe sets Dx, Dy ∈ A such that N |= x ∈ j(Dx), y ∈ j(Dy). However, themodel N evaluates the membership in ground model Borel sets correctly, sox ∈ Dx, y ∈ Dy. Both points x, y also belong to BV [H] and V [H]. Find filtersHx, Hy ⊂ Coll(ω, κ) generic over the models V [x] and V [y] respectively suchthat V [x][Hx] = V [y][Hy] = V [H]. If the points x, y ∈ X were E-equivalent inV [H], then the formula φ(·, [x]E , ~v) = φ(·, [y]E , ~v) would have to define both xand y in the model V [H] by the forcing theorem, and this is impossible. Thus,V [H] |= ¬x E y. Since j(E) ⊂ E ∩ V [H], we conclude that N |= ¬x j(E) y asdesired.

Theorem 3.4.5. Suppose that X is a Polish space, I a σ-ideal on X suchthat the quotient forcing PI is proper. Let x be a PI-generic point. Then,2ω ∩ V [x]E = {f(x) : f : X → 2ω is a partial Borel function coded in V whichis E-invariant on its domain}.

Proof. For the first part, if f : B → 2ω is a Borel function which is constant oneach E-equivalence class, then certainly B f(xgen) ∈ V [x]E . Namely, f(xgen)

will be definable from [xgen]E as the unique value of f on B ∩ [xgen]E . Notethat the constant property of f survives to every forcing extension by coanalyticabsoluteness. This proves the right-to-left inclusion.

For the opposite inclusion, proceed as in the third case of the trichotomytheorem. Let λ ∈ κ be large regular cardinals. Suppose that 〈B, p〉 ∈ PI ×Coll(ω, κ) is a condition that forces y ∈ V [xgen]E ∩ 2ω. Strengthening thecondition we may assume that there is a formula φ and a sequence ~v ∈ V ofparameters such that it is forced that y is the unique element of 2ω satisfyingφ(y, [xgen]E , ~v). Since V [xgen]E ⊂ V [xgen], we may also strengthen the condition


if necessary to find a Borel function f : X → 2ω such that it is forced that y =f(xgen). We will now find a Borel I-positive set C ⊂ B on which the function gis E-invariant. This will certainly complete the proof of the proposition, sincethen 〈C, p〉 forces that y is the image of the generic real by an E-invariantfunction.

Let H ⊂ Coll(ω, κ) be a generic filter, and in V [H] find the elementaryembedding j : V → N as in Fact 2.5.1; in particular, M = j′′Hλ is a countableset in the model N . Working in the model N , let C ′ = {x ∈ j(B) : x is Pj(I)-generic over M}; this is a Borel j(I)-positive set. We will show that in N , thefunction j(f) is j(E)-invariant on the set C ′; the desired set C is then obtainedby the elementarity of the embedding j. Thus, let x0, x1 ∈ C ′ be j(E)-equivalentpoints in the model N . The two points are E-equivalent in V [H] as j(E) ⊂ E,and they are also PI -generic over V . Find filters H0, H1 ⊂ Coll(ω, κ) genericover V [x0] and V [x1] respectively such that V [H] = V [x0][H0] = V [x1][H1].By the forcing theorem, it must be the case that the formula φ(·, [x0]E , ~v) =φ(·, [x1]E , ~v) must define a point y ∈ 2ω which is the joint value of f(x0) andf(x1) in the model V [H]. Now, by the definitions in the model N , the pointsx0, x1 are in j(B) and therefore in the domain of the function j(f). Sincej(f) ⊂ fV [H], it follows that y must be the unique value of j(f)(x0) and j(f)(x1)in the model N . We conclude that in the model N , the function j(f) is j(E)-invariant on the set C ′, and the theorem follows.

Let I be a σ-ideal on a Polish space X such that the quotient poset PI isproper. Let x ∈ X be a V -generic point for PI . On general forcing grounds [22,Lemma 15.43], it must be the case that V [x]E is a generic extension of V , andV [x] is a generic extension of V [x]E . We do not know which poset is responsiblefor the former extension, even though there is a canonical candidate isolated inthe next section which works in all cases computed in this book. In the commoncase where V [x]E has the same reals as V , we can compute the poset responsiblefor the latter extension.

Definition 3.4.6. Let I be a σ-ideal on a Polish space X, let E be an analyticequivalence relation, and let x ∈ X be a PI -generic point. In the model V [x],let Ix be the collection of ground model coded Borel sets B such that for everycardinal κ, Coll(ω, κ) forces that B contains no point that is simultaneouslyPI -generic over V and E-equivalent to x.

It should be clear that I ⊂ Ix, Ix ∈ V [x]E , and if V [x]E contains no new realsthen Ix is a σ-ideal of Borel sets there. Moreover, if 2ω ∩ V [x]E = 2ω ∩ V , theideal Ix has the ergodicity property in V [x]E : any pair of Ix-positive Borel setscontains a pair of E-connected points. This follows from the fact that in somelarge collapse, each of the sets in the pair contains some points equivalent to x;in particular, such points are equivalent to each other, and analytic absolutenesstransports this feature back to V [x]E .

Theorem 3.4.7. Suppose that X is a Polish space, I a σ-ideal on X such thatthe quotient forcing PI is proper. Let x be a PI-generic point. If 2ω ∩ V =2ω ∩ V [x]E, then V [x]E = V [Ix] and V [x] is a PIx-extension of V [x]E.

54 CHAPTER 3. TOOLS

Proof. We will first show that x is a PIx generic point over V [x]E . Move backto the ground model, and assume for contradiction that a condition 〈B, p〉 ∈PI ×Coll(ω, κ) forces that D ⊂ PIx is an open dense set in the model V [xgen]Esuch that the point xgen does not belong to any of its elements. Strengtheningthe condition if necessary, we may find a formula φ(u, [xgen]E , ~v) with parameters

in V ∪ {[xgen]E} defining the set D: D = {C ∈ PIxgen : φ(C, [xgen]E , ~v)}. Ofcourse, the condition 〈B, p〉 forces B ∈ PIxgen , so there must be a strengtheningB′ ⊂ B, p′ ≤ p and a Borel set C ⊂ B such that 〈B′, p′〉 C ∈ D. Note thatC ∈ PI must be an I-positive Borel set in the ground model.

Find mutually generic filters G ⊂ PI , H ⊂ Coll(ω, κ) with B′ ∈ G, p′ ∈ H,let xgen ∈ X be the point associated with the filter G, use the definition ofthe ideal Ixgen to find a point y ∈ C ∩ V [G,H] which is PI -generic over V andequivalent to xgen, and use the homogeneity features of the poset Coll(ω, κ) as inFact 2.3.1 to find mutually generic filters G ⊂ PI , H ⊂ Coll(ω, κ) such that y isthe generic point associated with G (so C ∈ G), p ∈ H, and V [G,H] = V [G, H].

Since the points xgen, y are E-equivalent, the definition of the set D is evalu-ated in the same way with either xgen or y plugged into the definition. Applyingthe forcing theorem to the filters G, H, there must be a condition in the filterG × H stronger than 〈B, p〉 which forces C ∈ D and xgen ∈ C. However, thisdirectly contradicts the statement forced by the weaker condition 〈B, p〉!

To show that V [x]E = V [Ix], note that Ix ∈ V [xgen]E , so the right to leftinclusion is immediate. For the opposite inclusion, assume that a ∈ V [x]Eis a set of ordinals, perhaps defined as a = {α : φ(α,~v, [x]E)} in the modelV [x,H], where ~v are parameters in the ground model and H ⊂ Coll(ω, κ). Letb = {α : ∃B ∈ PVI , B /∈ Ix B Coll(ω, κ) φ(α, ~v, [xgen]E)} ∈ V [Ix] and showthat a = b. It is immediate that a ⊂ b since the filter defined by the PI -genericpoint x ∈ X has empty intersection with Ix. On the other hand, if α ∈ b aswitnessed by a set B then there is a PI -generic point y ∈ B ∩ [x]E over V , bythe usual homogeneity argument there is a V [y]-generic filter Hy ⊂ Coll(ω, κ)such that V [y][Hy] = V [x][H], by the forcing theorem V [y][Hy] |= φ(α,~v, [x]E),and therefore α ∈ a. The theorem follows.

3.4b The model V [[x]]E

The model V [x]E has a larger, less well understood, possibly choiceless com-panion V [[x]]E . In spirit, this larger model is the intersection of all modelscontaining V and a point E-equivalent to x. In formal language,

Definition 3.4.8. Let E be an analytic equivalence relation on a Polish spaceX. Let x ∈ X be a V -generic point. The model V [[x]]E consists of all setsa ∈ V [x] such that the model V [x] satisfies: for every ordinal κ, Coll(ω, κ)forces that for every y ∈ X with x E y, a ∈ V [y].

Theorem 3.4.9. Let E be an analytic equivalence relation on a Polish spaceX, and let x ∈ X be a V -generic point.


1. V [[x]]E is a model of ZF with V [x]E ⊆ V [[x]]E ⊆ V [x];

2. if x, y ∈ X are V -generic E-equivalent points, then V [[x]]E = V [[y]]E;

Proof. Start with the second item and the special case where the pair 〈x, y〉 isV -generic. Let φ(x) be the definition of the model V [[x]]E as described above.We will argue that for every ordinal κ, Coll(ω, κ) forces that for every pointy ∈ X which is E-equivalent to x, the formula φ(y) defines the same class inV [y] as φ(x) defines in V [x]. We will show that {a ∈ V [x] : V [x] |= φ(a, x)} ⊂{a ∈ V [y] : V [y] |= φ(a, y)}; the other inclusion has the same proof. Supposethat a ∈ V [y] and V [y] |= φ(a, y). Then a ∈ V [x], since otherwise V [x] forms acounterexample to V [y] |= φ(a, y) inside some large common generic extensionof both models V [y] and V [x]. Also, V [x] |= φ(a, x), because otherwise in somecommon collapse forcing extension of V [x] and V [y], there is a counterexampleto φ(a, x) which is also a counterexample to φ(a, y). The general case of (2)now follows just as in the proof of Theorem 3.4.2(3).

To show that V [[x]]E is a model of ZF, argue that L(V [[x]]E) = V [[x]]E .Indeed, for every ordinal κ it is the case that Coll(ω, κ) forces that L(V [[x]]E)is contained in every model V [y] where y E x, since as argued in the previousparagraph, the models V [y] all contain V [[x]]E as a class. But then, everyelement of L(V [[x]]E) belongs to V [[x]]E by the definition of V [[x]]E .

To show that V [x]E is a subset of V [[x]]E , let a ∈ V [x]E be a set of ordinalsand let V [G] be a generic extension of V inside a generic extension of V [x]containing a point y ∈ X equivalent to x; we must argue that a ∈ V [G]. Let Pbe a poset generating that extension, and let κ = |P(P )|. There is a commonColl(ω, κ) extension V [H] of both V [G] and V [x]. Then a is definable in V [H]from parameters in V and perhaps the additional parameter [x]E = [y]E . SinceV [H] is a homogeneous extension of the model V [G] and all parameters of thedefinition are themselves definable from parameters in V [G], it must be the casethat a ∈ V [G] as desired, and the proof is complete.

Theorem 3.4.10. Let I be a σ-ideal on a Polish space X such that the quotientforcing PI is proper. Let E be an analytic equivalence relation on X, and letx ∈ X be a PI-generic point. Then 2ω ∩ V [[x]]E = {f(x) : f is a Borel functioncoded in V which on its domain attains only countably many values on eachequivalence class}.

An important observation complementary to the wording of the theorem: thestatement that a given Borel function f attains only countably many values oneach E-equivalence class is Π1

1 in the code for the function and the equivalencerelation, and therefore it is invariant under forcing by Shoenfield’s absoluteness.To see this, note that every E-equivalence class is analytic, its image under fis analytic, and so if countable it contains only reals hyperarithmetic in anyparameter that can define it as an analytic set. Thus, f attains only countablymany values on each E-equivalence class if and only if for every pair of pointsx, y ∈ dom(f), either x, y are not equivalent or f(x) is hyperarithmetic in y, thecode for f and the code for E. This is a coanalytic formula by the usual codingof ∆1

1 sets as in [27, Theorem 2.8.6].

56 CHAPTER 3. TOOLS

Proof. First, observe that if f : B → 2ω is a Borel function attaining onlycountably many values on each equivalence class, and x ∈ X is any point in aforcing extension of the ground model V , then the model V [x] contains someenumeration of the countable set f ′′([x]E ∩B) by Shoenfield absoluteness, andtherefore contains all elements of this countable set. This proves the right-to-leftinclusion.

For the opposite inclusion, suppose that B ⊂ X is an I-positive Borel setforcing y ∈ V [[xgen]]E ∩ 2ω. Let M be a countable elementary submodel ofa large structure containing B and the name y, and let C ⊂ B be the BorelI-positive set of all PI -generic points over M . Let f : C → 2ω be the functiondefined by f(x) = y/x; thus f is a Borel function and C f(xgen) = y. We willprove that f attains only countably many values at every E-equivalence classin C. By Theorem 3.4.9 applied to the transitive isomorph of M , whenever

x0, x1 ∈ C are E-related points, then V [[x0]]M [x0]E = V [[x1]]

M [x1]E . The model

V [[x0]]M [x0]E is countable, and it must contain the real y/x1. Thus the countable

model V [[x0]]M [x0]E contains all reals y/x where x ∈ C is a point E-related to

x0, and the proof is complete.

The axiom of choice may fail in the model V [[x]]E , and the study of itsfailure may shed light on the canonization properties of the equivalence relationE. The previous claim puts a slight restriction on the type of choice failure onecan get:

Corollary 3.4.11. Let I be a σ-ideal on a Polish space X such that the quotientposet PI is proper, let E be an analytic equivalence relation on X, and let x ∈ Xbe a PI-generic point over V . Then V [[x]]E satisfies the following sentence:every set A ⊂ ω × 2ω with nonempty vertical sections has a subset A′ ⊂ A withnonempty vertical sections such that ℵ1 6≤ |A′|.

Example 3.4.22 provides a model of ZF where the above sentence fails.

Proof. Let B ∈ PI force A ⊂ ω × 2ω is a set in V [[xgen]]E with nonemptyvertical sections. Let M be a countable elementary submodel of a large structurecontaining B, and let C be the set of PI -generic points over M in B. A genericityargument together with the previous claim will show that for every numbern ∈ ω, the set C is covered by a countable collection of Borel sets {Din :i ∈ ω} such that every Din is in the model M and there is a Borel functionfin : Din → 2ω which takes at most countably many values at each E-class andDin 〈n, fin(xgen)〉 ∈ A. Let A′ be the name for the set of all pairs 〈n, y〉 suchthat y = fin(x) for some i ∈ ω and some point x E xgen. It is immediate thatA′ ∈ V [[x]]E . Also, the set A′ is countable in V [xgen] and so V [[xgen]]E cannotsatisfy ℵ1 ≤ |A′|.

The structure of the model V [[x]]E turns out to be much easier to understandin the case where the equivalence E is reducible to the orbit equivalence of aPolish group action. If the acting group is a closed subgroup of S∞, thenthe model is in fact equal to V (a) for some hereditarily countable and fairly


canonically chosen set a as proved in Theorem 4.3.11. In the case of a generalPolish group action, we have less information: the model V [[x]]E is a HODtype model like V [x]E , and the difference between the two models resides in thehereditarily countable sets. Let x ∈ X be a PI -generic point over V , and let Zbe the set of all sets in V [[x]]E that are hereditarily countable in V [x]. Let κ bea suitably large cardinal, let H ⊂ Coll(ω, κ) be a filter generic over V [x], andin the model V [x][H] form the model V [[[x]]]E of the sets hereditarily definablefrom parameters in V ∪ Z and from the parameter [x]E and Z.

Proposition 3.4.12. The set Z is definable in V [x][H] from the parameter [x]E.Moreover, the model V [[[x]]]E does not depend on the choice of the cardinal κor the generic filter H.

Proof. For the first sentence, work in the model V [x][H] and recall the definitionof the set Z: z ∈ Z iff z ∈ V [[x]]E and z is hereditarily countable in V [x]. Thefirst conjunct depends only on the equivalence class [x]E by Theorem 3.4.9(2).The second conjuct is equivalent to φ(z) =“for every y ∈ [x]E , z is hereditarilycountable in V [y]”, which again clearly depends only on [x]E as required. How-ever, the equivalence of φ(z) with hereditary countability of z in V [x] needs anargument.

Clearly, φ(z) implies the hereditary countability of z in V [x]. On the otherhand, assume that z ∈ V [[x]]E is a set hereditarily countable in V [x]; we mustverify that z is hereditarily countable in every model V [y] where y E x. As xis PI -generic over V , a density argument will show that there must be a BorelI-positive set B ⊂ X coded in V , a Borel function f : B → 2ω×ω coded inV such that trcl(z) is equal to the transitive isomorph of the relation f(x),and B the transitive isomorph of f(xgen) ∈ V [[xgen]]E , and there must be acountable elementary submodel M ∈ V of some large structure containing f,Bsuch that x is a PI -generic point over M . Now let y ∈ X be an arbitrary pointE-related to X. By the analytic absoluteness between V [y] and V [x][H], theremust be a point x′ ∈ V [y] which is in B E-related with y and PI -generic overM–in the model V [x][H], x′ = x is such a point. Now, the models M [[x]]E andM [[x′]]E coincide by Theorem 3.4.9(2) applied to the model M . The transitiveclosure of z (as the transitive isomorph of f(x)) belongs to M [[x]]E and thereforeto M [[x′]]E , and the model M [[x′]]E belongs to V [y] and is countable there–itis a subset of the model M [x′] and a generic extension of a countable model isagain countable. All in all, V [y] |= z is hereditarily countable.

The second sentence is proved in the same way as Theorem 3.4.2.

Theorem 3.4.13. Let I be a σ-ideal on a Polish space X. Let E be an analyticequivalence relation on the space X, Borel reducible to an orbit equivalence of aPolish group action. Let x ∈ X be a PI-generic point over V . Then V [[x]]E =V [[[x]]]E.

The argument will go through for equivalence relations somewhat more compli-cated than just orbit equivalence relations, such as E1. For equivalence relationsof the complexity EKσ and beyond, the argument breaks down entirely and we

58 CHAPTER 3. TOOLS

must replace it by ad hoc fixes specific to the given pair I, E. We do not havean example of a situation where the conclusion of the theorem fails.

Corollary 3.4.14. If in addition to the assumptions of Theorem 3.4.13, 2ω ∩V [[x]]E = 2ω ∩ V [x]E holds, then V [[x]]E = V [x]E.

Proof. If V [[x]]E and V [x]E contain the same reals, then they also contain thesame hereditarily countable-in-V [x] sets. To see this, assume for contradictionthat a ∈ V [[x]]E is a set of minimal rank with countable transitive closure suchthat a /∈ V [x]E . Then a ⊂ V [x]E , there is b ∈ V [x]E which is countable inV [x]E such that a ⊂ b and there is an enumeration π : ω → b in V [x]E ; theset π−1a is in V [[x]]E and therefore in V [x]E , and so a ∈ V [x]E , contradictingthe assumption. Therefore, V [[[x]]]E = V [x]E and by the theorem, V [[x]]E =V [[[x]]]E = V [x]E as desired.

Corollary 3.4.15. If in addition to the assumptions of Theorem 3.4.13, clas-sifiable by countable structures→Ismooth holds, then V [[x]]E = V [x]E.

Proof. By virtue of the previous corollary, it is enough to show that V [[x]]Eand V [x]E agree on the reals. Suppose that B ∈ PI is a condition forcingy ∈ V [[xgen]]E ∩ 2ω. Let M be a countable elementary submodel of a largestructure containing B, and let F be the equivalence relation on the Borel I-positive set C ⊂ B of all PI -generic points over M in B defined by x0 F x1 ifV [[x0]]M [x0] = V [[x1]]M [x1]. It is not difficult to verify that this is an equivalencerelation classifiable by countable structures. By Theorem 3.4.9(2) applied tothe transitive isomorph of M , it is even the case that E ⊆ F . Now use theassumption of the corollary to find a Borel I-positive set D ⊂ C on whichF is smooth, reduced to the identity by some Borel function g : D → 2ω.Since E ⊆ F , the function g is constant on E-equivalence classes and so D g(xgen) ∈ V [xgen]E by Theorem 3.4.5. Now, if x ∈ D is a PI -generic point overV , the model V [x]E contains g(x), and by an absoluteness argument contains

also V [[x]]M [x]E , which in turn contains the point y/x.

Proof. For the proof of Theorem 3.4.13, let x be a PI -generic point over V andlet W = V [[[x]]]E .

For the inclusion W ⊂ V [[x]]E , assume for contradiction that there is a seta ∈ W which does not belong to V [[x]]E , and let a be an ∈-minimal such set.We will reach a contradiction by showing that a belongs to every model V [y]where y ∈ X is a V [x]-generic point E-related to x; this implies that a ∈ V [[x]]Eby the definitions. Let φ be a formula defining the set a in some large collapseextension of V [x] from some parameters ~v ∈ V and [x]E ; we can in fact findsuch a formula that defines a in every large collapse extension of V [x] just as inthe proof of Theorem 3.4.2. Let λ be a cardinal larger than the size of the posetadjoining the point y over the model V [x]. Let b ∈ V [y] be the set of all sets csuch that V [y] |= Coll(ω, λ) φ([y]E , ~v, c); we will show that b = a. And indeed,let Hx, Hy ⊂ Coll(ω, κ) be filters generic over V [x] and V [y] respectively suchthat V [x][Hx] = V [y][Hy]. In these two equal models, the formula φ must define


the same set and the set defined is a. It follows that b = a ∩ V [y]. However, allelements of a are in V [[x]]E by the minimality choice of a, so they are in V [y].It follows that b = a ∩ V [y] = a as desired.

For the other inclusion, suppose that a ∈ V [[x]]E is a subset of W ; we mustshow that a ∈ W . By a genericity and properness argument, there must be inV a countable elementary submodel M of a large structure such that x is PI -generic over M and M contains a name for the set a. Let M [x] = {σ/x : σ ∈Mis a PI -name}. We have to show that the intersection M [x] ∩ V [[x]]E does notdepend on the particular choice of x in its equivalence class:

Claim 3.4.16. In every forcing extension of the model V [x], whenever x′ ∈X is a point PI-generic over M and V , E-related to x, it is the case thatM [x] ∩ V [[x]]E = M [x′] ∩ V [[x]]E.

Proof. Suppose first that there is a poset Q ∈ M and a filter H which is Q-generic over both M and V such that x, x′ both belong to M [H]. Then byProposition 2.3.4(3), M [H] ∩ V [x] = M [x] and M [H] ∩ V [x′] = M [x′]. Now,since the model V [[x]]E = V [[x′]]E is in both models V [x] and V [x′], this meansthat M [x] ∩ V [[x]]E = M [H] ∩ V [[x]]E = M [x′] ∩ V [[x]]E as required.

Now, in the case that the pair 〈x, x′〉 is not as assumed in the first paragraph,we will find another point x′′ ∈ X which is E-equivalent to both x and x′ andsuch that each of the pairs 〈x, x′′〉 and 〈x′, x′′〉 is as in the first paragraph.Then the previous argument shows that M [x] ∩ V [[x]]E = M [x′′] ∩ V [[x]]E =M [x′] ∩ V [[x]]E as required. The existence of x′′ will be obtained from theassumption that E is Borel reducible to an orbit equivalence of a Polish groupaction. On the other hand, if EKσ were Borel reducible to E then such a pointx′′ may not exist.

To find the point x′′, let f : X → Y be a Borel map reducing the equivalencerelation E to the orbit equivalence relation of a continuous action of some Polishgroup G, all of this in V and M . Let h ∈ G be a group element in V [x, x′] suchthat h · f(x) = f(x′). Let g ∈ G be a PJ -generic point over the model V [x, x′],where J is the meager ideal on the Polish group G. Then gh is a PJ -generic pointover V [x, x′] as well, since the right multiplication by h is a selfhomeomorphismof the group G. Since the notion of a maximal antichain in PJ is absolutebetween wellfounded models of set theory, the point gh is PJ -generic over bothV [x] and M [x], and the point g is PJ -generic over both V [x′] and M [x′]. Lety = gh · f(x) = g · f(x′); so y ∈M [x][gh]∩M [x′][g]. By coanalytic absolutenessbetween the models V [x], V [y], and M [y], there must be a point x′′ ∈ X ∩M [y]such that f(x′′) is orbit equivalent to y. This point x′′ works: it is E-equivalentto both x and x′′, and each of the pairs 〈x, x′′〉 and 〈x′, x′′〉 belongs to somePI × PJ extension of both M and V as required in the first paragraph.

Claim 3.4.17. M [x] ∩W ∈W and a ∩M [x] ∈W .

Proof. Let N = M [x] ∩W . Move to some large collapse extension V [H] of themodel V [x]. As per the claim and the inclusion W ⊂ V [[x]]E , N is the unique

60 CHAPTER 3. TOOLS

possible value for M [x′] ∩W as x′ ranges over all points in V [H] that are PI -generic over M and V and E-equivalent to x. This provides a definition of theset N that will put it into W .

Let π : N → N be the inverse of the Mostowski collapse of the model N .Clearly, N , π ∈ W and since W ⊂ V [[x]]E , it is the case that both of theseobjects are in V [[x]]E . Thus, π−1a ∈ V [[x]]E . Moreover, N , π−1a are bothhereditarily countable sets in the model V [x] and therefore they belong to theset Z, and to the model W . Thus, the set π′′π−1a = a ∩N belongs to the setN .

Now, by the elementarity of the model M [x], there is only one subset of Wwhich is an element of M [x] and whose intersection with M [x]∩W is a∩M [x],and it is the set a. The claim now shows that in every forcing extension ofV [x], the set a is the unique subset c ⊂ W such that for every point x′ PI -generic over M and PI , E-related to x, it is the case that c ∈ M [x′] andc ∩M [x′] = a ∩M [x]. This yields a definition of the set a from the parametersM ∈ V and a ∩M [x] ∈W , and so a ∈W .

One consequence of the proof is that in the model V [x], whenever a ∈ V [[x]]Eis a set then the set [a]ℵ0 ∩ V [[x]]E ⊂ [a]ℵ0 is stationary. This does not holdfor general intermediate choiceless extensions, as the model in Example 3.4.22shows.

3.4c Comparison with the symmetric models of ZF

Let I be a σ-ideal on a Polish space X, and let G be a Borel group acting in aBorel I-preserving way on the space X. Then for every Borel set B ⊂ X andevery g ∈ G the set g · B = {g · x : x ∈ B} is Borel again, since it is a Borelinjective image of B, and B ∈ I ↔ g · B ∈ I. It follows that the group acts onthe poset PI by automorphisms. The action then can be immediately extendedto the Boolean model V PI : by induction on the rank of the PI -name τ , defineg · τ as the set {〈g ·B, g · σ〉 : 〈B, σ〉 ∈ τ}.

In such a situation, one can define symmetric models of ZF (typically withoutthe axiom of choice) as in [22, Lemma 15.51 ff]. Suppose that F is a normalfilter of subgroups of G: a collection of subgroups of G closed under superset,finite intersection, and conjugation. For a name τ , define supp(τ), the supportof τ , to be the stabilizer of τ , say that the name is F -symmetric if its supportis in F , and hereditarily F -symmetric if the supports of all names mentioned inτ are symmetric. Then, for a PI -generic point x ∈ X over V , one can form thesymmetric model V [x]G,F = {τ/x : τ is a hereditarily F -symmetric PI -name}.We have

Fact 3.4.18. [22, Lemma 15.51] The symmetric model is a model of ZF.

In this section we address the question whether the models V [x]E , V [[x]]E in-troduced in the previous section, where E is the orbit equivalence relation on X


generated by the action of the group G, have any relationship with the symmet-ric models obtained from the action. The general answer appears to be highlystructured.

Theorem 3.4.19. Let G be a closed subgroup of S∞, acting continuously on aPolish space X. Let F be the filter generated by clopen subgroups of G, and let Ibe the σ-ideal of meager subsets of X. Then PI forces V [[xgen]]E = V [xgen]G,F .

Note that the closed subgroups of S∞ are characterized as those Polish groupspossessing a neighborhood basis around the unit consisting of clopen subgroups[14, Theorem 2.4.1]. Note also that every continuous group action is an actionby homeomorphisms, and thefore preserves the σ-ideal I. A number of classi-cal choiceless models of set theory are obtained in a way compatible with thetheorem.

Example 3.4.20. [22, Example 15.52] Let S∞ act on (2ω)ω by composition.The resulting model V [[xgen]]E is the classical Cohen model for the failure ofthe axiom of choice. The prime ideal theorem holds in that model by the workof Halpern and Levy; in particular, there is an ultrafilter on ω in the model.

Example 3.4.21. Let K be the countable group of al eventually constant in-finite binary sequences that are eventually constant, with coordinatewise bi-nary addition and discrete topology. Let G = Kω act on (2ω)ω by coor-dinatewise addition. The resulting model V [[xgen]]E does not contain anynonprincipal ultrafilters on ω. Consider the ω-sequence a of sets given bya(n) = {[xgen]E0 , [1 − xgen]E0}. An elementary symmetricity argument showsthat a ∈ V [[xgen]]E and at the same time there is no choice function f ∈ V [[xgen]]that would choose one element from each pair a(n). It follows that V [[xgen]]Ehas no nonprincipal ultrafilter on natural numbers–such an ultrafilter wouldimmediately yield a choice function.

Example 3.4.22. Jech [22, Example 15.59] provides a different symmetric sub-model of the Cohen extension in which there is no nonprincipal ultrafilter on ω.Let G be the group (2ω)ω acting on X = G by translation, and let F be the filterof those subgroups of G containing some Gn = {x ∈ G : x � n is a constant zerofunction}. The symmetric model M certainly does not fall into our context–note that the orbit equivalence relation has only one class, and the groups inthe filter F are closed and not open. In fact, the model M is not of the formV [[x]]E for any equivalence relation; it is an example of a (choiceless) submodelof the Cohen forcing extension which is not of the form V (a) for some countabletransitive set a. To see that, consider the set A ⊂ ω × 2ω, A = {〈n, y〉 : y is atranslate of the n-th entry of the generic sequence by a ground model elementof 2ω}. The set A belongs to the model M , and whenever B ⊂ A is a set inM with nonempty vertical sections, a simple symmetricity argument shows thatone of its sections Bn is relatively open in An. The set of ground model elementsof 2ω can be injectively mapped into any nonempty relatively open subset ofAn, and so ℵ1 ≤ |B| in the model M . Finally, compare this observation withCorollary 3.4.11.

62 CHAPTER 3. TOOLS

Proof. We shall first show that the models V [[xgen]]E and V [xgen]G,F are thesame as far as transitive sets countable in V [xgen] are concerned. Since V [[xgen]]E =V (a) for some transitive countable set a ∈ V [x] as proved in Theorem 4.3.11,this shows that V [[xgen]]E ⊆ V [xgen]G,F . We will conclude the argument byshowing that every intermediate model W of ZF, V [[xgen]]E ⊆ W ⊆ V [xgen],is either equal to V [[xgen]]E or else contains some transitive set countable inV [xgen] which is not in V [[xgen]]E .

The comparison in the case of countable transitive sets passes through adescriptive set theoretic characterization. If R ⊂ ω×ω is a wellfounded relationsatisfying the axiom of extensionality, let most(R) be the unique countabletransitive set such that the membership relation on it is isomoprhic to R.

Claim 3.4.23. Let B ⊂ X be a Borel nonmeager set. Let f : B → P(ω × ω)be a Borel function such that for every x ∈ B, the relation f(x) is wellfoundedand satisfies the axiom of extensionality. The following are equivalent:

1. there is a Borel I-positive set C ⊂ B and a clopen subgroup H ⊂ G suchthat for every x ∈ C and every h ∈ H, either hx /∈ C or most(f(x)) =most(f(gx));

2. there is a Borel I-positive set C ⊂ B forcing most(f(xgen)) ∈ V [[xgen]]E.

3. there is a Borel I-positive set C ⊂ B forcing most(f(xgen)) ∈ V [xgen]G,F .

Proof. (1) certainly implies (2). As H has countable index in G, for every V -generic point x ∈ X, the model V [x] contains a countable set a ⊂ [x]E ∩C thatcontains representatives of all H-suborbits of [x]E intersecting the set C, and soit contains all the countably many hereditarily countable sets most(f(y)) : y ∈[x]E ∩ C. Thus, the set most(f(xgen)) belongs to every forcing extension of Vcontaining an E-equivalent of xgen, and so belongs to V [[xgen]]E .

(2) implies (1). Consider the poset PJ where J is the meager ideal onthe Polish space X × G. The poset PJ adds points xgen ∈ X and ggen ∈ G;the point xgen ∈ X is PI -generic over V by the Kuratowski-Ulam theorem.

Consider the condition D0 = C × G ∈ PJ . Since D0 f(xgen) ∈ V [[xgen]]E ,there must be a Borel function f and a stronger condition D1 ⊂ D0 forcing inPJ that most(f(xgen)) = most(f(ggen · xgen)). Let M be a countable elementarysubmodel of a large structure and let D2 be the set of PJ -generic points overM in the set D1; this is still a nonmeager Borel set, and the forcing theoremshows that for every 〈x, g〉 ∈ D2 it is the case that most(f(x)) = most(f(g ·x)).Let U ⊂ X and V ⊂ G be basic open neighborhoods such that D is comeagerin U × V . Since the acting group G is a closed subgroup of S∞, the unit hasa neighborhood basis consisting of clopen subgroups, and so there must be apoint g0 ∈ G and a clopen subgroup H ⊂ G such that Hg0 ⊂ V . We claim thatthe set C ′ = {x ∈ X: for comeagerly many g ∈ V , 〈x, g〉 ∈ D} together withthe subgroup H will work as in (1).

First of all, the set C ′ ⊂ C is nonmeager by Kuratowski-Ulam theorem.Since the meager ideal is Borel on Borel, the set C ′ is also Borel–[33, Theorem


16.1]. Now suppose x ∈ C ′ and h1 ∈ H are such that y = h1 · x ∈ C ′. Sincethe sets A0 = {h ∈ H : 〈x, hg0〉 ∈ D} and A1 = {h ∈ H : 〈h1 · x, hg0〉 ∈ D}are comeager in H, there must be a group element h ∈ A0 ∩ A1 · h1. Then,both pairs 〈x, g0h〉 as well as 〈y, g0hh

−11 〉 are in the set D. This means that

most(f(x)) is isomorphic to most(f(g0h · x)) and most(f(y)) is isomorphic tomost(f(g0hh

−11 · y). However, g0hh

−11 · y = g0hh

−11 h1 · x = g0h · x and so

most(f(x)) and most(f(y)) are isomorphic as desired.To see why (3) implies (1), let τ be a symmetric PI -name for a countable

transitive set with support containing some clopen subgroup H ⊂ G. Finda Borel function fτ : X → P(ω × ω) such that PI forces τ to be equal tomost(fτ (x)). For every group element h ∈ H it is the case that h · τ = τ andso the set {x ∈ X : most(fτ (x)) = most(fτ (h · x))} contains a comeager set.Thus, the Kuratowski-Ulam theorem implies that the Borel set D = {〈x, h〉 :most(fτ (x)) = most(fτ (h · x))} is comeager in X × H and the set Cτ = {x ∈X : Dx is comeager in H} is comeager. As in the previous proof then, for everyx ∈ Cτ and every h ∈ H, either h ·x /∈ Cτ or else most(fτ (x)) = most(fτ (h ·x)).

Now let C, f be as in (3). Strengthen C if necessary to find a symmetricname τ such that C most(f(xgen)) = τ and let H be the support of τ . Theset {x ∈ C : most(f(x) = most(fτ (x))}∩Cτ is Borel, nonmeager, and it has theproperties required in (1).

To see the final implication (1) to (3), note that by thinning out the setC as in (1) we may assume that the rank of most(f(x)) is the same for allx ∈ C; call this ordinal α(f, C). By transfinite induction on α(f, C), for all f, Csimultaneously, argue that (1) implies that there is a hereditarily symmetricname τ of rank α(f, C) such that C τ = most(f(xgen)). Suppose this hasbeen proved for all ordinals less than α(f, C). Let τ be the set of all pairs〈D,σ〉 such that there is an ordinal β ∈ α(f, C) such that σ is a hereditarilysymmetric name of rank ≤ β and D forces σ to be some element of the uniqueset f(y) where y is some element of the set C in the H-orbit of xgen of rank≤ β. It is clear that all names appearing in τ are hereditarily symmetric, andmoreover the name τ itself is symmetric with support H. Thus, τ is hereditarilysymmetric, and it is enough to show that C τ = most(f(xgen)). Suppose forcontradiction that some condition C0 ⊂ C forces the inequality. Let χ be aname for some element of the set most(f(xgen)) \ τ . Since this is a name foran element of a set in the model V [[xgen]]E , it is itself forced to be an elementof that model of rank β < α(f, C). Since (2) implies (1), there is a Borelfunction fχ, a condition C1 ⊂ C0 and a clopen subgroup H1 ⊂ H as in (1), such

that C1 χ = most(fχ(xgen)). The induction hypothesis yields a hereditarilysymmetric name ψ of rank ≤ β which is forced by C1 to be equal to χ. Thepair 〈C1, ψ〉 belongs to τ , and this contradicts the choice of χ.

It follows that the models V [[xgen]]E and V [xgen]G,F agree on transitive setscountable in V [x]. To obtain the full agreement, we have to study the modelsof ZF intermediate between V [[x]]E and V [x]. First, we verify that V [x] is aforcing extension of V [[x]]E and compute the poset.

64 CHAPTER 3. TOOLS

Fix a PI -name a for some transitive countable set such that V [[xgen]]E =V (a). Fix a Borel function f : X → P(ω × ω) such that it is forced thata = most(f(xgen)) Let x ∈ X be a PI -generic point over V . Let Q be the posetof all pairs 〈B, f〉 such that B ⊂ X is a basic open set, e is a finite functionfrom ω to a/x, and in some large forcing extension there is a PI -generic pointy ∈ X over V such that y ∈ B, most(f(y)) = a/x, and the unique Mostowskiisomorphism extends e. The ordering on Q is coordinatewise extension.

Claim 3.4.24. The point x together with the unique isomorphism between f(x)and the set a are Q-generic objects over V [a].

Proof. An important part of the proof is checking that the poset Q is in factnonempty. Choose a large enough cardinal κ and look at Coll(ω, Vκ) extensionV (a)[z] of the model V (a). The model V (a)[z] satisfies the axiom of choice:since a ∈ Vκ, it is a model of the form V [∈] where ∈ is the relation on ω definedby n ∈ m if z(n) ∈ z(m), and adjoining a single real to a model of choiceproduces again a model of choice. By analytic absoluteness between V (a)[z]and V [x] conclude that V (a)[z] contains a PI -generic point x′ ∈ X over V suchthat a = most(f(x′)), since the model V [x] contains such a point, namely x.Thus indeed, 〈X, 0〉 ∈ Q.

First check that the intersection of the first coordinates of conditions in aQ-generic filter over V (a) is forced to be a PI -generic point; let ygen be thename for this point. To see this, let 〈B, e〉 ∈ Q be a condition and C ⊂ X be aclosed nowhere dense set; it will be enough to find an extension of the condition〈B, e〉 whose first coordinate is disjoint from C. This is simple though; just finda large forcing extension of V (a) such that it contains a PI -generic point x′ ∈ Bover V as in the definition of the poset Q, find a basic open set B′ ⊂ B suchthat x′ ∈ B and B′ ∩ C = 0, and consider the condition 〈B′, e〉 ≤ 〈B, e〉.

The check that Q forces the union of the second coordinates in the genericfilter to be the unique isomorphism between a and most(f(ygen)) is similar.Now, V (a) |= Q V [ygen] = V (a)[ygen] and by the forcing theorem, back in

V there must be a condition in PI that forces the generic xgen to be Q-generic

over the model V (most(f(xgen))). This must be the largest condition: if somecondition in PI forced the opposite, then the whole argument above could berepeated below it, reaching a contradiction. By the forcing theorem again, thepoint x is indeed Q-generic over V (a) as desired.

Let σ ∈ V [[x]]E be a Q-name for a subset of V [[x]]E and a condition q ∈ Qforcing σ /∈ V [[x]]E . We will find a set N ∈ V [[x]]E satisfying the axiom ofextensionality, countable in V [x], such that q σ ∩ N /∈ V [[x]]E . Thus, writingπ : N → N for the inverse of the Mostowski collapsing map of N to a transitiveset N in V [[x]]E , it is the case that q π−1σ /∈ V [[x]]E ; note that π−1σ is asubset of a transitive set countable in V [x]. It will then follow from the claimand the forcing theorem immediately that whenever b ⊂ V [[x]]E is a set inV [x]E which does not belong to V [[x]]E , then the model V [[x]]E [b] containssome transitive set countable in V [x] that does not belong to V [[x]]E . Thetheorem will follow as per the first paragraph of this proof.

3.5. ASSOCIATED FORCINGS 65

Back in the ground model, choose a countable elementary submodel M of alarge structure containing a PI -name for the name σ. Write M [x] = {τ/x : τ is aPI -name in the model M}; this is a countable set in the model V [x], containinga and Q as subsets and σ as an element. The following claim is proved as inCorollary 3.4.17.

Claim 3.4.25. M [x] ∩ V [[x]]E ∈ V [[x]]E.

Now, let N = M [x] ∩ V [[x]]E ; this is the set we wanted. Suppose for con-tradiction that there is a condition r ∈ Q, r ≤ q, forcing σ ∩ N ∈ V [[x]]E .Strengthening r further if necessary, we may assume that there is b ∈ V [[x]]Esuch that r σ ∩ N = b. Now, r ∈ N . There must be a set c ∈ V [[x]]E suchthat r does not decide the membership of c in σ. By elementarity of the modelM [x] inside the model V [x], such a set c should exist in M [x] and therefore inN . Now both c ∈ b and c /∈ b are impossible, the final contradiction.

3.5 Associated forcings

3.5a The poset PEI

In all special cases investigated in this book, the intermediate extension V [xgen]Eof Definition 3.4.1 is generated by a rather natural regular subordering of PI ,denoted by PEI .

Let X be a Polish space and I a σ-ideal on it. We would like to definePEI to be the partial ordering of I-positive E-saturated Borel sets ordered byinclusion. The problem here is that E-saturations of Borel sets are in generalanalytic, and in principle there could be very few Borel E-saturated sets. Thereare two possibilities for fixing this. The preferred one is to work only with σ-ideals I satisfying the third dichotomy [67, Section 3.9.3]: every analytic set iseither in the ideal I or it contains an I-positive Borel subset. Then the quotientPI is dense in the partial order of analytic I-positive sets, and we can define PEIas the ordering of analytic E-saturated I-positive sets.

Another possibility is to prove that in a particular context, E-saturationsof many Borel sets are Borel. This is for example true if E is a countableBorel equivalence relation. It is also true if the poset PI is bounding and theequivalence E is reducible to EKσ . There, if B ⊂ X is a Borel I-positive setand f : B → Πn(n+ 1) is the Borel function reducing E to EKσ , the boundingproperty of PI can be used to produce a Borel I-positive compact set C ⊂ Bon which the function f is continuous, and then [C]E is Borel. However, thequestion whether every analytic I-positive E-saturated set has a Borel I-positiveE-saturated subset seems to be quite difficult in general, even if one assumesthat every I-positive analytic set contains a Borel I-positive subset.

Definition 3.5.1. (The correct version) Let X be a Polish space and I a σ-idealon it such that every analytic I-positive set contains a Borel I-positive subset.PEI is the poset of I-positive E-saturated analytic sets ordered by inclusion. If

66 CHAPTER 3. TOOLS

C ⊂ X is a Borel I-positive set then PE,CI is the poset of relatively E-saturatedanalytic subsets of C ordered by inclusion.

By a slight abuse of notation, if every I-positive analytic set contains a BorelI-posiive set, we will write PI for the poset of all analytic I-positive sets orderedby inclusion, so that PEI ⊂ PI . The previous discussion seems to have little tooffer for the solution of the main problem associated with the topic of this book:

Question 3.5.2. Is PE,CI a regular subposet of PI for some Borel I-positiveset C ⊂ X?

Note that there is a natural candidate for the projection from the posetPI to PEI , namely the E-saturation. In all specifica cases investigated in thisbook, the answer to the previous question turns out to be positive, the set C isinvariably equal to the whole space, and the saturation is the projection on adense set of conditions in PI . We will include one rather simple case criterionfor regularity that will be used throughout the book.

Definition 3.5.3. Let I be a σ-ideal on a Polish space X, E an analytic equiv-alence relation on X, and let B ⊂ X be an analytic set. We say that B is anE-kernel if for every analytic set C ⊂ B, [C]E ∩B ∈ I implies [C]E ∈ I.

Theorem 3.5.4. Let I be a σ-ideal on a Polish space X such that every I-positive analytic set has a Borel I-positive subset, and let E be an analyticequivalence relation on X. The poset PEI is regular in PI if and only if everyBorel I-positive set has an analytic I-positive subset which is also an E-kernel.

The main point here is that in the cases encountered in this book, the kernelsare typically very natural closed sets. Observe that in the case that saturationsof sets in I are still in I, every analytic set is a kernel:

Corollary 3.5.5. Suppose that I is a σ-ideal on a Polish space X such thatevery I-positive analytic set has a Borel I-positive subset, and E is an analyticequivalence relation on X. If E-saturations of sets in I are still in I, then theposet PEI is regular in PI .

Proof. Suppose that every I-positive Borel set contains a Borel I-positive kernel;we must show that PEI is regular in PI . Let A ⊂ PEI be a maximal antichain;we must show that A is also maximal in PI . Suppose not, and let B ∈ PI bean analytic set incompatible with all elements of the antichain A, and thin itout if necessary to a kernel. Then, for every set C ∈ A, B ∩ C ∈ I, and so[B∩C]E ∈ I by the assumptions. However, [B∩C]E = [B]E ∩C since the set Cis E-invariant, and so [B]E is a condition in PEI incompatible with all elementsof the antichain A, contradicting its maximality in PEI .

On the other hand, suppose that PEI is regular in PI , and B ⊂ X is a Borel I-positive set. To produce an I-positive E-kernel in it, consider the open dense setD ⊂ PEI consisting of those analytic E-invariant sets with I-small intersectionswith B and those sets which do not have an E-invariant I-positive analyticsubset with I-small intersection with B. Since the poset PEI is regular in PI ,


there must be a set B′ ∈ D compatible with B, so B′ ∩ B /∈ I. Note that thismeans that no E-invariant I-positive subset of B′ has I-small intersection withB. We claim that B ∩B′ is the requested E-kernel. And indeed, if C ⊂ B ∩B′is an analytic set such that [C]E ∩ B ∩ B′ ∈ I, then [C]E ⊂ B′ has I-smallintersection with the set B and therefore has to be I-small!

The poset PEI is frequently ℵ0-distibutive, and yields the model V [xgen]E .The following two theorems provide criteria for this to happen.

Definition 3.5.6. The equivalence E on a Polish space X is I-dense if for everyI-positive Borel set B ⊂ X there is a point x ∈ B such that [x]E ∩ B is densein B.

Proposition 3.5.7. If the poset PI has the continuous reading of names, PEIis a regular subposet of PI , and E is I-dense, then PEI is ℵ0-distributive.

Proof. Since PI is proper, so is PEI , and it is enough to show that PEI doesnot add reals. Suppose that B ∈ PI forces that τ is a name for a real in thePEI -extension. Let M be a countable elementary submodel of a large enoughstructure, and let C ⊂ B be the I-positive set of all M -generic points for PI .If x ∈ C is a point, write Hx ⊂ PEI for the M -generic filter of all sets D ∈ PEIsuch that [x]E ⊂ D. It is clear that the map x 7→ Hx is constant on equivalenceclasses, and so is the Borel map x 7→ τ/Hx. Thin out the set C if necessaryto make sure that the map x 7→ τ/Hx is continuous on C. By the densityproperty, there is a point x ∈ C such that [x]E ∩C is dense in C. Thus, the mapx 7→ τ/Hx is continuous on C and constant on a dense subset of C, thereforeconstant on C. The condition C clearly forces τ to be equal to the single pointin the range of this function.

Note that typically the poset PEI is fairly simply definable. If suitable largecardinals exist, then the descending chain game on the poset PEI is determined.Thus, when the poset is ℵ0-distributive, Player I has no winning strategy inthis game, so it must be Player II who has a winning strategy and the posetis strategically σ-closed. It is not entirely clear whether there may be cases inwhich PEI is ℵ0-distributive but fails to have a σ-closed dense subset. In manysituations, one can find forcing extensions in which PEI is even ℵ1-distributiveor more.

In all situations described in the later chapters of this book, the forcing PEIgenerates the V [xgen]E extension of Theorem 3.4.4. To prove the appropriategeneral theorem, we need a definition:

Definition 3.5.8. Let I be a σ-ideal on a Polish space X and let E be anequivalence relation on X. We say that E is I-homogeneous if for every pairof I-positive analytic sets B,C ⊂ X, either there is a Borel I-positive subsetB′ ⊂ B such that [B′]E ∩ C ∈ I, or there is a Borel I-positive subset B′ ⊂ Band an injective Borel map f : B′ → C such that f ⊂ E and f preserves theideal I: images of I-small sets are I-small, and images of I-positive sets areI-positive.

68 CHAPTER 3. TOOLS

This property may not always be completely straightforward to verify, but it issatisfied in all natural cases; in particular, it holds whenever E is a countableBorel equivalence relation.

Theorem 3.5.9. Suppose that I is a σ-ideal on a Polish space X such that thequotient forcing PI is proper and every analytic I-positive set contains a BorelI-positive subset. If E is I-homogeneous, then the model V [xgen]E is forced tobe equal to the PEI -extension.

Proof. Let G ⊂ PI be a generic filter and H = G ∩ PEI . It is clear thatV [H] ⊂ V [xgen]E , since H is definable from [xgen]E in every further forcingextension as the collection of those Borel I-positive E-saturated sets coded inthe ground model containing the calss [xgen]E as a subset. The other inclusionis harder.

Suppose that φ(α,~v, [xgen]E) is a formula with an ordinal parameter, otherparameters in V , and another parameter [xgen]E , and let κ be a cardinal. Adensity argument shows that there must be a condition B ∈ PI such that thecondition 〈P, 1〉 ∈ PI×Coll(ω, κ) decides the truth value of φ, and H contains acondition below the pseudoprojection of B to PEI . We will show that for everytwo such conditions, the decision of the truth value must be the same. Thus, theset of ordinals defined by φ in the PI ×Coll(ω, κ) must be already in the modelV [H], defined as the set of those ordinals α such that for some condition B ∈ PIwhose pseudoprojection to PEI belongs to the filter H, 〈B, 1〉 φ(α,~v, [xgen]E).

So suppose for contradiction that there are conditions C0, C1 ∈ PI andB ∈ PEI such that B is below the pseudoprojection of both C0, C1 into PEI , and〈C0, 1〉 φ(α,~v, [xgen]E) and 〈C1, 1〉 ¬φ. Since B is below the pseudoprojec-tions, the first item of the assumptions is excluded for B,C in both cases C = C0

or C = C1; indeed, the set [B′]E would be an extension of B in PEI incompatiblewith the condition C0 or C1. So the second case must happen, and strengthen-ing twice, we get a Borel I-positive set B′ ∈ PI below B and Borel injectionsf0 : B′ → C0 and f1 : B′ → C1 as postulated in the assumptions. Since thesefunctions preserve the ideal I and are subsets of the equivalence E, B′ forcesin the forcing PI that the points xgen ∈ B′, f0(xgen) ∈ C0 and f1(xgen) ∈ C1

are E-equivalent V -generic points for the poset PI , giving the same generic ex-tension. Thus, if G ⊂ PI is a V -generic filter containing the condition B′ andK ⊂ Coll(ω, κ) is a V [G]-generic filter, the forcing theorem applied to f0(xgen)and f1(xgen) implies that V [G,K] |= φ(α,~v, [f0(xgen)]E)∧¬φ(α,~v, [f1(xgen)]E),which is impossible in view of the fact that f0(xgen) E f1(xgen).

As a final remark regarding the poset PEI , its definition also called for the

consideration of relativized posets PE,CI for Borel I-positive sets C. The reasonis that even in quite natural cases, one has to pass to a condition in order to findthe requested regularity, as the following example shows. Compare this also toTheorem 4.2.1 and its proof.

Example 3.5.10. Consider X = ωω × 2 and the σ-ideal I generated by setsB ⊂ X such that {y ∈ ωω : 〈y, 0〉 ∈ B} is compact and {y ∈ ωω : 〈y, 1〉 ∈ B}


is countable. The quotient forcing is clearly just a disjoint union of Sacks andMiller forcing. Consider the equivalence relation E on X defined as the equalityon the first coordinate. Then E has countable classes, indeed each of its classeshas size 2, and PEI is not a regular subposet of PI . To see that, consider amaximal antichain A in Sacks forcing consisting of uncountable compact sets,and let A = {C × 2 : C ∈ A}. It is not difficult to see that A is a maximalantichain in PEI , but the set ωω×1 ⊂ X is Borel, I-positive, and has an I-smallintersection with every set in A.

3.5b The reduced product posets

Another forcing feature of the arguments in this book is the use of a reducedproduct of quotient posets. There is a general definition of such products:

Definition 3.5.11. Let I be a σ-ideal on a Polish space X, and let E be aBorel equivalence relation on X. The reduced product PI ×E PI is definedas the poset of those pairs 〈B,C〉 of Borel I-positive sets such that for everysufficiently large cardinal κ, in the Coll(ω, κ) extension there are E-equivalentpoints x ∈ B and y ∈ C, each of them V -generic for the poset PI . The orderingis that of coordinatewise inclusion.

The left coordinates of conditions in a reduced product generic filter form aPI -generic filter, and similarly for the right coordinates. To see this, assume that〈B,C〉 is a condition in PI×EPI and D ⊂ PI is a dense set. The poset Coll(ω, κ)forces that there exists a pair of V -generic E-equivalent points x ∈ B, y ∈ C.Find a condition p ∈ Coll(ω, κ) and conditions B′ ⊂ B and C ′ ⊂ C in the denseset D such that p x ∈ B′, y ∈ C ′ and observe that 〈B′, C ′〉 is still a conditionin the reduced product PI ×E PI and it clearly forces that both filters on theleft hand side and the right hand side meet the dense set D. Thus, the reducedproduct generic filter is given by two points xlgen, xrgen ∈ X, each of which isV -generic for the poset PI . These two points are typically not E-related. Thegeneral definition of the reduced product is certainly puzzling, and in all specialcases, a good deal of effort is devoted to the elimination of the Coll(ω, κ) forcingrelation from it. The reduced products are always used in the intermediate caseof the trichotomy 3.4.4:

Proposition 3.5.12. If I is a σ-ideal on a Polish space X such that the quotientforcing PI is proper, and E is a Borel equivalence relation on X which is I-ergodic, then PI ×E PI = PI × PI .

Proof. Let B ⊂ X be a Borel I-positive subset such that every two Borel I-positive subsets C0, C1 ⊂ B contain a pair of equivalent points. We will showthat the reduced product below the condition 〈B,B〉 consists of all pairs ofBorel I-positive subsets of B. In other words, for every pair 〈C0, C1〉 of Borel I-positive subsets of B there is a forcing extension in which there are E-equivalentpoints x0 ∈ C0, x1 ∈ C1, both V -generic for the poset PI . Proceed as in thethird case of the trichotomy theorem 3.4.4 to find a generic extension V [K] in

70 CHAPTER 3. TOOLS

which there is an elementary embedding j : V → N into a possibly illfoundedmodel N which nevertheless contains M = j′′Hc+ as a countable set. Workingin N , let D0 = {x ∈ jC0 : x is M -generic for Pj(I)}, and similarly for D1. Inthe model N , these are Borel j(I)-positive sets by Fact 2.6.1, they are bothsubsets of jB, and so they must contain j(E)-related points x0 ∈ D0, x1 ∈ D1

by the ergodicity. Now, the model V [K] sees x0 ∈ C0, x1 ∈ C1, and these twopoints are PI -generic over V . Moreover, since j(E) ⊂ EV [K], the model alsosees x0 E x1 and we are finished.

Proposition 3.5.13. Let I be a σ-ideal on a Polish space X and E is ananalytic equivalence relation on X.

1. The reduced product PI ×E PI forces V [xlgen] ∩ V [xrgen] = V [xlgen]E =V [xrgen]E;

2. if the model V [xlgen]E contains no new reals then the pair 〈xlgen, xrgen〉 isforced to be PIxlgen × PIxrgen -generic over V [xlgen]E.

Proof. First argue that the models V [xlgen]E and V [xrgen]E are forced to beequal. Indeed, if φ is a formula with a ground model parameter p, α is anordinal, κ is a large enough cardinal and 〈B,C〉 is a condition in the reducedproduct, it cannot be the case that B Coll(ω, κ) φ(α, p, [xgen]E) and C Coll(ω, κ) ¬φ(α, p, [xgen]E): in a Coll(ω, κ) extension, there would be V -generic points x ∈ B and y ∈ C with x E y by the definition of the reducedproduct, and homogeneity arguments would provide Coll(ω, κ)-generic filtersGx, Gy such that V [G] = V [x][Gx] = V [y][Gy]. Then, the forcing theorem onthe x and y sides would indicate that V [G] |= φ(α, p, [x]E) and φ(α, p, [y]E),which is certainly impossible as x E y. Thus, the reduced product forces thatthe same formulas must define the same sets of ordinals in the models V [xlgen]Eand V [xrgen]E –so these two models must be the same.

The previous paragraph shows that the model V [xlgen]E = V [xrgen]E is asubset of V [xlgen] ∩ V [xrgen]. To see the opposite inclusion, suppose that acondition 〈B0, C0〉 forces some set τ of ordinals to be in the intersection ofV [xlgen] and V [xrgen]; we must show that it belongs to the model V [xlgen]E .Strengthening the condition if necessary, one can find PI -names τl and τr suchthat 〈B0, C0〉 τ = τl/xlgen = τr/xrgen. Since the poset PI ×E PI below thecondition 〈B0, C0〉 forces the left generic to be PI -generic over V , there must bea Borel I-positive set B1 ⊂ B0 such that in a large collapse extension, wheneverx ∈ B1 is a PI -generic point over V then there is a filter H ⊂ PI ×E PI genericover V containing 〈B0, C0〉 and such that x is the left half of the pair given bythe filter H.

Claim 3.5.14. In a large collapse extension, whenever x, y are PI-genericpoints over V , both in B1 and E-related, then τl/x = τl/y.

Proof. Move to a large collapse extension V [G] and suppose for contradictionthat x, y ∈ B1 are two E-related PI -generic points over V such that τl/x 6= τl/y.Choose an ordinal α such that α ∈ τl/x \ τl/y.


Back to V . By the forcing theorem, there must be conditions Bx, By ⊂ B1

in PI such that x ∈ Bx, y ∈ By, Bx α ∈ τl and By α /∈ τl. Moreover,Bx can be chosen to force that large collapse forces that there is a PI -genericpoint over V in the set By which is E-related to xgen. It follows that wheneverC2 ⊂ C0 is a Borel I-positive set such that 〈Bx, C2〉 ∈ PI ×E PI , then even〈By, C2〉 ∈ PI ×E PI . To see this, by the definitions in the collapse extensionV [G] there must be points x′ ∈ Bx , z ∈ C1 which are both PI -generic over Vand E-related. By the choice of the condition Bx, there must be also a pointy′ ∈ By in V [G] which is PI -generic over V and E-related to x′. Then the pair〈y′, z〉 ∈ E witnesses the fact that 〈By, C2〉 ∈ PI ×E PI .

In V [G] again, find a filter H ⊂ PI ×E PI generic over V such that x is itsleft half and 〈B0, C0〉 ∈ H. There must be a condition 〈B2, C2〉 ∈ H such thatB2 ⊂ Bx and C2 α ∈ τr. But then, 〈By, C2〉 is a condition in the reducedproduct such that By α /∈ τl and C2 α ∈ τr, contrary to 〈B0, C0〉 forcingτl/xlgen = τr/xrgen.

Now, any condition 〈B2, C2〉 in the reduced product forcing xlgen ∈ B1

also forces τl/xlgen ∈ V [xlgen]E : the set τl/xlgen is defined in a large collapseextension as the unique set of the form τl/x where x ∈ B1 ∩ [xlgen]E is a pointPI -generic over V . We conclude that V [xlgen]∩V [xrgen] ⊂ V [xlgen]E as required.

To prove (2), assume that the model V [xlgen] = V [xrgen] does not containany new reals. The first paragraph of the present proof shows that the idealsI xlgen and I xrgen are forced to be equal and we will denote them by I∗. Notethat any condition 〈B,C〉 forces itself into the product PI∗ ×PI∗ since it forcesxlgen ∈ B and xrgen ∈ C. Thus the generic filter on PI ×E PI is also a filteron PI∗ × PI∗ . We will show that it is in fact a PI∗ × PI∗ -generic filter overV [xlgen]E .

Suppose for contradiction that 〈B0, C0〉 ∈ PI ×E PI is a condition forcingthe filter on PI∗×PI∗ not to be V [xlgen]E-generic, missing a dense set D definedby a formula φ. Since V [xlgen]E is forced to contain no new reals or Borel sets,there must be a condition 〈B1, C1〉 ≤ 〈B0, C0〉 in the reduced product forcingsome pair 〈B2, C2〉 ≤ 〈B0, C0〉 into the product PI∗ × PI∗ and the dense set D.We will reach a contradiction by producing, in some large collapse extension,pairs 〈x1, y1〉 and 〈x2, y2〉, both reduced product generic over V , one of themmeeting the condition 〈B1, C1〉, the other meeting the condition 〈B2, C2〉, suchthat x1 E x2. The definition of the set D yields the same set whether definedfrom [x1]E or [x2]E , and so the filter associated with 〈x2, y2〉 does meet the setD (in the condition 〈B2, C2〉), contrary to what was forced by the condition〈B0, C0〉.

Let κ be a suitably large cardinal. Choose (PI ×E PI) × Coll(ω, κ)-namesσ and τ such that 〈B1, C1, 1〉 forces σ, τ to be PI -generic points over V in thesets B2, C2 respectively, E-equivalent to xlgen. By the forcing factorization,Fact 2.3.2, there must be an I-positive Borel set B3 ⊂ B2 such that in a largecollapse extension, for every point x ∈ B3 PI -generic over V there is a filterG ⊂ (PI ×E PI) × Coll(ω, κ), generic over V , such that x = τ/G. This meansthat for every condition B4 ⊂ B3, it is the case that the pair 〈B4, C2〉 belongs to

72 CHAPTER 3. TOOLS

the reduced product: in a large collapse extension, one can pick a point x ∈ B4

PI -generic over V , then a filter G ⊂ (PI ×E PI)×Coll(ω, κ) generic over V suchthat τ/G = x, and then σ/G is a point in C2 which is PI -generic over V and E-equivalent to x. Since the reduced product forces that the left coordinate of thegeneric pair is a PI -generic point over V , the forcing factorization, Fact 2.3.2,applied again shows there must be an I-positive Borel set B5 ⊂ B3 such thatin a large collapse extension, for every PI -generic point x ∈ B5 over V there isa filter reduced product generic over V containing the condition 〈B5, C2〉 suchthat the left coordinate of the generic pair is exactly x.

Move to a large collapse forcing extension of V and pick a point x2 ∈ B5

PI -generic over V . There is a point y2 ∈ X such that the pair 〈x2, y2 is PI×EPI -generic over V . There is also a filter G ⊂ (PI×E PI)×Coll(ω, κ)-generic over Vsuch that τ/G = x2. Let 〈x1, y1〉 be the pair associated with G. By the choiceof the name τ , it is the case that x1 E x2, and we are done.

Chapter 4

Classes of equivalencerelations

4.1 Smooth equivalence relations

In the case that the equivalence relation E is smooth, the model V [xgen]E asdescribed in Theorem 3.4.4 takes on a particularly simple form.

Proposition 4.1.1. Let I be a σ-ideal on a Polish space X such that the quo-tient forcing PI is proper, and suppose that E is a smooth equivalence relationon the space X. Then V [xgen]E = V [f(xgen)] for every ground model Borelfunction f reducing E to the identity.

Proof. Choose a Borel function f : X → 2ω reducing the equivalence E to theidentity. Let G ⊂ PI and H ⊂ Coll(ω, κ) be mutually generic filters, and letxgen ∈ X be a point associated with the filter G. First of all, f(xgen) ∈ 2ω isdefinable from the equivalence class [xgen]E : it is the unique value of f(x) for allx ∈ [xgen]E . Thus, V [f(xgen)] ⊂ V [xgen]E . On the other hand, the equivalenceclass [xgen]E is also definable from f(xgen), the model V [G,H] is an extensionof V [f(xgen)] via a homogeneous notion of forcing Coll(ω, κ), and therefore by[22, Lemma 26.7] V [xgen]E ⊂ V [f(xgen)].

The total canonization of smooth equivalence relations has an equivalent re-statement with the quotient forcing adding a minimal real degree. It is necessarythough to discern between the forcing adding a minimal real degree, and theforcing producing a minimal extension. In the former case, we only ascertainthat V [y] = V or V [y] = V [xgen] for every set y ⊂ ω in the generic extension.In the latter case, this dichotomy in fact holds for every set y of ordinals in theextension. For example, the Silver forcing adds a minimal real degree, while itproduces many intermediate σ-closed extensions, as shown in Section 5.8.

Proposition 4.1.2. Let X be a Polish space and I be a σ-ideal on X such thatthe quotient forcing PI is proper. The following are equivalent:

73

74 CHAPTER 4. CLASSES OF EQUIVALENCE RELATIONS

1. PI adds a minimal real degree;

2. I has total canonization of smooth equivalence relations.

Proof. First, assume that the canonization property holds. Suppose that B ∈ PIforces that τ is a new real; we must show that some stronger condition forcesV [τ ] = V [xgen]. Thinning out the set B if necessary we may find a Borel

function f : B → 2ω such that B τ = f(xgen). Note that singletons musthave I-small f -preimages, because a large preimage of a singleton would be acondition forcing τ to be that ground model singleton. Consider the smoothequivalence relation E on B, E = f−1id. Since it is impossible to find a BorelI-positive set C ⊂ B such that E � B = ee, there must be a Borel I-positive setC ⊂ B such that E � C = id, in other words f � C is an injection. But then,C V [τ ] = V [xgen] since xgen is the unique point x ∈ C such that f(x) = τ .

Now suppose that PI adds a minimal real degree, and let B ∈ PI be aBorel I-positive set and E a smooth equivalence relation on B with I-smallclasses. We need to produce a Borel I-positive set on which E is equal to theidentity. Let f : B → 2ω be a Borel function reducing E to the identity. Notethat since f -preimages of singletons are in the ideal I, f(xgen) is a name for

a new real and therefore it is forced that V [xgen] = V [f(xgen)]. Let M bea countable elementary submodel of a large structure, and let C ⊂ B be theBorel I-positive set of all M -generic points in B for the poset PI . The set f ′′Cis Borel. Consider the Borel set D ⊂ f ′′C ×C defined by 〈y, x〉 ∈ D if and onlyif f(x) = y. The set D has countable vertical sections: If f(x) = y then by theforcing theorem applied in the model M it must be the case that M [x] = M [y]and thus the vertical section Dy is a subset of the countable model M [y]. Usethe uniformization therem to find countably many graphs of Borel functionsgn : n ∈ ω whose union is D. The sets rng(gn) : n ∈ ω are Borel since thefunctions gn are Borel injections, and they cover the whole set C. Thus there isa number n ∈ ω such that the set rng(gn) is I-positive. On this set, the functionf is an injection as desired.

4.2 Countable equivalence relations

Borel countable equivalence relations are a frequent guest in mathematical prac-tice. Their treatment in this book is greatly facilitated by essentially completeanalysis of the associated forcing extensions.

Theorem 4.2.1. Suppose that I is a σ-ideal on a Polish space X such thatthe quotient forcing is proper. Suppose that E is a countable Borel equivalencerelation on X. Then for every Borel I-positive set B ⊂ X there is a Borel I-positive set C ⊂ B such that PE,CI is a regular subposet of PI below C. Moreover,

C forces the model V [xgen]E to be equal to the PE,CI extension. The remainder

forcing PI � C/PE,CI is weakly homogeneous and c.c.c.

4.2. COUNTABLE EQUIVALENCE RELATIONS 75

Here, PE,CI is the poset of I-positive Borel subsets of C which are relativelyE-saturated inside C. The set C ⊂ X may not be the whole space, as Exam-ple 3.5.10 shows.

Proof. Use the Feldman–Moore theorem 2.2.5 to find a Borel action of a count-able group G on B whose orbit equivalence relation is exactly E.

Claim 4.2.2. There is a Borel I-positive set C ⊂ B such that the E-saturationof any I-small Borel subset of C has I-small intersection with C.

Proof. Let M be a countable elementary submodel of a large structure and letC be the set of all M -generic elements of the set B for the poset PI . We claimthat this set works.

Suppose for contradiction that there is an I-small Borel subset D of C whoseE-saturation has an I-positive intersection with C. Note that [D]E is a Borelset again as E is a countable Borel equivalence relation. Thinning out the setC ∩ [D]E we may find a Borel I-positive subset C ′ and a group element g ∈ Gsuch that gC ′ ⊂ D.

Now consider the set O = {A ∈ PI : if there is a Borel I-positive subset of Awhose shift by g is I-small, then A is such a set}. Clearly, this is an open densesubset of PI below B in the model M , and therefore there is A ∈ M ∩ O suchthat A∩C ′ /∈ I. Now g(A∩C ′) ⊂ D and therefore this set is in the ideal I, andby definition of the set O it must be the case that gA ∈ I. But as gA ∈ I ∩M ,it must be the case that C ∩ gA = 0, which contradicts the fact that g(A ∩ C ′)is a nonempty subset of C.

Fix a set C ⊂ B as in the claim. The following is an immediate consequenceof Theorem 3.5.5 and the previous claim.

Claim 4.2.3. PE,CI is a regular subposet of PI below C.

Claim 4.2.4. The remainder forcing PI � C/PE,CI is weakly homogeneous and

c.c.c.

Proof. Let G ⊂ PI be a generic filter containing the condition C with its asso-ciated generic point xgen, and let H = G∩PE,CI . Note first that the remainderforcing consists of exactly those Borel sets D ⊂ C such that their E saturationbelongs to the filter H, or restated, those Borel sets containing some element ofthe equivalence class of xgen.

For the c.c.c. suppose that {Dα : α ∈ ω1} is a putative antichain in theremainder poset in the model V [H]. Since the equivalence class of xgen is count-able and ℵ1 is preserved in V [G], there must be α 6= β ∈ ω1 such that Dα, Dβ

contain the same element of the equivalence class. It follows that Dα ∩Dβ is alower bound of Dα, Dβ in the remainder forcing.

For the weak homogeneity, return to the ground model. Suppose thatD0, D1, D2 ⊂ C are Borel I-positive sets, D2 is E-saturated and forces in PE,CI

both D0, D1 to the remainder forcing. By the countable additivity of the idealI there must be an element g ∈ G and a positive set D0 ⊂ D0 ∩D2 such that


gD0 ⊂ D1–if for each g ∈ G the set Ag = {x ∈ D0 ∩ D2 : gx ∈ D1} wasin the σ-ideal I, so would be their union, and the saturation of the I-positiveset D0 ∩D2 \

⋃g Ag below D0 would force D2 not to belong to the remainder

forcing. The map π defined by π(A) = gA preserves the ideal I, preserves the

E-saturations and therefore the condition [D0]E ∈ PE,CI forces that it inducesan automorphism of the remainder poset moving a subset of D0 to a subset ofD1. The homogeneity of the quotient forcing follows.

The fact that PE,CI generates the V [xgen]E model now follows directly fromTheorem 3.5.9; the group action provides the necessary automorphisms of theposet PE,CI .

Inspecting the proofs in the previous chapter, the fact that the model V [xgen]Eis obtained by the forcing PEI are invariably proved in a significantly tighter fash-ion: it is proved that in the PI extension, the equivalence class of the genericreal is the only one which is a subset of all E-invariant Borel sets in the genericfilter. This means that, in the notation of the previous proof, the filter H isinterdefinable with [xgen]E in the PI -extension V [G] and they therefore haveto define the same HOD model. This, however, does not have to be the casein general even in the very restrictive case that the forcing PI adds a mini-mal real degree. Consider the case where PI is the Silver forcing and E is theequivalence relation on 2ω defined by x E y if x, y differ on finite set of entries,and moreover, the set of their differences has even cardinality. Comparing Ewith E0, we see that E ⊂ E0 and every E0 equivalence class decomposes intoexactly two E classes. Note that the E0-saturation of any Silver cube is equalto its E-saturation, and therefore the posets PE0

I and PEI are co-dense. If Gis the Silver generic filter and H = G ∩ PEI then the E0-class of the Silver realbelongs to all sets in H, and it splits into two E-classes, each of which has theproperty that it is a subset of all sets in the PEI -generic filter. Moreover, neitherof these classes is ordinally definable from H or parameters in V . To see this,suppose that a condition p in the Silver forcing forces that a formula φ definesthe E-equivalence class of xgen from the parameters G ∩ PE0

I and ~v ∈ V . Letx ∈ 2ω be a Silver real meeting the condition p, and let x be equal to x exceptfor one entry that is not in the domain of p. Then x is again a Silver real, thegeneric filters associated with x and x are different, but still they yield the sameextension and the same intersection with PE0

I . Thus, the formula φ applied toeither of them yields the same set of reals. This is impossible though as ¬x E x.

Let us now return back to the general treatment of Borel countable equiv-alence relations. In the common case that the forcing PI adds a minimal realdegree, there is much more information available about the V [xgen]E model.

Theorem 4.2.5. Suppose that I is a σ-ideal on a Polish space X such that thequotient forcing is proper and adds a minimal real degree. Suppose that E is anessentially countable equivalence relation on X which does not simplify to smoothon any Borel I-positive set. Then the model V [xgen]E is forced to contain no

4.2. COUNTABLE EQUIVALENCE RELATIONS 77

new countable sequences of ordinals, and it is the largest intermediate extensionin the PI-extension.

Proof. Let f : X → Y be a Borel reduction of E to a Borel countable equivalencerelation F on a Polish space Y . Let B ⊂ X be an I-positive Borel set. By thetotal canonization of smooth equivalences we may pass to a Borel I-positive setC on which the function is injective and therefore E itself has countable classes,and by the previous theorem we may assume that the poset PE,CI is a regularsubposet of PI below C.

We will first argue that the poset PE,CI is ℵ0-distributive. The forcing isa regular suposet of a proper forcing, therefore proper, and so it is enough toshow that it does not add reals. Suppose that τ is a PE,CI -name for a real, andD ⊂ C is a Borel I-positive set. We must produce a Borel I-positive subset ofD forcing a definite value to τ . Let M be a countable elementary submodel of alarge enough structure, and let A ⊂ D be the Borel I-positive subset consistingof all PI -generic points over M . The map x 7→ Kx assigning to each genericpoint the corresponding PE,CI generic filter for M , is E-invariant: Kx is the setof all relatively E-saturated Borel subsets containing x, which depends only onthe E-equivalence class of a point x ∈ C. Thus the map x 7→ τ/Kx is Borel andalso E-invariant. Since the forcing PI adds a minimal real degree, there is aBorel I-positive set A′ ⊂ A such that this map is either one-to-one or constanton A′. The former possibility cannot occur–since the function is E-invariant,it would have to be the case that E � A′ = id, contradicting the assumptionthat E cannot be reduced to a smooth equivalence on a positive Borel set. Thelatter possibility forces τ to be equal to the constant value of the function onA′, as desired.

Now we need to argue that the model V [xgen]E is the largest intermediatemodel between V and V [G]; in other words, if U is a model of ZFC such thatV ⊆ U ⊆ V [G], then either U = V [G] or U ⊆ V [xgen]E . Suppose that U = V [Z]is an intermediate model of ZFC, for a set Z ⊂ κ for some ordinal κ. Fix aname Z for the set Z. If there is a countable set a ⊂ κ in V such that Z ∩a /∈ Vthen V [G] = V [Z] by the minimal real degree assumption. Suppose then thatfor every countable set a ∈ V , it is forced that Z ∩ a ∈ V . Consider the namesfor countably many sets {Z/y : y ∈ [xgen]E and y is PI -generic over V }. Theremust be a countable set b ⊂ κ such that if any two of these sets differ at someordinal, then they differ at an ordinal in b. By properness, this set b has to becovered by a ground model countable set a ⊂ κ. Suppose that D ⊂ C in PI is acondition which forces this property of the set a, and moreover decides the valueof Z ∩ a to be some definite set c ∈ V . Then D forces that Z is definable from[xgen]E , D, c, and Z in the model V [G] by the formula ”Z is the unique value

of Z/y for all V -generic points y ∈ [xgen]E ∩D such that Z/y ∩ a = c”. Notethat the same formula defines Z in every further generic extension of V [G]: theequivalence class [xgen]E is countable in V [G] and therefore evaluated the samein V [G] as in the further extension by coanalytic absoluteness, and thereforethe quantifiers in φ range over the same set whether applied in V [G] or furtherextension. Thus, V [Z] ⊂ V [xgen]E !


An important heuristic consequence of the last sentence of this theorem: ifthe poset PI adds a minimal real degree, we cannot use the model V [xgen]Eto discern between the various nonsmooth countable equivalence relations onthe underlying Polish space X, because they all happen to generate the samemodel. This is apparently not to say that every two such countable equivalencerelations must be similar or of the same complexity at some I-positive Borelset.

Corollary 4.2.6. If the forcing PI is proper, nowhere c.c.c. and adds a minimalforcing extension, then I has total canonization for essentially countable Borelequivalences.

Proof. Looking at the previous two theorems, if the poset PI adds a minimal realdegree and E is an essentially countable equivalence relation on the underlyingPolish space, then PI decomposes into an iteration of an ℵ0-distibutive and ac.c.c. forcing. If PI is nowhere c.c.c. then the first extension of the iterationmust be nontrivial, contradicting the assumptions.

In the next section, we will be able to extend the conclusion of the corollary tototal canonization of equivalences classifiable by countable structures, but thetreatment of the essentially countable equivalences is a necessary preliminarystep towards that goal.

The final observation in this section concerns the simplification within theclass of countable equivalence relations. The class of Borel countable equivalencerelations is very complex and extensively studied. It is possible that manyforcings have highly nontrivial intersection of their spectrum with this class.In the fairly common case that the forcing preserves Baire category, all thatcomplexity boils down to E0 though.

Theorem 4.2.7. Whenever I is a σ-ideal on a Polish space X such that theforcing PI is proper and preserves Baire category, then every essentially count-able Borel equivalence relation is reducible to E0 on a positive Borel set.

Proof. Suppose B ∈ PI is an I-positive Borel set and f : B → Y is a Borelmap into a Polish space with a countable equivalence relation E on it. In [31,Theorem 12.1], Miller and Kechris show that there is a Borel set D ⊂ ωω×Y ×Ysuch that for every z ∈ ωω the section Dz is a Borel hyperfinite equivalencerelation included in E, and for every y ∈ Y the set Ay = {z ∈ ωω : [y]E = [y]Dz}is comeager in ωω. Now the set D′ ⊂ B×ωω of all pairs x, z such that z ∈ Af(x),has comeager vertical sections. Since the forcing PI preserves the Baire property,it must be the case that there is a point z ∈ ωω such that the correspondinghorizontal section C ⊂ B is I-positive. But then, Dz = E on f ′′C and thereforethe pullback f−1E is on C reducible to a hyperfinite equivalence relation Dz

and therefore to E0.

4.3. EQUIVALENCES CLASSIFIABLE BY COUNTABLE STRUCTURES79

4.3 Equivalences classifiable by countable struc-tures

This is a class of equivalence relations very common in mathematical practice.

Definition 4.3.1. An equivalence relation E on a Polish space X is classifiableby countable structures, cbcs if there is a Borel map f : X → ΠnP(ωn) suchthat elements x, y ∈ X are E-related if and only if f(x), f(y) are isomorphic asrelational structures on ω.

An isomorphism of structures on ω is naturally induced by a permutation ofω, and therefore every equivalence relation classifiable by countable structuresis Borel reducible to an orbit equivalence generated by a Polish action of aclosed subgroup of the infinite permutation group S∞. The converse implicationalso holds by a result of Becker and Kechris [27, Theorem 12.3.3]: every orbitequivalence relation generated by a Polish action of the infinite permutationgroup or its closed subgroups is classifiable by countable structures.

The behavior of Borel equivalence relations classifiable by countable struc-tures is best analyzed via the notion of Friedman-Stanley jump. If E be anequivalence relation on a Polish space Y , then E+ is the equivalence relation onY ω defined by ~x E+ ~y if the E-saturations of rng(~x) and rng(~y) are the same. IfE is a Borel equivalence relation then E+ is again Borel and it is strictly higherthan E in the Borel reducibility hierarchy. By induction on α ∈ ω1 define Borelequivalence relations Fα by setting F1 = id on the Cantor space, Fα+1 = F+

α

and Fα is the disjoint union of Fβ : β ∈ α for limit ordinals α.

Fact 4.3.2. 1. Every cbcs Borel equivalence relation is Borel reducible tosome Fα for some α ∈ ω1.

2. For every ordinal α ∈ ω1, Fα is Borel reducible to equality of transitivecountable sets of rank exactly ω + α.

Proof. The first item can be found for example in [27, Theorem 12.5.2]. Tointerpret the second item correctly, consider the Polish space X of all binaryrelations on ω with the relation of isomorphism E and consider the Borel setBα = {x ∈ X : x satisfies the axiom of extensionality and x has rank exactlyω + α}. The second item then says that Fα is Borel reducible to E � Bα. Theproof of the second item proceeds by induction on α, and the entirely routineargument is left to the reader.

The equivalence F2 occupies a central position among the cbcs relations. Itis the equivalence relation of (2ω)ω that connects two sequences x, y if rng(x) =rng(y). While it is not necessarily the case that every cbcs equivalence relationcan be simplified to one reducible to F2 on a positive set, still the canoniza-tion properties of F2 extend to the whole class of equivalences classifiable bycountable structures:

Theorem 4.3.3. Let I be a σ-ideal on a Polish space X such that the quotientposet PI is proper. Then


1. if ≤ F2 →Iessentially countable then cbcs→I essentially countable;

2. if ≤ F2 →Ismooth then cbcs→Ismooth;

3. if I has total canonization for equivalences Borel reducible to F2 then ithas total canonization for equivalence relations classifiable by countablestructures.

Proof. Since essentially countable equivalence relations are Borel reducible toF2, the second and third item abstractly follow from the third. The proof ofthe first item uses two claims of independent interest.

Claim 4.3.4. Assume that ≤ F2 →Iessentially countable. Then, for everyBorel equivalence relation E on a Polish space Y , ≤B E →Iessentially countableimplies ≤B E+ →Iessentially countable.

Proof. Let B ⊂ X be a Borel I-positive set, and let F be a Borel equivalencerelation on B reducible to E+ by a Borel function f : B → Y ω. Let M be acountable elementary submodel of a large structure and let C = {x ∈ B : x isM -generic for the poset PI}. This is a Borel I-positive set; we will show thatF � C ≤B F2. Since equivalence relations reducible to F2 can be simplified toessentially countable ones on a Borel I-positive set by the assumption, this willfinish the proof of the claim.

For every n ∈ ω, let Gn be the equivalence relation defined by x0 Gn x1 iff(x0)(n) E f(x1)(n). The set Dn = {A ⊂ B : A is a Borel I-positive set suchthat Fn � A is essentially countable} is dense in PI below B. Enumerate thecountable set Dn ∩M as {Amn : m ∈ ω}, for each n,m find a Borel reductionfmn ∈ M of Gn � Amn to a countable Borel equivalence relation Emn on a Polishspace Y mn . It is easy to arrange that Y mn are pairwise disjoint subspaces of 2ω.Note that for every number n, C ⊂

⋃mA

mn , and the sets (fmn )′′(Amn ∩ C) are

all Borel by Fact 2.6.2(3).Let R ⊂ C × 2ω be the set of all pairs 〈x, y〉 such that there are numbers

m,n, k such that y ∈ [(fmn )′′(Amn ∩ C)]Emn and for all (some) z ∈ Amn ∩ Csuch that fmn (z) Emn y, it is the case that f(z)(n) = f(x)(k). This is a Borelset with nonempty countable vertical sections, so it is the union of graphs ofcountably many Borel functions gn : C → 2ω. Let g : C → (2ω)ω be definedby g(x)(n) = gn(x). It will be enough to show that for all points x0, x1 ∈ C,x0 F x1 if and only if rng(g(x0)) = rng(g(x1)). This is immediate, however.The vertical sections Rx depend only on the [rng(f(x))]E , and this proves theleft-to-right implication. For the opposite implication, note that if ¬x0 F x1

and n ∈ m is such that f(x0)(n) /∈ [rng(f(x1)]E , then for any number m suchthat x0 ∈ Amn it is the case that fmn (x0) ∈ Rx0 while fmn (x0) /∈ Rx1 .

Claim 4.3.5. Given a countable collection of analytic equivalence relations onBorel domains, every equivalence relation on a Borel I-positive set Borel re-ducible to their disjoint sum is reducible to one of them on an I-positive set.


Proof. This is just a countable additivity argument.

Now, by induction on α ∈ ω1 argue that ≤B Fα →Iessentially countable,using the previous two claims on successor and limit steps respectively. If Eis an arbitrary (possibly analytic) cbcs equivalence relation on an I-positiveBorel set B, first use Theorem 4.4.1 to simplify E to Borel on a Borel I-positivesubset, use Fact 4.3.2 to argue that E then must be reducible to some Fα fora countable ordinal α, and use the first sentence of this paragraph to completethe argument.

As in the previous sections, we want to connect canonization properties ofcbcs equivalence relations with forcing properties of the quotient posets. Forthe first theorem, the following new forcing property will be instrumental.

Definition 4.3.6. A poset P has the separation property if it forces that forevery countable set a ⊂ 2ω and every point y ∈ 2ω \ a there is a ground modelcoded Borel set B ⊂ 2ω separating y from a: y ∈ B and a ∩B = 0.

Theorem 4.3.7. Suppose that I is a σ-ideal on a Polish space X such that PIis proper.

1. If PI has the separation property then cbcs→Ismooth;

2. if I has total canonization for essentially countable Borel equivalence re-lations, then PI has the separation property.

Proof. For the first item, the previous theorem shows that it is enough to showthat the separation property implies that ≤B F2 →Ismooth. Let B ⊂ X be aBorel I-positive set and let E be a Borel equivalence relation on B reducibleto F2 by a Borel function f : B → (2ω)ω. By the separation property, eachelement of the set rng(f(xgen)) is separated from the rest by a ground modelcoded set. Let M be a countable elementary submodel of a large structure, letC = {x ∈ B : x is M -generic for PI}, let {Dn : n ∈ ω} be an enumerationof Borel sets in M , and let g : C → 2ω be the function defined by g(x)(n) =1 if rng(f(x)) ∩ Dn contains exactly one element. We claim that g reducesE � C to the identity. Indeed, if x0 E x1, then rng(f(x0)) = rng(f(x1))and g(x0) = g(x1). On the other hand, if ¬x0 E x1 then there will be anumber m such that f(x0)(m) /∈ rngf(x1), and a number n ∈ ω such thatrng(f(x0)) ∩Dn = {f(x0)(m)}; so g(x0)(n) = 1. Now, either g(x1)(n) = 0 andthen we are done. Or, g(x1)(n) = 1, the set rng(f(x1))∩Dn contains exactly oneelement necessarily distinct from f(x0)(m). In this case, find a basic open setO separating f(x0)(m) from this unique element, let n′ be a number such thatDn′ = Dn ∩O, and conclude the proof by observing that g(x0)(n′) 6= g(x1)(n′).

For the second item, suppose that the assumptions hold and B a ⊂ 2ω

is a countable set. We will find a stronger condition D ⊂ B and a countableground model list of Borel sets {Bn : n ∈ ω} such that D forces {Bn : n ∈ ω}to separate the set a in the sense that every element of a belongs to at leastone Borel set on the countable list while no Borel set on the list contains more


than one element of a. This certainly implies the separation property. Withoutloss of generality we may assume that V ∩ a = 0 is forced as well. (If not, useproperness to enclose the countable set a∩V into a ground model countable set,and separate it from the rest by a countable collection of singletons.) Thinningout the set B we may assume that there are Borel functions {fn : n ∈ ω} on theset B such that B a = {fn(xgen) : n ∈ ω}. Let M be a countable elementarysubmodel of a large enough structure, and let C ⊂ B be the I-positive Borelset of all M -generic points for PI in the set B.

Observe that each of the functions fn is countable-to-one on the set C:since the poset PI adds a minimal real degree by Proposition 4.1.2, the setDn ⊂ PI consisting of those conditions on which the function fn is one-to-one is dense in PI , and C ⊂

⋃(M ∩ Dn). This means that the equivalence

relation E generated by the relation x0Ex1 ↔ ∃n0∃n1 fn0(x0) = fn1

(x1) hascountable equivalence classes; i t is easily verified that it is Borel as well. Bythe canonization assumption, there must be then a Borel I-positive set D ⊂ Csuch that E � D = id. This means that for distinct points x0, x1 ∈ D thecountable sets {fn(x0) : n ∈ ω} and {fn(x1) : n ∈ ω} are disjoint; in particular,the functions fn are all injections on D. Let fn = (fn � D) \

⋃m∈n fm and let

Bn = rng(fn). The sets Bn are ranges of Borel injections and as such they areBorel, and for every point x ∈ D their list separates the set {fn(x) : n ∈ ω}. Itfollows that D the sets in {Bn : n ∈ ω} separate the set a.

Theorem 4.3.8. Let I be a σ-ideal on a Polish space X such that the poset PIis proper.

1. If PI adds only a finite collection of real degrees then cbcs→Iessentiallycountable;

2. if I has total canonization for essentially countable Borel equivalence re-lations then I has total canonization for equivalence relations classifiableby countable structures;

3. if PI is nowhere c.c.c. and adds a minimal forcing extension then I hastotal canonization for equivalence relations classifiable by countable struc-tures.

The assumption of the first item is to be interpreted in the obvious way: ifG ⊂ PI is a generic filter then in the extension V [G], there is a finite set b ⊂ 2ω

such that for every point y ∈ 2ω there is z ∈ b such that V [y] = V [z].

Question 4.3.9. Can the assumption in (1) be weakened to “if PI adds only acountable collection of real degrees”?

The main reason for asking this question are the results of Section 5.17. Anumber of quotient forcings in this book are shown to add a single real degree.The quotient forcings adding a finite set of real degrees of size greater thanone typically come from finite products (Section 5.4 or 5.16), and thereforeform quite a small class. On the other hand, the countable support iteration of


countable length commonly adds only countable collection of real degrees, onefor each ordinal in the length of the iteration, by Corollary 5.17.14. Thus, thepositive answer to the question would yield a stronger canonization theorem forthe countable support iteration ideals.

Proof of Theorem 4.3.8. For the first item, by Theorem 4.3.3, it is enough toshow ≤B F2 →Iessentially countable. Let B ⊂ X be a Borel I-positive set andlet E be a Borel equivalence relation on B reducible to F2 by a Borel functionf : B → (2ω)ω. Let a be the name for the set rng(f(xgen)). It is clear fromthe definitions that a ∈ V [[xgen]]E ; in fact, the following theorem will show thatV [a] = V [[xgen]]E .

The first observation: the degree assumption shows that V [[xgen]]E containsan enumeration of the set a in ordertype ω. To see this, note that there is a finiteset b ⊂ a such that a ⊂

⋃z∈b V [z], each model V [z] contains a wellordering of

its 2ω, and these wellorderings can be combined to give a wellordering of the seta. Since the set a is countable in the PI -extension and the poset PI preservesω1, this wellordering will have ordertype that is countable in V , and thereforecan be permuted into an ordering of ordertype ω in the model V [[x]]E .

Let z ∈ (2ω)ω be a name for an enumeration of the set a in the modelV [[xgen]]E . Thinning out the set B if necessary, we may find a Borel functiong : B → (2ω)ω such that B z = g(xgen). Using Theorem 3.4.10, we may thinout the set B further to ensure that the function g attains only countably manyvalues at each E-class. Use Fact 2.6.2 to thin out the set B further to makesure that rng(f(x)) = rng(g(x)) for all x ∈ B, and that g′′B is a Borel set. Thedefinitions immediately imply that g reduces E to F2 � g′′B and that F2 � g′′Bhas countable classes. This concludes the proof of the first item.

The second item follows immediately from the first one; note that the to-tal canonization for essentially countable equivalence relations contains totalcanonization for smooth equivalence relations in its statement, which impliesthat PI adds a single real degree by Proposition 4.1.2. The third item is theconjunction of the second and Corollary 4.2.6.

Corollary 4.3.10. The following σ-ideals have total canonization for equiva-lence relations classifiable by countable structures:

1. the Laver ideal on ωω;

2. the σ-ideal generated by sets of finite Hausdorff measure of a fixed dimen-sion, on every compact metric space X;

3. the σ-ideal generated by sets of continuity for a fixed Borel function on aPolish space X;

4. the σ-ideal of sets of zero capacity for a fixed subadditive stable Ramseycapacity.


Proof. We must verify that the quotient posets are proper and generate a min-imal forcing extension. This follows from work done previously and indepen-dently elsewhere.

The first case, the Laver ideal, is separate from the others in that the asso-ciated σ-ideal is not Π1

1 on Σ11 [67, Proposition 3.8.15]. However, the quotient

poset is isomorphic to Laver forcing of Section 5.13 and the Laver forcing gen-erates a minimal forcing extension by a theorem of Groszek [16].

All the named σ-ideals have been studied in the corresponding sections of[67, Chapter 4]. They are game ideals, the quotient forcings are also < ω1-proper, and with the exception of the nondominating ideal they are Π1

1 on Σ11.

Therefore, by Theorem 3.2.3 and Theorem 4.3.8, it is only necessary to verifythat they generate no intermediate c.c.c. extensions; Theorem 2.6.9 shows thatit is just enough to rule out definable c.c.c. reals in the extension. This mustbe verified for each case separately.

In the case of the Hausdorff measure ideals and Ramsey capacity ideals, thequotient poset PI does not add any independent reals by [67, Theorem 4.3.25]and [67, Theorem 4.4.8]. This rules out any c.c.c. reals by Fact 2.6.6. In thecase of the σ-continuity ideal, the poset PI preserves Baire category, thereforethe only intermediate c.c.c. real that comes into question is a Cohen real byFact 2.6.6 again. The Cohen real is impossible though since the poset preservesouter Lebesgue measure [67, Section 4.2.3]. Steprans showed earlier and in adifferent language that the poset PI has the Laver property and therefore doesnot add Cohen reals [59].

While the model V [x]E can acquire many forms in the case of equivalencerelations classifiable by countable structures, the model V [[x]]E is easy to com-pute.

Theorem 4.3.11. Let I be a σ-ideal on a Polish space X such that the quotientposet PI is proper, and let E be a cbcs equivalence relation on X. Then PI forcesV [[xgen]]E = V (a) for some hereditarily countable set a.

Proof. Let B ⊂ be a Borel I-positive set. Since E is reducible to an orbitequivalence relation induced by a Polish action of S∞, Theorem 4.4.1 showsthat there is a Borel I-positive subset C ⊂ B such that E � C is a Borelequivalence relation. By Fact 4.3.2, E � C is Borel reducible to the equality ofcountable transitive sets of some fixed countable rank α. Let Y be the space ofbinary relations on ω, let F be the isomorphism relation on X, let D ⊂ Y bethe Borel set of those relations that satisfy the axiom of extensionality and arewellfounded of rank α. Let f : C → D be a Borel reduction of the equivalencerelation E � C to F � D.

Let a be the PI -name for the transitive isomorph of the relation f(xgen),and argue that C V [[xgen]]E = V (a). It is clear that the set a belongsto every model V [y] if y is some V [xgen]-generic element of X E-related toxgen: the model V [y] contains some element of the set C E-related to y byanalytic absoluteness between the models V [y] and V [x][y], and writing z ∈ V [y]

4.4. ORBIT EQUIVALENCE RELATIONS 85

for this element it must be the case that the transitive isomorph of f(z) isisomorphic, and therefore equal, to the transitive isomorph of f(xgen) = a.Thus, V (a) ⊆ V [[xgen]]E .

For the opposite inclusion, in the model V (a) consider the poset P that addsa bijection between ω and HV

c+ ∪ a with finite approximations. Let H ⊂ P be afilter generic over V [xgen]. The model V (a)[H] satisfies the axiom of choice; it isequal to V [z] for some binary relation on ω that is isomorphic to the ∈ relationon HV

c+ ∪ a. By the analytic absoluteness between the models V [xgen][H] andV (a)[H], there is a point y ∈ V (a)[H] such that y is PI -generic over V andthe transitive isomorph of f(y) is exactly a. The forcing theorem shows thatthere must be a P -name y ∈ V (a) for such a point. Now let H0, H1 ⊂ P bemutually generic filters over V (a). A mutual genericity argument shows thatV (a) = V (a)[H0] ∩ V (a)[H1], and both of the models on the right side of thisequality contain points E-related to xgen, namely y/H0 and y/H1. It followsthat V [[xgen]]E ⊆ V (a)[H0] ∩ V (a)[H1] = V (a).

4.4 Orbit equivalence relations

Orbit equivalence relations of Polish group actions form a quite special subclassof analytic equivalence relations. In our context, they can be simplified to Borelequivalence relations by passing to a Borel I-positive Borel set:

Theorem 4.4.1. Suppose that X is a Polish space, I a σ-ideal on it such thatthe quotient poset PI is proper, B ∈ PI an I-positive Borel set, and E anequivalence relation on the set B reducible to an orbit equivalence relation ofa Polish action. Then there is a Borel I-positive set C ⊂ B such that E � Cis Borel. If the ideal I is c.c.c. then in fact the set C can be found so thatB \ C ∈ I.

Note that the equivalence E may have been analytic non-Borel in the beginning.

Proof. Suppose that a Polish group G acts on a Polish space Y , and f : B → Yis a Borel reduction of E to the orbit equivalence relation EG of the action. By[4, Theorem 7.3.1], the space Y can be decomposed into a union of a collectionAα : α ∈ ω1 of pairwise disjoint invariant Borel sets such that EG � Aα is Borelfor every ordinal α; moreover, this decomposition is absolute between varioustransitive models of ZFC with the same ℵ1.

Let M be a countable elementary submodel of a large enough structureand let C ⊂ B be the Borel I-positive subset of all M -generic points for PIin the set B. This is a Borel I-positive set by the properness of the quotientposet PI , and if I is c.c.c. then in fact B \ C ∈ I. For every point x ∈ C,M [x] |= f(x) ∈

⋃{Aα : α ∈ M ∩ ω1}. Absoluteness considerations imply that

f ′′C ⊂⋃α∈ω1∩M Aα. The equivalence relation EG restricted on the latter set

is Borel, and so E � C is Borel.

This feature is quite special to orbit equivalence relations, and elsewhere itmay not hold. Consider the following example:


Example 4.4.2. Let K be an analytic non-Borel ideal on ω and the equivalencerelation EK on X = (2ω)ω defined by ~xEK~y if {n ∈ ω : ~x(n) 6= ~y(n)} ∈ K.Consider the ideal I on X associated with the countable support product ofSacks forcing. Then the equivalence EK is non-Borel, and since EK ≤ EK � Bfor every Borel I-positive set B ⊂ X, it is also the case that EK � B is non-Borel.

4.5 Hypersmooth equivalence relations

Equivalence relations reducible to E1 are connected to decreasing sequences ofreal degrees in the generic extension.

Theorem 4.5.1. Suppose that X is a Polish space and I is a σ-ideal such thatthe forcing PI is proper. The following are equivalent:

1. E1 is in the spectrum of I;

2. some condition forces that in the generic extension there is an infinitedecreasing sequence of real degrees;

3. some condition forces that in the generic extension there is a strict infinitedecreasing sequence of real degrees.

Here, a decreasing sequence of V -degrees is a sequence 〈xn : n ∈ ω〉 of infinitebinary sequences such that whenever m ∈ n then V [xn] ( V [xm]. A strict de-creasing sequence is a decreasing sequence such that moreover for every numberm, the sequence 〈xn : m ∈ n〉 belongs to V [xm].

Proof. Suppose first that E1 is in the spectrum of I, as witnessed by a BorelI-positive set B ⊂ X an equivalence relation E on B, and a Borel reductionf : B → (2ω)ω of E to E1. We claim that B forces that in the generic extension,for every number n ∈ ω there is m ∈ ω such that f(xgen) � [n, ω) /∈ V [f(xgen) �[m,ω)]; so (2) holds.

Suppose for contradiction that some condition C ⊂ B forces the oppositefor some definite number n ∈ ω. Let M be a countable elementary submodelof a large enough structure containing C and f , and let D ⊂ C be the BorelI-positive set of all M -generic points in the set C. Consider the Borel setZ ⊂ (2ω)ω\n defined by z ∈ Z if and only if for every number m > n there is aBorel function g ∈M such that g(z � [m,ω)) = z. It is not difficult to see thatE1 restricted to Z has countable classes, since there are only countably manyfunctions in the model M . Now, whenever x ∈ D then f(x) � [n, ω) ∈ Z by theforcing theorem, and for x, y ∈ D, f(x) E1 f(y) if and only if f(x) � [n, ω) E1

f(y) � [n, ω). This reduces E � D to a Borel equivalence relation with countableclasses, which contradicts the assumption that E1 is embeddable into it.

Now, in the generic extension V [x] define a function h ∈ ωω by induction:h(0) = 0, and h(n + 1) is the least number m ∈ ω such that f(x) � [h(n), ω) /∈V [f(x) � [m,ω)]. The function h clearly belongs to all models V [f(x) � [m,ω)],

4.5. HYPERSMOOTH EQUIVALENCE RELATIONS 87

and therefore the sequence f(x) � [h(n), ω) : n ∈ ω forms the required strictlydecreasing sequence of real degrees.

To show (2) implies (3), suppose that some condition forces 〈xn : n ∈ ω〉 tobe the decreasing sequence in 2ω. The strict decreasing sequence is obtainedby a simple adjustment. Just replace 〈xn : n ∈ ω〉 by a decreasing sequence〈xn : n ∈ ω〉 such that V [xn] |= xn+1 is the ≤n-least point below which thereis an infinite decreasing sequence of degrees and V [xn+1] 6= V [xn]. Here, ≤n isany well-ordering of 2ω ∩ V [xn] simply definable from a wellordering of V andxn; for example, let Q be the least poset of size ≤ c in V for which xn is generic,and then turn any well-order of Q-names for reals into a well-order of V [xn].

To prove (3) implies (1), Fact 2.6.2 yields a Borel I-positive set B and Borelfunctions fn : B → 2ω, for each n ∈ ω, representing the points on the strictdecreasing sequence. We claim that the product of these functions, g : B →(2ω)ω defined by g(x)(n) = fn(x), has the property that E1 is Borel reducibleto the pullback E = g−1E1 restricted to any Borel I-positive set. This willprove the theorem. Suppose for contradiction that there is a Borel I-positiveset C ⊂ B such that E1 6≤ E � B; by the dichotomy 2.2.10 it must be the casethat E � C ≤ E0 via some Borel map g : C → 2ω. Move to the forcing extensionV [xgen] for some PI -generic point xgen ∈ C.

Claim 4.5.2. For every number n ∈ ω, g(xgen) ∈ V [fn(xgen)].

Proof. Working in V [fn(xgen)], we see that there is a point x ∈ C such that〈fm(x) : m > n〉E1〈fm(xgen) : m > n〉, since such an x exists in the modelV [xgen] and analytic absoluteness holds between the two models. Note thatthe sequence 〈fm(xgen) : m > n〉 is in the model V [fn(xgen)]. Now all suchnumbers x must be mapped by g into the same E0-equivalence class. This classis countable and contans g(xgen). The claim follows.

Now in the model V [g(xgen)], there is a point x ∈ C such that g(x) = g(xgen),since there is such a point x in the model V [xgen] and analytic absoluteness holdsbetween the two models. The sequence 〈fn(x) : n ∈ ω〉 must be E1-equivalent tothe sequence 〈fn(xgen) : n ∈ ω〉; let us say that they are equal from some m on.But then, fm(x) = fm(xgen) ∈ V [g(xgen)] ⊂ V [fm+1(xgen)], contradiction!

Corollary 4.5.3. If the forcing PI adds just a finite set of real degrees then E1

does not belong to the spectrum of I.

This corollary applies immediately to many σ-ideals, among others the smoothσ-ideal, the ideal associated with the product of two Miller forcings and theLaver ideal.


Chapter 5

Classes of σ-ideals

5.1 σ-ideals σ-generated by closed sets

Following the exposition of [67], the σ-ideals that should be easiest to deal withare those σ-generated by closed sets. Regarding the canonization properties ofσ-ideals in this class, on one hand there is the σ-ideal of countable sets which hasthe Silver property as an immediate corollary of Silver’s theorem; on the otherhand, there is the σ-ideal of meager sets, which has a very complicated spectrum.Thus, the treatment of this class must make finer distinctions between variousσ-ideals. Still, there is a wealth of relevant general information available:

Theorem 5.1.1. Suppose that I is a σ-ideal on a Polish space X, σ-generatedby closed sets. Then

1. every analytic I-positive set contains a Gδ I-positive subset;

2. every Borel function on an analytic I-positive domain is continuous on aGδ I-positive subset;

3. the σ-ideal I is Π11 on Σ1

1 if and only if the collection of closed sets in I iscoanalytic in the standard Borel space F (X) if and only if I is σ-generatedby a coanalytic collection of closed sets.

4. if I is a Π11 on Σ1

1 then I has the selection property;

5. the poset PI is < ω1-proper and preserves Baire category;

6. moreover, if I is Π11 on Σ1

1 then every intermediate extension of the PIextension is generated by a single Cohen real.

Proof. (1) is a result of Solecki [53]. To prove (2), choose a Borel function f onan analytic I-positive set A ⊂ X. First, use (1) to thin out the set A to a Gδset whose intersections with open sets are either I-positive or empty. It is notdifficult to see that every closed set C ∈ I must be relatively meager in such

89

90 CHAPTER 5. CLASSES OF σ-IDEALS

A. Then, use a classical theorem [33, Theorem 8.38] to find a dense Gδ subsetB ⊂ A on which f is continuous and argue that such B must be I-positive bythe previous sentence.

For (3), it is clear that if I is Π11 on Σ1

1 then the set of closed sets in I iscoanalytic. On the other hand, if I is σ-generated by a coanalytic collectionof closed sets then it is Π1

1 on Σ11 by [33, Theorem 35.38]. (5) is contained

in [67, Theorem 4.1.2] and (6) is in [67, Theorem 4.1.7]. That leaves (4) tobe proved. The selection property follows from Theorem 3.1.4 if the σ-ideal isgenerated by a Borel collection of closed sets. However, there are importantσ-ideals generated by closed sets without such a Borel basis, and the argumentfrom Theorem 3.1.4 must be restated to work for them.

Let D ⊂ 2ω×X be an analytic set all of whose vertical sections are I-positive.We will show

Claim 5.1.2. There is a perfect set P ⊂ 2ω and a continuous function g :P × ωω → X such that for every z ∈ P , the range of the function gz = g(z, ·) :ωω → X is a subset of Dz, and the images of nonempty open sets under thisfunction are I-positive.

Suppose that the claim has been obtained. Choose an Fσ-universal Fσ setF ⊂ P × X, and use the nonmeager section uniformization theorem to find aBorel function h : P → ωω such that for every z ∈ P , if g−1

z Fz ⊂ ωω is meagerthen h(z) /∈ g−1

z Fz. Let f : P → X be the function defined by f(z) = g(z, h(z)).This function is certainly Borel, and for each z ∈ P the value f(z) belongs toDz. We must show that rng(f) /∈ I. Indeed, suppose that C ∈ I is a set, andcover it by the union of closed sets in I, C ⊂

⋃n Cn. There is z ∈ Z such that

Fz =⋃n Cn. Note that the preimages g−1

z Cn ⊂ ωω are closed sets with emptyinterior by the choice of the function g, so they aremeager and so is their uniong−1z Fz. Then the point f(z) is in rng(f) \

⋃n Cn, showing that the set rng(f)

is I-positive.

The proof of the claim is standard. Let k : ωω → 2ω × X be a continuousfunction whose image is the analytic set D. For every z ∈ 2ω let Az = f−1Dz;remove from Az all the relatively open sets with I-small k-image, obtaining aclosed set Az = [Tz] for some pruned tree Tz ⊂ ω<ω. Note that the f -images ofnonempty relatively open subsets of Az are I-positive, and for every t ∈ ω<ω,the statement t ∈ Tz is analytic since I is Π1

1 on Σ11. The map z 7→ Tz is σ(Σ1

1)-measurable, therefore Baire measurable, and so continuous on a dense Gδ-set.For every z there is a map π : ω<ω → ω<ω such that π preserves extension andlength of sequences and for every t ∈ ω<ω, π′′{u : t ⊂ u} = {v ∈ Tz : π(t) ⊂ v}.An application of a Jankov-von Neumann uniformization theorem 2.1.6 yieldsa σ(Σ1

1)-measurable map z 7→ πz, which is then continuous on further denseGδ subset. Let P ⊂ 2ω be a perfect subset of this dense Gδ set, and let g :P × ωω → X be the map defined by g(z, y) = f(

⋃n πz(y � n)). It is immediate

to check that the function g has the desired properties.

5.1. σ-IDEALS σ-GENERATED BY CLOSED SETS 91

The classification of possible intermediate forcing extensions of the PI ex-tension yields an immediate attractive corollary:

Corollary 5.1.3. If I is a Π11 on Σ1

1 σ-ideal σ-generated by closed sets, thenthe following are equivalent:

1. PI does not add Cohen reals;

2. I has total canonization for equivalence relations classifiable by countablestructures;

3. I has the Silver property for Borel equivalence relations classifiable bycountable structures.

Proof. (1) implies (2) by Theorem 4.3.8 and the fact that the only possible in-termediate forcing extension is a single Cohen real extension. Note that theposet PI must be nowhere c.c.c.; if it had a c.c.c. part, then that part would bein fact isomorphic to Cohen forcing by Fact 2.6.6(3) (2) implies (3) by Propo-sition 3.3.2 and the selection property proved above. Finally, (3) implies (1):if PI did add Cohen reals, all of the equivalence relations in the spectrum ofthe meager ideal would also show up in the spectrum of I, and this includesequivalence relations such as E0. This of course immediately contradicts theSilver property for such equivalence relations.

The corollary suggests an obvious question:

Question 5.1.4. Let I be a Π11 on Σ1

1 σ-ideal σ-generated by closed sets, suchthat the quotient poset PI does not add Cohen reals. Does it follow that I hastotal canonization for analytic equivalence relations?

The work in this section should be viewed as an effort to give a positiveanswer to this question by considering certain special classes of σ-ideals first.We will show that calibrated σ-ideals and certain other σ-ideals generalizing theσ-ideal generated compact subsets of the Baire space have the free set property,and therefore do have total canonization for analytic equivalence relations. Fur-thermore, Corollary 5.2.12 in Section 5.2 shows that if a σ-ideal σ-generated byclosed sets does not add an infinitely equal real (a condition slightly strongerthan not adding a Cohen real) then it has total canonization and Silver propertyfor the equivalence relations reducible to EKσ or =J for an analytic P-ideal J .This still leaves a good number of interesting open special cases.

Now, the spectrum of the meager ideal is extremely complicated and it istreated as a separate case in Section 5.1c.

5.1a Calibrated ideals

In this section, we will prove a general canonization theorem for a class of σ-ideals widely encountered in abstract analysis:


Definition 5.1.5. [30] A σ-ideal on a Polish space X is calibrated if for everyclosed I-positive set C ⊂ X and countably many closed sets {Dn : n ∈ ω} inthe ideal I there is a closed set C ′ ⊂ C disjoint from all the sets {Dn : n ∈ ω}.

One wide class of calibrated σ-ideals generated by compact sets consistsof those σ-ideals with the covering property, i.e. those where every I-positiveanalytic set contains a compact I-positive subset. This includes the σ-idealof countable sets; the σ-ideal generated by sets of finite packing measure ofsome fixed dimension [67, Section 4.1.6]; the σ-ideal generated by the sets ofmonotonicity of a given continuous function [66, Lemma 2.3.7]; the σ-ideal gen-erated by H-sets [42], [69]; or the σ-ideal generated by compact sets of extendeduniqueness [28].

Another class of calibrated σ-ideals consists of those σ-ideals σ-generated byall closed sets in a fixed larger σ-ideal with the covering property. This includesthe σ-ideal σ-generated by closed Lebesgue measure zero sets; the σ-ideal σ-generated by the compact sets on which E0 is smooth; the σ-ideal σ-generatedby closed σ-porous sets; or the σ-ideal σ-generated by closed sets of capacity 0for a fixed subadditive capacity.

Other σ-ideals σ-generated by closed sets may satisfy calibration for moresophisticated reasons, such as the σ-ideal σ-generated by closed sets of unique-ness [8]. Some σ-ideals may fail calibration but satisfy some weakenings of itwhich still allow our argument to go through without significant changes. Suchis the case of stratified calibration and the σ-ideal σ-generated by closed sets offinite dimension in the Hilbert cube.

A simple example of a σ-ideal which is not calibrated is the meager ideal.A more sophisticated class of non-calibrated σ-ideals are the ones discussed inthe next section. They in fact remain non-calibrated no matter what the choiceof Polish topology on the underlying space as long as the Borel structure ispreserved.

All the σ-ideals discussed above have a coanalytic set of generators andtherefore are Π1

1 on Σ11.


1 σ-ideal on a compact metrizable space X,σ-generated by compact sets. If I is calibrated, then I has the free set property.

Corollary 5.1.7. For such a calibrated σ-ideal I, the total canonization foranalytic equivalence relations and the Silver property for Borel equivalence rela-tions both hold.

Proof of Theorem 5.1.6. Let B ⊂ X be a Borel I-positive set and let D ⊂ B×Bbe an analytic set with I-small vertical sections. We must find a Borel I-positiveset C ⊂ B such that D ∩ (C × C) ⊂ id.

Use the Burgess–Hillard theorem [33, Theorem 35.43] to cover the set Dwith countably many Borel sets with I-small compact vertical sections. Thefunctions fn : B → K(X) for n ∈ ω assigning to each point its correspondingvertical section are Borel by [33, Theorem 28.8]–so we have that for every x ∈ B,fn(x) ∈ I and Dx ⊂

⋃n fn(x). Thinning out the set B if necessary, we may


assume that the functions fn are all continuous–Theorem 5.1.1 (2). Furtherthinning out the set B, we may assume that it is a Gδ set such that intersectionswith open sets are either I-positive or empty by Solecki’s theorem [53]. LetB =

⋂nOn for some open sets On ⊂ X.

Construct a suitable tree representation of the set B. By tree inductionon t ∈ ω<ω, build open sets Ot ⊂ X such that s ⊂ t implies Ot ⊂ Os, thediameters of the sets Ot tend to zero in some fixed complete compatible metricfor the space X, Ot ⊂ O|t|, Ot ∩ B 6= 0, and for every neighborhood O withnonempty intersection with Ot ∩ B there is m ∈ ω such that Otam ⊂ O. Forevery point y ∈ ωω, the intersection

⋂nOy�n is a singleton and its single element

g(y) ∈ X belongs to the set B. The function g : ωω → X is easily checked tobe continuous.

Pick a dense set a ⊂ ωω. Use the calibration assumption to find compactsets Kt included in both Ot and the closure of B that are I-positive and disjointfrom all sets {fn(g(y)) : n ∈ ω, y ∈ a}. We will find a certain tree T ⊂ ω<ω

which splits densely often and

(*) for every splitnode t ∈ T , every x ∈ Kt and every neighborhood P of xthere is m such that tam ∈ T and Otam ⊂ P .

Claim 5.1.8. Whenever T satisfies (*) then g′′[T ] is an I-positive set.

Proof. Let {Hn : n ∈ ω} be compact subsets of X in the σ-ideal I; we mustproduce a point y ∈ ωω such that g(y) /∈

⋃nHn. To do that, by induction

on n ∈ ω build an increasing sequence 〈tn : n ∈ ω〉 of splitnodes tn such thatOtn+1 ∩ Hn = 0; then y =

⋃n tn will be as required. Now suppose that tn

has been constructed. Since Ktn is not a subset of Hn, there must be a pointx ∈ Ktn \ Hn. Use (*) to find a number m such that Otam ⊂ X \ H and lettn+1 be some splitnode of T extending tanm. The induction can proceed.

Of course, the point of the construction of T will be that the set g′′[T ]consists of pairwise E-independent elements. Then, it is an I-positive analyticset which has an I-positive Gδ-subset C by a theorem of Solecki [53]. The setC will be as required.

Let 〈〈ti, Pi〉 : i ∈ ω〉 be an enumeration of the product ω<ω × O, whereO is some fixed countable basis of the space X, with infinite repetitions. Byinduction on i ∈ ω build finite sets Ti ⊂ ω<ω and open sets Ui ⊂ ωω so that

I1 T0 ⊂ T1 ⊂ . . . and U0 ⊃ U1 ⊃ . . . . The set Ti is closed under theoperation of taking the longest common initial segment. Moreover, Ti+1

is an end-extension of Ti, and for every t ∈ Ti+1 \ Ti it is the case thatevery infinite extension of t in T belongs to Ui;

I2 for every t ∈ Ti, for every neighborhood P with nonempty intersectionwith Kt there is m ∈ ω and y ∈ Ui such that tam ⊂ y and Otam ⊂ P ;

I3 if (a) ti ∈ Ti and Pi has nonempty intersection with Kti then Ti+1 =

Ti ∪ {s} for some node s ∈ ω<ω such that some m ∈ ω, tai m ⊂ s andOtam ⊂ Pi. Otherwise (b) Ti+1 = Ti;


I4 if I3(a) holds, then if y, z ∈ Ui+1 and s ∈ ω<ω is a sequence with s ⊂ yand s 6⊂ z then g(z) /∈

⋃n∈i fn(g(y)).

After the induction has been performed, let S be the downward closure of⋃i Ti. The first item shows that [S] ⊆

⋂i Ui; in fact, an equality can be arranged

here. It also implies that⋃i Ti is exactly the set of splitnodes of the tree S, and

the third item implies that S satisfies (*). Now suppose that y, z are two distinctbranches through S and n ∈ ω, and argue that g(z) /∈ fn(g(y)). There has tobe i ∈ ω larger than n and such that the longest initial segment of y in Ti is ti,I3(a) holds, the node s ∈ Ti+1 \ Ti is still an initial segment of y, and moreoverti is not an initial segment of z. But then, I4 shows that g(z) /∈ fn(g(y)) asdesired!

The induction is started with T0 = {0} and U0 = [T ]. Suppose Ti, Ui havebeen obtained. If I3(b) holds then let Ti+1 = Ti and Ui+1 = Ui. If I3(a) occurs,

first use I2 to find a point y ∈ Ui such that tai m ⊂ y for a number m which is notused on any of the sequences in the set Ti and such that Otam ⊂ Pi. Adjustingy slightly, we may find it in the dense set a. The compact set

⋃n∈i fn(g(y)) is

disjoint from all the compact sets Kt for t ∈ Ti, and therefore it has an openneighborhood P such that its closure is still disjoint from all of them. Use thecontinuity of the functions fn and g to find an initial segment s of y such thatevery infinite extension y′ of s belongs to the set Ui, and

⋃n∈i fn(g(y′)) ⊂ P .

Let Ti+1 = Ti ∪ {s} and Ui+1 = {z ∈ Ui : s ⊂ z or g(z) is not in the closureof P}. It is not difficult to verify that the induction hypotheses continue tohold.

The σ-ideals with the covering property form a very special subclass of thecalibrated ideals. For them, we can even prove the mutual genericity propertyas well as the rectangular Ramsey property, and this is utilized in Section 5.16dealing with their products. The total canonization of analytic equivalencerelations could be proved in this subclass simply by quoting Theorem 3.4.4and observing that the rectangular Ramsey property prevents the nontrivialergodicity in Case 1 of that theorem from ever occurring. The general calibratedσ-ideals have much more complicated behavior, and typically both the mutualgenericity and the rectangular property will fail. In one especially common case,we can find a Borel function on pairs that on any Borel positive rectangle coversa large set.

Theorem 5.1.9. Let µ be the usual Borel probability measure on 2ω, and letI be the σ-ideal σ-generated by the closed sets of µ-mass 0. There is a Borelfunction f : (2ω)2 → 2ω such that for every pair B,C ⊂ 2ω of analytic I-positivesets, the range f ′′(B × C) contains a nonempty open set.

Proof. Use the concentration of measure, Fact 2.7.2, to find numbers {ni : i ∈ ω}such that for every i ∈ ω, ni < ni+1 and, considering the Hamming cube 2ni+1\ni

with the normalized counting measure µi and the normalized Hamming distancedi, the di−1/8-neighborhood of any set of µi-mass > 2−i−1 has µi-mass greaterthan 1/2. Extend the definition of the distance di to 2ω and finite binary


sequences of length greater than ni+1 by setting di(s, t) = di(s � [ni, ni+1), t �[ni, ni+1)). For points x, y ∈ 2ω, let 〈ij : j ∈ ω〉 be an increasing enumerationof the numbers i such that x is di − 1/4-close to y. Define f(x, y)(j) = 0 ifi2j+1 = i2j + 1. We claim that this function f : (2ω)2 → 2ω works.

In order to show this, the following claim will be useful:

Claim 5.1.10. Let i ∈ ω, let s ∈ 2ni be a sequence and let B ⊂ 2ω be a Borelset of relative mass > 2−i in [s].

1. if j > i and u ∈ 2nj\ni is a binary sequence, then there is an extensions′ ⊃ s such that s′ ∈ 2nj , B has relative mass at least 2−j inside [s′], andfor every i ≤ k < j, dk(s′, u) > 1/4;

2. if t ∈ 2ni is another sequence and C ⊂ 2ω is a Borel set with relativemass > 2−i in [t], j > i and z ∈ 2j\i is a binary sequence, then there areextensions s′ ⊃ s and t′ ⊃ t of length nj such that for every i ≤ k < j,dk(s′, t′) < 1/4 if and only if z(k) = 0.

Proof. Both parts of the claim are obtained by induction on j − i, and it isclearly enough to resolve the case j = i+ 1. For (1), note that the set b = {v ∈2ni+1\ni : B has relative mass > 2−i in [sav]} has µi-mass at least 2−i by theFubini theorem. The 1/4-neighborhood of b and the 1/4-neighborhood of theflip of b–both of these sets have µi-mass > 1/2, therefore they have a nontrivialintersection. If w is a point in the intersection, there is a point v0 ∈ b within1/4-distance of w and a point v1 ∈ b within 1/4-distance of the flip of w. Suchpoints v0, v1 must have distance at least 1/2, so at least one of them is di-fartherthan 1/4 from the sequence u; say it is v0. The sequence s′ = sav0 ∈ 2ni+1 isas required.

For (2), consider the sets b = {v ∈ 2ni+1\ni : B has relative mass > 2−i−1 in[sav]} and c = {v ∈ 2ni+1\ni : C has relative mass > 2−i−1 in [sav]}; both of thehave µi-mass at least 2−i−1. There are sequences v0 ∈ b and v1 ∈ c of di-distance> 1/4, proved just as in the previous paragraph. There are also sequences v0 ∈ band v1 ∈ c of di-distance < 1/4, since the di − 1/8-neighborhoods of b, c haveboth mass greater than 1/2 and therefore intersect. The conclusion of item (2)immediately follows.

Now let B,C ⊂ 2ω be analytic I-positive sets. Thinning out the sets B,C ifnecessary, we may assume that both areGδ and their intersections with open setsare either empty or I-positive; write B =

⋂nOn and C =

⋂n Pn, intersections

of open sets. The closures of B and C have positive µ-mass, and by the Lebesguedensity theorem, there must be finite binary sequences s0, t0 of equal length suchthat the closures of the sets B and C have relative masses > 1/2 in the respectiveopen neighborhoods [s0], [t0]. Let 〈ij : j ∈ j0〉 be an increasing enumeration ofthe numbers i such that t0 is di − 1/4-close to s0. To simplify the notationassume that j0 is an even number. Let u0 : j0/2 → 2 be the function definedby u(j) = 0 if i2j+1 = i2j + 1. We will show that for every z ∈ 2ω extending uthere is x ∈ B ∩ [s0] and y ∈ C ∩ [t0] such that f(x, y) = z.

By induction on k ∈ ω build finite binary sequences sk, tk so that


I1 s0 ⊂ s1 ⊂ . . . and similarly on the t side. Moreover, [sk+1] ⊂ Ok and[tk+1] ⊂ Pk;

I2 there are numbers lk such that sk, tk are both of length lk, and the closuresof the sets B and C are sets of relative mass > 2−j0−k in the respectiveopen neigborhoods [sk], [tk];

I3 there are numbers i2(j+k) and i2(j+k)+1 such that they are the only num-bers i between lk and lk+1 such that sk+1 is di − 1/4-close to tk+1, andz(j0 + k) = 0 iff these numbers are successive.

In the end, we will set x =⋃k sk and y =

⋃k tk. I1 will imply that x ∈ B and

y ∈ C, and I3 will imply that f(x, y) = z as required.To perform the induction, suppose that sk, tk have been found. First, use the

Lebesgue density theorem to find l′ > lk and an extension s′ ⊃ sk in 2nl′ suchthat [s′] ⊂ Ok and B has relative mass in [s′] larger than 2−l

′. Use Claim 5.1.10

(1) to find an extension t′ ⊃ tk in 2nl′ such that C ∩ [t′] has relative mass > 2−l′

and for every l ≤ i < l′ it is the case that di(s′, t′) > 1/4. Now switch the

roles of s, t: find a number l′′ > l′ and extensions t′′ ⊃ t′ and s′′ ⊃ s′ in 2nl′′

such that both sets B,C have relative mass > 2−l′′

in [s′′], [t′′] respectively,[t′′] ⊂ Pk, and for every lk ≤ i < l′′ it is the case that di(s

′′, t′′) > 1/4. Finally,use Claim 5.1.10 (2) on s′′, t′′ to find their extensions sk+1, tk+1 that satisfy theinduction hypothesis at k + 1.

5.1b The σ-compact ideal and generalizations

The calibration feature of a σ-ideal I is highly dependent on the choice of thetopology for the underlying Polish space X. Moreover, passing to an I-positiveBorel set B ⊂ X, it may be difficult to recognize the calibration just fromthe structure of the σ-ideal below B–even though the total canonization orSilver property certainly survive such a restriction. In this section, we will dealwith a class of σ-ideals σ-generated by closed sets that cannot be presentedas calibrated σ-ideals restricted to a condition, no matter what the choice ofthe topology or the Borel positive set. The most prominent representative ofthis class is the σ-ideal I on ωω σ-generated by compact sets. Its dichotomyproperties are well known, and its canonization properties have been thoroughlystudied by Otmar Spinas.

Fact 5.1.11. (Hurewitz, [32]) Whenever A ⊂ ωω is an analytic set, exactly oneof the following is true: either A ∈ I or A contains all branches of a superperfecttree. The ideal I is Π1

1 on Σ11.

Fact 5.1.12. [55] The ideal I has the free set property. [57] The ideal I hastotal canonization for Borel graphs.

As every Π11 on Σ1

1 σ-ideal σ-generated by closed sets, the ideal I has theselection property, its free set property implies total canonization, and so inconjunction with Proposition 3.3.2 this yields


Corollary 5.1.13. The ideal I has the Silver property. Restated, for everyBorel equivalence relation E on ωω, either the space ωω is covered by countablymany E-equivalence classes and countably many compact sets, or it has a closedsubset homeomorphic to ωω consisting of E-inequivalent elements.

We will now generalize Spinas’s results to partially ordered sets of infinitelybranching trees with various measures of the size of branching. For an idealK on a countable set a let P (K) be the poset of all trees T ⊂ a<ω such thatevery node of T extends to a splitnode of T , and every splitnode t ∈ T the set{i ∈ a : tai ∈ T} is not in the ideal K. Thus for example the Miller forcing isP (Frechet ideal on ω). The computation of the ideal I(K) associated with theforcing gives a complete information. Let X = aω with the product topology,where a is taken with the discrete topology. For every function g : a<ω → Kconsider the closed set Ag = {x ∈ aω : ∀n x(n) ∈ g(x � n)} ⊂ X, and theσ-ideal I(K) on the space X σ-generated by all these closed sets. The followingis obtained by a straightforward generalization of the proof of Fact 5.1.11.

Fact 5.1.14. Let A ⊂ X be an analytic set. Exactly one of the followinghappens:

• A ∈ I(K);

• there is a tree T ∈ P such that [T ] ⊂ A.

Moreover, if the ideal K is coanalytic then the σ-ideal I(K) is Π11 on Σ1

1.

Thus the map T 7→ [T ] is a dense embedding of the poset P (K) into PI(K). Itis clear now that the poset P (K) is proper and preserves Baire category sincethe corresponding ideal is generated by closed sets [67, Theorem 4.1.2]. If theideal K is nonprincipal then the forcing adds an unbounded real. Other forcingproperties depend very closely on the position of the ideal K in the Katetovordering. The poset may have the Laver property (such as with K equal to theFrechet ideal), or it may add a Cohen real (such as in K equal to the nowheredense ideal on 2<ω). Similar issues are addressed in [48, Section 3]. We firstlook at a very well behaved special case, generalizing the result of [55].

Theorem 5.1.15. Suppose that K is an Fσ ideal on a countable set. ThenI(K) has the free set property.

Proof. Fix an Fσ-ideal K on ω and use Mazur’s theorem [37] to find a lowercontinuous submeasure φ on P(ω) such that K = {a ⊂ ω : φ(a) < ∞}. Theargument now proceeds with a series of claims.

Claim 5.1.16. The forcing P (K) has the weak Laver property.

Here, a forcing P has the weak Laver property if for every condition p ∈ P ,every name f for a function in ωω forced to be dominated by a ground modelfunction there is a condition q ≤ p, an infinite set b ⊂ ω, and sets {cn : n ∈ b} ofthe respective size n such that q ∀n ∈ b f(n) ∈ cn. This is a forcing propertyappearing prominently in connection with P-point preservation [68].


Proof. Let p ∈ P (K) be a condition, g ∈ ωω a function and f a P (K)-name fora function in ωω pointwise dominated by g. By induction on n ∈ ω constructtrees Un ∈ P (K) and their finite subsets un ⊂ Un, numbers mn and finite setscn ⊂ ω so that

I1 T = U0 ⊃ U1 ⊃ . . . and {t} = u0 ⊂ u1 ⊂ . . . where t is the shortestsplitnode of T ;

I2 un is an inclusion initial set of splitnodes of Un and for every node t ∈ un,the set of its immediate successors that have some successor in un+1 hasφ-mass at least n;

I3 Un+1 f(mn) ∈ cn and |cn| ≤ mn.

The induction is not difficult to perform. Given Un, un, let mn = |un| andfor every splitnode t ∈ un and every immediate successor s of t that has nosuccessor in un, thin out the tree Un � s to decide the value of f(mn). Since thisvalue must be smaller than g(mn), there are only finitely many possible valuesfor this decision and so, thinning out the set of immediate successors further ifnecessary, the decision can be assumed to be the same for all such immediatesuccessors of t, yielding a number kt ∈ ω. Let cn = {kt : t ∈ un} and let Un+1

be the thinned out tree. Finally, find un+1 ⊂ Un+1 satisfying I2.In the end, U =

⋂n Un is a tree in the poset P (K) forcing ∀n f(mn) ∈ c(n)

as desired.

Claim 5.1.17. Whenever a ⊂ ω is a K-positive subset and T ∈ P (K) forcesb ⊂ ω is an element of K, then there is a condition S ≤ T and a K-positiveground model set c ⊂ a such that S c ∩ b = 0.

Proof. This is in fact true for any forcing with the weak Laver property in placeof P (K). Thinning out the condition T if necessary we may find a number m ∈ ωsuch that T φ(b) < m. Find disjoint finite sets an ⊂ a with φ(an) > 2mn,and use the weak Laver property to find a condition S ≤ T , an infinite setb ⊂ ω and sets Bn ⊂ P(an) for every n ∈ b such that all sets in Bn have φ-mass< m, |Bn| < n, and S b ∩ an ∈ Bn. Then for every number n ∈ b, the setcn = an \

⋃Bn has mass at least nm, and it is forced by S to be disjoint from

b. Thus the set c =⋃n∈b cn works as desired.

Now, suppose that D ⊂ ωω × ωω is a Borel set with I(K)-small verticalsections, and T ∈ P (K) is a tree. We must find a tree S ⊂ T such that[S]×[S]∩D ⊂ id. Thinning out the tree T if necessary we may assume that thereis a continuous function f : [T ]×ω<ω → K such that for every branch x ∈ [T ] thesection Dx is included in the I(K)-small set {y ∈ ωω : ∀∞n y(n) ∈ f(x)(y � n)}.Let {tn : n ∈ ω} be an enumeration of ω<ω with infinite repetitions, and byinduction construct trees Un ∈ P (K) and their finite subsets un ⊂ Un so that

I1 T = U0 ⊃ U1 ⊃ . . . and {t} = u0 ⊂ u1 ⊂ . . . where t is the shortestsplitnode of T ;


I2 un is an inclusion initial set of splitnodes of Un;

I3 if tn ∈ un then un+1 = un together with some set of splitnodes extendingtn. All the new splitnodes differ from all the old ones and among each otheralready at their |tn|-th entry. Moreover the set {i ∈ ω : ∃t ∈ un+1 t

an i ⊂ t}

has φ-mass at least n. If tn /∈ un then un+1 = un;

I4 if tn ∈ un and t 6= tn is another node in un, then for every node s ∈ un+1\uand every x ∈ [Un+1 � s] it is the case that f(x)(t) is disjoint from the set{i ∈ ω : tai ∈ Un+1 but no node of un extends tai}.

This is not difficult to do using the previous claim. In the end, the tree S =⋂n Un belongs to P (K), and its set of splitnodes is exactly

⋃n un. We must

show that [S] × [S] ∩ D ⊂ id. To see this, suppose for contradiction thatx 6= y ∈ [S] are two branches and y ∈ Dx. This means that there is a numberm0 such that for every m > m0, y(m) ∈ f(x)(y � m). Find a number n ∈ ωsuch that tn ∈ un is an initial segment of x, un+1 \un contains still longer initialsegment of x, and the longest node t ∈ un which is an initial segment of y is oflength greater than m0. Then I4 shows that y(|t|) /∈ f(x)(t), contradicting thechoice of m0!

Corollary 5.1.18. For Fσ ideals K, the ideal I(K) has the Silver property.

The previous result cannot be generalized to much more general ideals. Amore or less canonical ideal on a countable set which is not a subset of an Fσ-ideal is K = Fin × Fin, the ideal on ω × ω generated by vertical sections andsets with all vertical sections finite. To simplify notation, restrict the σ-idealI(K) to the closed subset X ⊂ (ω×ω)ω of those functions x such that for everyn < m, the numbers in the pair x(n) are smaller than both numbers in the pairx(m).

Theorem 5.1.19. The ideal I(K) does not have the free set property. Thereis a Borel function f : X × X → 2ω such that for Borel I(K)-positive setsB,C ⊂ X, the image f ′′(B × C) has nonempty interior.

This theorem would not be particularly exciting if the poset PI(K) added aCohen real. However, it is not difficult to show that it preserves outer Lebesguemeasure. The results of the following section then imply that the σ-ideal I(K)has the Silver property for equivalences Borel reducible to EKσ , Ec0 , or for Borelequivalences classifiable by countable structures.

Proof. Say that points x, y ∈ X are entangled at n,m if the pairs x(n), y(m)are not comparable in the natural coordinatewise ordering of pairs of naturalnumbers. Note that if x, y are entangled at n0,m0 and at n1,m1, then eithern0 ∈ n1,m0 ∈ m1 or n1 ∈ n0,m1 ∈ m0; so the locations where the pair x, y isentangled are naturally linearly ordered. Now let f(x, y)(i) = 0 if at the i-thlocation n,m where x, y are entangled, x(n)(0) > y(m)(0), and let f(x, y) = 1otherwise.


To see that the function f works as required in the statement of the theorem,let B,C ⊂ X be I(K)-positive subsets, and let S, T ⊂ (ω × ω)<ω be trees suchthat [S] ⊂ B, [T ] ⊂ C, and K-positively splitting nodes are cofinal in both Sand T . Let s0, t0 be the first K-positively splitting nodes in S, T respectively, leta be the finite set of positions at which they are entangled, and let u : |a| → 2be the binary sequence defined by u(j) = 0 if at the j-th position (n,m) ofentanglement of s0, t0, it is the case that s0(n)(0) > t0(m)(0). We will showthat whenever v ∈ 2ω extends u then there are points x ∈ [S], y ∈ [T ] such thatf(x, y) = v.

By induction on i ∈ ω build splitnodes si, ti of S, T respectively so that

I1 s0 ⊂ s1 ⊂ . . . and t0 ⊂ t1 ⊂ . . . ;

I2 si, ti are entangled exactly at pairs in a and the pairs (|sj |, |tj |) for j ∈ i

I3 v(|a|+ i) = 0 if and only if si+1(|si|)(0) > ti+1(|ti|)(0).

If this succeeds, the points x =⋃i si ∈ [S] and y =

⋃i ti ∈ [T ] will have

f(x, y) = v as required. The induction itself is easy. Suppose that si, ti havebeen constructed, and for simplicity assume that v(|a| + i) = 0. Use the factthat ti is K-positively splitting node of the tree T to find a number k ∈ ω suchthat there are infinitely many l such that tai 〈k, l〉 ∈ T . Let si+1 be any splitnodeof S extending si such that si+1(|si|)(0) > k. Let ti+1 be any splitnode of thetree T extending ti such that ti+1(|ti|)(0) = k and ti+1(|ti|)(1) is greater thanall numbers appearing in the pairs in rng(si+1). This concludes the inductionstep.

For the failure of the free set property, let D = {〈x, y〉 : f(x, y) ends with atail of 1’s}. The vertical sections of D are I-small: whenever x ∈ X is a point,let A ⊂ ω × ω be the set of all pairs 〈k, l〉 such that 〈n,m〉 ∈ rng(x) and n > kimplies m > l. The set A has finite vertical sections and as such is in the idealK. The vertical section Dx consists of those y ∈ Y such that for all but finitelymany i ∈ ω, y(i) ∈ A, and therefore Dx ∈ I(K). No I(K)-positive Borel setB ⊂ X can be free for D, since the set f ′′(B×B) has a nonempty interior, andas such must contain a pair of points 〈x, y〉 such that f(x, y) end with a tail of1’s.

It is not too difficult to show that the quotient poset PI preserves outerLebesgue measure, and so by the results of Section 5.2, it has total canoniza-tion at least below EKσ , =J for analytic P-ideals J , and below equivalencesclassifiable by countable structures.

5.1c The meager ideal

The spectrum of the meager ideal is quite complicated; we only point out severalfairly simple features.

Proposition 5.1.20. E0 is in the spectrum of the meager ideal.


No other countable Borel equivalence relations belong to the spectrum as byTheorem 4.2.7, essentially countable→Ireducible to E0.

Proof. It is enough to show that for every Borel nonmeager set B ⊂ 2ω, E0 ≤E0 � B. By the Baire category theorem, there is a finite sequence s ∈ 2<ω suchthat B is comeager in Os, B ⊂ Os ∩

⋂n Pn where Pn ⊂ 2ω is open dense. By

induction on n ∈ ω build pairs {t0n, t1n} of distinct finite binary sequences of thesame length such that for every number n ∈ ω, writing t for the concatenation

sa(ti(0)0 )a(t

i(1)1 )a . . .a (t

i(n)n ), we have Ot ⊂ Pn no matter what the choices of

i(0), i(1), . . . i(n). Now let f : 2ω → B be the continuous function for which

f(x) is defined as the concatenation of s with all tx(n)n : n ∈ ω. It is not difficult

to see that f is the required reduction of E0 to E0 � B.

Proposition 5.1.21. E2 is in the spectrum of the meager ideal.

Proof. Consider the E2 equivalence on 2ω. The σ-ideal J σ-generated by Borelgrainy subsets of 2ω consists of meager sets only, as shown in Claim 5.6.10.Thus, whenever B ⊂ 2ω is a Borel non-meager set then B /∈ J and therefore E2

is Borel reducible to E2 � B by Fact 5.6.1. The proposition follows.

Proposition 5.1.22. The spectrum of Cohen forcing is cofinal in ≤B and itincludes EKσ .

Proof. In the Section 5.7 on the infinite countable support product of Sacksforcing, we prove that its associated ideal on (2ω)ω consists of meager sets, andits spectrum is cofinal in ≤B . Thus, the equivalence relations exhibited in thatproof will also yield the same feature for Cohen forcing. These equivalencerelations include EKσ , among others.

To place F2 in the spectrum of Cohen forcing, we will make a couple ofpreliminary remarks. Let X = (2ω)ω and let I be the meager ideal on X. It iswell known that the poset P = PI is in the forcing sense equivalent to the finitesupport product of countably many Cohen forcings, adding a point xgen ∈ X.The following definition and claim are well-known and we omit the proofs.

Definition 5.1.23. Let M be a transitive model of set theory and a ⊂ 2ω bea set. We will say that a is a Cohen-symmetric set for M if no finite tuple ofdistinct elements of a belongs to any meager set coded in M , and moreover,a ⊂ 2ω is dense. A finite collection {ai : i ∈ n} of subsets of 2ω is mutuallyCohen-symmetric over M if no finite tuple of elements of

⋃i ai belongs to any

meager set coded in M , and each of the sets is dense.

Claim 5.1.24. Let M be a countable transitive model of set theory, and let{ai : i ∈ n} be a collection of subsets of 2ω. The following are equivalent:

1. the sets in {ai : i ∈ n} are mutually Cohen symmetric over M ;


2. there is a sequence 〈xi : i ∈ n〉 of points in the space X which is Pn-genericover M and for every i ∈ n, rng(xi) = ai.

Moreover, if (1) or (2) hold, if {yi : i ∈ n} are any one-to-one enumerationsof the sets ai and B ⊂ Xn is a Borel nonmeager set coded in M , the set{~g ∈ Sn∞: the sequence 〈yi ◦ gi : i ∈ n〉 is Pn-generic over M and belongs to B}is nonmeager in the product Sn∞.

Corollary 5.1.25. In the poset Pn, the sentences of the forcing language men-tioning only rng(xi) : i ∈ n and some elements of the ground model are decidedby the largest condition.

Corollary 5.1.26. If M is a transitive model of set theory and bi : i ∈ n arenonempty subsets of X such that every finite tuple of distinct elements of

⋃i bi

is Pn-generic over the model M , then the sets⋃x∈bi rng(x) : i ∈ n are mutually

Cohen-symmetric.

Theorem 5.1.27. Let E be an analytic equivalence relation on X = (2ω)ω suchthat F2 ⊂ E. Either E has a comeager equivalence class, or F2 ≤B E.

Proof. There are several distinct cases.Case 1. Suppose first P×P x0 E x1. In such a case, let M be a countable

elementary submodel of a large structure and let B = {x ∈ X : x is M -genericfor P}. This is a Borel comeager subset of X. Any two elements of B must beE-related. To prove this, for points x0, x1 ∈ B pick a point y ∈ X such that yis both M [x0]- and M [x1]- generic for the poset P , use the forcing theorem toargue that M [x0, y] |= x0 E y and M [x1, y] |= x1 E y, use analytic absolutenessfor wellfounded models to conclude that x0 E y E x1, and finally argue thatx0 E x1 by transitivity of E. The first case of the conclusion of the theoremhas just been verified.

Case 2. Suppose instead P×P×P rng(x0)∪rng(x1) E rng(x0)∪rng(x2).Again, let M be a countable elementary submodel of a large structure and letB = {x ∈ X : x is M -generic for P} and argue that any two elements of B mustbe E-related. To see this, let x0, x1 ∈ B and pick a point y ∈ X such that y isboth M [x0]- and M [x1]- generic for the poset P . Divide a = rng(x0) into twodense sets a0, a1, similarly for b = rng(x1) = b0 ∪ b1 and c = rng(y) = c0 ∪ c1.By the claim and the forcing theorem applied in M , we can now argue thata = a0 ∪ a1 E a0 ∪ c0 E c0 ∪ c1 E c0 ∪ b0 E b0 ∪ b1 = b, and by transitivity a E bas desired. The first case of the conclusion of the theorem is satisfied.

Case 3. As another case, suppose that P×P rng(x0) E rng(x0)∪rng(x1).In this case again, we get an E-comeager equivalence class similarly to theprevious argument.

Case 4. Now assume that all of the previous cases failed. Let M be acountable elementary submodel of a large structure, and let f : 2<ω × 2ω → 2ω

be a continuous function such that s ⊂ f(s, y) and no finite tuple of dis-tinct elements of rng(f) belongs to any meager set in M . Such a functionis not difficult to construct. Now let g : X → X be the function defined byg(x)(π(m, k)) = f(s(m), x(k)) where π : ω × ω → ω is a fixed bijection and


〈s(m) : m ∈ ω〉 is a fixed enumeration of 2<ω. We claim that the Borel functiong reduces F2 to E.

To see this, it is clear that the function g is an F2-homomorphism, andso F2-equivalent points are sent to E-equivalent points. We must prove thatif x0, x1 ∈ X are not F2-equivalent then their g-images are not E-equivalent.There are several cases depending on the inclusion comparison of rng(x0) andrng(x1); we will handle the case when the intersection and rng(x0) \ rng(x1),rng(x1)\ rng(x0) are all nonempty sets. In such a case, write a0 = {f(s, y) : y ∈rng(x0) ∩ rng(x1), s ∈ 2<ω}, a1 = {f(s, y) : y ∈ rng(x0) \ rng(x1), s ∈ 2<ω} anda2 = {f(s, y) : y ∈ rng(x1) \ rng(x0), s ∈ 2<ω}. These three sets as well as theirunion are Cohen-symmetric over M , by Claim 5.1.24 and the failure of Claim 2we see that ¬a0 ∪ a1 E a0 ∪ a2, in other words ¬x0 E x1. The other cases aredealt with similarly.

Corollary 5.1.28. Whenever {Gn : n ∈ ω} is a collection of analytic equiva-lences on a fixed Polish space Y such that F2 is not Borel reducible to any ofthem, then F2 is not reducible to

⋂nGn either.

This should be compared with the case of Eω0 =⋂nE

n0 , where En0 is the equiv-

alence on (2ω)ω connecting two points if their entries on the first n coordinatesare E0 connected. The equivalences forming the intersection are each reducibleto E0 while Eω0 is known not to be even essentially countable.

Proof. Suppose for contradiction that f : (2ω)ω → Y is a Borel function re-ducing F2 to

⋂nGn. Write Gn for the pullback of Gn by the function f and

observe that these are all analytic equivalence relations containing F2 as a sub-set, to which F2 is not reducible. Thus, each of the equivalence relations Gnhas a comeager equivalence class Cn ⊂ (2ω)ω. The set C =

⋂n Cn is still

comeager, and so it contains non-F2-equivalent elements x0, x1. Now, the pointsf(x0), f(x1) are Gn-related for every n, and so

⋂nGn-related. This contradicts

the assumption that f was a reduction of F2 to⋂nGn.

Proposition 5.1.29. F2 is in the spectrum of the meager ideal.

Proof. Proceed as in the fourth case of the above proof with E = F2. Clearly,the first three cases are impossible. Let B ⊂ X be a Borel nonmeager set, andlet M be a countable elementary submodel of a large structure containing B.Let f : X → X be a Borel function preserving F2 such that rng(f) consistsof enumerations of Cohen-symmetric sets for M . Let D ⊂ X × S∞ be the set{〈x, g〉 : f(x)◦g ∈ B}. By Claim 5.1.24, the vertical sections of this Borel set arenonmeager. By Fact 2.1.4, the set D has a Borel uniformization g : X → S∞.Plainly, the function x 7→ f(x) ◦ g(x) is a Borel reduction of F2 to F2 � B.

The previous treatment also yields a useful uniformization-type theorem forBorel reductions of F2:

Corollary 5.1.30. Let E be a Borel equivalence relation on a Polish space Y ,and let A ⊂ (2ω)ω × Y be an analytic set such that


1. for every x ∈ (2ω)ω enumerating a dense subset of 2ω, the vertical sectionAx is nonempty;

2. whenever x0, x1 ∈ (2ω)ω enumerate the same dense subset of 2ω andy0, y1 ∈ Y are points such that 〈x0, y0〉, 〈x1, y1〉 ∈ A, then y0 E y1;

3. whenever x0, x1 ∈ (2ω)ω enumerate distinct dense subsets of 2ω, then theirvertical sections are not E-related.

Then, F2 ≤B E.

Proof. The set A has a Baire measurable uniformization f : (2ω)ω → Y byJankov-von Neumann theorem 2.1.6. The function f is defined on a comeagerset by (1), and so it is continuous on a comeager subset B ⊂ (2ω)ω. We justproved that F2 ≤B F2 � B, and the function f witnesses F2 � B ≤B E.

5.2 Porosity ideals

In [67, Section4.2], the class of porosity ideals was introduced and its mainproperties investigated.

Definition 5.2.1. Let X be a Polish space, and let U be a countable collectionof its Borel subsets. An abstract porosity is a function por with domain P(U) andrange consisting of Borel subsets of X, such that a ⊂ b implies por(a) ⊂ por(b).Given an abstract porosity por, a set A ⊂ X will be called porous if it is coveredby a set of the form por(a)\

⋃a for some a ⊂ U . The associated porosity σ-ideal

is the σ-ideal on X σ-generated by porous sets.

Example 5.2.2. Every σ-ideal I σ-generated by closed sets is a porosity ideal.Just let U be a basis for the underlying Polish space, and for a ⊂ U , let f(a) =X \

⋃a if this is a set in I, and let f(a) = 0 otherwise.

Example 5.2.3. Let f : X → Y be a Borel function and let I be the σ-idealσ-generated by those sets A ⊂ X on which the function is continuous. ThenI is a porosity ideal. To see this, fix countable bases BX and BY for the twoPolish topologies on X and Y , let U = {u(O,P ) : O ∈ BX , P ∈ BY } whereu(O,P ) = {x ∈ O : f(x) /∈ P} and let por(a) = {x ∈ X : ∀O ∈ BX x ∈O → ∃P ∈ BY f(x) ∈ P ∧ u(O,P ∈ a}. It is not difficult to see that I isindeed the porosity σ-ideal associated with this porosity function. The forcingproperties of the poset PI do not really depend on the choice of the functionf . Either, the whole space X belongs to the σ-ideal I, and then the posetPI is trivial. Or, X /∈ I, and then the poset PI is densely isomorphic to oneobtained from a specific function, the Pawlikowski function [6], [67, Section4.2.3]. The combinatorial form of the resulting poset was investigated earlierand independently by Steprans [59] and Sabok [45].

5.2. POROSITY IDEALS 105

Example 5.2.4. Let X be a compact metrizable space with a complete metricd, and consider the σ-ideal I σ-generated by metrically σ-porous sets. There isa great number of variations in its definition [63], but in all cases it is a porosityideal. To describe one version, for a set A ⊂ X and a point x ∈ X define theporosity of A at x as the limsup of numbers r(A, x, δ)/δ as δ (a positive realnumber) tends to zero, where r(A, x, δ) is the supremum of all radii of open ballsthat are included in the open ball centered at x of radius δ, that are disjointfrom the set A. Call a set porous if it has nonzero porosity at all of its points,and let I be the σ-ideal σ-generated by porous sets. It is not difficult to seethat this σ-ideal fits into our framework. Just let the set U consist of all openballs of rational radius centered at a point in a fixed countable dense subset ofX, and for a ⊂ U let por(a) be the set of all points x ∈ X where the limsup ofnumbers s(a, x, δ)/δ as δ tends to zero, is greater than zero. Here, s(a, x, δ) isthe supremum of the radii of balls in a which are subsets of the ball of radius δaround the point x.

Fact 5.2.5. Let X be a Polish space and I be a porosity σ-ideal on it. Then

1. the quotient poset PI is < ω1-proper and preserves Baire category;

2. if the generating porosity function is coanalytic then I is Π11 on Σ1

1;

3. if the generating porosity function is Borel then I is a game ideal;

4. if the generating porosity is coanalytic then every intermediate extensionof the PI-extension is given by a single Cohen real.

Here, to evaluate the complexity of the porosity function por, consider thecountable set U with discrete topology, the space P(U) with the product topol-ogy, identify por with the set {〈a, x〉 ∈ P(U) × X : x ∈ por(a)} ⊂ P(U) × X,and let the words “Borel”, “coanalytic” etc. above refer to the complexity ofthis set. Note that in Examples 5.2.4 and 5.2.3, the porosities are Borel.

Proof. (1) is proved in [67, Theorem 4.2.]. (2) is proved in [67, Theorem 3.8.7].For the third item, the Borel basis for I is naturally indexed by the spaceP(U)ω. The scrambling tool of Definition 3.1.1 is obtained either abstractly,by an application of Theorem 3.1.4, or by a direct construction of the integergame, as is done in the proof of [67, Theorem 4.2.3(3)]. (4) is in essence [67,Theorem 4.2.7], and it will not be needed here.

By Example 5.2.2, the class of porosity σ-ideals includes the meager ideal,and therefore we cannot expect a general canonization theorem for this class.We will prove a canonization theorem showing that the Cohen real is the maintroublemaker here.

Theorem 5.2.6. Let I be a porosity σ-ideal on a Polish space X such that theposet PI does not add an infinitely equal real. Then I has total canonization forequivalence relations Borel reducible to EKσ , or =J for an analytic P-ideal J ,and for equivalence relations classifiable by countable structures.


Here, an infinitely equal real is a point of ωω in the generic extension which hasinfinite intersection with every point in the ground model. Certainly, a Cohenreal is an infinitely equal real. It is a longstanding open problem in forcingtheory whether a forcing adding an infinitely equal real adds also a Cohen real;in other words, if the hypothesis of our theorem is in fact equivalent to PInot adding a Cohen real. On a heuristic level, every iterable forcing propertythat rules out adding a Cohen real also rules out adding infinitely equal real.For example, if a poset P is bounding then it does not add an infinitely equalreal (any point x ∈ ωω in the extension is dominated by a ground model pointy ∈ ωω and then x ∩ y = 0), and if P preserves outer Lebesgue measure then itdoes not add an infinitely equal real (if an : n ∈ ω is a fast increasing sequenceof finite subsets of ω, each equipped with uniform probability measures µn, theproduct

∏n an is equipped with the Borel probability measure µ =

∏n µn. If

x ∈ ωω is a point in the extension then the set B = {y ∈∏n an : x ∩ y 6= 0} is

a set of µ-mass < 1 and so its complement in∏n an contains a ground model

element y which then has an empty intersection with x).The most efficient path towards the proof of Theorem 5.2.6 uses a Boolean

game.

Definition 5.2.7. Let I be a σ-ideal on a Polish space X. The symmetriccategory game G (or G(I) if confusion is possible) between players I and IIproceeds in the following way. In the beginning, Player II indicates an initialI-positive Borel set Bini ⊂ X. After that, the players alternate for ω manyrounds, in round n Player I plays a collection {Bni : i ∈ ω} of countably manyI-positive Borel sets, and Player II responds with a collection {Cni : i ∈ ω} ofI-positive Borel sets such that Cni ⊂ Bni for every i, n ∈ ω. Player I wins if inthe end, the result of the play, the set D = Bini ∩

⋂n

⋃i C

ni is an I-positive set.

The symmetric category game seems to be a close relative of the Boolean gamesof [67, Section 3.10]. In contradistinction to the games described there, thisgame does not seem to have a classical forcing equivalent. It is neverthelessvery useful for the present task.

Theorem 5.2.8. Let I be a σ-ideal on a Polish space X.

(1) If I is Π11 on Σ1

1 then the symmetric category game is determined;

(2) if Player I has a winning strategy in the game then the quotient poset PIis proper.

If Player II has no winning strategy, then

(3) PI preserves Baire category;

(4) for every uncountable regular κ and every subset of κ in the PI-extension,either the set contains a ground model stationary subset or its complementcontains a ground model infinite set;

(5) every intermediate extension is given by a single Cohen real;


(6) if PI adds no infinitely equal real then I has total canonization for equiv-alence relations Borel reducible to EKσ or to =J for an analytic P-ideal Jon ω or classifiable by countable structures.

This greatly simplifies the proofs of [67, Theorem 4.1.6 and 4.1.7]. An attentivereader will notice how Theorem 3.4.4 connecting equivalence relations and in-termediate models, and the last two items above play together. If PI adds noinfinitely equal real then it does not add a Cohen real, by the fifth item thereare no nontrivial intermediate models between the ground model and the PI ex-tension, and then Theorem 3.4.4 severely restricts the behavior or equivalencerelations on Borel I-positive sets. The last item then (partially) confirms theresulting suspicions.

Proof. For item (1) of the theorem, under the assumption that the σ-ideal isΠ1

1 on Σ11 we will unravel the symmetric category game to a Borel game. To

this end, let A ⊂ 2ω × X be an analytic set universal for analytic subsets ofX, and let D ⊂ (2ω)ω × ωω be a closed set such that its projection is theanalytic set {z ∈ (2ω)ω :

⋂nAz(n) /∈ I}. Now define the unraveled version Gu

of the symmetric category game G: Player II begins with a point zini ∈ 2ω suchthat Azini is Borel and I-positive. After that, at round n Player I indicates acountable list {Bni : i ∈ ω} of Borel I-positive sets, Player II responds with alist {Cni : i ∈ ω} of their Borel I-positive subsets, and Player I reacts with apoint zn ∈ 2ω such that the set Az(n) is Borel, I-positive and a subset of

⋃i C

ni .

At some rounds of his choice, Player I in addition indicates a natural numberm(n). In the end, Player I wins if he produced infinitely many of the additionalnumbers, and the point 〈zini, z(n) : n ∈ ω, 〈m(n) : n ∈ ω〉〉 belongs to D.

The game Gu has a payoff set for Player I which is Gδ in the wide tree ofall possible plays: a point in the space (2ω)ω × ωω is continuously producedin the play and Player I must make sure that the production process will becomplete and ends with a point in a fixed closed set. Thus, by [35], the gameGu is determined. The game Gu is obviously harder for Player I than G, so ifPlayer I has a winning strategy in Gu, then he has a winning strategy in G. Wemust prove that a winning strategy σu for Player II in the game Gu yields awinning strategy σ in G for Player II.

To describe the winning strategy σ in the game G, we will let Player II playso that his first move is Azini , where zini is the first move of the strategy σu,and then in each round n he will on the side produce the following objects:

• for every function v : n+ 1→ n+ 1∪ {pass} whose numerical values staybelow the identity, a play pv;

• a point z(n) ∈ 2ω;

• pv is a play of the game Gu of n+1 rounds played according to the strategyσu. In his last move in this play, Player I indicates the point z(n) and thenumber (or pass) v(n). In the game G, Player II plays a list {Cni : i ∈ ω}so that its union is equal to Az(n);


• u ⊂ v → pu ⊂ pv.

This is not difficult to do. Suppose that the play has proceeded to round n.Player I makes his move, a countable list {Bni : i ∈ ω} of Borel I-positivesets. Enumerate all functions v as in the first item above by 〈vk : k ∈ K〉and by induction on k ∈ K + 1 construct a decreasing sequence of countablelists {Bni (k) : i ∈ ω} so that Bni (0) = Bni , and the list {Bni (k + 1) : i ∈ ω} isthe answer of strategy σ after Player I extends the play pvk�n with the move{Bni (k) : i ∈ ω}. In the end, let zn ∈ 2ω be a point such that Azn =

⋃iB

ni (K),

and let pvk = pavk�n{Bni (k) : i ∈ ω}a{Bni (k + 1) : i ∈ ω}avk(n)az(n).

In the end of a play of the game G played according to this strategy, Player IIhas to win. Suppose for contradiction that he lost, and so Cini ∩

⋃n

⋃i C

ni /∈ I;

then there must be a function y ∈ ωω such that 〈zini, z(n) : n ∈ ω, y〉 ∈ D. Letf : ω → ω ∪ {pass} be some function whose numerical values stay below theidentity and in their sequence form the point y. A brief review of the definitionsshows that the play

⋃n pf�n is a counterplay against the strategy σu in the

game Gu that Player I won, contradicting the choice of the presumably winningstrategy σu.

For the properness, suppose that Player I has a winning strategy σ in thegame G, B ∈ PI is an arbitrary condition and M is a countable elementarysubmodel of a large structure containing B and σ. Consider a play agains thestrategy σ in which all moves of Player II are in the model M , Bini = B andthe sets Bni belong to the n-th open dense subset of PI in M in some fixedenumeration. The outcome of the play is a master condition for the model M .For the preservation of Baire category, it is enough to show that the poset PIdoes not add an eventually different real by ???. Let B ∈ PI be an arbitrarycondition and y a name for an element of ωω. We will find a condition C ⊂ Band a point z ∈ ωω such that C y ∩ z is infinite. Consider the followingstartegy for Player II. In the beginning, he indicates Bini = B. Fix a bijectionπ : ω → ω × ω and, at round n, after Player I chooses his list {Bni : i ∈ ω},Player II finds the sets Cni to decide the value of y(π−1(i, n)) to be some definitenumber z(π−1(i, n)), for each i, n. The outcome of the play is a condition C ⊂ Bwith the required properties.

For item (4), let B ∈ PI be an arbitrary condition, κ an uncountable regularcardinal, and c a PI -name for a subset of κ. Assume that no condition forces aground model stationary subset of κ to be a subset of c. Consider the followingstrategy for Player II in the symmetric category game. He plays Bini = B. Atround n, after Player I provided a list {Bni : i ∈ ω} of Borel I-positive sets, theassumptions imply that for each i ∈ ω the set bi = {α ∈ κ : ∃C ⊂ Bni C α /∈ c}must contain a club–otherwise its complement is a stationary subset forced byBni to be a subset of c. Thus, there is an ordinal αn ∈

⋃i bi larger than the

previously chosen ordinals αk : k ∈ n and conditions Cni ⊂ Bni , each forcingαn /∈ c, and the list {Cni : i ∈ ω} will be Player II’s next move. Since thisstrategy is not winning for Player II, there will be a counterplay against it whoseresult C ⊂ B is I-positive. It is not difficult to see that C {αn : n ∈ ω} ⊂ cis an infinite ground model set as desired.


For item (5) of the theorem, suppose that A is a complete subalgebra of thecompletion of the poset PI . We will show that if the weight of A is uncountablebelow every condition, then the models V [H] and V [H ∩A] are the same whereH is any generic filter on (the completion of) PI . Let B ∈ PI be an arbitrarycondition. Consider the strategy σ in the game G for Player II with the followingdescription. His first move is Bini = B. At round n, after Player I chooses hislist {Bni : n ∈ ω}, consider the projections {ani : i ∈ ω} of the sets on this listto the complete Boolean algebra A. Since the weight of A is uncountable, thiscollection has a disjoint refinement {bni : n ∈ ω} by a result of Balcar [2]. LetCni ⊂ Bni be any set forcing bni into the generic filter, with diameter < 2−n

in some complete compatible metric for the underlying Polish space X. Notethat the sets {Cni : i ∈ ω} must be pairwise incompatible as {bni : i ∈ ω} are.The strategy σ given by this description cannot be winning for Player II, andso there is a counterplay against it that Player I wins, and the outcome of theplay, a Borel set C ⊂ B, is I-positive. Now, the set C forces the following: forevery n ∈ ω there is exactly one i ∈ ω such that Cni ∈ H, and this is exactlythe only i ∈ ω such that bni ∈ H. Therefore, the collection {Cni : i, n ∈ ω} ∩His in the model V [H ∩A]. In the model V [H], its intersection is nonempty andcontains only the generic point xgen. By the analytic absoluteness between themodels V [H] and V [H ∩A], the intersection must be nonempty in V [H ∩A] aswell, and so xgen ∈ V [H ∩A] as required.

For item (6) of the theorem, we will first treat the case of equivalence rela-tions Borel reducible to EKσ . Recall the metric d on ωω defined by d(x, y) =supn |x(n)− y(n)|; this is a metric that can attain infinite values, and EKσ con-nects points with finite distance. The metric d will be naturally extended tofinite sequences of natural numbers. Let B be a Borel I-positive set and let E bean equivalence relation on the set B, with a Borel reduction f : B → dom(EKσ )of E to EKσ . The following claim is central.

Claim 5.2.9. Suppose that the equivalence classes of E are in the σ-ideal I.Whenever n ∈ ω and {Bj : j ∈ ω} is a countable collection of Borel I-positivesubsets of B, then there are Borel I-positive sets {Cj : j ∈ ω} such that

1. Cj ⊂ Bj;

2. if i 6= j are distinct natural numbers and x ∈ Ci, y ∈ Cj are points, thend(f(x), f(y)) > n.

Once the claim is proved, the canonization for equivalence relations reducibleto EKσ follows immediately. Suppose that E has I-small equivalence classes;we must produce a Borel I-positive selector. We will describe a strategy forPlayer II: he starts with the set B = Bini and at round n answers with sets Cnjof diameter < 2−n such that for i 6= j and x ∈ Cnj , y ∈ Cni it is the case thatd(f(x), f(y)) > n. This is possible by Claim 5.2.9. Since this is not a winningstrategy, there must be a counterplay by Player I in which the result C ⊂ Bis I-positive. This is the desired Borel I-positive set such that E � C = id: if


x 6= y are distinct points in the set C, it must be the case that d(f(x), f(y)) > nfor every number n and therefore ¬x E y.

To prove the claim, it is enough to show a weaker version in which i in thesecond item is equal to 0 and then apply the claim repeatedly. For the weakerversion, first use the assumption that the equivalence relation E has small classesto find, for every j > 0, countably many finite sequences {tkj : k ∈ ω} of natural

numbers such that for k 6= l, d(tkj , tlj) > 2n and the set {x ∈ Bj : tkj ⊂ f(x)}

is I-positive. Now, if x ∈ B0 is a PI -generic point, for every j > 0 there canbe at most one k ∈ ω such that d(f(x), tkj ) ≥ n; let g be a name for a functionfrom ω \{0} to ω that for each j identifies such k if it exists. Since the poset PIis assumed not to add an infinitely equal real, there must be a Borel I-positiveset C0 ⊂ B and a ground model function h ∈ ωω such that C0 h ∩ g = 0;thinning out the set C0 if necessary we may assume that for every j > 0 and

every x ∈ C0 it is the case that d(f(x), th(j)j ) > n. The weaker version of the

claim will be satisfied if we set Cj = {x ∈ Bj : tkj ⊂ f(x)}. This concludes theargument in the case of equivalence relations Borel reducible to EKσ .

To prove the theorem for equivalence relations Borel reducible to =J foran analytic P-ideal J on ω, first use a theorem of Solecki [54] to find a lowersemicontinuous submeasure φ on ω such that J = {a ⊂ ω : limk φ(a \ k) = 0}.Let B be a Borel I-positive set, let E be an equivalence relation on B, and letf : B → P(ω) be a Borel function reducing E to =J . There are two distinctcases.

Case 1. For every ε > 0 and every Borel I-positive set B′ ⊂ B there is aBorel I-positive set C ′ ⊂ B′ such that for every x, y ∈ C ′, φ(f(x)∆f(y)) < ε.In this case, we will find a Borel I-positive set C ⊂ B and an alternative Polishtopology τ on C such that E � C is a closed relation in the topology τ × τ . By???, the relation E � D is smooth, and then the work below EKσ above willconclude the canonization in this case.

To find the set C, consider the following strategy of Player II. He starts withB = Bini, and at round n he plays sets Cnj so that for every x, y ∈ Cnj it is thecase that φ(f(x)∆f(y)) < 2−n. Since this is not a winning strategy, there mustbe a counterplay whose result C ⊂ B is an I-positive set. Consider a Polishtopology τ on D which generates the same Borel structure as the original one,and which makes all the sets Cnj ∩ C open. We claim that E � C is a closedrelation in the topology τ × τ . Suppose that x, y ∈ C are E-unrelated points,so there is n ∈ ω such that limk φ((f(x)∆f(y)) \ k) > 2−n. Let i, j be numberssuch that x ∈ Cn+2

i , y ∈ Cn+2j . The choice of the sets Cn+2

i , Cn+2j together

with the subadditivity of the submeasure φ shows that E ∩ Cn+2i × Cn+2

j = 0.

Since the sets Cn+2i ∩ C,Cn+2

j ∩ C are open in the new topology, we producedan open neghborhood of the point 〈x, y〉 /∈ E which is disjoint from E, showingthat the complement of E � D is open, and E � D is closed in the new topologyas required.

Case 2. If Case 1 fails for some B′ ⊂ B and ε > 0, proceed just like inthe argument for equivalence relations reducible to EKσ below the set B′. Thedetails are quite obvious and left to the reader.


This concludes the argument for the equivalence relations Borel reducible to=J for an analytic P-ideal J . The total canonization for equivalence relationsclassifiable by countable structures, use Theorem 4.3.8. Note that total can-onization for essentially countable equivalence relations follows from the EKσcase, as essentially countable equivalence relations are Borel reducible to EKσ .

The previous theorem does allow for a uniform treatment of many forcingnotions, but in some sense it asks more questions than it answers.

Question 5.2.10. Is it true that if Player I has a winning strategy in thesymmetric category game then the σ-ideal I has the selection property? IfPlayer I has a winning strategy and PI adds no infinitely equal real, does itfollow that I has total canonization for analytic equivalence relations?

Proof of Theorem 5.2.6. In view of the previous theorem, it is enough to showthat if I is a a porosity σ-ideal then Player I has a winning strategy in thesymmetric category game G(I). Let U be a countable collection of Borel setsand por the porosity function that define the σ-ideal I. Let Player I play sothat at round n+1, the collection {Bn+1

i : i ∈ ω} includes some I-positive Borelsubset of each set of the form Cnj ∩ u where j ∈ ω, u ∈ U , and Cnj ∩ u /∈ I,and some I-positive Borel subset of each set of the form Cnj \ (por(a) ∪

⋃a)

where a = {u ∈ U : Cnj ∩ u ∈ I}. Note that the sets mentioned last areindeed I-positive: the set Cnj \

⋃a is I-positive since it is obtained from Cnj

by subtracting countably many I-small sets, and the set Cnj \ (por(a) ∪⋃a)

is obtained from the I-positive set Cnj \⋃a the generator por(a) \

⋃a of the

σ-ideal I. As the play progresses, Player I will also create an increasing chain ofcountable elementary submodels Mn of some large structure and he will makesure that the sets Cnj are all elements of the model Mn and the sets Bnj belongto the first n many open dense sets of PI in the models Mm : m ∈ n under somefixed enumeration of the models Mm : m ∈ n.

To show that the previous paragraph indeed constitutes a description ofa winning strategy for Player I, let D = Bini ∩

⋂n

⋃i C

in be the result of a

play according to this strategy; we must show that D /∈ I. To this end, let{an : n ∈ ω} be a countable collection of subsets of U and find a point x ∈D ∩

⋃n(por(an) \

⋃an). To find x, by induction on n find numbers jn ∈ ω so

that

I1 B0j0⊃ B1

j1⊃ . . . ;

I2 Bnjn ∩ (por(an) \⋃an) = 0.

This is not difficult to do. At step n, let b = {u ∈ U : Cnjn ∩ u ∈ I}. If

an 6⊂ b, then find u ∈ an \ b, and choose jn+1 such that Bn+1jn+1

⊂ Cnjn ∩ u–this is

possible by the description of the strategy–, otherwise find jn+1 so that Bn+1jn+1

⊂Cnjn \ (por(b) ∪

⋃b–this is again possible by the description of the strategy. in

both cases, item I2 is satisfied, in the latter case because por(an) ⊂ por(b)


holds, as implied by an ⊂ b. In the end, the intersection⋂nB

njn

is nonemptyand contains a single point which is generic over the union of the models Player Iplayed for the poset PI . The unique point x in that intersection has the desiredproperties.

Corollary 5.2.11. If I is a porosity σ-ideal associated with a Borel porosityfunction such that PI does not add an infinitely equal real, then it has the Silverproperty for Borel equivalence relations in classes named in Theorem 5.2.6.

Corollary 5.2.12. If f : X → Y is a Borel function between Polish spacesand I is the σ-ideal on X σ-generated by the sets on which f is continuous,then I has the Silver property for Borel equivalence relations in classes namedin Theorem 5.2.6.

Theorem 5.2.13. Let I be a porosity σ-ideal associated with a Borel notion ofabstract porosity. If the quotient PI is bounding, then I has total canonizationfor analytic equivalence relations, and the Silver property for Borel equivalencerelations.

Proof. [67, Theorem 5.2.6] shows that Π11 on Σ1

1 σ-ideals whose quotients areproper, bounding, and preserve Baire category have the rectangular property.Theorem 3.2.5 then shows that I has total canonization for analytic equivalencerelations. Finally, Proposition 3.3.2 yields the Silver property for I!

Corollary 5.2.14. The σ-ideal I of σ-porous sets has total canonization foranalytic equivalence relations, and the Silver property for all Borel equivalencerelations.

Proof. It is necessary to verify the bounding property for the σ-ideal I. [39]showed that every analytic I-positive set contains a compact I-positive subset;for the case of zero-dimensional spaces there is even a simpler determinacyargument of Diego Rojas [43]. To get the bounding property, we must in additionprove the continuous reading of names: every Borel function f : B → 2ω on anI-positive Borel set can be restricted to an I-positive compact set on which itis continuous. This was proved in [64] even though there is no statement of thisexact result in that paper. In order to extract it, consider Lemma 4.1 there anduse Theorem 4.5 to the σ-ideal of subsets of X×2ω with σ-porous projection tothe X coordinate to find a compact subset of the graph of f with an I-positiveprojection to X.

Note that the Silver property of I is a fairly short and quotable result.However, its current proof is extremely heavy on technology of several sorts:mathematical analysis and overspill techniques in the work of Zeleny, Zajıcek,and Pelant, determinacy of Borel integer games, the original Silver dichotomy,and a number of forcing arguments.

Question 5.2.15. Does the σ-ideal of σ-porous sets have the free set property?Does it have a coding function?

5.3. HALPERN-LAUCHLI FORCING 113

Question 5.2.16. Does the σ-continuity ideal have the rectangular Ramseyproperty? How about the total canonization for analytic equivalence relations?The free set property? Does it have a coding function?

5.3 Halpern-Lauchli forcing

The Halpern-Lauchli forcing P consists of those trees T ⊂ 2<ω such that there isan infinite set aT ⊂ ω such that t ∈ T is a splitnode if and only if |t| ∈ aT . Theordering is that of reverse inclusion. The terminology alludes to the Halpern-Lauchli theorem, which for partitions of 2<ω provides homogeneous sets thatare exactly sets of splitnodes of trees in P . The poset P is quite similar tothe Sacks or Silver forcing notions, and the standard fusion arguments showthat it is proper and has the continuous reading of names. Consequently, thecomputation of the associated ideal can be found in [67, Proposition 2.1.6]. LetX = 2ω, and let I be the σ-ideal generated by those Borel sets A such that forno tree T ∈ P , [T ] ⊂ A. Then I is a Π1

1 on Σ11 σ-ideal, every positive analytic

set contains a positive Borel subset, and the map T 7→ [T ] is a dense embeddingfrom P to PI .

This forcing serves as a good example refuting several natural conjectures.It has the free set property, but not the mutual generics property. It has totalcanonization, but not the Silver dichotomy. There is a natural σ-closed regularsubforcing, which then cannot be obtained as PEI from any Borel equivalencerelation E on X.

Proposition 5.3.1. I has the free set property.

Proof. We will need a sort of a reduced product of the Halpern-Lauchli forcing.Let P×rP be the poset of all pairs 〈T,U〉 ∈ P×P such that aT = aU . Similarlyto the usual product, the reduced product adds points xlgen, xrgen ∈ X which arethe intersections of all trees on the left (or right, respectively) side of conditionsin the generic filter. Still, the difference between P ×r P and P × P shouldbe immediately apparent. A more detailed analysis will show that P forcesthat the set {aT : T ∈ G} is a generic filter for the poset P(ω) modulo finite,and the reduced product is equivalent to the two step iteration of P(ω) modulofinite followed with the product of two copies of the remainder forcing P/(P(ω)modulo finite).

We will need a computation of the σ-ideal associated with the product P ×rP . Let I ×r I be the collection of all Borel subsets B ⊂ 2ω × 2ω such that thereis no pair 〈T,U〉 ∈ P ×r P such that [T ]× [U ] ⊂ B.

Claim 5.3.2. I ×r I is a σ-ideal of Borel sets, and the map 〈T,U〉 7→ [T ]× [U ]is a dense embedding of P ×r P into PI×rI .

In particular, if D ⊂ 2ω × 2ω is a Borel set with I-small vertical sections, itcannot contain a reduced rectangle, therefore it is I×r I-small, and the reducedproduct forces its generic pair not to belong to the set D.


Proof. Suppose that T,U ∈ P ×r P , and [T ] × [U ] =⋃nBn is a countable

union of Borel sets. We must find a pair of trees T , U ∈ P ×r P such thatthe set [T ] × [U ] is a subset of one of the Borel sets in the union. Strengthenthe condition 〈T,U〉 if necessary to find a specific number n ∈ ω such that〈T,U〉〈xlgen, xrgen〉 ∈ Bn. Let M be a countable elementary submodel of alarge enough structure, and let 〈Dm : m ∈ ω〉 be an enumeration of all opendense subsets of P×rP in the model M . By induction on m ∈ ω build conditions〈Tm, Um〉 ∈M in the reduced product so that

I1 T0 = T , the conditions Tm ∈M ∩ P form a decreasing sequence, and thefirst m+1 splitting levels of Tm are equal to the first m+1 splitting levelsof Tm+1, and the same on the U side;

I2 for every choice of nodes t ∈ Tm+1, u ∈ Um+1 just past the m-th splittinglevel, the condition 〈Tm+1 � t, Um+1 � u〉 ∈ P ×r P is in the set Dm.

The induction is elementary, at each step making a pass through all pairs 〈t, u〉as in the second item to handle them all. In the end, the pair 〈T =

⋂m Tm, U =⋂

n Um〉 ∈ P ×r P is a condition such that every pair 〈x, y〉 ∈ [T ] × [U ] is M -generic for the reduced product below the condition 〈T,U〉. The forcing theoremimplies that M [x, y] |= 〈x, y〉 ∈ Bn, and by analytic absoluteness 〈x, y〉 ∈ Bn.In other words, [T ]× [U ] ⊂ Bn as desired.

An essentially identical argument yields

Claim 5.3.3. Suppose that M is a countable elementary submodel of a largeenough structure and T ∈M ∩P is a tree. There is a tree U ∈ P , U ⊂ T , suchthat every two distinct elements of U are M -generic for the reduced product.

The proposition immediately follows. Let D ⊂ X×X be a Borel set with allvertical sections in the ideal I; clearly, such a set belongs to the σ-ideal I ×r I.Let T ∈ P . Let M be a countable elementary submodel of a large enoughstructure containing both D and T . Let U ⊂ T be a tree in P such that everytwo distinct points of [U ] are reduced product generic for M . An absolutenessargument immediately shows that ([U ]× [U ]) ∩D ⊂ id.

Proposition 5.3.4. I does not have the mutual generics property.

Proof. Suppose that M is a countable elementary submodel of a large structure,and suppose for contradiction that there is a tree T ∈ P such that [T ] consistssolely of points mutually generic for P × P over M . For every point x ∈ [T ]let gx ⊂ P ∩ M be the filter generic over M , generated by x: gx = {S ∈P ∩M : x ∈ [S]}. Similarly, let hx ⊂ (P(gw) modulo finite)∩M be the filtergeneric over M generated by x: hx = {aS : S ∈ gx}. The key point: the setB = {x ∈ [T ] : ∃S ∈ gx aT 6⊂ aS modulo finite} is relatively meager in [T ]. Ifnot, by the Baire category theorem there would be a tree S ∈ P ∩M and a nodet ∈ T such that for comeagerly many points x ∈ [T ] passing through t, S ∈ gxand aT \ aS contains some elements above |t|. Take two extensions t0, t1 ∈ T of

5.4. THE E0-IDEAL AND ITS PRODUCTS 115

the tree T that split at some level in aT \ aS and points x0, x1 ∈ [T ] extendingthem respectively such that S ∈ gx0

, gx1. However, this means that t0 ∩ t1 is a

splitnode of S, contradicting the choice of the two nodes t0, t1.Since the set B is meager, we can pick two distinct ponts x0, x1 /∈ B and

observe that the set aT ⊂ ω diagonalizes both of the filters hx0 , hx1 . In partic-ular, hx0 does not contain a set with finite intersection with some set in hx1 ,as would be the case if the filters gx0

, gx1were generic over M for the product

P × P !

Proposition 5.3.5. I does not have the selection property.

Proof. Find a Borel injection f : 2ω → [ω]ℵ0 whose range consists of pairwisealmost disjoint sets. Let D ⊂ 2ω × 2ω be the Borel set of all pairs 〈x, y〉 suchthat y(n) = 0 whenever n /∈ f(x). It is fairly obvious that the vertical sectionsof the set D are pairwise disjoint I-positive sets. It turns out that D is thesought counterexample to the selection property.

Indeed, suppose that T ∈ P is a tree such that the closed set [T ] ⊂ 2ω iscovered by the sections of the set D, and visits each section in at most onepoint. There must be distinct points y0, y1 ∈ [T ] such that the set a = {n :y0(n) = y1(n) = 1} is infinite. If x ∈ 2ω is such that y0 ∈ Dx, it must be thecase that a ⊂ f(x), and the same for y1. However, since the sets in the rangeof f are pairwise almost disjoint, there can be only one such point x ∈ 2ω, andboth y0, y1 must belong to Dx, contradicting the choice of the tree T ∈ P .

Proposition 5.3.6. PI regularly embeds a nontrivial σ-closed forcing.

Proof. If G ⊂ PI is a generic filter, then H ⊂ P(ω) mod finite, H = {aT :T ∈ G} is a generic filter as well. The function T 7→ aT is the associatedpseudoprojection from P to P(ω) mod finite.

5.4 The E0-ideal and its products

Let E0 be the equivalence on 2ω defined by x E0 y iff x∆y is a finite set. This isa basic example of a nonsmooth Borel equivalence relation. Let I be the σ-idealon the space X = 2ω generated by Borel partial E0-selectors; this turns outto be exactly the σ-ideal of Borel sets on which E0 is smooth. The purposeof this section is to show that this ideal and its finite products possess a verystrong canonization property: while it is obviously not possible to canonize theequivalence relation E0 on an I-positive set to anything simpler, this is the onlyobstacle and all other analytic equivalence relations simplify to either id or eeor E0 on a Borel I-positive set.

The ideal I and its quotient poset PI have been studied in [67]. In thespirit of seeking connections between various fields of set theory, we will brieflydescribe a creature forcing style presentation of the quotient forcing here. Definea creature c = 〈c(0), c(1)〉 to be an ordered pair of distinct finite binary sequencesof the same length. A composition of a finite sequence of creatures 〈cj : j ∈ m〉is another creature, a pair of distinct binary sequences each of which is obtained


by concatenation of all sequences cj(b(j)) : j ∈ m. for some function b ∈ 2m.The partial order P consists of pairs p = 〈tp,~cp〉 where ~cp denotes an ω-sequenceof creatures and tp is a finite binary string. The order is defined by q ≤ p if thereis a decomposition of ω into finite consecutive intervals 〈in : n ∈ ω〉 such that

tq = tap sa0 sa1 . . . where sm ∈ ~cp(m) for every m ∈ i0, and the creatures ~cq(n)

are obtained as a composition of creatures in the set {~cp(m) : in ≤ m < in+1}.For a condition p ∈ P the set [p] ⊂ 2ω consists of all concatenations of tp withcp(j)(x(j)) for all j ∈ ω, as x varies over elements of 2ω.

Fact 5.4.1. [67, Section 4.7.1]

1. Every analytic subset of 2ω either is in the ideal I or contains a subset ofthe form [p] for some p ∈ P , and these two options are mutually exclusive;

2. the σ-ideal I on 2ω is Π11 on Σ1

1;

3. I does not have the selection property;

4. the forcing PI is proper, bounding, preserves Baire category and outerLebesgue measure, and it adds no independent reals.

Thus, the quotient poset has a dense subset that is naturally isomorphic to theforcing P described above. Note that for every condition p ∈ P there is a naturalhomeomorphism f : 2ω → [p] defined by f(x) = the concatenation of tp withthe sequences cp(j)(x(j)) for all i ∈ ω. The homeomorphism f preserves theE0 equivalence and therefore f -images and preimages of sets in I are again inI. This proves the homogeneity of the poset PI and allows us to reduce variousjobs on an arbitrary Borel I-positive set to the same work on the whole space2ω by pulling the context back along the natural homeomorphism.

Regarding the canonization properties of the σ-ideal I, the equivalence re-lation E0 clearly belongs to the spectrum of I essentially by its definition. Wewill show that E0 is the only obstacle to canonization for I:

Theorem 5.4.2. Every Borel equivalence relation on an I-positive Borel setsimplifies to id or to ee or to E0 on I-positive Borel subset.

We will prove a canonization result for the E0 ideal in a much strongerform, a form that deals with arbitrary finite products of the E0 ideal. Thecomputation of the product ideal is instructive and yields complete information.Fix a number n ∈ ω and write Xn = (2ω)n.

Definition 5.4.3. A set B ⊂ Xn is an E0-box if it is of the form∏i∈n[pi] where

each pi is a condition in the poset P . If this is the case, we write B(i) = [pi].

Note that if B is an E0-box then there is a natural homeomorphism of Xn

with B–namely the product of the natural homeomorphisms between 2ω andB(i) for each i ∈ n. This homeomorphism preserves the E0 equivalence in eachcoordinate. It will allow us to reduce work on an arbitrary E0-box to workon the whole space Xn by simply pulling back the context to Xn along thehomeomorphism.


Definition 5.4.4. The σ-ideal In on Xn is σ-generated by all analytic setsA ⊂ X such that there is i ∈ n such that each section of A in the i-th coordinateconsists of pairwise E0-inequivalent elements of 2ω.

Fact 5.4.5. Let n ∈ ω.

1. In is a Π11 on Σ1

1 σ-ideal consisting of meager sets;

2. every analytic subset of Xn either is in I or contains an E0-box as a subset,and these two options are mutually exclusive;

3. the quotient poset PIn is proper, bounding and preserves Baire category.

In particular, the quotient poset PIn has a dense subset naturally isomorphicto the product of n copies of the poset P .

Proof. (1) immediately follows from the Kuratowski-Ulam theorem and the factthat Borel E0-independent subsets of 2ω are meager. (2) was proved in [66,Lemma 2.3.58]. (3) follows from the fact that PIn is a product of n many copiesof the poset PI , that poset is proper, bounding, and preserves Baire category,and the conjunction of these properties is preserved under products by [67,Theorem 5.2.6]. A direct argument for (3) is of course possible as well.

Theorem 5.4.6. Let n ∈ ω be a number. Let Bi : i ∈ n be Borel subsets of 2ω

on which E0 is not smooth, and let E be a Borel equivalence relation on∏iBi.

Then there are Borel sets Ci ⊂ Bi on which E0 is still not smooth, such thatE �

∏i Ci =

∏i Fi where each Fi is one of id, ee, or E0.

Proof. The strategy of the proof can be described as follows. First, we treat acouple of special cases: the smooth equivalence relations, the equivalence rela-tions that are a subset of the product of E0’s (this uses a partition theorem),and the conjunction of these two cases can be cranked up to cover the case ofessentially countable equivalence relations. After that, a brief induction argu-ment on the dimension of the product reduces the general case to the case of anessentially countable equivalence relation, concluding the proof.

The following remnant of the free set property will be useful at two distinctplaces in the proof:

Claim 5.4.7. Let n ∈ ω be a number.

1. Let R ⊂ (2ω)n × 2ω be a meager relation. Then there is a E0-box C ⊂(2ω)n such that for every x ∈ C and every y ∈ C(0), 〈x, y〉 ∈ R impliesx(0) E0 y.

2. Let B ⊂ (2ω)n be an E0-box. Let R ⊂ B × B(0) be a relatively meagerrelation. Then there is a E0-box C ⊂ (2ω)n such that for every x ∈ C andevery y ∈ C(0), 〈x, y〉 ∈ R implies x(0) E0 y.


Proof. We will start with the first item. Let Ok : k ∈ ω be a collection of opendense subsets of (2ω)n × 2ω such that their intersection is disjoint from R. Byinduction on l ∈ ω build finite binary sequences t(0, i, l) and t(1, i, l) for i ∈ nso that

• for a fixed number l the sequences are pairwise distinct and have all thesame nonzero length;

• whenever u(i) : i ∈ n and v are finite binary sequences such that each u(i)was obtained by concatenation of t(b(i, k), i, k) : k ∈ l for some functionb : n × l → 2 and v was obtained by concatenation of some t(c(k), 0, k) :k ∈ l for some function c : l → 2 such that c(l − 1) 6= b(0, l − 1), then forevery x ∈ (2ω)n such that ∀i ∈ n u(i) ⊂ x and every y ∈ 2ω such thatv ⊂ y it is the case that 〈x, y〉 ∈

⋂k∈lOk.

This is easy to do. In the end, let pi ∈ P be the condition 〈0, 〈t(0, i, l), t(1, i, l)〉 :l ∈ ω〉. It is clear that the E0-box C =

∏i∈n[pi] has the required properties.

The second item can be immediately reduced to the first. Let pi : i ∈ nbe the conditions in P such that [pi] = B(i), let fi : 2ω → [pi] be the naturalhomomorphism, and apply the first item to the relation (f0 × f1 × . . . fn ×f0)−1R ⊂ (2ω)n × 2ω.

Lemma 5.4.8. Let n ∈ ω be a number, let B ⊂ Xn be an In-positive Borel set,and let E be a smooth equivalence relation on B. There is a set a ⊂ n and aBorel I-positive set C ⊂ B such that E � C = the equality of entries at the seta.

Proof. The proof goes by induction on n ∈ ω. Let f : B → 2ω be a Borelfunction reducing the equivalence relation E to the identity. Thinning out theset B if necessary, we may assume that the function f is continuous. Inscribingan E0-box and pulling back along a canonical homeomorphism of Xn with thebox, we can assume that the set B is in fact equal to the whole space Xn. Thereare two cases.

Case 1. There is a Borel In-positive set B′ ⊂ B and an index i ∈ n suchthat whenever x, y ∈ B′ are two sequences whose entries are the same exceptpossibly at the entry i, then f(x) = f(y); without loss of generality assume thati = n − 1. In such a case, the functional value of f on an entry from x ∈ B′depends only on x � n − 1, and so f induces a smooth equivalence relationE′ on the projection of B′ into the first n − 1 many coordinates. Inscribe anE0-box into B′ and use the induction hypothesis to thin out the first n − 1many coordinates of the box so as to canonize the equivalence relation E′ to theprescribed form. The equivalence relation E restricted to the resulting E0-boxC ⊂ Xn is of the prescribed form.

Case 2. If Case 1 fails, then we will produce an E0-box C ⊂ B such thatE � C = id. By induction on l ∈ ω build finite binary sequences t(0, i, l) andt(1, i, l) for i ∈ n so that


I1 for a fixed number l the sequences are pairwise distinct and have all thesame nonzero length; write d(i, l) for the set of all finite sequences u ob-tained by concatenation of t(b(i, k), i, k) : k ∈ l;

I2 there is an injective function gl :∏i∈n d(i, l) → 2ml for some natural

number ml ∈ ω such that for every choice of sequences u(i) ∈ d(i, l), it isthe case that

∏i∈n[u(i)] ⊂ f−1[gl(〈u(i) : i ∈ n〉].

The induction step is easy, using the case assumption and the continuity ofthe function f repeatedly. In the end, let p(i) ∈ P be conditions for every i ∈ ndescribed by tp(i) = 0 and cp(i)(l) = 〈t(0, i, l), t(1, i, l)〉 for every l ∈ ω. The item[I2] of the induction hypothesis immediately implies that if x, y ∈

∏i∈n[p(i)]

are distinct points then f(x) 6= f(y). Thus∏i∈n[p(i)] is the desired E0-box on

which E is equal to the identity.

Lemma 5.4.9. Let n ∈ ω be a number, and let E ⊂ En0 be a Borel equivalencerelation on an In-positive Borel set B ⊂ (2ω)n. There is a set a ⊂ n and anI-positive Borel set C ⊂ B such that E � C =

∏i∈n Fi, where Fi = id if i ∈ a,

and Fi = E0 if i /∈ a.

Proof. Most of the work necessary for the proof is performed in an abstractand well known context of canonization on products of finite sets. We recordthe finite results first. To compactify the notation, the following usage willprevail. The number n ∈ ω denotes the dimension of the product. n sequenceswill be adorned with an arrow (such as ~a) and simple set theoretic operations

and relations will be applied coordinatewise. Thus ~a ⊂ ~b means that for eachi ∈ n, ~a(i) ⊂ ~b(i) etc. If ~a is a sequence of sets, then ~x ∈ ~a means that∀i ∈ n ~x(i) ∈ ~a(i), in other words the sequence ~a is identified with the product∏i∈n ~a(i).

Claim 5.4.10. For every sequence ~k ∈ ωn there is a sequence ~l ∈ ωn such thatfor every sequence ~b of finite sets of size ~l and every partition of the set [~b]2 into

two parts there is a sequence ~c ⊂ ~b of size ~k such that the set [~c]2 is a subset ofone piece of the partition.

Claim 5.4.11. For every sequence ~k ∈ ωn there is a sequence ~l ∈ ωn such thatfor every sequence ~b of finite sets of size ~l and every equivalence relation E on ~b,there is a sequence ~c ⊂ ~b of size ~k and a set a ⊂ n such that for every ~x, ~y ∈ ~c,~x E ~y if and only if ∀i ∈ a ~x(i) = ~y(i).

Claim 5.4.12. For every sequence ~k ∈ ωn there is a sequence ~l ∈ ωn suchthat for every sequence ~b of finite sets of size ~l and every equivalence relationE on 2× ~a there is a sequence ~c ⊂ ~b of size ~k such that either for every ~x ∈ ~c,〈0, ~x〉 E 〈1, ~x〉 holds (i.e. the equivalence relation does not depend on the firstcoordinate on 2× ~c) or for every ~x, ~y ∈

∏~c, ¬0ax E 1a~y holds (i. e. the first

coordinate decomposes 2× ~c into two E-unrelated blocks).


Proof. To establish a suitable terminology for the proof, a configuration is a pairco = 〈co(0), co(1)〉 of sequences in 2n. If ~c is an n-sequence of pairs of naturalnumbers, then co ◦~c is the pair 〈~x, ~y〉 of sequences defined by ~x(i) = the smallerelement of ~c(i) if co(0) = 0 and ~x(i) = the larger element of ~c(i) if co(0) = 1,and similarly for ~y and co(1). A pair 〈~x, ~y〉 has configuration co if there is asequence ~c such that 〈~x, ~y〉.

Choose ~l so large that the succession of the following homogenization argu-ments yields a sequence of size ~k. Let ~b be an n-sequence of finite sets of size~l.

First, consider the partition p : ~b → 2 defined by p(~x) = 0 if 0a~x E 1a~x.

Shrinking the sequence ~b, we get a sequence which is entirely included in onepiece of the partition. If this partition piece has value 0 then we are in the“either” clause of the claim, and we are finished. So assume that the partitionpiece has value 1.

For every configuration co consider the partition pco : [~b]2 → 2 defined by

p(~x) = 0 if 0a(co ◦ ~x)(0) E 1a(co ◦ ~x)(1). Find a sequence ~c ⊂ ~b of size ~k suchthat [~c]2 is homogeneous for all the 2n+1 many partitions obtained in this way.We claim that the homogeneous value will be 1 for all the partitions. Thenclearly ~c is a sequence satisfying the “or” clause of the claim, completing theproof.

Assume for contradiction that for some configuration co the homogenousvalue is 0. Find sequences ~x, ~y ∈ ~c with this configuration such that for everyi ∈ n, if ~x(i) = ~y(i) then there are some elements of c(i) both below and above~x(i), and if ~x(i) 6= ~y(i) then there are some elements of ~c(i) strictly between ~x(i)and ~y(i). Find ~z ∈ ~c such that for every i ∈ n, if ~x(i) = ~y(i) then ~x(i) = ~z(i),and if ~x(i) 6= ~y(i) then ~z(i) is strictly between ~x(i) and ~y(i). The homogeneityimplies that 0a~x E 1a~y, 0a~x E 1a~z, and 0a~z E 1ay, since the ordered pairs〈x, y〉, 〈x, z〉 and 〈z, y〉 all have the same configuration co. The transitivity of therelation E implies that 0a~z E 1az. This, however, contradicts the homogeneityassumption made in the third paragraph of the present proof.

Now we are ready to tackle the infinite situation. Let B ⊂ (2ω)n be an In-positive Borel set, and let E be a Borel equivalence relation which is a subset ofthe product of E0’s. Let g : B → 2(ω<ω)n be the function defined by g(~x)(~t) = 0if ~x rew ~t E ~x. Shrinking the set B if necessary we may assume that the functiong is continuous. First, inscribing an E0-box into B and pulling back throughthe natural homeomorphism between (2ω)n and the box if necessary we mayassume that in fact the set B is the whole space (2ω)n. To cut the notationalclutter, below we identify 2ω with P(ω) via the characteristic functions.

By induction on j ∈ ω build n-sequences ~bj and ~ej of finite subsets of ω andnumbers mj , mj so that

I1 m0 < m0 < m1 < m1 < . . . the sets on the sequence ~bj are all subsetsof the interval (mj , mj) and the sets on the sequence ~ej are all subsets ofthe interval (mj ,mj+1);


I2 for any pair ~c, ~d of sequences of sets below mj either (a) for every ~x ∈(P(ω)n such that each set on the sequence ~ej is an initial segment of the

corresponding set on ~x, ~c ∪ ~x E ~d ∪ ~x holds, or (b) for every such ~x,

~c∪ ~x E ~d∪ ~x fails. Thus the relation Fj on P(mj)n, connecting ~c and ~d if

(a) above occurs, is an equivalence relation;

I3 the sizes of the sets ~bj grow sufficiently fast as indicated by the needs ofthe argument to follow.

For every sequence ~d of sets below mj , let E~d,j be the equivalence relation on∏~bj defined by ~u Ed ~v iff ~d∪{~u} Fj ~d∪{~v}. By a repetitive use of the previous

two claims at every j ∈ ω, shrink the sets on the sequences ~bj if necessary sothat

I4 for every sequence ~d below mj there is a set a~d,j ⊂ n such that E~d,j �∏~bi

is just the equality at the set a~d;

I5 for sequences ~c, ~d below mj , either (a) for every ~u ∈ bj ~c ∪ {~u} Fj ~d ∪ {~v}holds, or (b) for every ~u,~v ∈ bj ~c ∪ {~u} Fj ~d ∪ {~v} fails.

For every j ∈ ω, let µj be the submeasure on the set∏~[bj ]2 defined by µj(h) = 0

if h = 0, µj(h) = 1 if h 6= 0 contains no subset of the form∏

[~cj ]2 for a sequence

of sets ~cj ⊂ ~bj each of size at least three, and µj(h) ≤ r if h can be decomposed

into r many sets of µj-mass The sequences ~bj must be chosen growing so fastthat

I6 the numbers µi(∏

[~bj ]2) tend to infinity fast enough so that the partition

theorem 2.7.6 may be applied for partitions into 2n many pieces.

This ends the inductive construction. Consider the partition p :∏j [~bj ]

2 ×ω → P(n) defined by p(k, 〈~dj : j ∈ ω〉) = a~d,k where ~d is the sequence of sets

defined as {max(~d0)}∪~e0 ∪{max(~d1(}∪ · · · ∪~ej−1. The partition theorem 2.7.6

yields sequences ~cj : j ∈ ω such that ~cj ⊂ ~bj is a sequence of sets of size at least

three, an infinite set w ⊂ ω and a set a ⊂ n such that whenever ~dj : j ∈ ω are

sequences of pairs with ~dj ∈ [~cj ]2, and k ∈ w, then p(k, 〈~dj : j ∈ ω〉) = a.

Finally, we are in a position to construct the sets Ci : i ∈ n such that eachCi ⊂ P(ω) is a closed set on which the equivalence relation E0 is nonsmooth, andsuch that E �

∏i Ci connects pairs ~x and ~y just in case that ∀i ∈ a ~x(i) = ~y(i)

and ∀i /∈ a ~x(i) E0 ~y(i). Let Ci consist of all sets of the form {k0} ∪ e0 ∪ {k1} ∪e1 ∪ . . . where kj is the largest element of the set ~cj(i) if j /∈ w, and kj is one ofthe two largest elements of the set ~cj(i) if j ∈ w. We must verify the advertisedproperties of the sets Bi : i ∈ n.

First of all, for each i ∈ n the equivalence relation E0 is not smooth on theset Ci. To see this, consider the map h : 2ω → B defined by h(z) = the uniqueelement of the set B such that for every l ∈ ω, if j is l-th element of the set


w ⊂ ω, h(z) contains the largest element of the set ~cj(i) if and only if x(l) = 1.This map reduces E0 to E0 � Ci, and therefore E0 � Ci is not smooth.

To verify the canonical form of the equivalence relation E of the set C =∏i∈n Ci, first choose an arbitrary n-sequence ~x ∈ C. The homogeneity proper-

ties of the collection 〈~cj : j ∈ ω〉 show that for every j ∈ w it is the case thata~x∩mj ,j = a. This observation together with a simple induction on j ∈ w showsthat if sequences ~x, ~y ∈ C have the same entries on all i ∈ a, then for all j ∈ ω~x∩ mj Fj ~y ∩ mj . If, on the other hand, the sequences ~x, ~y differ at some entryi ∈ a, then find the least j ∈ w such that the sequences ~x∩ mj and ~y∩ mj differat some entry i ∈ a, observe that this is also the least j ∈ w such that ~x ∩ mj

and ~y ∩ mj are not Fj-related, and use I5(b) to argue by induction on all largernumbers k ∈ w that (~x ∩ mk) Fk (~y ∩ mk) fails.

Now assume in addition that the sequences ~x, ~y ∈∏i∈nBi satisfy ∀i ∈

n ~x(i) E0 ~y(i). If for every i ∈ a, ~x(i) = ~y(i), then choose j ∈ w so large thatall differences between the sequences ~x and ~y occur below mj . The previousparagraph together with the definition of the equivalence relation Fj shows that~x E ~y holds. If, on the other hand, there is a number i ∈ a such that ~x(i) 6= ~y(i),then again look at the same large j ∈ w. This time, the previous paragraphand the definition of the relation Fj shows that ~x E ~y fails. The proof iscomplete.

The previous lemmas can be easily cranked up to cover the general case ofanalytic equivalence relations with countable classes, and then to the essentiallycountable equivalence relations.

Lemma 5.4.13. Let n ∈ ω be a number, and let E be an analytic equivalencerelation with countable classes on an In-positive Borel set B ⊂ (2ω)n. Thereis a set a ⊂ n and an I-positive Borel set C ⊂ B such that E � C =

∏i∈n Fi,

where Fi = id if i ∈ a, and Fi = E0 if i /∈ a.

Proof. For every i ∈ n let Ri ⊂ (2ω)n× 2ω be the relation consisting of all pairs〈x, y〉 such that x ∈ B and there is x′ ∈ B E-related to x such that x′(m) =y. These are analytic relations with countable vertical sections, and thereforerelatively meager in every E0-box. A repeated application of Claim 5.4.7 thenyields an E0-box B′ ⊂ B such that for every x E x′ ∈ B′ and every i ∈ n,〈x, x′(i)〉 ∈ R implies x(i) E0 x′(i). In other words, the equivalence E is asubset of the product of E0’s on B′. Lemma 5.4.9 then yields the final BorelIn-positive Borel set C ⊂ B′.

Lemma 5.4.14. Let n ∈ ω be a number, and let E be an essentially countableequivalence relation on an In-positive Borel set B ⊂ (2ω)n. There is a partitionn = a ∪ b ∪ c and an I-positive Borel set C ⊂ B such that E � C =

∏i∈n Fi,

where Fi = id if i ∈ a, Fi = E0 if i ∈ b, and Fi = ee if i ∈ c.

Proof. Let f : B → Y be a Borel function reducing E to some Borel equivalencerelation F on a Polish space Y with countable classes. First apply Lemma 5.4.8


to the smooth equivalence E′ induced by f to find a Borel In-positive set B′ ⊂ Band a set c ⊂ n such that E′ � B′ =

∏i∈n Fi where Fi = id if i /∈ c and Fi = ee

if i ∈ c. This is to say that the values f(x) depend only on x � n \ c and that inan injective way. Ergo, the equivalence relation F induces a Borel equivalencerelation with countable classes on the set B′′ = the projection of B′ into thecoordinates outside the set c. An application of Lemma 5.4.13 concludes theproof.

Now we are ready to complete the proof of Theorem 5.4.6. By inductionon n ∈ ω we will prove the statement “for every m ∈ ω, for every analyticequivalence relation E on a Borel Im+n-positive set B ⊂ (2ω)m × (2ω)n, thereis a Borel Im+n-positive subset C ⊂ B such that for every z ∈ (2ω)m, E � Czis essentially countable”. The theorem will follow from the case m = 0 andLemma 5.4.14 above.

Suppose that m,n are given. Use the induction hypothesis to thin out B sothat for every index i ∈ n every z ∈ (2ω)m, and every y ∈ 2ω, the equivalencerelation E � {x ∈ B : x � m = z ∧ x(m + i) = y} is essentially countable. LetB = B0 and consider the relation R0 on B0 × B0(m) defined by 〈x, y〉 ∈ R0 ifthere is x′ ∈ B0 such that x′ � m = x � m and x′(m) = x(m).

Case 1. If the relation R0 is not meager, then use the Kuratowski-Ulamtheorem to find y ∈ B0(m) such that the horizontal section Ry0 is nonmeager inB0–thus, Ry0 ⊂ B0 is an analytic Im+n-positive set. Use uniformization 2.6.2(3)to find a Borel Im+n-positive set C ⊂ Ry0 and a Borel function f : C → B0

such that for every x ∈ C, f(x) � m = x � m, f(x)(m) = y, and f(x) E x.Now, for every z ∈ (2ω)m, the function f reduces E � Cz to the equivalencerelation E � {x ∈ B : x � m = z, x(m) = y}. This latter equivalence relation isessentially countable and in this case we are done.

Case 2. If the relation R0 is meager, then by Claim 5.4.7 there is a boxB1 ⊂ B0 such that for every x ∈ B1 and every y ∈ B1(m), 〈x, y〉 ∈ R0 impliesx(m) E0 y. Now repeat the whole argument at the coordinate m + 1 with therelation R1 ⊂ B1 × B1(m + 1) defined by 〈x, y〉 ∈ R1 if there is x′ ∈ B1 suchthat x′ � m = x � m and x′(m + 1) = x(m + 1), and by induction on i ∈ nrepeat it at every coordinate i. If at any stage of the induction we end up inCase 1, then we are done. Otherwise, we end up with a box C = Bn ⊂ B suchthat for every i ∈ n and every x0, x1 ∈ Bn with x0 � m = x1 � m and x0 E x1 itis the case that x0(m + i) E0 x1(m + i). This means that for every z ∈ (2ω)m

the equivalence relation E � Cz is a subset of product of E0’s. The classes of Ezare already countable, but we still have to deal with the possibility that Ez isanalytic and not Borel. An application of Lemma 5.4.9 shows that we can thinout the box C even further so that for every z ∈ (2ω)m, E � Cz takes one of thefinitely many prescribed Borel forms, and they all have countable equivalenceclasses. The proof is complete.


5.5 The E1 ideal

E1 is the equivalence relation on the Polish space X = (2ω)ω defined by x E1 yif for all but finitely many n ∈ ω, x(n) = y(n). This is the canonical example ofa Borel equivalence relation not Borel reducible to an orbit equivalence relationof a Polish group action. In view of the Kechris-Louveau dichotomy 2.2.10, itis natural to define IE1 to be the σ-ideal on X σ-generated by Borel sets Bsuch that E1 � B is essentially countable. Clearly, E1 cannot be canonizedto anything simpler on a Borel I-positive set. We will prove a canonizationtheorem saying that among the hypersmooth equivalence relations this is theonly obstacle ???.

From the forcing point of view, the σ-ideal IE1is somewhat awkward as

the quotient poset PIE1is not proper [66, Lemma 2.3.33], [27, ???]. At the

same time, there is a closely related σ-ideal whose quotient poset is proper andpreviously independently studied. It is the iteration of Sacks forcing along thereverse ω ordering at which a number of authors arrived from different anglesand at different times [17], [26], [?]. The associated ideal was computed in [67,Section 5.4]. The connection between the E1 equivalence and reverse iterationsshould not be surprising in view of Theorem 4.5.1. Here, we show that there isa natural collection of σ-ideals between the E1 ideal and the reverse iterationideal that allow uniform treatment.

5.5a The σ-ideals and their dichotomies

Definition 5.5.1. Let S ⊂ [ω]ℵ0 be a nonempty analytic set. The σ-ideal ISon X = (2ω)ω is σ-generated by the sets Az where z ranges over all elements of2ω and Az = {x ∈ X : ∀s ∈ S ∃n ∈ s x(n) ∈ ∆1

1(z, (ω, n)}.

Note that the generators of the σ-ideal IS are coanalytic. This is in contradis-tinction to other σ-ideals considered in this book, where the generators aretypically Borel or analytic, but it will cause only very minor annoyance. Itwill turn out that IE1 = I[ω]ℵ0 , while the reverse Sacks iteration ideal is equalto I{ω}. Before we can arrive at this conclusion though, we must show thatIS-positive analytic sets come in a specific form. That is the subject of thefollowing definition and theorem.

Definition 5.5.2. Let s ⊂ ω be an infinite set, with an increasing enumerationπ : ω → s. An s-keeping homeomorphism is a continuous injective map f :X → X such that for every x0, x1 ∈ X and every i ∈ ω, (a) if x0(i) 6= x1(i)then f(x0)(π(j)) 6= f(x1)(π(j)) for all j ≤ i and (b) if x0(j) = x1(j) for allj > i then f(x0)(n) = f(x1)(n) for all n > π(i). An s-cube is the range of astrict s-keeping homeomorphism. If S ⊂ P(ω) is a collection of infinite sets, anS-cube is a s-cube for some s ∈ S. A reverse cube is an ω-cube.

Note that if f is an s-keeping homeomorphism for an infinite s ⊂ ω then f isa one-to-one continuous reduction of E1 to E1 � rng(f) by the two demands(a) and (b). [26] used a definition of a projection-keeping homeomorphism that

5.5. THE E1 IDEAL 125

differs in an insignificant technical detail from our definition of an ω-keepinghomeomorphism.

Theorem 5.5.3. Let S ⊂ [ω]ℵ0 be a nonempty analytic set.

1. IS is a Π11 on Σ1

1 σ-ideal.

2. An analytic subset of X is IS-positive if and only if it contains a strict S-cube. In fact, an analytic subset A ⊂ X×ωω has an IS-positive projectionif and only if there is an S-cube C and a continuous function g : C → ωω

such that g ⊂ A.

3. If A ⊂ X is an analytic IS-positive set and {An : n ∈ ω} are analyticsubsets of X, then there is an S-cube C ⊂ A such that for every n ∈ ω,An ∩ C is a Borel set.

The quotient poset PIS is typically not proper. The statements (2) and (3)above may be viewed as the remnants of the bounding property and properness,respectively

Proof. The difficult part is inscribing an S-cube into an IS-positive analyticset. After that, a couple of routine applications of Kuratowski-Ulam theoremcomplete the proof.

So let S ⊂ [ω]ℵ0 be an analytic set and A ⊂ X × ωω be an analytic set.In order to promote the similarity with the canonization arguments in the nextsection, we will divide the treatment into two cases, even though here one ofthe cases is nearly trivial.

Case 1. There is a parameter z ∈ 2ω, a set s ∈ S and a point 〈x, y〉 ∈ Asuch that S,A are both Σ1

1(z) and for every n ∈ s, x(n) /∈ ∆11(x � (n, ω), z).

In this case, we will inscribe a strict S-cube C into p(A) and find a continuousfunction g : C → ωω such that g ⊂ A. The argument uses a version of Gandy-Harrington forcing. To simplify notation, assume that the parameter z is in fact0; otherwise just insert the parameter in suitable places below.

First, a couple of simple difinitions regarding sets of finite sequences. If~k ∈ ω<ω write 2

~k =∏i∈dom(~k) 2

~k(i). The expression ~k ≤ ~l means that dom(~k) ⊆dom(~l) and ∀i ∈ dom(~k) ~k(i) ≤ ~l(i). If ~k ≤ ~l and σ ∈ 2

~l then σ � 2~k ∈ 2

~k is

the sequence 〈σ(i) � ~k(i) : i ∈ dom(~k)〉. Finally, define the poset Q to consist of

tuples q = 〈~k = kq, F = Fq, T = Tq〉 such that ~k ∈ ω<ω, T ⊂ S is a nonempty

Σ11 set, F is a function with dom(F ) = 2

~k and rng(F ) consisting of nonemptyΣ1

1 subsets of A and there is a witness to q ∈ Q. Here, a witness is a tuple

〈s, 〈xσ, yσ〉 : σ ∈ 2~k〉 such that

• s ∈ Tq with an increasing enumeration π : ω → s, 〈xσ, yσ〉 ∈ F (σ) and

whenever i ∈ dom(~k) and σ ∈ 2~k then xσ(π(i)) /∈ ∆1

1(xσ � (π(i), ω));

• moreover, if σ, τ ∈ 2~k are sequences and i ∈ dom(~k) is a number then (a)

if σ(i) 6= τ(i) then xσ(π(j)) 6= xτ (π(j)) for all j ≤ i and (b) if σ(j) = τ(j)for all j > i then xσ(n) = xτ (n) for all n > π(i).


The stem of the witness 〈s, 〈xσ, yσ〉 : σ ∈ 2~k〉 is the common value of xσ �

(π(max dom(~k)), ω) for σ ∈ 2~k. The ordering on Q is defined by r ≤ q if

~kq ≤ ~kr, for every σ ∈ 2~kr Fr(σ) ⊂ Fq(σ � 2

~kq ), and finally Tr ⊂ Tq. Note thatQ is a countable notion of forcing.

Let M be a countable elementary submodel of a large structure containingall the above objects, and let G ⊂ Q be a generic filter over the model M . Therewill be a unique point sgen in the intersection

⋂q∈G Tq, and for each x ∈ X there

will be a unique point 〈f(x), h(x)〉 in the intersection⋂q∈G Fq(x �

~kq). We willshow that f is a sgen-keeping homeomorphism, its range will be an s-cube C,and the map g : C → ωω defined by g = h ◦ f−1 is the desired continuousuniformization of the set A on C. For all of this, several density arguments willbe necessary; they will be supported by the following series of claims.

Claim 5.5.4. Whenever q ∈ Q and D is a dense subset of the Gandy-Harrington

forcing on X × ωω then there is r ≤ q such that for every σ ∈ 2~kr , Fr(σ) ∈ D.

This is to assure that for every x ∈ X, the set {Fq(x � ~kq) : q ∈ G} is a Gandy-Harrington-generic filter over M , and therefore its intersection will contain aunique point 〈f(x), h(x)〉.

Proof. Enumerate 2~kq as {σi : i ∈ n} and by induction on i ≤ n form a de-

creasing chain of conditions q = q0 ≥ q1 ≥ q2 ≥ . . . , all with the same ~k and Tcoordinates, such that for every i ∈ n, Fqi+1

(σi) ∈ D. The condition r = qn willbe as required.

Suppose that qi has been found. Consider the set B = {〈x, y〉 ∈ Fqi(σi) :there is a witness to qi ∈ Q whose σi-th coordinate is 〈x, y〉}. This is a nonemptyΣ1

1 set and therefore has a subset B′ ⊂ B in the dense set D. Let Fqi+1 be equalto Fqi except Fqi+1

(σi) = B′. This completes the induction step.

The same type of argument, except simpler, yields

Claim 5.5.5. Whenever q ∈ Q and D is a dense subset of the Gandy-Harringtonforcing on [ω]ℵ0 , there is r ≤ q such that Tr ∈ D.

This is to assure that the filter {Tq : q ∈ G} is Gandy-Harrington-generic overM and therefore its intersection will contain a unique point sgen ∈ S.

Claim 5.5.6. Whenever q ∈ Q and ~l ≥ ~kq, there is an extension r ≤ q such

that ~l = ~kr.

Proof. In order to give a concise proof of the claim, we first need to show thatany condition q ∈ Q has a sufficiently rich collection of witnesses. Let q ∈ Qand write ~k = ~kq. Let 〈s, 〈xσ, yσ〉 : σ ∈ 2

~kq 〉 be a witness to q ∈ Q with

stem x, let π : ω → s be an increasing enumeration, let i = max dom(~k), andlet a ⊂ 2ω be a countable set. We claim that there is an improved witness:

a tuple 〈s, 〈x′σ, y′σ〉 : σ ∈ 2~k〉 with the same stem as the original one such

that for every j ≤ i x′σ(π(j)) /∈ a. To see why this is true, note that the set


{〈x′σ(π(j)) : σ ∈ 2~k, j ≤ i〉 : 〈s, x′σ : σ ∈ 2

~k〉 is a witness to q ∈ Q with stemx} is a nonempty Σ1

1(x) set: its definition does not depend on s as only the the

first dom(~k) many elements of s are used in the definition. The tuples in thisset contain no ∆1

1(x) element, and apply Proposition 2.1.10.

Back to the proof of the claim. First of all, it is easy to extend q to rsuch that dom(~kr) is an arbitrary given number larger than dom(~k): simply

let ~kr = ~k concatenated with some zeroes and Fr(σ) = Fq(σ � dom(~k)). So

we can assume that dom(~l) = dom(~k). It is also enough to consider the case

where there is a single number i ∈ dom(~k) such that ~l(i) = ~k(i) + 1 and the

other values of those sequences are equal. We must show that r = 〈~l, Fr, Tq〉 is

a condition in the poset Q, where Fr(σ) = Fq(σ � ~k).

To prove this, we must find a witness for r ∈ Q. Choose 〈s, 〈xσ, yσ〉 : σ ∈ 2~k〉

witnessing the fact that q ∈ Q. For every τ ∈∏j>i 2

~k(j) and every u ∈ 2~k(i)

let qτ,u be the condition defined by ~kqτ,u = ~k � i, Tqτ,u = Tq and Fqτ,u(η) =Fq(η

auaτ). This is a condition in Q as witnessed by 〈s, 〈xηauaτ , xηauaτ 〉 : η ∈2~k�i〉. Use the first paragraph to find, for every τ and u, two such witnesses

〈xτ,u0η , yτ,u0

η 〉 and 〈xτ,u1η , yτ,ubη 〉 : η ∈ 2

~k�i for qτ,u ∈ Q with the same stem andthe same s coordinate that use distinct points at coordinates π(j) : j ≤ i. For

σ ∈ 2~l, σ = ηa(uab)aτ for some bit b ∈ 2 let xσ = xτ,ubη , yσ = yτ,ubη and check

that 〈s, 〈xσ, yσ〉 : σ ∈ 2~l〉 witness the fact that r ∈ Q.

Claim 5.5.7. For every q ∈ Q there is r ≤ q with ~kq = ~kr, and a map π :

dom(~kq)→ ω such that

1. π is an initial part of the increasing enumeration of every element of Tr;

2. for every pair σ, τ ∈ 2~k of sequences, every i such that σ(i) 6= τ(i) and

every j ≤ i, the projections of Fr(σ) and Fr(τ) to the π(j)-th componentof their first coordinate are disjoint.

This is to assure that f has the property (a) of the definition of an sgen-keepinghomeomorphism.

Proof. Let ~k = ~kq. Pick a witness 〈s, 〈xσ, yσ〉 : σ ∈ 2~k〉 to q ∈ Q. The condition

r ∈ Q will be constructed so that it has the same witness. Let π : ω → s

be the increasing enumeration, and for every σ ∈ 2~k and every j ∈ dom(~k)

find a basic open set Oσ,j ⊂ 2ω containing xσ(π(j)) such that if xσ(j) 6= xτ (j)

then Oσ,j ∩ Oτ,j = 0. Improve the condition q ∈ Q to r so that ~kr = ~k,~Tr = {t ∈ Tq : π � dom(~k) is an initial segment of the increasing enumeration

of t}, and Fr(σ) = {〈x, y〉 ∈ Fq(σ) : ∀j ∈ dom(~k) x(π(j)) ∈ Oσ,j}. It is notdifficult to check that r ≤ q is a condition with the same witness as q, and it isas required by the (a) definitory property of the witness.


The rest of the proof of Case 1 is just a routine recitation of elementary densityarguments.

Case 2. If Case 1 fails, then let z ∈ 2ω be a parameter such that S,A areboth Σ1

1(z) and note that then p(A) is a subset a generator of the σ-ideal IS ,namely the set {x ∈ X : ∀s ∈ S ∃n ∈ s x(n) ∈ ∆1

1(x � (n, ω), z)}. We concludethat p(A) ∈ IS .

The rest of the argument for Theorem 5.5.3 is now easy. For (2), supposethat A ⊂ X × ωω is an analytic set. If p(A) is IS-positive, then the aboveargument provides an S-cube C and a continuous function g : C → ωω suchthat g ⊂ A. If, on the other hand, p(A) contains an S-cube C, a range ofan s-keeping homeomorphism f , then C and so p(A) must be IS-positive. Tosee this, note that whenever z ∈ 2ω is a parameter and n ∈ s, then the setBn,z = {x ∈ X : f(x) ∈ ∆1

1(f(x) � (n, ω), z)} is meager by Kuratowski-Ulam–ithas countable sections in the ni-th coordinate where i is such that n is i-thelement of the set s. This means that the f -preimages of the generators of theσ-ideal IS are meager (the generator associated with z has f -preimage which isa subset of the union

⋃nBn,z) and so C = rng(f) /∈ IS .

For (1), suppose that A ⊂ 2ω×X is an analytic set, and choose a parameterz ∈ 2ω such that A,S are both Σ1

1(z). The case 1 above then shows that forevery y ∈ 2ω, the set Ay is IS-positive if and only if there is x ∈ X and s ∈ Ssuch that 〈y, x〉 ∈ A and for every n ∈ s, x(n) /∈ ∆1

1(x � (n, ω), z, y). This is aΣ1

1(z) condition by ???.Finally, for (3) suppose that A /∈ IS is an analytic set. Thinning out the

set A if necessary, we may assume that it is an S-cube, with an s-keepinghomeomorphism f : X → A for some s ∈ S. Now let {An : n ∈ ω} be countablymany analytic sets. Their f -preimages are still analytic, they have the Baireproperty, and there is a dense Gδ subset B ⊂ X such that in it they are relativelyclopen. The proof of (2) above shows that f ′′B ⊂ A is an IS-positive set, andin it all sets {An : n ∈ ω} are relatively clopen. (3) follows.

There are two interesting special cases that we will consider here. If S ={ω} then the quotient poset PIS has a dense subset naturally isomorphic tothe reverse iteration of Sacks forcing, which is defined to consist of reversecubes ordered by inclusion. A more interesting case occurs when S = [ω]ℵ0 ; itcorresponds to the σ-ideal IE1 and the Kechris-Louveau dichotomy 2.2.10.

Corollary 5.5.8. Let S = [ω]ℵ0 . The ideals IS and IE1contain the same

analytic sets. In other words, for every analytic set A ⊂ X, A /∈ IS iff E1 isBorel reducible to E1 � A iff E1 � A is not Borel reducible to E0.

Proof. Suppose that A /∈ IS ; we will conclude that E1 is Borel reducible toE1 � A. Use Theorem 5.5.3 to find an infinite set s ⊂ ω and an s-keepinghomeomorphism f : X → A whose compact range C is a subset of A. Clearly,f is a continuous injective reduction of E1 to E1 � C as required.

Suppose that A ∈ IS ; we must conclude that E1 � B is Borel reducible toE0 for some Borel superset B ⊃ A. For simplicity assume that A is Σ1

1. Use


the first reflection theorem 2.1.2 to find a Borel set B ⊃ A such that for everyx ∈ B the set {n : x(n) /∈ ∆1

1(x � (n, ω))} is finite. Use Kreisel selection [27,Theorem 2.4.5] to find a Borel function h : B → ω such that this finite setis always bounded by h(x). Let i : B → X be the Borel function assigningeach x ∈ B the sequence equal to x above h(x) and equal to the zero sequencebelow it. Review the definitions to observe that the E1-equivalence classeson rng(i) are countable. Use the first reflection theorem to produce a Borelsuperset C ⊃ rng(i) on which the E1 classes are countable. Now, E1 � C is ahypersmooth equivalence relation with countable classes, and by [27, Theorem8.1.1(iii)], such an equivalence is Borel reducible to E0 via some Borel functionj : C → 2ω. The composition j ◦ i : B → 2ω reduces E1 � B to 2ω as desired.

Corollary 5.5.9. (Kechris-Louveau) If E is a hypersmooth equivalence relationon a Polish space Y then either E ≤B E0 or E1 ≤B E.

Proof. Fix Borel maps fn : n ∈ ω, fn : Y → 2ω such that for every y0, y1 ∈ Y ,y0 E y1 iff the set {n : fn(y0) = fn(y1)} ⊂ ω is co-finite. Let g : Y → X be themap defined by g(y) = 〈fn(y) : n ∈ ω〉, and look at the set A = rng(g). The setA ⊂ X is analytic; we discern the cases where A ∈ J or not.

If A /∈ J then by Theorem 5.5.3 there is an s-cube C ⊂ A for some infiniteset s ⊂ ω, a continuous map g : C → Y such that f ◦ g = id, and an s-keepinghomeomorphism h : X → C whose range C is. Review the definitions to seethat g ◦ h : X → Y is a continuous injective reduction of E1 to E � A.

If A ∈ J then there is a Borel set B ⊃ A such that E1 � B is Borel reducibleto E0 via a map h : B → 2ω. It is immediate that h ◦ f is a reduction of E toE0.

Corollary 5.5.10. Let S = {ω}. The poset PIS has a dense subset naturallyisomorphic to the Sacks iteration along ω with the inverted ordering.

Proof. Up to an insignificant technical detail, the iteration isolated in [26] con-sists of ω-cubes ordered by inclusion. Theorem 5.5.3 precisely states that thisis a dense subset of the ordering PIS .

5.5b Canonization results

The canonization properties of reverse cubes are unclear in general. However,the hypersmooth equivalences do canonize down to a simple irreducible set ofunavoidable obstacles.

Theorem 5.5.11. Let B ⊂ X = (2ω)ω be a Borel set such that E1 � B is notessentially countable. Let E be a hypersmooth equivalence relation on B. Thenthere is a compact set C ⊂ B such that E1 � C is still not essentially countable,and E � C equals to id or to ee or to E1.


Theorem 5.5.12. Let B ⊂ X = (2ω)ω be a reverse cube, and let E be ahypersmooth equivalence relation on B. Then there is a reverse cube C ⊂ Bsuch that E � C equals to id or to ee or to E1, or to idn for some n ∈ ω.

Here, idn is the smooth equivalence relation connecting points x, y ∈ X if forall m ≥ n, x(m) = y(m)–so that E1 =

⋃n idn. Theorem 5.5.11 is a formal

consequence of Theorem 5.5.12.The arguments below follow the same pattern. There is a split into cases,

in one of the cases we use a Gandy-Harrington forcing just like in the previoussection, and in the other a fusion argument appears. Thus, we need to makepreliminary fusion definitions. Let X = (2ω)ω and let I be the σ-ideal of analyticsets which contain no reverse cube.

Definition 5.5.13. Let B,C ⊂ X be reverse cubes and n ∈ ω. We writeC ≤n B if C ⊂ B and C is idn-invariant inside B. A fusionsequence is asequence 〈Cj : j ∈ ω〉 of reverse cubes such that there are increasing numbers〈nj : j ∈ ω〉 such that Cj+1 ≤nj Cj .

Proposition 5.5.14. Suppose that 〈Cj : j ∈ ω〉 is a fusion sequence of reversecubes. Then

⋂j Cj /∈ I.

Proof. It is clear that C =⋂j Cj is a nonempty compact set. To show that

C /∈ I, we will show that the generating sets of the ideal I are relatively meagerin C. Let z ∈ 2ω be an arbitrary parameter, let n ∈ ω and consider the coana-lytic set B = {x ∈ C : x(n) ∈ ∆1

1(x � (n, ω), z)}. Let D ⊂ 2ω × (2ω)(n,ω) be thenonempty compact projection of the set C into (2ω)[n,ω). The horizontal sectionsof the set D are all perfect: if nonempty, every such horizontal section is equalto a corresponding section in the cube Cn since C ≤n Cn, and the section in thecube Cn is perfect by the definition of a projection-keeping homeomorphism.The horizontal sections corresponding to the set B are countable, and there-fore meager in the horizontal sections of the set D. By the Kuratowski-Ulamtheorem, B is meager in C as desired.

As a useful warm-up, we will reprove the canonization of smooth equivalencerelations achieved in [26].

Theorem 5.5.15. For every reverse cube A ⊂ X and a smooth equivalence Eon A there is a reverse cube C ⊂ A such that E � C = idn for some n ∈ ω orE � C = ee.

Proof. Let A ⊂ X be an I-positive Borel set and E a smooth equivalencerelation. Let f : A → 2ω be the Borel reduction of E to the identity. Passingto an I-positive subset if necessary, we may assume that f � A is continuous.

Case 1. Suppose that there exists a parameter z ∈ 2ω such that f ∈ Π01(z)

and there is a set A ∈ Σ11(z) with A ⊂ A, a point x ∈ A and a number m ∈ ω

such that for every n ∈ ω, x(n) /∈ ∆11(x � (n, ω)), and f(x) /∈ ∆1

1(x � [m,ω), z).In this case, we will use a variation of the Gandy-Harrington poset form theprevious section to find a reverse cube C ⊂ A such that E � C = idm−1.


Assume for simplicity z = 0 and choose x,m as described. Let Q be the

poset of all pairs q = 〈~k = ~kq, F = Fq〉 such that for every σ ∈ 2~k, F (σ) is

a nonempty Σ11 subset of A and there is a witness to q ∈ Q. This is a tuple

〈xσ : σ ∈ 2~k〉 such that for every σ ∈ 2

~k, xσ ∈ F (σ) and for every n ∈ ω,

xσ(n) /∈ ∆11(x � (n, ω), and for every pair σ, τ ∈ 2

~k, (a) if σ(i) 6= τ(i) andj ≤ i then xσ(j) 6= xτ (j), (b) if i is the largest number that σ(i) 6= τ(i) then

xσ � (i, ω) = xτ � (i, ω), and (c) if σ � dom(~k) \ m 6= τ � dom(~k) \ m, then

f(xσ) 6= f(xτ ). The ordering on Q is defined by r ≤ q if ~kq ≤ ~kr and for every

σ ∈ 2~kr , Fr(σ) ⊂ Fq(σ � ~kq). The case assumption yields the starting point:

Q 6= 0.The argument now follows closely that of Theorem 5.5.3. Let M be a count-

able elementary submodel of a large enough structure and let G ⊂ Q be a filtergeneric over M . For every x ∈ X consider the filter {Fq(x � ~k) : q ∈ G} onthe Gandy-Harrington forcing on X. It turns out that this filter is Gandy-Harrington generic over M , and so its intersection contains a unique elementg(x) ∈ A. The function g : X → A is a projection-keeping homeomorphism.Moreover, if x0, x1 ∈ X are points such that x0 � [m,ω) 6= x1 � [m,ω), thenf(x0) 6= f(x1). The whole treatment can be nearly literally copied from the pre-vious section. Note the application of Proposition 2.1.10 in one of the densityarguments.

Let C = rng(g); this is a reverse cube such that for all x0, x1 ∈ C withx0 � [m,ω) 6= x1 � [m,ω) it is the case that f(x0) 6= f(x1). Now, if thenumber m ∈ ω was chosen to be smallest possible, the reverse cube C is suchthat for every x ∈ C, f(x) ∈ ∆1

1(x � [m,ω)). Now choose any reasonableeffective enumeration of ∆1

1(·) singletons by natural numbers such as in [27,Corollary 2.8.4]. For every natural number e, the set Be = {x ∈ C : e codesa ∆1

1(x � [m,ω))-singleton and that singleton is equal to f(x)} is coanalytic,and C ⊂

⋃eBe. By Theorem 5.5.3 (3), we can thin out the cube C so that

all these sets Be have Borel intersection with C. By Theorem 5.5.3(2) and theσ-additivity of the σ-ideal IS we can then further thin out C so that it is whollyincluded in one of these sets. After that, E � C = idm as required.

Case 2. If Case 1 fails, we will construct a fusion sequence C0 ≥0 C1 ≥C2 ≥ . . . so that for every j ∈ ω, for every x ∈ Cj , the functional value f(x)depends only on x � (j, ω). In the end, let C =

⋂j Cj ; this is a reverse cube

such that f is constant on E1 classes of C. Since the equivalence relation E1 isdense in C2 and the function f is continuous, this means that f � C is constantand E � C = ee as desired.

The construction of Cj proceeds by induction. Suppose that Cj has beenconstructed. Find z ∈ 2ω such that Cj ∈ Σ1

1(z). The case assumption showsthat for every x ∈ Cj such that for every n ∈ ω, x(n) /∈ ∆1

1(x � (n, ω)),f(x) ∈ ∆1

1(x � [j + 1, ω), z). Just as above, one can use the Π11 coding of ∆1

1

sets and Theorem 5.5.3(3) to find a subcube Cj+1 so that f(x) depends only onx � [j + 1, ω). Since f(x) depends only on x � [j, ω) for x ∈ Cj , we can find thesubcube Cj+1 so that Cj+1 ≤j Cj . This completes the induction step and theproof.


Proof of Theorem 5.5.12. Let A ⊂ X be a reverse cube. Let E =⋃Fn for some

inclusion increasing sequence of smooth equivalence relations, and let fn : A→2ω be Borel functions reducing each Fn to the identity.

Case 1. There is a parameter z ∈ 2ω such that the functions fn are all Π01(z),

a set A ∈ Σ11(z) with A ⊂ A, a number m ∈ ω, and a point x ∈ A such that

∀n x(n) /∈ ∆11(x � (n, ω), z) and ∀i fi(x) /∈ ∆1

1(x � (m,ω), z). We will assumethat m is minimal possible, we will make the harmless assumption that A = Aand that thinning out if necessary, that ∀x ∈ A ∀n x(n) /∈ ∆1

1(x � (n, ω), z)holds. Here we use the Gandy-Harrington forcing to extract a reverse cubeC ⊂ A such that on it E � C ⊆ idm, and then thin it out further so that on iteither E � C = idm or E � C = idm−1.

Assume for simplicity that the parameter z is in fact 0. Let Q be the posetof all pairs q = 〈~kq = ~k, Fq = F 〉, where ~k ∈ ω<ω and Fq is a map that assigns

each sequence σ ∈ 2~k a nonempty Σ1

1 subset of A. Moreover, there must be a

witness to q ∈ Q, and this is a tuple 〈xσ : σ ∈ 2~k〉 such that for every σ ∈ 2

~k

it is the case that xσ ∈ F (σ) and ∀n x(n) /∈ ∆11(x � (n, ω), z), and for every

pair σ, τ ∈ 2~k, (a) if σ(i) 6= τ(i) and j ≤ i then xσ(j) 6= xτ (j), (b) if i is

the largest number that σ(i) 6= τ(i) then xσ � (i, ω) = xτ � (i, ω), and (c) if

σ � (m,dom(~k)) 6= τ � (m, dom(~k)) then fn0(xσ) 6= fn1

(xτ ) for every n0, n1 ∈ ω.

The ordering on Q is the usual one: r ≤ q if ~kq ≤ ~kr and for every σ ∈ 2~kr ,

Fr(σ) ⊂ Fq(σ � ~kq). The case assumption shows that Q 6= 0.The treatment of the forcing follows the previous proofs in this section. In

the claim dealing with splitting of witnesses, the full strength of ??? is used.Let M be a countable elementary submodel of a large enough structure andlet G ⊂ Q be a filter generic over M . For every x ∈ X consider the filter{Fq(x � ~k) : q ∈ G} on the Gandy-Harrington forcing on X. It turns out thatthis filter is Gandy-Harrington generic over M , and so its intersection containsa unique element g(x) ∈ A. The function g : X → A is a projection-keepinghomeomorphism, and whenever x0, x1 ∈ X are points such that x0 � [m,ω) 6=x1 � [m,ω) and n0, n1 ∈ ω are natural numbers, then fn0

(x0) 6= fn1(x1). So,

writing C = rng(g), we see that this is a reverse cube on which E ⊆ idm.By the minimal choice of the number m, for every x ∈ C there is a number

n ∈ ω such that fn(x) ∈ ∆11(x � [m − 1, ω)). Use Theorem 5.5.3(3) to thin

out C if necessary so that the coanalytic sets On,k = {x ∈ C : is a code for a∆1

1(x � [m− 1, ω)) singleton which is equal to fn(x)} are relatively Borel in C,and then use Theorem 5.5.3(2) to thin out the reverse cube C further so that itis a subset of one of these sets, say On0,k0 . It follows that idm−1 ⊆ E � C. Wewill now thin out C further to fully canonize E � C.

Write Cm for the projection of C to (2ω)[m,ω) and similarly for Cm−1; theseare reverse cubes in the appropriate sense. Note that for all n > n0 and x ∈ C,the functional value fn(x) depends on x � [m− 1, ω) only, and write fn for thefunction from Cm−1 to 2ω. Use Theorem 5.5.3(2) to thin out C so that thefunctions fn : n ∈ ω are all continuous on Cm−1.

Consider the sets D0 = {〈x, P 〉 : x ∈ Cm, P ⊂ 2ω is a nonempty perfectset, {x} × P ⊂ Cm−1, and for some n ≥ n0, fn is constant on {x} × P} and


D1 = {〈x, P 〉 : x ∈ Cm, P ⊂ 2ω is a nonempty perfect set, {x} × P ⊂ Cm−1,and for every n ≥ n0, fn is one-to-one on {x} × P}. Since the functions fn arecontinuous, both of these sets are analytic in (2ω)[m,ω) ×K(2ω). Note that theunion of projections of D0 and D1 covers the whole cube Cm: for every x ∈ Cm,either there is n ≥ n0 such that the function fn has an uncountable fiber onthe set {y ∈ 2ω : yax ∈ Cm} and then x belongs to the projection of D0, orelse the fibers are countable and then there is a perfect set on which all of thesefunctions are injective, for example by Mycielski’s theorem ???. Thus, we canthin out Cm into a reverse subcube if necessary and find a continuous functiong : Cm → K(2ω) such that (a) g ⊂ D0 or (b) g ⊂ D1. In either case, it ispossible to thin out C further so that for every x ∈ C, x(m−1) ∈ g(x � [m,ω)).In case (a), we get E � C = idm, in case (b) we get E � C = idm−1 as desired.

Case 2. There is a reverse cube C0 ⊂ A, a parameter z ∈ 2ω and a numbern such that for every x ∈ C0 and every m ∈ ω, f(x) ∈ ∆1

1(x � (m,ω), z). In thiscase, a fusion construction similar to Case 2 of the proof of Theorem 5.5.15 willyield a subcube C ⊂ C0 such that fn � C is constant, and therefore E � C = ee.

Case 3. Suppose now that both Cases 1 and 2 fail. We will build a fusionsequence whose limit, a reverse cube C ⊂ X, will have E � C = E1. Byinduction on j ∈ ω build numbers mj , nj ∈ ω and cubes Cj so that m0 <m1 < m2 < . . . , Cj ≥mj Cj+1, and fnj � Cj = idmj . If this succeeds, letC =

⋂j Cj . This is a reverse cube; the assumptions immediately imply that

n0 < n1 < . . . and E � C = E1. To perform the induction, suppose Cj ,mj , njhave been constructed. Let z ∈ 2ω be such that Cj ∈ Σ1

1(z). Since Case 1fails, for every x ∈ Cj such that ∀i x(i) /∈ ∆1

1(x � (i, ω), z) there must be nsuch that fn(x) ∈ ∆1

1(x � (mj , ω), z). Use Theorem 5.5.3(3) to thin out Cj sothat the coanalytic set {x : fn(x) ∈ ∆1

1(x � (mj , ω), z)} are relatively Borel foreach n ∈ ω, and then use Theorem 5.5.3(2) to thin out Cj to be a subset of oneof these sets, for n = nj+1. These surgeries on the reverse cube Cj may haveresulted in a change of the parameter z from which Cj is defined, so find a newparameter z′ ∈ 2ω coding z and such that Cj is Σ1

1(z′). Since Case 2 fails, theremust be x ∈ Cj such that for every i ∈ ω, x(i) /∈ ∆1

1(x � (i, ω), z′) and for somem > mj , fnj+1

(x) /∈ ∆11(x � (m,ω), z′). Let mj+1 > mj be the least number

m for which such a point x exists. The proof of Case 1 of Theorem 5.5.15shows that there is a subcube Cj+1 ⊂ C ′j such that Enj+1 � Cj+1 = idmj ;since Enj ⊂ Enj+1

and Enj � Cj = idmj , we can find such a subcube so thatCj+1 ≤mj Cj . This concludes the induction step and the proof.

Proof of Theorem 5.5.11. Let A ⊂ X be an analytic set such that E1 � A isnot essentially countable, and let E be a hypersmooth equivalence on A. UsingTheorem 5.5.3(2), thinning out the set A if necessary we may assume that Ais an s-cube for some infinite set s ⊂ ω, with an s-keeping homeomorphismf : X → A. Let F = f−1E be the (still hypersmooth) pullback of E, anduse the previous theorem to find a reverse cube D ⊂ X with a strict ω-keepinghomeomorphism g : X → D such that F � D is either equal to E1 or to ee or toidn for some n ∈ ω. In the first two cases, the composition f ◦ g is an s-keepinghomeomorphism and on its range C, the equivalence E is equal to either E1


or to ee. In the last case, there is an obvious ω \ n-keeping homeomorphismh : X → D, and then f ◦ g ◦ h is an s′-keeping homeomorphism (for s′ = s\thefirst n-many elements of s) such that on its range C, E � C = id.

5.5c Borel functions on pairs

The methods of this section make it possible to identify coding functions for theσ-ideals discussed above. This makes it very unlikely that any reduced productforcing type of argument could be used to produce further canonization results.

Theorem 5.5.16. Let I be the σ-ideal on X consisting of Borel sets with noreverse subcube. There is a Borel function f : X2 → 2ω such that for everyreverse cube C ⊂ X, f ′′C2 = 2ω.

Proof. Let f(x, y)(m) = x(m)(k) where k is the least number such that x(m)(k) 6=y(m)(k) if such exists; if such a k does not exist then let f(x, y)(m) =trash. Thisfunction works.

Let C ⊂ X be a reverse cube and let z ∈ 2ω be an arbitrary point. Wehave to produce points x, y ∈ C such that f(x, y) = z. Choose a parameterz′ ∈ 2ω coding both C and z and consider the following Gandy-Harrington typeforcing Q. A condition in Q is a pair q = 〈Aq, Bq〉 of Σ1

1(z′) subsets of C suchthat there is an witness to q ∈ Q: this is a pair 〈xq, yq〉 such that xq ∈q A,yq ∈ Bq, for every n ∈ ω xq(n) /∈ ∆1

1(z′, x � (n, ω)) and similarly on y-side, andfinally, for some number n ∈ ω (the split point of the witness), for every m ≥ nxq(m) = yq(m) holds and for every m < n, xq(m) 6= yq(m) and xq(m)(k) = z(k)hold, where k is the least number such that xq(m)(k) 6= yq(m)(k). The orderingis that of coordinatewise inclusion. Note that Q is a countable poset.

Let M be a countable elementary submodel of a large structure and letG ⊂ Q is a filter generic over M . The intersection of the first coordinates ofconditions in the generic filter contains a unique point x ∈ X and the secondcoordinates all intersect in a singleton as well, {y}. We will argue that f(x, y) =z.

The arguments transfers from the previous subsections in all details, exceptfor one. For any condition q ∈ Q we must produce witnesses with arbitrarilylarge split points. Suppose 〈xq, yq〉 is a witness with split point n; we mustproduce a witness with split point n+ 1. For every m ∈ n let k(m) be the leastnumber k such that xq(m)(k) 6= yq(m)(k) and consider the set D = {u ∈ 2ω :there are points xu ∈ A, yu ∈ B such that for all m ∈ ω, xu(m) /∈ ∆1

1(z′, xu �(m,ω)), similarly on the yu side, for all m > n xu(m) = yu(m) = xq(m),xu(n) = yu(n) = u, and for all m < n, xq(m) � k(m) + 1 ⊂ xu(m) andyq(m) � k(m)+1 ⊂ yu(m)}. This is a Σ1

1(z′, xq � [n+1, ω) set containing a non-∆1

1(z′, xq � [n+ 1, ω) point, namely xq(n). Therefore, the set D is uncountableand contains two distinct points u, u′ ∈ 2ω. Let k be the least number such thatu(k) 6= u′(k); say u(k) = z(n). Use the definition of the set D to get points x′qand y′q such that for all m > n x′q(m) = y′q(m) = xq(m), x′q(n) = u and y′q(n) =u′, and for all m < n, xq(m) � k(m)+1 ⊂ x′q(m) and yq(m) � k(m)+1 ⊂ y′q(m).The pair 〈x′q, y′q〉 is the desired witness with split point n+ 1.


Theorem 5.5.17. Let I be the σ-ideal on X consisting of Borel sets on whichE1 is reducible to E0. There is a Borel function f : X2 → ωω such that forevery I-positive Borel set B ⊂ X, the set f ′′B2 ⊂ ωω is dominating.

Proof. Let f : X2 → ωω be the function defined by f(x, y)(n) = min{k : forevery m ∈ n either x(m) = y(m) or x(m) � k 6= y(m) � k}. This function works.

In order to prove this, let B ⊂ X be a Borel I-positive set, and z ∈ ωω

be a function. We must find points x, y ∈ B such that f(x, y)(n) > z(n) forall but finitely many n ∈ ω. In order to do so, use Corollary 5.5.8 to find aninfinite set s ⊂ ω with the increasing enumeration π : ω → s and a strict s-keeping homeomorphism g : X → B. Let z′ ∈ 2ω be a point coding z, s, and g.Consider the poset Q of all pairs q = 〈Aq, Bq〉 of Σ1

1(z′) sets such that there isa witness for q ∈ Q, and this is a pair 〈x, y〉 such that x ∈ Aq, y ∈ Bq, for everyn ∈ ω g(x)(π(n)) /∈ ∆1

1(z′, x � (n, ω)), similarly on the y-side, and for somenumber n ∈ ω, for all m ≥ n x(m) = y(m), and for all m < n, it is the casethat f(x)(π(m)) 6= f(y)(π(m)) and writing k for the least number such thatf(x)(π(m)(k) 6= f(y)(π(m))(k), k > z(l) for all l ≤ π(m+ 1). The ordering onQ is that of coordinatewise inclusion.

Just as in all the previous similar arguments, if M is a countable elementarysubmodel of a large structure and G is a Q-generic filter over M , the first andsecond coordinates of the conditions in the filter give some points x, y ∈ B. Itis then straightforward to show that f(x, y)(n) > z(n) for all n > π(0).

Corollary 5.5.18. Let J be a σ-ideal on a Polish space X such that the quotientforcing PJ is proper. If E1 is in the spectrum of the σ-ideal J then J has a codingfunction.

Proof. By Theorem 4.5.1, if E1 is in the spectrum then there is a conditionB ∈ PJ and names 〈yn : n ∈ ω〉 for elements of the Cantor space such thatB ∀n yn /∈ V [〈ym : m > n〉]. Using the properness of the quotient poset,thinning out the condition B if necessary it is possible to find Borel functionsgn : B → 2ω representing the names yn for every n ∈ ω, and combine thesefunctions to g : B → (2ω)ω. Observe that if C ⊂ B is a J-positive Borelset then the analytic set g′′C ⊂ (2ω)ω is I-positive: whenever z ∈ 2ω is aparameter, M is a countable elementary submodel of a large enough structurecontaining z, J, C, and x ∈ C is a PI -generic point over M , it is the case for everyn ∈ ω that gn(x) /∈ M(〈gm(x) : m > n〉) by the forcing theorem. Therefore,gn(x) /∈ ∆1

1(z, gm(x) : m > n) and so g(x) does not belong to the generatorset of the σ-ideal J associated with the parameter z. Now, let h be the codingfunction for J as isolated in Theorem 5.5.16. Clearly, h ◦ g is a coding functionfor I.


5.6 The E2 ideal

E2 is the equivalence relation on X = 2ω defined by xE2y if Σ{ 1n+1 : x(n) 6=

y(n)} <∞. It is a basic example of a turbulent equivalence relation, and there-fore it is not classifiable by countable structures [27, Theorem 13.9.1]. There isan associated σ-ideal and a quotient proper notion of forcing. For xE2y defined(x, y) = Σ{ 1

n+1 : x(n) 6= y(n)}, otherwise let d(x, y) =∞. Thus, d is a metricon each equivalence class. Kanovei [27, Definition 15.2.2] defined the collectionof grainy subsets of X as those sets A ⊂ 2ω such that there is a real numberε > 0 such that there is no finite sequence of elements of A, the successiveelements of which have d-distance at most ε and the two endpoints are at adistance at least 1 from each other. Let I be the σ-ideal generated by the Borelgrainy sets.

Fact 5.6.1. [27, Section 15.2] The ideal I is Π11 on Σ1

1, and every analyticI-positive set has a Borel I-positive subset. A Borel set B ⊂ X is in the idealI if and only if the equivalence E2 � B is essentially countable if and only if E2

is not reducible to E2 � B.

The features of the quotient forcing were surveyed in [27, 67]. Here, we willprovide a more thorough treatment:

Theorem 5.6.2. The forcing PI is proper, bounding, preserves Baire categoryand outer Lebesgue measure, and adds no independent reals.

The canonization properties of the ideal I seem to be difficult to isolate.Clearly, the ideal was defined so as to have the equivalence relation E2 in thespectrum. This is reflected in the following:

Theorem 5.6.3. E2 is in the spectrum of I. Moreover, PE2

I is a regular ℵ0-distributive subposet of PI , and it generates the model V [xgen]E2 .

No other features of the spectrum are obvious.We will prove

Theorem 5.6.4. I has total canonization for equivalences classifiable by count-able structures.

In particular, this shows that the forcing PI adds a minimal real, improving [27,Theorem 15.6.3].

Question 5.6.5. Are there any equivalence relations in the spectrum of I otherthan E2?

Proof of Theorem 5.6.2. We will need some preliminary notation and observa-tions. Abuse the notation a little and include I-positive analytic sets in it; byFact 5.6.1, Borel sets will be dense in it anyway. For points x, y ∈ 2ω and anumber k ∈ ω write dk(x, y) =

∑{ 1n+1 : n ≥ k, x(n) 6= y(n)} < ε; this is a

notion of pseudodistance between sets satisfying triangle inequality which canbe infinite or zero for distinct points. For sets A,B ⊂ 2ω let dk(A,B) be theassociated Hausdorff distance: the infimum of all ε such that for every x ∈ Athere is y ∈ B such that dk(x, y) < ε and vice versa, for every x ∈ B there isy ∈ A such that dk(x, y) < ε. The following claims will be used repeatedly:


Claim 5.6.6. If A,B ⊂ 2ω are analytic sets such that A /∈ I and ∀x ∈ A ∃y ∈B x E2 y then B /∈ I.

Proof. If B ∈ I, then B can be enclosed in a countable union B′ of Borel grainysets, and E2 � B′ can be reduced to some countable equivalence relation E on2ω via a Borel function f : B′ → 2ω. The set A can be thinned out to a BorelI-positive set A′ ⊂ A. Consider the relation R ⊂ A′ × 2ω where 〈x, y〉 ∈ Rif ∃z ∈ B′ x E2 z ∧ f(z) = y. This is an analytic relation such that non-E2-equivalent points in A′ have disjoint vertical sections, and the union of verticalsections of points belonging to the same fixed E2 class is countable. These twoare Π1

1 on Σ11 properties of relations, and so by the first reflection theorem, R

can be enclosed in a Borel relation R′ with the same properties. The Novikovuniformization theorem for Borel sets with countable sections yields a Boreluniformization g : A′ → 2ω of R′. This is a Borel function under which distinctE2-classes have disjoint countable images. By a result of Kechris, [27, Lemma7.6.1], E2 � A′ must then be essentially countable.

Claim 5.6.7. If {Ai : i ∈ n} are I-positive analytic sets with dk(Ai, Ai+1) < εfor all i ∈ n− 1 and D ⊂ PI is an open dense set, then there are analytic sets{A′i : i ∈ n} such that A′i ⊂ Ai, A′i ∈ D, and dk(A′i, A

′i+1) < ε for all i ∈ n− 1.

Note that in the conclusion, the dk-distances between the various sets {A′i : i ∈n} must be smaller than (n− 1)ε.

Proof. By induction on i ∈ n find sets Aij ⊂ Aj : j ∈ n such that at stage i, the

dk-distances are still < ε and Aii ∈ D. The sets An−1j : j ∈ n will then certainly

work as desired. The induction is easy. Given the sets {Aij : j ∈ n}, strengthen

Aii+1 to a set B in the dense set D. Then, for every j ∈ n, write Ai+1j = {x ∈

2ω : ∃~y ∈ (2ω)n ∀l ∈ n − 1 dk(~y(l), ~y(l + 1)) < ε and ∀l ∈ n ~y(l) ∈ Ail and~y(i+1) ∈ B and ~y(j) = x}. These are all analytic sets with dk(Ai+1

j , Ai+1j+1) < ε,

Ai+1i+1 = B, and by the previous claim they must all be I-positive since the set

B is. The induction can proceed.

Claim 5.6.8. Suppose that {Ai : i ∈ n} are I-positive analytic sets with d-distance < ε, and 1 > δ > 0, γ > 0 are real numbers. Then there are I-positiveanalytic sets Abi : i ∈ n, b ∈ 2, binary sequences tbi ∈ 2<ω of the same length ksuch that

1. |d(t0i , t1i )− δ| < 2ε and d(t0i , t

0j ), d(t1i , t

1j ) are both smaller than 2ε;

2. Abi ⊂ Ai∩ [tbi ], and the dk-distances of the sets {Abi : i ∈ n, b ∈ 2} are < γ.

Proof. For each choice of k and ~t = 〈tbi : i ∈ n, b ∈ 2〉 as in the first item of theclaim let B~t ⊂ (2ω)n×2 be the analytic set of all tuples 〈xbi : i ∈ n, b ∈ 2〉 suchthat tbi ⊂ xbi , xbi ∈ Ai, and the dk-distances between points on the tuple are < γ.If for some ~t, the projection of B~t into the first coordinate is I-positive, thenthe projections into any coordinate are I-positive analytic sets by Claim 5.6.6,and they will work as {Abi : i ∈ n, b ∈ 2} required in the claim.


So it will be enough to derive a contradiction from the assumption that theprojections of the sets B~t into the first coordinate are always in the ideal I.These are countably many I-small analytic sets, so they can be enclosed in aBorel I-small set C ⊂ 2ω. The set A0 \ C is analytic and I-positive, and so itcontains an ε-walk whose endpoints have d-distance > 1, so it has to containtwo points x0

0, x10 with |d(x0

0, x10) − δ| < ε. Choose points xbi : i > 0, b ∈ 2 in

the sets Ai so that d(x00, x

0i ), d(x1

0, x1i ) are always smaller than ε, find a number

k such that the dk-distance of these points is pairwise smaller than γ, and let~t = 〈xbi � k : i ∈ n, b ∈ 2〉. This sequence should belong to the set B~t, but itsfirst coordinate should not belong to the projection of B~t. This is of course acontradiction.

This ends the preliminary part of the proof. For the properness of PI , let Mbe a countable elementary submodel of a large enough structure and let B ∈ PIbe a Borel I-positive set. We need to show that the set C = {x ∈ B : x isM -generic} is I-positive. To do that, we will produce a continuous injectiveembedding of the equivalence E2 into E2 � C. Fix positive real numbers εnwhose sum converges. Enumerate the open dense subsets of PI in the model Mby {Dn : n ∈ ω}. By induction on n ∈ ω build functions πn : 2n → 2<ω as wellas sets {At : t ∈ 2n} so that

I1 if m ∈ n then πn extends πm in the natural sense: for every t ∈ 2m andevery s ∈ 2n, πm(t) ⊂ πn(s). Also As ⊂ At, and As ∈ PI ∩M ∩Dn is anI-positive analytic set;

I2 all sequences in the range of πn have the same length kn ∈ ω, andπn approximately preserves the d-distance: |d(u, v) − d(πn(u), πn(v))| <Σm∈nεm;

I3 the sets {Au : u ∈ 2n} have pairwise dkn -distance smaller than εn/2.

The induction step is handled easily using the previous two claims. In theend, for every sequence y ∈ 2ω the sets {At : t ⊂ y} generate an M -genericfilter, so they do have a nonempty intersection which must contain a singlepoint π(y) =

⋃n πn(x � n). It is immediate from I2 that the map π : 2ω → C is

a continuous injective reduction of E2 to E2 � C and so C /∈ I as desired.For the bounding part, we must show that for every I-positive Borel set

B ⊂ 2ω and every Borel function f : B → 2ω there is a compact I-positivesubset of B on which the function f is continuous. This is obtained by arepetition of the previous argument. Just note that the image of the map πobtained there is compact, and if the open dense set Dn consists of sets B′ onwhich the bit f(x)(n) is constant, then f � rng(π) is continuous.

For the preservation of outer Lebesgue measure, it is enough to argue thatthe ideal I is polar in the sense of [67, Section 3.6.1]; i.e., it is an intersection ofnull ideals for some collection of Borel probability measures. In other words, weneed to show that every Borel I-positive set B ⊂ 2ω carries a Borel probability


measure that vanishes on the ideal I. This uses a simple claim of independentinterest:

Claim 5.6.9. The usual Borel probability measure µ on 2ω vanishes on the idealI, meaning that every set in I has µ-mass zero.

Proof. Argue by repeatedly applying Steinhaus theorem [3, Theorem 3.2.10] tothe compact Cantor group 〈2ω,+〉. Let B ⊂ 2ω be a Borel non-null set; we mustshow that B is not grainy; in other words, given ε > 0, we must find an ε-walkin B whose endpoints have distance at least 1. Fix ε > 0 and by induction oni ∈ ω build Borel non-null sets Bi ⊂ 2ω and finite sets ai ⊂ ω so that

I1 B = B0 ⊃ B1 ⊃ . . . ;

I2 the sets ai are pairwise disjoint and for every i ∈ ω it is the case thatε/2 <

∑n∈ai

1n+1 < ε;

I3 Bi+1∆χai ⊂ Bi where χai is the characteristic function of ai.

To perform the induction step, suppose that the set Bi+1 and ai do not existto satisfy the requirements in I1, I2 and I3. This means that for every finite setb ⊂ ω such that I2 is satisfied with b = ai, the set Cb = {x ∈ Bi : x∆χb ∈ Bi} isLebesgue null, and the set B′i = Bi \

⋃b Cb is Borel and Lebesgue positive. Now

use Steinhaus theorem to argue that B′i∆B′i contains an open neighborhood of

0, and so contains the characteristic function of some set b satisfying the seconditem. This is of course a contradiction.

In the end, choose j ∈ ω such that jε2 > 1 and a point x ∈ Bj+1. The

points x, x∆χaj , x∆χaj∆χaj−1. . . form the required ε-walk whose endnodes

have distance greater than 1.

If B ⊂ 2ω is a Borel I-positive set and π : 2ω → B is a continuous one-to-onereduction of E2 to E2 � B then the transported measure µ on B must vanishon the sets in the ideal I as well.

For the preservation of Baire category, it is enough to argue that every BorelI-positive set B ⊂ 2ω there is a Polish topology on B generating the same Borelstructure as the usual one, such that all sets in the ideal I are meager in thistopology. Then, a reference to Kuratowski-Ulam theorem and/or [67, Corollary3.5.8] will finish the argument. Just as in the treatment of Lebesgue measure,it is enough to show

Claim 5.6.10. The ideal I consists of meager sets.

The proof of the claim is the same as in the measure case, given that Stein-haus theorem holds for Baire category too.

Proof of Theorem 5.6.4. One can argue directly by a demanding fusion argu-ment via the separation property of the poset PI , which was our original wayof dealing with the challenge. There is a simple proof that uses the existence


of a forcing with E2 in the spectrum and total canonization for equivalencesclassifiable by countable structures.

Let B ⊂ 2ω be a Borel I-positive set and E an equivalence on B that isclassifiable by countable structures. Fix a Borel function f with domain Breducing E to isomorphism of countable structures. Fix a function g : 2ω → Breducing E2 to E2 � B. Fix a σ-ideal J on a compact space Y such that J hastotal canonization for equivalences classifiable by countable structures and E2

is in the spectrum of J as witnessed by a Borel equivalence F on Y , reduced toE2 by a Borel function h : Y → 2ω–see Theorem 5.14.4.

Consider the equivalence relation G on Y defined by y0 G y1 if the structuresfgh(y0) and fgh(y1) are isomorphic. Use the total canonization feature of theσ-ideal J to find a Borel J-positive set D ⊂ Y such that the function g ◦ h isone-to-one and the equivalence relation G � D is either equal to identity or toD2. The set C = gh′′D ⊂ B is a one-to-one image of a Borel set and thereforeBorel. It is also I-positive, since E2 Borel reduces to F � D which Borel reducesto E2 � C via g ◦ h. The equivalence relation E � C is either equal to identityor to C2 depending on the behavior of G � D. The theorem follows.

Proof of Theorem ??. The first sentence follows from the analysis of the idealI immediately. The Borel sets in I are exactly those sets B for which E2 � B isessentially countable, and so by the E2 dichotomy 5.6.1, E2 is Borel reducibleto E2 � B for every Borel I-positive set B.

For the regularity of PE2

I in PI , just notice that E2-saturations of analyticI-small sets are I-small by Claim 5.6.6, and then apply Corollary 3.5.5. For theℵ0-distributivity, first verify that E2 is I-dense and then apply Proposition 3.5.7.To see the I-density, note that the fusion process from the previous proofs startswith an arbitrary analytic I-positive set B and inscribes into it a compact I-positive set C ⊂ B such that the equivalence class of any point in C is dense inC.

To show that the poset PEI generates the V [xgen]E2model, first verify the

I-homogeneity of the equivalence relation E2 and then apply Theorem 3.5.9.For the I-homogeneity, just observe that if B,C ⊂ 2ω are analytic I-positivesets with the same E2-saturation, then one can thin B if necessary to get afunction f : B → C whose graph is a subset of E2. Then, Claim 5.6.6 showsthat the function f transports I-positive sets to I-positive sets, as required inthe definition of I-homogeneity.

Theorem 5.6.11. There is a coding function for the σ-ideal I.

Proof. The target space for the coding function will be R+. Let f(x, y) =Σ{1/(n+ 1) : x(n) 6= y(n)} if the indicated sum is finite, and let f(x, y) =trashotherwise. This function works.

For suppose that B ⊂ 2ω is a Borel I-positive set. A closer look at thefusion argument in Theorem 5.6.2 shows that it yields numbers {ni : i ∈ ω} anda function g : 2<ω → 2<ω such that

• 0 = n0 < n1 < n2 < . . . , for every i ∈ ω and every t ∈ 2i g(t) ∈ 2ni , andt ⊂ s ∈ 2<ω implies g(t) ⊂ g(s);

5.7. PRODUCTS OF PERFECT SETS 141

• for every i ∈ ω and every s, t ∈ 2i+1, write ε(s, t) = Σ{ 1/n+ 1 : ni ≤ n <

ni+1, g(s) 6= g(t)}. It is the case that |ε(s, t) − 1i | ≤ 2−i−2 whenever the

last entries on s, t differ, and ε(s, t) ≤ 2−i−2 if the last entries of s, t arethe same;

• for every x ∈ 2ω, g(x) =⋃i g(x � i) ∈ B.

Now suppose that r ∈ R is a positive real number. By induction on m ∈ ωbuild numbers im ∈ ω and binary sequences sm, tm ∈ 2im so that

I1 sm ⊂ sm+1, tm ⊂ tm+1;

I2 2−im−1 < r− d(g(sm), g(tm)) < 2−im , except for 0 where d(g(s0), g(t0)) <r − 2−i0 .

In the end, let x = limm g(sm) and y = limm g(tm); it immediately follows fromI2 that d(x, y) = r. The induction starts with choosing i0 so that r > 2−i0

and letting s0 = t0 be any binary sequence of length i0. Suppose that sm, tmhave been successfully found. Find infinite binary sequences z0, z1 extendingsm such that d(z0, z1) = r − 2−im−1 − d(g(sm), g(tm)). Let z2 be an infinitebinary sequence extending tm such that for every j > im, z2(m) = z1(m).It follows from the second property of the function g above that r − 2−im <d(g(z0), g(z2)) < r. Find im+1 so large that r − 2−im < d(g(z0 � im+1), g(z2 �im+1)) < r − 2−im+1 , let sm+1 = z0 � im+1, tm+1 = z2 � im+1, and continue tothe next step of the induction!

Corollary 5.6.12. If E2 is in the spectrum of a σ-ideal J then there is a codingfunction for J .

Proof. Let X be the Polish space on which J lives. Let B ⊂ X be a BorelJ-positive set such that there is an equivalence relation E on B Borel reduciblevia some function g : B → 2ω to E2, such that for every J-positive Borel setC ⊂ B, E2 ≤B E � C. Let h : (2ω)2 → 2ω be a coding function for the E2-idealas found in Theorem 5.6.11; we claim that h ◦ g is a coding function for J .Indeed, if C ⊂ B is a Borel J-positive set then g′′C ⊂ 2ω is an analytic set onwhich the equivalence relation E2 is not reducible to a countable one, thereforethe set g′′C is positive in the sense of the E2-ideal and the set h′′(g′′C)2 hasnonempty interior. The corollary follows.

5.7 Products of perfect sets

In this section, we will provide several canonization and anticanonization resultsfor infinite products of perfect sets on the underlying Polish space X = (2ω)ω.It is fairly clear that the equivalence relation E1 maintains its complexity onevery such infinite product. On the other hand, equivalence relations far awayfrom E1–such as the equivalence relations classifiable by countable structures–do


simplify all the way to smooth by passing to a suitable product of perfect sets.The whole situation is closely related to the forcing properties of the infiniteproduct of Sacks forcing.

The infinite product of Sacks forcing is the poset P of all ω-sequences p ofperfect binary trees, ordered by coordinatewise inclusion. The computation ofthe associated ideal yields a complete information. For a condition p ∈ P write[p] = {x ∈ X : for all n ∈ ω, x(n) is a branch of the tree p(n)}. Let I be thecollection of those Borel subsets of X = (2ω)ω which do not contain a subsetof the form [p] for some p ∈ P . The following Fact is a conjunction of therectangular property of Sacks forcing [67, Theorem 5.2.6] and [67, Proposition2.1.6].

Fact 5.7.1. I is a Π11 on Σ1

1 σ-ideal of Borel sets. Every positive analytic setcontains a Borel positive subset.

In particular, the map p 7→ [p] is an isomorphism between P and a dense subsetof the poset PI . The forcing properties of the poset P are well known.

Fact 5.7.2. The poset P is proper, bounding, preserves Baire category and outerLebesgue measure, and it adds no independent reals.

The spectrum of the ideal I is very complicated. Already the understandingof smooth equivalence relations (i.e. names for reals) seems to be out of reach.We will include several basic results. First, consider the obvious equivalence E1

on X.

Theorem 5.7.3. E1 is in the spectrum of I. Moreover, the poset PE1

I is regularin PI , it is ℵ0-distributive, and its extension is equal to V [xgen]E1

.

It can be shown that the reduced product PI ×E1 PI is proper, and it adds adominating real. The model V [[xgen]]E1

is equal to V [xgen]E1.

Proof. To see that E1 is indeed in the spectrum, if B ⊂ X is an I-positiveset, use Fact 5.7.1 to find a sequence p ∈ P of perfect trees such that [p] ⊂ Band let πn : 2ω → [p(n)] be a homeomorphism for every number n. Thenπ : x 7→ 〈πn~x(n) : n ∈ ω〉 is a Borel reduction of E1 to E1 � B.

To prove that PE1

I is a regular ℵ0-distributive subposet of PI , we will arguethat every set of the form [p] for p ∈ P is an E1-kernel and then apply Theo-rem 3.5.4. Suppose that B ⊂ [p] is an analytic set such that [B]E1 /∈ I; we mustconclude that [B]E1 ∩ [p] /∈ I. Just find a condition q ∈ P so that [q] ⊂ [B]E1 ,and use the σ-additivity of the σ-ideal I to thin out the trees on the sequence q ifnecessary and find a number n such that ∀x ∈ [q] ∃y ∈ B x � (n, ω) = y � (n, ω).But then, for every number i > n it must be the case that q(i) ⊂ p(i). Thus,the set [B]E1 ∩ [p] contains the product Πi≤np(i)×Πi>nq(i), and it is I-positive.

To show that the poset PE1

I yields the model V [xgen]E1, we must show that

for analytic I-positive sets B,C ⊂ X, either there is I-positive Borel B′ ⊂ Bsuch that [B′]E1

∩ C ∈ I, or there is an I-positive Borel B′ ⊂ B and an I-preserving Borel injection f : B′ → C with f ⊂ E1. After that, Theorem 3.5.9completes the proof.


Thus, fix the sets B,C and find a condition p ∈ P with [p] ⊂ B. If [p]E1∩C ∈I, then we are done, otherwise find a condition q ∈ P with [q] ⊂ [p]E1

∩C. Theremust be a number n0 ∈ ω such that ∀n ≥ n0 q(n) ⊂ p(n). Consider the setB′ = Πn∈n0

[p(n)] × Πn≥n0[q(n)], fix homeomorphisms πn : [p(n)] → [q(n)] for

every n ∈ n0, and consider the map f : B′ → C defined by f(x)(n) = πn(x)(n)if n ∈ n0 and f(x)(n) = x(n) for n ≥ n0. It is not difficult to see that thefunction f is a Borel injection with the required properties. As an interestingaside, note that in V [G], [xgen]E1

is the only equivalence class contained in allsets in the filter K and so K and [xgen]E1

are interdefinable elements. This is tosay, in the ground model, if B ⊂ X is an I-positive Borel set and f : B → X isa Borel function such that ∀x ∈ B ¬f(x) E1 x, then there is an I-positive Borelset C ⊂ B such that [C]E1 ∩f ′′C = 0. Since the poset PI is bounding, there areincreasing functions g, h ∈ ωω and an I-positive Borel set C ⊂ B such that forevery point x ∈ C and every number i ∈ ω there are numbers n ∈ [g(i), g(i+1))and k ∈ h(i) such that f(x)(n)(k) 6= x(n)(k). Thinning the set C further ifnecessary we may assume that C = [p] for some condition p ∈ P such thatwhenever n ∈ [g(i), g(i+ 1)) then the first splitnode of the tree p(n) is past thelevel h(i). It is not difficult to see that such a set C ⊂ B works as desired.

Theorem 5.7.4. The spectrum of I is cofinal among the Borel equivalencerelations under ≤B. It contains EKσ .

Proof. Let J be a Borel ideal on ω, then the equivalence relation EJ , the moduloJ equality of sequences in (2ω)ω is a Borel equivalence, it is in the spectrum ofthe ideal I for the same reason as E1, and the equalities of this type are cofinalin ≤B by Fact 2.2.12. There is an Fσ ideal J such that equality modulo J isbireducible with EKσ , and so EKσ is in the spectrum of I as well.

The equivalences from the previous proof are never reducible to orbit equiv-alences of Polish group actions, which leaves open the possibility of a strongcanonization theorem for I and orbit equivalences. The following two canoniza-tion theorems should be viewed as important stepstones to that so far unattainedgoal.

First, set up a fixed framework for fusion arguments. Let p ∈ P . A fusionsequence below p is a sequence 〈un, pn : n ∈ ω〉 such that un is a sequence oflength n whose entries are finite binary trees with n splitting levels, pn is acondition in P below p, and for every m ∈ ω the tree pn(m) end-extends un(m).Moreover, we require that pn+1 ≤ pn and for every m ∈ n, the tree un+1(m)end-extends the tree un(m). It is clear that the conditions {pn : n ∈ ω} in afusion sequence share a lower bound, the condition q ∈ P such that for everym ∈ ω, q(m) =

⋃m<n un(m). In this context, we will write [un] for the set of

all sequences t of length n such that t(m) is an endnode of un(m). For everyt ∈ [un] the condition pn � t is defined as that sequence r such that for all m ∈ n,r(m) =the set of all nodes of pn(m) compatible with t(m), and for all m ≥ n,r(m) = pn(m). For every x ∈ [pn] let x � un be the unique element of un] suchthat x ∈ pn � t.


Theorem 5.7.5. Equivalences classifiable by countable structures→Ismooth.In other words, for every equivalence relation on X classifiable by countablestructures there is an infinite product of perfect sets on which E is smooth.

Proof. We will start with a simple claim.

Claim 5.7.6. Let p ∈ P be a condition and {f, gm : m ∈ ω} be Borel functionsfrom [p] to 2ω. At least one of the following two options holds:

1. either there is a condition q ≤ p such that for every x, y ∈ [q], ∀n >0 x(n) = y(n) implies f(x) = f(y);

2. or there is a condition q ≤ p such that for every x, y ∈ [q], x(0) 6= y(0)implies f(x) 6= gm(y) for every m ∈ ω.

Proof. Thinning the condition p if necessary, we may assume that the functionsf, g are continuous. Suppose that the first item fails. Build a fusion sequence〈un, pn : n ∈ ω〉 below p such that for every n ∈ ω and every t, s ∈ [un], ift(0) 6= s(0) then f ′′[pn � t] ∩ g′′m[pn � s] = 0 for all m ∈ ω. It is clear that in theend, the lower bound q of the fusion sequence will be the condition required inthe second item.

To perform the induction step, find any sequence un+1 of finite trees belowpn that can serve as the next entry of the fusion sequence. We must show howto thin out pn to pn+1 and preserve the induction hypothesis.

Suppose that t, s ∈ [un+1] are two sequences such that ~t(0) 6= ~s(0) andchoose a number m ∈ n. Use the assumption to find points x0, x1 ∈ [pn � ~t]which differ at the first coordinate only, such that f(x0) 6= f(x1); perhapsf(x0)(i) = 0 while f(x1)(i) = 1 for some i ∈ ω. Use the continuity of thefunction f to find finite sequences t0, t1 of the same length that differ only at0-th coordinate, x0 ∈ [pn � t0], every x ∈ [pn � t0] has f(x)(i) = 0, and similarlyfor subscript 1. Consider the sequence s which is defined by s(j) = s(j) ift(j) 6= s(j) and s(j) = t(j) otherwise; so s ≥ s. The continuity of the function gon [pn] shows that strengthening s if necessary, it must be the case that eithergm(x)(i) = 0 for all y ∈ [pn � s], or gm(x)(i) = 1 for all y ∈ [pn � s]. Suppose theformer case holds. Thinning out pn below s so that pn � s = pn � s and below tso that pn � t ≤ pn � t1 (which is possible to do simultaneously) we achieve thatf ′′[pn � t] ∩ g′′m[pn � s] = 0.

The condition pn+1 ≤ pn is now obtained by repeating the thinning pro-cedure in the previous paragraph for all pairs of sequences t, s ∈ [un+1] whichdiffer at the first coordinate and all numbers m ∈ n. The induction hypothesiscontinues to hold.

The canonization of equivalences classifiable by countable structures is achievedvia the separation property of Definition 4.3.6 and Theorem 4.3.7. To prove theseparation property, suppose that p ∈ P is a condition forcing z ∈ 2ω anda ⊂ 2ω is a countable set not containing z. We must produce a condition q ≤ pand a compact set C ⊂ 2ω such that q z ∈ C and a ∩ C = 0. Thinning outthe condition p if necessary, we can find continuous functions f, gn : [p] → 2ω


for n ∈ ω such that p z = f(xgen) and a = {gn(xgen) : n ∈ ω}. It will beenough to find a condition q ≤ p such that f ′′[q] is disjoint from

⋃n g′′n[q]; then,

q together with the set C = f ′′[q] will have the required properties.To find the condition q, build a fusion sequence 〈un, pn : n ∈ ω〉 below p

such that

I1 for every t ∈ [un], if there is a condition r ≤ pn+1 such that r � t 6= 0 andthe value of f(x) does not depend on x(n) for x ∈ [r � t], then pn+1 issuch a condition;

I2 for every t ∈ [un], if there is a condition r ≤ pn+1 such that ∀s ∈ [un] r �s 6= 0 and for all x ∈ [r � t] and every y ∈ [r], x(n) 6= y(n) impliesf(x) 6= fm(y) for all m ∈ ω, then pn+1 is such a condition.

The induction process is trivial. Let q ∈ P be a lower bound of the fusionsequence, and let x, y ∈ [q]. We must prove that f(x) 6= gm(y) for all m ∈ ω.There are two distinct cases.

Case 1. There is no number n ∈ ω such that, writing t = x � un, it isthe case that x(n) 6= y(n) and the assumption of item (I1) fails for t and n. Inthis case, write zn ∈ [q] for the sequence obtained from y by replacing its firstn entries with the corresponding entries of x and argue that the value f(zn)does not depend on n. To show that f(zn) = f(zn+1), discern two subcases. Ifx(n) = y(n) then zn = zn+1 and f(zn) = f(zn+1) certainly holds. On the otherhand, if x(n) 6= y(n), the case assumption shows that the assumption of item(I1) holds and implies exactly that f(zn) = f(zn+1). Now note that z0 = y andlimn zn = x. Since the function f is continuous, it follows that f(y) = f(x) andf(x) = f(y) 6= gm(y) for all m ∈ ω by the initial assumptions on the functionsf, gm : m ∈ ω.

Case 2. If Case 1 fails then there must be the witnessing number n. Writet = x � un. Since the assumption of item (I1) fails for t, Claim 5.7.6 showsthat the assumption of item (I2) holds. Since x(n) 6= y(n), we conclude thatf(x) 6= gm(y) for all m ∈ ω as required. The proof is complete.

Theorem 5.7.7. Whenever J is an analytic P-ideal on ω, then ≤B=J→Ismooth.In other words, for every equivalence relation E on X Borel reducible to =J thereis an infinite product of perfect sets such that E is smooth on it.

Proof. Use a result of Solecki [54] to find a lower semicontinuous measure φ onω such that J = {a ⊂ ω : limn φ(a \ n) = 0}. The proof of the following claimis parallel to Claim 5.7.6:

Claim 5.7.8. Let p ∈ P , let f, g : [p]→ 2ω be Borel functions and let a ⊂ ω bea finite set. At least one of the following happens:

1. either either for every ε > 0 there is a condition q ≤ p such that for allx, y ∈ [q] with ∀n /∈ a x(n) = y(n) it is the case that φ(f(x)∆f(y) < ε;

2. or there is a condition q ≤ p and ε > 0 such that for every pair x, y ∈ [q]such that ∀n ∈ a x(n) 6= y(n) it is the case that ¬f(x) =J f(y).


Let p ∈ P and let f : [p] → 2ω be a Borel function reducing E � [p] to=J . Thinning out the condition p we may assume that the function f is infact continuous. We will construct a fusion sequence 〈un, pn : n ∈ ω〉 below psubject to a fairly nasty induction hypothesis. The aim of the construction is tofind equivalence relations Fn on [un] for every n ∈ ω such that E restricted tothe lower bound q of the fusion sequence is Borel reducible to the equivalencerelation F =

∏n Fn modulo finite on the set

∏n[un]. The equivalence relation

is a modulo finite product of equivalence relations on finite sets, and as such itis Borel reducible to E0. The previous theorem can then be applied to showthat thinning out q further if necessary, E � q is smooth. The reduction ofE � q to F will be attained through the natural continuous one-to-one functiong : [q]→

∏n[un] defined by g(x)(n) = x � un

It is time to expose the induction hypothesis on the fusion sequence. Forevery n ∈ ω there will be a number kn > n and a real number εn > 0 so that

I1 the numbers kn are increasing, the numbers εn decrease towards 0 withεn+1 < εn/8;

I2 for every pair s, t ∈ [un], writing a = {m ∈ n : t(m) = s(m)}, either (a)for all x ∈ pn � t and y ∈ pn � s such that x � a ∪ [n, ω) = y � a ∪ [n, ω)x E y fails, or (b) for all such x, y it is the case that x E y holds;

I3 for every pair s, t ∈ [un] such that clause (I2)(a) holds, for all x, y as abovelim supk φ(f(x)∆f(y) \ k) > εn holds;

I4 for every pair s, t ∈ [un] such that clause (I2)(b) holds, for all such x, yφ(f(x)∆f(y) \ kn) < εn/8 holds;

I5 for all t ∈ [un] and all a ⊂ n, if there is a condition r ≤ pn � t such that forevery x, y ∈ [r], x � a∪ [n, ω) = y � a∪ [n, ω) implies φ(f(x)∆f(y)) < εn/8then pn � t is such a condition;

I6 for all t ∈ [un] and all a ⊂ n, if there is a condition r ≤ pn such thatfor all s ∈ [un], r � s 6= 0 and such that for every x ∈ [r � t], y ∈ [r],∀m ∈ n \ a x(m) 6= y(m) implies ¬x E y, then pn � t is such a condition;

I7 for all t ∈ [un] and all a ⊂ n, if there is a condition r ≤ pn+1 � t suchthat for every x, y ∈ [r], x � a ∪ [n + 1, ω) = y � a ∪ [n + 1, ω) impliesφ(f(x)∆f(y)) < εn/8 then pn+1 � t is such a condition;

I8 for all t ∈ [un] and all a ⊂ n, if there is a condition r ≤ pn+1 such thatfor all s ∈ [un], r � s 6= 0 and such that for every x ∈ [r � t], y ∈ [r],∀m ∈ (n + 1) \ a x(m) 6= y(m) implies ¬x E y, then pn+1 � t is such acondition.

While the list of induction hypotheses may sound daunting, it is in fact veryeasy to get. Suppose that un, pn, kn, εn have been obtained. First, strengthenpn to pn+1 ≤ pn satisfying the last two items. Let un+1 to be the sequence ofthe finite trees recording pn+1(m) up to n+ 1-st splitting level, for m ∈ n+ 1.


Now run through the list of all pairs s, t ∈ [un+1], using the σ-additivity ofthe σ-ideal I each time to gradually strengthen pn+1 so as to satisfy item (I2).Run through the list again, gradually thinning out pn+1 again and using theσ-additivity of the σ-ideal to find real numbers εn+1(s, t) > 0 such that ifI2(a) holds for s, t then item (I3) is satisfied with εn+1(s, t) in place of εn. Letεn+1 > 0 be some real number smaller than all the numbers thus obtained. Runthrough the list again, thinning out pn+1 and using σ-additivity of the ideal Iagain to find numbers kn+1(s, t) such that if the claus (I2)(b) holds for s, t then(I4) is satisfied with kn+1(s, t) in place of kn. Let kn+1 be some number largerthan all numbers thus obtained. Finally, run through the list of all t ∈ [un+1]and a ⊂ n+ 1, gradually strengthening the condition pn+1 again so as to satisfy(I5) and (I6). This completes the induction step.

Now suppose that the fusion has been performed. For each n ∈ ω, let Fn bethe relation on [un] defined by s Fn t if (I2)(b) is satisfied for s, t. A brief checkis in order:

Claim 5.7.9. Fn is an equivalence relation.

Proof. The only nontrivial point is the transitivity. Let t0 Fn t1 Fn t2; we wantto show that t0 Fn t2. Write a01 = {m ∈ n : t0(m) = t1(m) and similarly fora02 and a12. Choose points x0 ∈ [pn � t0], x1 ∈ [pn � t1] and x2 ∈ [pn � t2] suchthat x0 � a01 ∪ [n, ω) = x1 � a01 ∪ [n, ω) and x1 � a12 ∪ [n, ω) = x2 � a12 ∪ [n, ω).By the assumptions, we have x0 E x1 E x2. Now amend x0 on a02 \ a01 sothat x0 = x2 on this set; the assumptions of case I2(a) are still satisfied andshow that x0 E x1. Similarly, amend x2 on a02 \ a12 so that x0 = x2 on thisset; it will be still the case that x1 E x2. In conclusion, x0 E x2, and sincex0 � a02 ∪ [n, ω) = x2 � a02 ∪ [n, ω), the pair x0, x2 contradicts I2(a) for t0, t2.It follows that I2(b) for t0, t2 must occur, in other words x0 Fn x1 holds.

Let q be a lower bound of the fusion sequence and suppose x, y ∈ [q]; wemust show that x E y if and only if g(x) F g(y). There are several cases:

Case 1. There is a number n such that, writing t = g(x)(n) and a = {m ∈n : x(m) = y(m)}, the assumption of case (I5) of the induction hypothesis failsfor t, a. In this case, we will conclude that both x E y and x F y fail. Indeed,Claim 5.7.8 shows that then the assumption of (I6) is satisfied and thereforex E y must fail. Also, for every sufficiently large number n ∈ ω, writinga = {m ∈ n : g(x)(n)(m) = g(y)(n)(m)}, it will be the case that (n \ a)∩ a = 0,and so the clause (I2)(a) is satisfied for t = g(x)(n) and s = g(y)(n) by theconclusion of (I6) at n, t, a. Thus, both g(x)(n) Fn g(y)(n) and x F y must fail.

Case 2. Same as case 1, except about item (I7) of the induction hypothesis.The argument and the conclusion are the same as in Case 1.

Case 3. If both Case 1 and Case 2 fail, then for every n let zn ∈ [q]be the sequence obtained by replacing the first n entries of x with the first nentries of y, and observe that φ(f(zn)∆f(zn+1)) < εn/2. To see this, discerntwo cases: if x(n) = y(n) then zn = zn+1 and the inequality certainly holds.If x(n) 6= y(n) then, since Case 2 failed at n, it must be that item (I7) of theinduction hypothesis occurred for g(y)(n) and the inequality again follows. Note


that since limn zn = y, the function f is continuous, and the submeasure φ islower semicontinuous, it follows that φ(f(zn)∆f(y)) < Σm≥nεm/2 < εn.

We will now show that if g(x)(n) Fn g(y)(n) for infinitely many n, thenx E y, and if ¬g(x)(n) Fn g(y)(n) for some n then ¬x E y. This will prove thereduction property of the function g on the pair x, y.

Case 3a. Suppose first that n is such that t = g(x)(n) Fn g(y)(n) = s.Writing a = {m ∈ n : x(m) = y(m)} and b = {m ∈ n : g(x)(m) = g(y)(m)}, itis obvious that a ⊆ b. Write z′n for the sequence obtained from x by replacing theentries in b\a with the corresponding entries of y. Since Case 1 fails, item (I5) ofthe induction hypothesis must hold for t, a, and we have φ(f(x)∆f(z′n)) < εn/4.Since t Fn s, clause (I2)(b) shows that φ((f(z′n)∆f(zn)) \ kn) < εn/4. Movingfrom f(x) to f(z′n) to f(zn) to f(y), we have that φ((f(x)∆f(y)) \ kn < εn/4 +εn/4 + εn/2 = εn. Now, if there are infinitely many numbers n such thatg(x)(n) Fn g(y)(n), we conclude that f(x) =J f(y) and x E y.

Case 3b. Suppose second that n is such that t = g(x)(n) Fn g(y) = s fails.As in the previous subcase, write a = {m ∈ n : x(m) = y(m)} and b = {m ∈ n :t(m) = s(m)}, and let z′n ∈ [q] be the sequence obtained from x by replacing itsentries in the set b \ a with the corresponding entries of y. Since ¬t Fn s holds,item (I3) shows that lim supk φ((f(z′n)∆f(zn)\k) > εn. However, just as in theprevious subcase, we have φ(f(x)∆f(z′n)) < εn/4 and φ(f(y)∆f(zn)) < εn/4 aswell. As φ is a submeasure, we conclude that lim supk φ((f(x)∆f(y)\k) > εn/2,in particular ¬x E y.

The results of this section obviously suggest a positive answer to the followingquestion, which nevertheless remains open.

Question 5.7.10. Is it true that for every equivalence relation E on (2ω)ω,reducible to the orbit equivalence of a Polish group action, there are perfectsets Pn ⊂ 2ω for each n ∈ ω such that E restricted to ΠnPn is smooth?

5.8 Silver cubes

In this section, we will consider the behavior of equivalence relations on Silvercubes. Recall that Silver forcing is the poset P whose elements p are just partialfunctions from ω to 2 with coinfinite domain. The ordering is that of reverseinclusion. For p ∈ P , write [p] for the set of all totalizations of the function p;thus [p] ⊂ 2ω is a compact set. A Silver cube is a set of the form [p] for somep ∈ P .

The calculation of the associated σ-ideal gives complete information. LetX = 2ω and let G be the Borel graph on X given by xGy if and only if thereis exactly one n such that x(n) 6= y(n). A set B ⊂ X is G-independent if notwo elements of B are G-connected. Let I be the σ-ideal generated by BorelG-independent sets. We have

Theorem 5.8.1. 1. Whenever A ⊂ X is an analytic set, either A ∈ I or Acontains a Silver cube, and these two options are mutually exclusive;

5.8. SILVER CUBES 149

2. the ideal I is Π11 on Σ1

1;

3. the ideal I does not have the selection property;

4. the forcing P is proper, bounding, preserves outer Lebesgue measure andBaire category, and adds an independent real.

Proof. (1) comes from [66, Lemma 2.3.37]. (2) follows from (1) and [67, Theorem3.8.9], or one can compute directly. (4) is well-known; the independent real a ⊂ω is read off the generic real xgen ∈ 2ω as a = {n ∈ ω : {m ∈ n : xgen(m) = 1}has even size}. Finally, the failure of the selection property is witnessed by theset D ⊂ 2ω ×X given by 〈y, x〉 ∈ D if x(3n) = x(3n+ 1) = y(n).

The theorem shows that the map p 7→ [p] is an isomorphism between P and adense subset of PI . The forcing properties of Silver forcing are well-known.

In the following proofs, we will repeatedly use fusion type arguments, andit will be useful to establish a fixed framework for them. A fusion sequencebelow a condition p ∈ P is a sequence 〈in, pn〉 such that p = p0 ≥ p1 ≥ . . .and 〈im : m ∈ n〉 is an increasing enumeration of the first n many points ofω \ dom(pn+1). It is clear that the conditions in a fusion sequence have a lowerbound, the condition q =

⋃n pn.

The canonization features of Silver cubes start with the total canonizationof smooth equivalence relations. This has been proved in a different languageby Grigorieff [15, Corollary 5.5] in a rather indirect fashion. We provide a directargument containing features useful later.

Theorem 5.8.2. The ideal I has total canonization for smooth equivalences.Restated, every smooth equivalence on a Silver cube is equal to either the identityor the full equivalence relation on a Silver subcube.

In other words, Silver forcing adds a minimal real degree.

Proof. The argument uses a claim of independent interest. Recall that if x ∈ 2ω

is an infinite binary sequence and m ∈ ω is a number, then x flip m is thebinary sequence obtained from x by flipping its m-th bit.

Claim 5.8.3. Whenever p ∈ P and {fj : j ∈ k} is a finite collection of contin-uous functions from p to 2ω such that preimages of singletons are I-small, thenthere is an arbitrarily large number m ∈ ω and a cube q ≤ p such that for everyx ∈ [q], both x, x flip m ∈ [p] and fi(x) 6= fi(x flip m) for all i ∈ k.

Proof. This is proved by induction on k. The case k = 0 is trivial, since thecollection of continuous functions in question is empty. Suppose that the claimhas been established for some k and fails for some {fj : j ∈ k + 1} and p ∈ P .We will reach a contradiction by producing a cube q ≤ p such that fk is constanton [q].

Build a fusion sequence 〈in, pn〉 below p so that for every n ∈ ω, the valueof fk(x) does not depend on x � an+1 for points x ∈ [pn+1], where an+1 = {il :


l ≤ n}. The continuity of the function fk implies that fk � [q] must then beconstant for any lower bound q of the fusion sequence.

The fusion sequence is built by induction on n. Suppose that pn and in−1

have been found. By the induction hypothesis at k, there is a subcube p′n ≤pn and a number in > in−1 such that for every x ∈ [p′n] it is the case thatx flip in ∈ [pn] and for all j ∈ k it is the case that fj(in) 6= fj(in+1). The setA = {x ∈ [p′n] : fk(x) 6= fk(x flip in+1)} is relatively open in [p′n]. If this setis nonempty, then it contains a Silver cube; however, this cannot happen sincethe claim is supposed to fail for {fj : j ∈ k + 1} and p. Ergo, A = 0. Now letpn+1 ∈ P be the function obtained from p′n by omitting the numbers i0, . . . infrom the domain of p′n.

We claim that for x ∈ [pn+1], the value of fk(x) does not depend on x � an.To see this, argue that fk(x) = fk(y) for every y ∈ [p′n] such that x agrees withy everywhere off the set an+1. Indeed, first amend y to y′ by flipping the entryy(in) if necessary to get an agreement with x(in) if necessary. By the definitions,we still have f(y′) = f(y) and since x, y′ ∈ [pn] and x, y′ agree everywhere off an,the induction hypothesis on the fusion sequence shows that f(x) = f(y′) = f(y)as desired. This completes the induction step in the construction of the fusionsequence and the proof.

Now suppose that B ⊂ X is an I-positive Borel set and f : B → 2ω isa Borel function with I-small preimages of singletons; we must produce an I-positive Borel subset of B on which f is one-to-one. Use the bounding propertyof Silver forcing and Fact 2.6.3 to get a Silver cube p ∈ P such that [p] ⊂ Band f � [p] is continuous. Build a fusion sequence 〈in, pn〉 below pn such that,writing an = {il : l ∈ n}, every two points x, y ∈ pn+1 that differ on the setan also have distinct f -images. It is clear that for any lower bound q of such afusion sequence, the function f � [q] is injective as required.

To perform the inductive step of the construction, suppose that in−1, pn havebeen found. For every h ∈ 2an let fh : [pn] → 2ω be the function defined byfh(x) = f(x rew h) where xh ∈ pn is the point obtained from x by rewriting theentries of x � an with h. These are continuous functions with I-small preimagesof singletons. By the claim, there is a cube p′n and a number in > in−1 suchthat fh(x) 6= fh(x flip in) for all h ∈ 2an and all x ∈ [p′n]. By the continuityof the functions fh, passing to a clopen subcube of p′n if necessary we can findnumbers mh ∈ ω and bits bh 6= ch such that for all x ∈ p′n, fh(x)(mh) = bh andfh(x flip in)(mh) = ch. Now let pn+1 be the subcube of pn obtained from p′nby omitting the numbers i0, . . . in from its domain.

To verify the properties of the cube pn+1, suppose that x, y ∈ [pn+1] arepoints that differ on the set an+1; we must show that they have different func-tional values. If they differ already on the set an then we are done by theinduction hypothesis. If x � an = y � an = h for some h ∈ 2an , then it must bethe case that x(in) 6= y(in). Find points x′, y′ ∈ [p′n] such that x = x′ rew h andy = y′ rew h and observe that f(x)(mh) = fh(x′)(mh) 6= fh(x′ flip in)(mh) =fh(y′)(mh) = f(y)(mh). This completes the induction step and the proof.


At the level of hyperfinite equivalence relations we hit the first nontrivialfeature of the spectrum of Silver forcing:

Theorem 5.8.4. E0 belongs to the spectrum of the Silver forcing. The posetPE0

I is regularly embedded in PI , it is ℵ0-distributive, and it yields the modelV [xgen]E0

of Definition 3.4.1.

The decomposition of Silver forcing into PE0

I and the remainder forcingobtained from this theorem and Theorem 4.2.1 is different from the widely usedGrigorieff decomposition of [15].

Proof. Look at the equivalence relation E0 on X. Whenever p ∈ P is a Silvercube and π : ω → ω \ dom(p) is a bijection then the map g : X → [p] defined byg(x) = f ∪ (x ◦ π−1) is a continuous reduction of E0 to E0 � [p], and thereforeE0 is in the spectrum. The remainder of the theorem is a consequence ofTheorem 4.2.1 as soon as we prove that E0-saturations of sets in I are still inI. The easiest way to see that is to consider the action of the countable groupof finite subsets of ω with symmetric difference on the space X where a finiteset acts on an infinite binary sequence by flipping all entries of the sequence onthe set. This action generates E0 as its orbit equivalence relation, and it alsopreserves the graph G and with it the σ-ideal I. Thus, E0-saturations of I-smallsets are still I-small.

As an interesting extra bit of information, the equivalence class [xgen]E0is

the only one in the extension V [H] which is a subset of all sets in the filter K,where H ⊂ PI is a generic filter and K = H ∩ PE0

I . In the ground model, thisis immediately translated to the following: If p ∈ P and f : [p]→ X is a Borelfunction such that ∀x ∈ [p] ¬f(x) E0 x, then there is a subcube q ≤ p such thatf ′′[q] ∩ [q]E0

= 0.Since the forcing PI is bounding, there must be a partition of ω into a

collection {In : n ∈ ω} of intervals and a cube q ≤ p such that for every x ∈ Cand every n ∈ ω there is a number m ∈ In with f(x)(m) 6= x(m), and thefunction f � [q] is continuous. Thin out the cube q if necessary so that infinitelymany of the intervals do not intersect the set ω \ dom(q). The cube [q] worksas desired.

There are many nonsmooth subequivalences of E0 that do not canonize toE0 or the identity on any Silver cube. This is in contradistinction to the featuresof the smooth ideal or the weak Milliken cubes. The simplest such an exampleis the equivalence relation E connecting x, y ∈ X if x, y differ on a finite setof entries, and the set is of even size. There seems to be such a variety ofsubequivalences of E0 that no nontrivial canonization theorem on Silver cubesis possible for them. However, this is the only obstacle for canonization in atleast two directions:

Theorem 5.8.5. If E is an equivalence on a Silver cube classifiable by countablestructures, then there is a Silver subcube on which E ⊆ E0 or E equals to thefull equivalence relation.


Proof. Suppose that B ⊂ 2ω is a Borel I-positive set and E is an equivalenceon it classifiable by countable structures. Since the Silver forcing extensioncontains a single minimal real degree, Theorem 4.3.8 implies that E simplifiesto an essentially countable equivalence relation on a positive Borel set. Sincethe Silver forcing extension preserves Baire category, Theorem 4.2.7 shows thatE restricted to a further I-positive subset is reducible to E0. Now we can dealwith this fairly simple situation separately.

Suppose that p ∈ P and E is an equivalence relation on [p] reducible to E0

via a Borel function f : [p] → 2ω. Passing to a subcube if necessary, we mayassume that f is continuous on [p]. Suppose that E has no I-positive equivalenceclasses. We must produce a subcube q ≤ p such that E ⊆ E0 on [q].

By induction on n ∈ ω, build a fusion sequence 〈in, pn : n ∈ ω〉 so that forevery x, y ∈ [pn+1] such that x(in) 6= y(in) there is a number m > n such thatf(x)(m) 6= f(y)(m). It immediately follows that if q is a lower bound of such afusion sequence, then E � [q] ⊆ E0. Suppose that pn, in−1 have been successfullyfound. Write an = {il : l ∈ n} and for every h ∈ 2a, let fh(x) = f(x rew h) asin the previous proof. By the σ-additivity of the σ-ideal I, it is possible to finda condition p′n ≤ pn such that

• for every h, k ∈ 2a, either (a) for every x ∈ [p′n], fh(x) E0 fk(x) holds or(b) for every x ∈ [p′n], fh(x) E0 fk(x) fails;

• there is a number m0 > n such that if h, k are such that (a) above holdsthen fh(x) = fk(x) on all entries above m0 for all x ∈ [p′n]; and for everyh ∈ 2an , for all x ∈ [p′n] the value of fh(x) � m0 is the same;

• if h, k are such that (b) above holds then there is a number mhk > n anddistinct bits bhk 6= chk such that for all x ∈ [p′n], fh(x)(mhk) = bhk andfk(x)(mhk) = chk.

Now let p′′n ≤ p′n be a condition such that for some number in > in−1,for every x ∈ [p′′n] and every h ∈ 2an , it is the case that x flip in ∈ [p′n]and fh(x) 6= fh(x flip in). Such a subcube exists by Claim 5.8.3. Use thecontinuity of the function f to pass to a relatively clopen subcube if necessaryand find numbers mh and distinct bits bh 6= ch such that for every x ∈ [p′′n] andevery h ∈ 2a it is the case that fh(x)(mh) = bh and fh(x flip in)(mh) = ch.Let pn+1 be the condition obtained from p′′n by removing the points {il : l ≤ n}from its domain. We claim that pn+1 works as required.

For this, let x, y ∈ pn+1 be points such that x(in) 6= y(in); we must showthat f(x) and f(y) differ above n. Let h = x � an and k = y � an, and findpoints x′, y′ ∈ [p′′n] such that x = x′ rew h and y = y′ rew k; so f(x) = fh(x′)and f(y) = fk(y′). There are two distinct cases according to whether (a) or(b) above holds for h, k. If (b) holds then clearly fh(x′)(mhk) 6= fk(y′)(mhk)and the conclusion follows. If, on the other hand, the clause (a) holds, thenfh(x′)(mh) 6= fh(x′ flip in)(mh) = fk(x′ flip in)(mh) = fk(y′)(mh) by thechoice of the number mh. This completes the proof.

Regarding further canonization properties of Silver cubes, Michal Doucha proved


Fact 5.8.6. [10] If E is an equivalence relation on a Silver cube, Borel reducibleto =J for some analytic P-ideal J , then there is a Silver subcube on whichE ⊆ E0 or E equals to the full equivalence relation.

Finally, there is a lack of canonization in the direction of general Kσ equivalencerelations, as witnessed by the following

Theorem 5.8.7. EKσ is in the spectrum of I. It is witnessed by a certainBorel equivalence relation E on X such that the poset PEI is regular in PI , it isℵ0-distributive and it yields the V [xgen]E extension.

Proof. Let c : 2ω → ωω be the continuous function defined by c(x)(n) = |{m ∈n : x(m) = 1}| and let E be the Borel equivalence relation on 2ω connectingx0, x1 ∈ 2ω if the function |c(x0)−c(x1)| ∈ ωω is bounded. Clearly, the functionh reduces the equivalence relation E to EKσ . To show that EKσ ≤B E, choosesuccessive finite intervals In ⊂ ω of length 2n+ 1 and for every function y ∈ ωωbelow the identity consider the binary sequence g(y) ∈ 2ω specified by thedemand that g(y) � In begins with y(n) many 1’s, ends with n− y(n) many 1’sand has zeroes in all other positions. It is not difficult to see that if y0, y1 ∈ ωωbelow the identity are two functions then |c(g(y0)) − c(g(y1)) � In| is boundedby |y0(n) − y1(n)| and the bound is actually attained at the midpoint of theinterval In. Thus, the function g reduces EKσ to E. To show that E ≤ E � [p]for every p ∈ P let π : ω → ω \ dom(p) be the increasing bijection, and observethat the map π : 2ω → [p] defined by π(x) = p ∪ (x ◦ π−1) reduces E to E � [p].Thus, the equivalence relation E shows that EKσ is in the spectrum.

The following definition will be helpful for the investigation of the poset PEI .Say that conditions p, q are n-dovetailed if between any two successive elementsof ω \dom(p) there is exactly one element of ω \dom(q) and min(ω \dom(p)) <min(ω \ dom(q)), and moreover, for every number m ∈ ω, the difference |{k ∈m : p(k) = 1}| − |{k ∈ m : q(k) = 1| is bounded in absolute value by n. Twoconditions are dovetailed if they are n-dovetailed for some n. It is clear that ifp, q are dovetailed and f : ω \ dom(p) → ω \ dom(q) is the unique increasingbijection, then the continuous function π : x 7→ h ∪ x ◦ f−1 from [p] to [q]preserves the σ-ideal I, and its graph is a subset of E. Moreover, if p, q ∈ P arearbitrary conditions and x ∈ [p] and y ∈ [q] are E-equivalent points such thatboth x \ p, y \ q take both 0 and 1 value infinitely many times, then there aredovetailed conditions p′ ≤ p and q′ ≤ q: just let a, b ⊂ ω be infinite subsets ofω \ dom(p) and ω \ dom(q) respectively such that between any two successiveelements of a there is at most one element of b and vice versa, and x � a and y � bboth return constantly zero value, and let p′ = x � (ω \ a) and q′ = y � (ω \ b).

For the regularity of PEI , by Theorem 3.5.4 it will be enough to show thatevery Silver cube is an E-kernel. So let [p] be a Silver cube for some p ∈ Pand let A ⊂ [p] be an analytic set such that [A]E /∈ I; we must conclude that[A]E ∩ [p] /∈ I. Choose a Silver cube [q] ⊂ [A]E and a point y ∈ 2ω such thatq ⊂ y and on ω \ dom(q), y attains both values 0 and 1 infinitely many times.Find x ∈ A such that x E y. Note that x must attain both values infinitelymany times on ω \ p: if it attained say 1 only finitely many times, there would


be no E equivalent point in [p] to the point y′ ∈ [q] which extends q with onlyzeroes. As in the previous paragraph, find dovetailed Silver cubes p′ ≤ p andq′ ≤ q, and conclude that [p′] ⊂ [p] ∩ [q]E as desired.

For the distributivity just note that E is I-dense and apply Proposition 3.5.7.To show that PEI yields the V [xgen]E extension, it is enough to prove that forall conditions p, q ∈ Q, either there is a condition p′ ≤ p such that [p′]E∩ [q] ∈ I,or there is a condition p′ ≤ p and an injective Borel map f : [p′] → [q] suchthat f ⊂ E and f preserves the ideal I; and then apply Theorem 3.5.9. So fixthe conditions p, q ∈ P . If [p]E ∩ [q] ∈ I then we are done; otherwise thin out qso that [q] ⊂ [p]E holds. As in the previous paragraph, there will be dovetailedSilver cubes p′ ≤ p and q′ ≤ q and a homeomorphism π : [p′] → [q′] whosegraph is a subset of E and which preserves the ideal I. Thus, the assumptionsof Theorem 3.5.9 are satisfied and PEI yields the V [xgen]E extension.

As an additional piece of information, in the PI -generic extension V [G] theequivalence class [xgen]E is the only one contained in every set in the filterH = G ∩ PEI . In the ground model, this translates to the statement thatwhenever p ∈ P and g : [p]→ 2ω is a Borel function such that ∀x ∈ [p] ¬g(x) Ex then there is a condition q ≤ p such that [q]E ∩ g′′[q] = 0. Since Silverforcing is bounding, strengthening p if necessary we may assume that g � [p] iscontinuous and there are successive intervals In : n ∈ ω in ω such that for everyx ∈ [p] and every n ∈ ω there is m ∈ In such that |f(x)(m) − f(g(x))(m)| >3n. Find a strengthening q ≤ p such that the set ω \ dom(q) visits everyinterval in the collection {In : n ∈ ω} in at most one point. To see that qworks as required, pick x, y ∈ q and argue that for every n ∈ ω there is msuch that |f(y)(m) − f(g(x))(m)| > n. Just find a number m ∈ In such that|f(x)(m) − f(g(x))(m)| > 3n and observe that |f(x)(m) − f(y)(m)| ≤ n sincethe set ω \ dom(q) has at most n elements below m.

Theorem 5.8.8. There is a coding function on Silver cubes.

Proof. Consider the function f : (2ω)2 → 2ω defined by f(x, y) = x ◦π−1 whereπ is the increasing enumeration of the set {n : x(n) 6= y(n)} if this set is infinite,and f(x, y) =trash if this set is finite. This function works.

Let p ∈ P be a condition. Whenever z ∈ 2ω is a binary sequence, let x, y ∈ 2ω

be the unique points such that g ⊂ x, y and z = x ◦ π−1 and 1 − z = y ◦ π−1

where π is the increasing enumeration of the complement of the domain of p.Clearly then, f(x, y) = z and f ′′[p]2 = 2ω.

5.9 Ramsey cubes

A central notion of Ramsey theory of Polish spaces is the following:

Definition 5.9.1. A Ramsey cube is a set of the form [a]ℵ0 where a ⊂ ω isinfinite. A dented Ramsey cube is the collection of all sets of the form c ∪ bwhere c ⊂ ω is a fixed finite set and b ranges over all elements of a fixed Ramseycube.

5.9. RAMSEY CUBES 155

The partition properties of Ramsey cubes are governed by Silver and Prikrytheorems:

Fact 5.9.2. For every Borel partition of a Ramsey cube into finitely manypieces, one of the pieces contains a Ramsey cube. For every Borel partition ofa Ramsey cube into countably many pieces, one of the pieces contains a dentedRamsey cube.

We first introduce the poset associated with the notion of a Ramsey cube,and its combinatorial presentation. Adrian Mathias defined the Mathias forcingP as the poset of all pairs p = 〈ap, bp〉 where ap ⊂ ω is finite and bp ⊂ ω isinfinite; thus, every condition p ∈ P is associated with a dented Ramsey cube[p] = {ap∪ b : b ∈ [bp]

ℵ0 . The ordering is defined by by q ≤ p if ap ⊂ aq, bq ⊂ bp,and aq \ ap ⊂ bp.

Let I be the σ-ideal on X = [ω]ℵ0 consisting of those sets A ⊂ X which arenowhere dense in the quotient partial order P(ω) modulo the ideal of finite sets.

Fact 5.9.3. 1. [36] Every analytic or coanalytic subset of [ω]ℵ0 is either inI or contains a dented Ramsey cube;

2. the map p 7→ [p] is an isomorphism of P with a dense subset of PI ;

3. [46] the σ-ideal I is Π12 on Σ1

1 and not ∆12 on Σ1

1;

4. the poset P is proper.

The canonization properties of Ramsey cubes are complex. The total canon-ization for smooth equivalence relations fails badly, as the Mathias forcing addsmany intermediate real degrees. Promel and Voigt [41] gave an explicit canon-ization formula for smooth equivalence relations which can be also derived fromearlier work of Mathias [36, Theorem 6.1]. For a function γ : [ω]<ℵ0 → 2 letΓγ : [ω]ℵ0 → [ω]≤ℵ0 be the function defined by Γγ(a) = {k ∈ a : γ(a ∩ k) = 0}and Eγ the smooth equivalence relation on [ω]ℵ0 induced by the function Γγ .

Fact 5.9.4. [36, 41] For every smooth equivalence relation E on [ω]ℵ0 there isa function γ : [ω]<ℵ0 → 2 and a Ramsey cube on which E = Eγ .

Here, we shall present a further canonization theorem, due to Adrian Math-ias, regarding countable Borel equivalence relations. There is an obvious obsta-cle for canonization there, the relation E0 as well as some of its subequivalences.It turns out that other countable equivalence relations can be simplified to theseon Silver cubes.

Theorem 5.9.5. (Mathias) For every countable Borel equivalence relation E on[ω]ℵ0 , there is a Ramsey cube on which E ⊆ E0. For every essentially countableequivalence relation E on [ω]ℵ0 , there is a Ramsey cube on which E is reducibleto E0.

Proof. The proof of Theorem 5.9.5 starts with two claims of independent inter-est.


Claim 5.9.6. 1. Let c ⊂ ω be an infinite set and f : [c]ℵ0 → P(ω) be a Borelfunction such that for every e ∈ [c]ℵ0 , f(e)\e is infinite. Then there existsan infinite set d ⊂ c such that for every e ∈ [d]ℵ0 , f(e) \ d is infinite.

2. Let f : [ω]ℵ0 → [ω]ℵ0 be a Borel function. The Mathias forcing forces thefollowing: either f(xgen) ⊂ xgen up to finitely many numbers, or there isa ground model superset d ⊃ xgen such that f(xgen) \ d is infinite.

Proof. For the first item, for an infinite set e ⊂ c, write eev for the set of itselements that are indexed by even numbers in its increasing enumeration, andeod for e \ eev. Partition [c]ℵ0 into two Borel parts: B = {e : f(eev) ∩ eod isinfinite}, and C = {f(eev) ∩ eod is finite}. By Galvin–Prikry theorem, theremust be a set d′ ⊂ c such that [d′]ℵ0 is either wholly contained in B or C. Letd = d′ev; we claim that the set d works as required.

Case 1. Suppose first that [d′]ℵ0 ⊂ B. Then for every infinite set e ⊂ dthere is e′ ⊂ d′ such that e′ev = e and e′od ⊂ d′od. Thus, f(e) ∩ d′od must beinfinite and so must be its superset, the set f(e) \ d. In this case, we are done.

Case 2. Now suppose that [d′]ℵ0 ⊂ C. Then for every infinite set e ⊂ d,it must be the case that f(e) ∩ (d′ \ e) is finite. If this set was infinite, find aset e′ ⊂ d′ such that e′ev = e and for any two successive elements of the set e,if there is an element of the set f(e) ∩ (d′ \ e) between them then the uniqueelement of e′ between them belongs to f(e)∩ (d′ \ e). Then clearly f(e′ev)∩ e′od

is infinite, contradicting the case assumption. Thus, the set f(e) \ d′ must beinfinite, as well as its superset f(e) \ d. The first item follows.

The second item is an elaboration of the first. Suppose B ∈ PI is a conditionforcing f(xgen) \ xgen is infinite. The set C = {x ∈ B : f(x) \ x is infinite} isstill I-positive and Borel, so thinning it out if necessary we may assume that itis a dented Ramsey cube. Using the first item, we can thin out this cube stillfurther to find d ⊂ ω such that all sets in the cube are subsets of d and theirf -images have infinitely many elements outside of d. This condition forces the”or” clause in the second item.

Claim 5.9.7. Let M be a countable elementary submodel of a large structureand let b ⊂ ω be an M -generic real for Mathias forcing. If e0, e1 ⊂ b are infinitesets then M [e0] = M [e1] if and only if the symmetric difference e0∆e1 is finite.

Proof. An infinite subset of a Mathias generic real is again Mathias generic[36], so it is the case that both e0, e1 are Mathias generic reals for the modelM . Clearly, if their symmetric difference is finite then they construct each otherand the models M [e0] and M [e1] are the same. For the opposite implication,suppose that e0 \ e1 is infinite and argue that e0 ∈M [e1] is impossible. Indeed,if e0 ∈M [e1], then e0 = f(e1) for some Borel function f ∈M . The second itemof Claim 5.9.6 shows that there is an infinite set d ∈ M such that e1 ⊂ d andf(e1) \ d is infinite. Since b is Mathias generic with infinite intersection withd, all but finitely many of its elements belong to d. But then, the set f(e1) \ bmust be infinite, and therefore cannot be equal to the set e0 which is a subsetof b.


Towards the proof of the theorem, suppose first that the Borel equivalencerelation E has in fact countable classes; it is then generated as an orbit equiva-lence relation of a Borel action of some countable group G. Let M be a countableelementary submodel of a large structure, and let b ⊂ a be a Mathias genericreal for M . If e ⊂ b is an infinite set, then the model M [e] contains all E-equivalents of e, since they are simply the images of e under the Borel action ofG. Claim 5.9.7 then shows that the E-class of e is a subset of the E0-class of e,and as e ⊂ b was arbitrary, E � [b]ℵ0 ⊆ E0.

The case of the general essentially countable equivalence relation uses thecanonization of smooth equivalence relations, as in Fact 5.9.4. Suppose thata ⊂ ω is an infinite set, F is a countable Borel equivalence relation on a Polishspace Y , f : [a]ℵ0 → Y is a Borel function, and E is the pullback of F by thefunction f . Use the canonization of smooth equivalence relations, Fact 5.9.4, tothin out the set a if necessary to find a function γ : [ω]<ℵ0 → 2 such that thesmooth equivalence relations associated with f and Γγ coincide at [a]ℵ0 . Thisallows us to replace the general case with the situation where f = Γγ and F isa Borel equivalence relation with countable classes on [ω]ℵ0 . Let C = Γ′′γ [a]ℵ0 .

By Silver’s theorem 5.9.2, there is an infinite set c ⊂ a such that [c]ℵ0 is eithera subset of C or disjoint from C. The latter case cannot happen, since the setΓγ(c) ⊂ c would contradict the homogeneity; thus, the former case rules. Bythe previous paragraph, there must be an infinite set d ⊂ c such that F � [d]ℵ0

is Borel reducible to E0. It follows that E � [d]ℵ0 is also reducible to E0, sincethe function Γγ Borel reduces it to F � [d]ℵ0 .

Beyond the realm of essentially countable equivalence relations, there is arich and chaotic spectrum of Borel equivalences on [ω]ℵ0 which maintain theircomplexity on every Ramsey cube.

Theorem 5.9.8. E0 is in the spectrum of Mathias forcing. The poset PE0

I isregularly embedded in PI , it is ℵ0-distributive and it yields the V [xgen]E0

model.

Proof. Identify subsets of ω with their characteristic functions and use this iden-tification to transfer E0 to the spaceX. The theorem follows from Theorem 4.2.1once we prove that E0 saturations of Borel I-small sets are I-small. However, itis clear that E0-saturation of a set nowhere dense in the algebra P(ω) modulofinite is again nowhere dense, because the saturation does not change the placeof the elements of the set in this algebra. The ℵ0-distributivity follows fromproposition 3.5.7, since the E0 orbit of any element in a dented cube is dense inthe dented cube.

In fact, it is not difficult to see that the decomposition of Mathias forcing as-sociated with the E0 equivalence is just the standard decomposition into P(ω)modulo finite followed with a poset shooting a set through a Ramsey ultrafilter.Note that in the Mathias extension, there are many E0-classes which are subsetof every E0-invariant set in the generic filter; every infinite subset of the generic


real will serve as an example. It turns out though that E0 is the only countableequivalence relation in the spectrum of Mathias forcing.

Theorem 5.9.9. The spectrum of the σ-ideal I is cofinal among Borel equiva-lence relations in ≤B; it includes EKσ .

Proof. Let K be a Borel ideal on ω containing all finite sets and let EK bethe equivalence relation on the space Y of increasing functions in ωω definedby y EK z if {n ∈ ω : y(n) 6= z(n)} ∈ K. For the cofinality of the spectrum,by Fact 2.2.12, it is enough to show that EK is in the spectrum of Mathiasforcing. Look at E, the relation on [ω]ℵ0 connecting two points if their increasingenumerations are EK-equivalent. Clearly, this is an equivalence relation Borelreducible to EK . On the other hand, if B ⊂ X is a Borel I-positive set, thenEK is Borel reducible to E � B. Just find a condition p in the Mathias forcingsuch that [p] ⊂ B and let π : ω → ap ∪ bp be the increasing enumeration. Toeach point y ∈ Y associate the set by = ap ∪ rng((π ◦ y) � ω \ |ap|) and observethat the map y 7→ by reduces EK to E � B. If K is an Fσ-ideal on ω such that=K is bireducible with EKσ , then also EK is bireducible with EKσ , showingthat EKσ is in the spectrum of I.

Theorem 5.9.10. F2 is in the spectrum of the Ramsey null ideal.

Proof. We will need a couple of definitions. For elements x, y ∈ [ω]ℵ0 , say thatx, y are dovetailed if they are disjoint and between any two successive elementsof x there is exactly one element of y and vice versa. Say that x, y are almostdovetailed if some of their finite modifications are. If x, y, z ∈ [ω]ℵ0 are pairwisealmost dovetailed, say that x is between y and z if for all but finitely manyn ∈ y, the smallest element of x above n is smaller than the smallest elementof z above n. A set of pairwise almost dovetailed points is dense in itself ifbetween any two elements of it there is another one. The following is nearlytrivial.

Claim 5.9.11. Let x, y, z ∈ [ω]ℵ0 be pairwise almost dovetailed. Then

1. x is either between y and z, or between z and y;

2. if x is between y and z, then x is not between z and y, and z is between xand y;

3. if x is between y and z, then the same holds of all the finite modificationsof the sets x, y, and z;

If a ⊂ [ω]ℵ0 is a countable collection of pairwise almost dovetailed points thenthere is a map π from a to the unit circle such that x is between y and z if thetriple 〈y, x, z〉 goes in the counterclockwise direction.

Let us now introduce a Mathias name for a somewhat canonical countableset of pairwise almost dovetailed points, dense in itself. Consider 2<ω orderedwith the dense lexicographic order <lex. Divide ω into successive intervals {In :


n ∈ ω} and find a function χ : ω → 2<ω such that χ � In is an order isomorphismbetween In and (2≤n, <lex). Finally, for every x ∈ [ω]ℵ0 and every s ∈ 2<ω, letxs ⊂ x be the image of the set {m : χ(m) = s} under the increasing enumerationof x. It is not difficult to see that the set {xs : s ∈ 2<ω} consists of pairwisealmost dovetailed points, and it is dense in itself.

To formulate the following claim, for sets y ⊂ x ⊂ ω say that y is thin in xif for every k ∈ y the next 2k many elements of x are not in y.

Claim 5.9.12. Let x0 ∈ [ω]ℵ0 , and let x1 ⊂ x0 be a set thin in x0. For everycountable set a consisting of pairwise almost dovetailed subsets of x1, dense initself, there is a set x2 ⊂ x0 such that the sets {xs2 : s ∈ 2<ω} and a are equalup to finite modifications of their elements.

Proof. First, use the density of the set a to build a bijection π : 2<ω → asuch that whenever s, t, u are three binary sequences listed in the lexicographicorder, then π(t) is between π(s) and π(u). We will find the point x2 ∈ [ω]ℵ0 sothat for every binary sequence s ∈ 2<ω, xs2 = π(s) up to a finite modification.Find numbers {mn : n ∈ ω} such that for every triple s <lex t <lex u ∈ 2≤n,everything works out correctly above mn:

• the sets π(s), π(t), π(u) are pairwise disjoint above mn;

• between any two successive elements of π(s) larger thanmn there is exactlyone element of π(u), and the same for the other two pairs

• for every m > mn in π(s), the smallest element of π(t) above m is smallerthan the smallest element of π(u) above mn.

Find intervals {Jk : k ∈ ω} of natural numbers such that whenever min(Jk) >mn, the interval Jk contains exactly one point from each set π(s) : s ∈ 2≤n listedin the lexicographic order, and for every s ∈ 2<ω, π(s) ⊂

⋃k Jk ∪m|s|. Now it

is not difficult to use the thinness of the set x1 in x0 to find a set x2 ⊂ x1 sothat, writing ξ : ω → x2 for its increasing enumeration, for every k ∈ ω, everynumber l ∈ Ik and every s ∈ 2<ω, if min(Jk) > m|s| and χ(l) = s then ξ(l) isthe unique point in π(s) ∩ Jk. The set x2 ⊂ x0 works as required.

Now let f : [ω]ℵ0 → ([ω]ℵ0)ω be any Borel function such that f(x) enumer-ates the set {xs : s ∈ 2<ω} and all finite modifications of elements of this set.Let E be the Borel equivalence relations connecting points x0, x1 if f(x0), f(x1)enumerate the same set. It will be enough to show that F2 is reducible to Erestricted to any dented Ramsey cube. To simplify the notation, consider justa Ramsey cube, a set [x0]ℵ0 for some infinite set x0 ⊂ ω. Let x1 ⊂ x0 be a setthin in x0. Choose a function π : 2<ω → x0 such that for finite binary sequencess, t ∈ 2<ω |s| < |t| implies π(s) < π(t), and if |s| = |t| and s is lexicographicallyless than t then again π(s) < π(t). The map π can be naturally extended toπ : 2ω → [ω]ℵ0 by π(y) = {π(y � m) : m ∈ ω}. It is not difficult to see that therange of π consists of pairwise almost dovetailed sets, and if a ⊂ 2ω is a denseset consisting of sequences that are not eventually constant, then π′′a is a set


dense in itself. Let B ⊂ (2ω)ω× [x0]ℵ0 be the set of all pairs 〈y, x〉 such that thesets π′′rng(y) and {xs : s ∈ 2<ω} are the same up to finite modifications of theirelements. If y ∈ (2ω)ω is an enumeration of a dense subset of 2ω consisting ofnot eventually constant sequences, then the vertical section By is nonempty bythe previous claim. By Corollary 5.1.30, F2 is Borel reducible to E � [x0]ℵ0 .

The whole treatment in the previous proof should be understood as a devel-opment of a symmetric Mathias extension, very much parallel to the symmetricCohen extension of Section 5.1c.

Definition 5.9.13. Let M be a transitive model of set theory. Say that aset a ⊂ [ω]ℵ0 is a symmetric Mathias set for M if a is closed under finitemodifications, any two elements of a are either finite modifications of each otheror otherwise almost dovetailed, x∪ y is a Mathias real for M for every x, y ∈ a,and finally if x, y ∈ a then there is z ∈ a that is between x and y.

The point of this definition is that just as in the case of symmetric Cohenor random extensions, the simple demands on the set a guarantee that thechoiceless model M(a) can be constructed as a submodel of a Mathias genericextension, and its theory is an element of the model M . The arguments in theprevious proof fairly easily give the following:

Proposition 5.9.14. Let M be a transitive model of set theory and a ⊂ [ω]ℵ0

be a symmetric Mathias set for M . Then in some forcing extension there is apoint x ∈ [ω]ℵ0 , M -generic for Mathias forcing, such that the closure of the set{xs : s ∈ 2ω} under finite modifications is equal to a.

5.10 Weak Milliken cubes

Definition 5.10.1. The symbol (ω)ω stands for the collection of all sequencesa of finite sets of natural numbers such that for every n ∈ ω, max(a(n)) <min(a(n + 1). A weak Milliken cube is a set of the form [a] = {

⋃n∈b a(n) :

b ∈ [ω]ℵ0} where a ∈ (ω)ω. A dented weak Milliken cube is a set of the form{c ∪ b : b ∈ [a]} for some finite set c ⊂ ω and a ∈ (ω)ω.

The partition theory of weak Milliken cubes is governed by Fact 2.7.5, inparticular we have

Fact 5.10.2. For every Borel partition of a weak Milliken cube into finitelymany pieces, there is a weak Milliken cube wholly contained in one of the piecesof the partition. For every Borel partition of a weak Milliken cube into countablymany pieces, there is a dented weak Milliken cube wholly contained in one pieceof the partition.

Regarding the canonization properties of weak Milliken cubes, one can im-mediately identify two obstacles to total canonization of Borel equivalence rela-tions: the equivalence Emin which connects two sets of natural numbers if theyhave the same minimum, and the equivalence E0. It turns out that these twoobstacles are to a great extent unique:

5.10. WEAK MILLIKEN CUBES 161

Theorem 5.10.3. Let E be an equivalence relation on [ω]ℵ0 which is Borelreducible to EKσ , or classifiable by countable structures, or Borel reducible to=J for some analytic P-ideal J on ω. Then there is a weak Milliken cube Csuch that E � C is equal to one of the following: id, ee, E0, Emin, E0 ∩ Emin.

Before we turn to the proof of the above canonization theorem, we take theopportunity to introduce a propoer forcing associated with weak Milliken cubesand its combinatorial presentation. Define the partial ordering P to consist ofall pairs p = 〈tp, ap〉 where tp ⊂ ω is a finite set and ap is an infinite sequence offinite subsets of natural numbers such that max(ap(n)) < min(ap(n+ 1)) for alln ∈ ω. Thus, every condition p ∈ P is associated with a dented weak Millikencube [p] = {tp ∪

⋃n∈b ap(n) : b ∈ [ω]ℵ0}. The ordering is defined by q ≤ p if

tp ⊂ tq and tq is the union of tp with some sets on the sequence ap, and all setson aq are unions of some sets on ap.

We will introduce a standard fusion-type terminology for the forcing P .q ≤ p is a direct extension of p if tq = tp. If a ∈ (ω)ω then a fusion sequencebelow a is a sequence 〈ui, bi : i ∈ ω〉 such that u0 = 0, b0 = a, and for everyi ∈ ω, ui is a finite union of sets on the sequence bi and bi+1 ≤ bi is a memberof (ω)ω with min(bi+1(0) > max(ui+1. If a ∈ (ω)ω is a sequence and m ∈ ωthen a\m is the sequence in (ω)ω obtained by erasing the first m entries on thesequence a.

Let I be the collection of all analytic subsets of [ω]ℵ0 containing no dentedweak Milliken cube as a subset. We have

Theorem 5.10.4. 1. The collection I is a σ-ideal and the map p 7→ [p] isan isomorphism of P with a dense subset of PI ;

2. Iis Π11 on Σ1

1;

3. I fails to have the selection property;

4. the forcing P is proper, preserves Baire category and outer Lebesgue mea-sure, and it adds no independent reals. It adds an unbounded real.

Proof. The proof starts with the identification of basic forcing properties of theposet P .

Claim 5.10.5. Let O ⊂ P be an open dense set and p ∈ P a condition. There isa direct extension q ≤ p such that for every n ∈ ω, the condition 〈tq∪aq(n), aq\n〉belongs to the set O.

Proof. Set up a simple fusion process to find a sequence b ≤ a such that forevery finite set u ⊂ ω, either 〈tp ∪

⋃n∈u a(n), b \ max(u)〉 ∈ O or else none

of direct extensions of this condition belong to O. Applying Fact 2.7.5 to thepartition π : (ω)ω → 2 given by π(d) = 0 if the former case holds for d(0), we geta homogeneous sequence c ⊂ b. The homogeneous color cannot be 1 since thecondition 〈tp, c〉 has an extension in the open dense set O. If c is homogeneousin color 0 then it confirms the statement of the claim with q = 〈tp, c〉.


Claim 5.10.6. If p ∈ P is a condition and [p] = B0 ∪ B1 is a partition intotwo Borel sets then there is a direct extension q ≤ p such that [q] is containedin one of the pieces of the partition.

Proof. This is an immediate consequence of Fact 2.7.5.

For the proof of properness of the poset P , let M be a countable elementarysubmodel of a large enough structure, let p ∈ P ∩M be a condition. We mustproduce a master condition q ≤ p. Enumerate the open dense sets of P inthe model M by 〈On : n ∈ ω〉 and use the claim and the elementarity of themodel M repeatedly to build a sequence p = p0, p1, p2, . . . of direct extensionsof p in the model M such that for every n ∈ ω, apn � n = apn+1 � n andfor every finite set u ⊂ ω with max(u) > n it is the case that the condition〈tp ∪

⋃m∈u apn+1

(m), apn+1\ max(u)〉 ∈ M belongs to the open dense set On.

The limit q ≤ p of the sequence of conditions 〈pn : n ∈ ω〉 will be the requiredmaster condition.

In order to show that I is a σ-ideal, suppose that {Bn : n ∈ ω} are Borel setsin the collection I and suppose for contradiction that there is a condition p ∈ Psuch that [p] ⊂

⋃nBn. Clearly p xgen ∈

⋃nBn and so, strengthening p if

necessary, we may find a single natural number n ∈ ω such that p xgen ∈ Bn.Let M be a countable elementary submodel of a large enough structure, and letq ≤ p be an M -generic condition described in the previous proof. It is immediatethat for every point x ∈ [q] the conditions 〈tq ∪ (q ∩

⋃i∈n aq(i)), aq \ n〉 : n ∈ ω

generate an M -generic filter gx ⊂ P ∩M . By the forcing theorem applied in themodel M , M [x] |= x ∈ Bn, and by Borel absoluteness, x ∈ Bn. Thus [q] ⊂ Bn,contradicting the assumption that Bn ∈ I.

In order to show that p 7→ [p] is an isomorphism between P and a densesubset of PI , it is necessary to show that if p, q ∈ P are conditions such that[q] ⊂ [p] then in fact q ≤ p. To simplify the discussion a bit, assume thattp = tq = 0. Suppose that q ≤ p fails, and find a number n ∈ ω such that aq(n)is not a union of sets on the sequence ap. In such a case, either there is anelement i ∈ aq(n) such that i /∈

⋃rng(a), in which case no set x ∈ [q] containing

i can be in [p]; or else there is a number m such that ap(m) ∩ aq(n) 6= 0 andap(m) \ aq(n) 6= 0. In the latter case find a point x ∈ [q] containing aq(n) asa subset and disjoint from the set ap(m) \ aq(n); such a point cannot belongto p] either. In both cases we have a contradiction with the assumption that[q] ⊂ [p]!

To show that the ideal I is Π11 on Σ1

1, let A ⊂ 2ω × [ω]ℵ0 be an analytic set,a projection of a closed set C ⊂ 2ω × [ω]ℵ0 × ωω. We must show that the set{y ∈ 2ω : Ay ∈ I} is coanalytic. Fix a point y ∈ 2ω and suppose that Ay /∈ I.This means that there is a condition p ∈ P such that [p] ⊂ Ay, by Shoenfieldabsoluteness this will be true in the P -forcing extension as well, and so theremust be a P -name τ for a point in ωω such that p 〈y, xgen, τ〉 ∈ C. By a fusionargument, there is a condition r ≤ p and a function g : [ω]<ℵ0 → ω<ω such thatthe following holds. For every b ∈ [ω]<ℵ0 , write rb ≤ r to be that conditionwhose trunk consists of tr and the union of i-th elements of the sequence ar for


all i ∈ b, and arb = ar \ max(b) + 1; we will have that for every b ∈ [ω]<ℵ0 ,dom(g(b)) = max(b) + 1 and rb g(b) ⊂ τ . In particular, the following formulaφ(y, r, g) holds: the function g respects inclusion, dom(g(b)) = max(b) + 1 andthe basic open neighborhood of the space 2ω×[ω]ℵ0×ωω given by the first max(b)elements of y in the first coordinate, the trunk trb in the second coordinate, andthe sequence g(b) in the third coordinate has nonempty intersection with theclosed set C. It is also clear that ψ(y, r, g) implies that [r] ⊂ Ay since thefunction g yields the requisite points in the set C witnessing that elements ofthe set [r] belong to Ay. We conclude that Ay /∈ I if and only if there is r ∈ Pand a function g : [ω]<ℵ0 → ω<ω such that ψ(y, r, g) holds, and this is ananalytic statement.

For the continuous reading of names, let p ∈ P and let f : [p] → ωω be aBorel function. Use Claim 5.10.5 to find a direct extension q ≤ p such that forevery number n ∈ ω and every set u ⊂ n + 1 with n ∈ u there is a number msuch that the condition 〈tp ∪

⋃i∈u aq(i), aq \ n+ 1〉 forces f(xgen)(n) = m. Let

M be a countable elementary submodel of a large structure and find a directextension r ≤ q as in the previous argument. The forcing theorem will thenshow that f � [r] is a continuous function.

For the preservation of Baire category, there is a simple fusion-type proof.A more elegant argument will show that the ideal I is the intersection of acollection of meager ideals corresponding to Polish topologies on the Polishspace [ω]ℵ0 that yield the same Borel structure, and use the Kuratowski-Ulamtheorem as in [67, Corollary 3.5.5]. This is the same as to show that for everycondition p ∈ P there is a Polish topology on [p] such that all sets in the idealI are meager with respect to it.

We will deal with the case of the largest condition p ∈ P ; there, tp = 0 andap(n) = {n}, so [p] = [ω]ℵ0 . We will show that every Borel nonmeager set inthe usual topology on the space [ω]ℵ0 is I-positive. Suppose that B ⊂ [ω]ℵ0 issuch a nonmeager set, containing the intersection O ∩

⋂On : n ∈ ω, where O is

a nonempty basic open set and On : n ∈ ω are open dense sets. Let r0 ∈ 2<ω

be a finite sequence such that whenever the characteristic function of a pointx ∈ [ω]ℵ0 contains r0 then x ∈ O. By a simple induction on n ∈ ω buildfinite sequences r0 ⊂ r1 ⊂ r2 ⊂ . . . such that for every point x ∈ [ω]ℵ0 whosecharacteristic function contains rn+1 \ rn, x ∈

⋂m∈nOm holds. In the end, let

t = {i ∈ dom(r0) : r0(i) = 1} and let a ∈ (ω)ω be the sequence defined bya(n) = {i ∈ dom(rn+1 \ rn) : rn+1(i) = 1}, and observe that [t, a] ⊂ B and soB /∈ I as required.

For the preservation of outer Lebesgue measure, suppose that A ⊂ 2ω is aset of outer Lebesgue mass > ε+ δ and p ∈ P is a condition forcing O to be asubset of 2ω of mass < ε. We must find a condition q ≤ p and a point z ∈ Asuch that q z /∈ O.

Find positive real numbers δu : u ∈ [ω]<ℵ0 that sum up to less than δ, andfind names Oj : j ∈ ω for basic open sets whose union O is. Using Claim 5.10.5and a simple fusion process, find a direct extension q ≤ p such that for everynonempty set u ∈ [ω]<ℵ0 , there is an open set Pu such that the conditiontp ∪

⋃i∈u aq(i)

⋃j∈max(u) Oj ⊂ Pu ⊂ O and the mass of O \ Pu is less than


δu. Clearly, the set R0 =⋃{Pu \ Pu∩max(u) : u ∈ [ω]<ℵ0 , |u| > 1} has mass at

most δ, and the set R1 = lim sup{P{n} : n ∈ ω} has size at most ε. Find apoint z ∈ A \ (R0 ∪R1), thin out the sequence q if necessary to make sure thatz /∈ P{n} for any n ∈ ω and conclude that q z /∈ O as desired.

Not adding independent reals is always a somewhat more delicate issue.Suppose that p ∈ P is a condition and y a name for an infinite binary sequence.A simple fusion process using Corollary 5.10.6 repeatedly at each stage yieldsa direct extension 〈tp, b〉 ≤ p such that for every n ∈ ω and every set u ⊂ nthe condition 〈tp ∪

⋃i∈u b(i), b \ n + 1〉 decides the value y(min(b(n)). Define

a partition π : ([ω]<ℵ0)2 → 2 by setting π(u, v) = 0 if the decision for u andmin(bmin(v)) was 0. This is clearly a clopen partition and therefore there is ahomogeneous sequence d ∈ (ω)ω by Fact 2.7.5. Let us say that the homogeneouscolor is 0. Consider the condition q ≤ p given by tq = tp ∪

⋃i∈d(0) b(i), and

aq(j) =⋃i∈d2j+1

b(i). It is not difficult to use the homogeneity of the sequence

d to show that the condition q forces y � {min(bmin(d(2j))) : j > 0} = 0.Finally, P certainly adds an unbounded real: the increasing enumeration of

the generic subset of ω cannot be bounded by any ground model function. Thefailure of the selection property of the ideal I is also simple. Let C ⊂ P(ω)be a perfect set consisting of pairwise almost disjoint sets, and consider the setD ⊂ C× [ω]ℵ0 consisting of all pairs 〈c, x〉 such that c ∈ C and x ⊂ c. It is clearthat D is a Borel set with I-positive vertical sections. If B ⊂ [ω]ℵ0 is a Borel setcovered by the vertical sections of the set D, visiting each vertical section in atmost one point, it must be the case that the elements of B are pairwise almostdisjoint, therefore B cannot contain a subset of the form [p] for any p ∈ P , andso B ∈ I.

The investigation of the spectrum of the ideal I begins with identifying theonly obvious feature:

Theorem 5.10.7. E0 is in the spectrum of the σ-ideal I. The poset PE0

I isregularly embedded in PI , it is ℵ0-distributive, and it yields the model V [xgen]E0 .

Proof. This is an immediate corollary of the work in Theorem 4.2.1 and thesimple observation that E0-saturations of I-small sets are still I-small.

As an additional piece of information, the equivalence class [xgen]E0 is forcedby PI to be the only E0 equivalence class in the intersection of all E0-invariantsets in the generic filter. Suppose that p is a condition and p y ⊂ ω is aninfinite set not E0 related to xgen. Strengthening the condition p if necessary

we may find a continuous function f : [p] → P(ω) such that p y = f(xgen).We must find a condition q ≤ p such that no element of [q] is E0-equivalent tof ′′[q]; then clearly q y /∈ [q]E0 and so the class [y]E0 is not a subset of allE0-invariant sets in the PI -generic filter.

To produce the condition q, first use a standard fusion process to strengthenp so that for every finite set u ∈ ω, the condition 〈tp ∪

⋃n∈u ap(n), ap \max(u)〉

identifies some number mu > max(ap(max(u) − 1)) such that mu ∈ xgen∆y,and some number ku > max(ap(max(u) − 1)) such that ku ∈ y. Thinning out


the sequence ap if necessary, assume that the numbers {ku,mu : u ⊂ n} are allsmaller than min(ap(n)). It is now not difficult to verify that the thinned outcondition q works as desired; that is, whenever u, v ⊂ ω are infinite sets, writingxu = tp ∪

⋃n∈u aq(n)) and xv = tq ∪

⋃n∈v aq(n) we must show ¬f(xu) E0 xv.

For every n ∈ u, we must produce a number k > max(aq(n− 1)) which belongsto the symmetric difference of f(xu) and xv. There are two cases:

• either n /∈ v. Then k = ku∩n+1 will work: as n ∈ u, k ∈ f(xu), whilexv contains no numbers in the interval (max(aq(n− 1)),min(aq(n+ 1))),where k belongs;

• or n ∈ v. In this case, k = mu∩n+1 will work. Either it is in the set aq(n),in which case it belongs to both xu, xv and does not belong to f(xu), orit is not in the set aq(n), in which case it does not belong to either xu, xvand does belong to f(xu).

The proof is complete.

Proof of Theorem 5.10.3. We will start with equivalence relations reducible toEKσ . Recall that dom(EKσ ) is the collection of all functions in ωω below theidentity equipped with the distance δ(x, y) = supn |x(n) − y(n)|; EKσ thenconnects the points x, y if δ(x, y) is finite. Let a ∈ (ω)ω be a sequence, and letE be an equivalence relation reducible to EKσ by a Borel function f : [a] →dom(EKσ ). Thinning out the sequence a we may assume that the function fis in fact continuous. Consider the partition (a)ω = B ∪ C placing a sequenceb ≤ a into the set B just in case that for every i there are points x, y ∈ [b(0), b\0]such that δ(f(x), f(y)) ≥ i. Using Milliken theorem ??? we may thin out thesequence a so that (a)ω ⊂ B or (a)ω ⊂ C. There is an appropriate split intocases.

Case 1. Assume first that (a)ω ⊂ B. In this case, we will first thin out ato c that E � [c] ⊆ E0, and then proceed with another partition argument. Toperform the thinning, by induction on i ∈ ω build a fusion sequence 〈ui, ci : i ∈ω〉 so that whenever j ∈ ω and t, s ⊂ j are two sets, then for every x ∈ [

⋃i∈s ui∪

uj , cj+1] and every y ∈ [⋃i∈t ui, cj+1] it is the case that δ(f(x), f(y)) ≥ j. It is

clear that the limit c of the fusion sequence will satisfy E � [c] ⊆ E0.Suppose that ui : i ∈ j and cj have been constructed. Enumerate the set

P(j) × P(j) by 〈〈sk, tk〉 : k ∈ 22j〉, and by induction on k build finite sets ukjand Milliken cubes ckj so that

I1 u0j = 0 and the set uk+1

j is a union of ukj with some sets on the sequence

ckj ;

I2 cj = c0j ≥ c1j ≥ . . . ;

I3 for every k there are numbers mk, lsk, l

tk so that for every x ∈ [

⋃i∈sk ui ∪

uk+1j , ck+1

j ] and every y ∈ [⋃i∈tk ui, c

k+1j ] it is the case that f(x)(mk) = lsk,

f(y)(mk) = ltk, and |lsk − ltk| > j.


This is not difficult to do using the case assumption. Suppose that ckj , ukj have

been found. The case assumption yields x0, x1 ∈ [⋃i∈sk ui ∪ u

k+1j , ckj ] such that

d(f(x0), f(x1)) > 2j. Find mk such that |f(x0)(mk)− f(x1)(mk)| > j. Find asequence ck+1

j ≤ ckj such that for every y ∈ [⋃i∈t ui, c

kj ] the value f(y)(mk) is

the same, equal to some rk. Then either f(x0)(mk) or f(x1)(mk) is farther thanj from f(y)(mk), say that it is x0, and put lk = f(x0)(mk). Use the continuityof the function f to find a set uk+1

j such that x0 ∈ [⋃i∈s ui∪ukj+1, c

kj ] and for all

points x in this dented cube it is the case that f(x)(mk) = lk. This concludesthe induction step.

After the induction on k has been performed, let uj , cj be the last set andthe last sequence arrived at in it. This completes the construction of the fu-sion sequence, and we now have E � (c)ω ⊂ E0, where c is the limit of thefusion sequence. Let (c)ω = B′ ∪ C ′ where b ∈ B′ if

⋃i∈ω b(i) is equivalent to⋃

i∈ω,i6=1 b(i). The Milliken theorem ??? can again be used to thin out c so thateither (c)ω ⊂ B′ or (c)ω ⊂ C ′. There are the corresponding two subcases.

Case 1a. Suppose that (c)ω ⊂ B′. In this case, first observe that E � [c]connects every pair of E0-equivalent points with the same minimum. To see this,use the case assumption to argue that for every infinite set e ⊂ ω and everyj > min(e) it is the case that

⋃i∈e c(i) is E-related to

⋃i∈e∪{j} c(i). Then,

whenever x, y ∈ [c] are E0-related points with the same minimum, it is possibleto transform one into another in finitely many steps described in the previoussentence, and the transitivity of the relation E shows that x and y are E-related.

Consider the partition of (c)ω into two Borel pieces B′′ and C ′′, puttingb ∈ B′′ if

⋃i∈ω b(i) is E-related to

⋃i∈ω,i6=0 b(i).The Milliken theorem again

provides a split into cases.

Case 1aa. Either there is d ≤ c such that (d)ω ⊂ B′′. In such a case,E � [d] = E0: whenever x, y ∈ [d] are E0-related, if their minimums are thesame then they are E-related by the preceding paragraph. If their minimumsare different, say min(x) < min(y), then there is a sequence b ≤ d such thatmin(b0) = min(x), min(b1) = min(y), and x E0

⋃i b(i). The homogeneity

assumption applied to b shows that x E⋃i∈ω b(i) E

⋃i∈ω,i6=0 b(i) E y as desired.

Case 1ab. Or, there is d ≤ c such that (d)ω ⊂ C ′′. In such a case,E � [d] = E0 ∩ Emin by a similar homogeneity argument as in Case 1aa.

Case 1b. Suppose now that (c)ω ⊂ C ′. In this case, we will thin out cto d such that E � [d] = id. By induction on i ∈ ω build a fusion sequence〈vi, di : i ∈ ω〉 so that whenever j ∈ ω and s, t ⊂ j are two distinct sets thenfor every x ∈ [

⋃i∈s vi, dj ] and every y ∈ [

⋃i∈s vi, dj ] such that x \ min(dj) =

y \min(dj) it is the case that x E y fails. The fusion d of this fusion sequencewill certainly have the required property that E � [d] = id: if x, y ∈ [d] are notE0-related then they are not E-related by the initial work in Case 1, and if theyare E0-related and distinct then they are not E-related by the assumption onthe fusion sequence past the last difference of the two points x, y.

Suppose that vi : i ∈ j and dj have been constructed. For every pair s, tof subsets of j define the partition of (dj)

ω into two Borel pieces Bs,t andCs,t by placing b ≤ dj into Bs,t if

⋃i∈s vi ∪

⋃i∈ω b(i) is E-related to

⋃i∈s vi ∪


⋃i∈ω,i6=0 b(i). Observe that there can be no sequence e ≤ dj such that (e)ω ⊂

Bs,t: ???Case 2. Suppose now that (a)ω ⊂ C. In this case, first observe that every

two points in [a] with the same minimum are E-related: if min(x) = min(y)then the homogeneity assumption applied to the sequence a\min(x) shows thatthe distance d(f(x), f(y)) must be finite. Now let F be the equivalence relationon ω relating i and j if for every x, y ∈ [c] such that min(x) = min(c(i)) andmin(y) = min(c(j)), x E y holds. There are two possibilities.

Case 2a. Either F has an infinite class, say e ⊂ ω. Let d ≤ c be thesequence consisting only of those entries of c indexed by elements of the sete ⊂ ω. It is clear that E � [d] = ee.

Case 2b. Or F has an infinite set consisting of pairwise inequivalent ele-ments, say e ⊂ ω. Again, let d ≤ c be the sequence consisting only of thoseentries of c indexed by elements of the set e ⊂ ω. It is clear that E � [d] = Emin.

This concludes the case of equivalence relations reducible to EKσ . Nowassume that a ∈ (ω)ω and E is an equivalence relation on [a] reducible to =J

for some analytic P-ideal J on ω. Use the theorem of Solecki ??? to find asubmeasure lower semicontinuous submeasure φ on ω such that J = {e ⊂ ω :limi φ(e \ i) = 0}. Let f : [a] → P(ω) be a Borel function reducing E to =J ;thinning out the sequence a we may assume that the function f is continuous.For every sequence b ≤ a and every j ∈ ω let ε(j, b) be the supremum of allnumbers of the form φ(f(x)∆φ(y)) as x, y vary over all points in [

⋃i∈j b(i), b \

j]. Note that for a fixed sequence b, the numbers ε(j, b) cannot increase asj increases, and so they converge to some real number ε(b) ≥ 0. Define thepartition (a)ω into two Borel pieces B and C by placing b ∈ B if ε(b) = 0. TheMilliken theorem gives us the first split into subcases:

Case 1. It is possible to thin out the sequence a so that (a)ω ⊂ B. In thiscase, we will find c ≤ a so that E � [c] becomes a relatively closed equivalencerelation. Closed equivalence relations are smooth by ???, and so a referenceto the previously treated case of equivalence relations reducible to EKσ willcomplete the argument here. To find c, by induction on i ∈ ω find nonemptyfinite sets ui such that min(ui+1) > max(ui), each ui is a union of finitely manyentries on a, and for every j ∈ ω, every set s ⊂ j and every x, y ∈ [a] suchthat the set

⋃i∈s ui ∪ uj is an initial segment of both x and y it is the case

that φ(f(x)∆f(y)) < 2−j . This is easily possible using the case assumptionrepeatedly. Let c = 〈ui : i ∈ ω〉. To see that the complement of E is relativelyopen in [c], let x, y ∈ [c] be E-unrelated points, and j ∈ ω be so large thatlimi φ((f(x)∆f(y)) \ i) > 2−j . Let s, t be initial segments of x, y such that forsome numbers ks, kt > j it is the case that uks ⊂ s and ukt ⊂ t. We claim thatthe relatively open sets [s] and [t] inside [c] are E-unrelated. Well, if x′ ∈ [s]∩ [c]and y′ ∈ [t] ∩ [c], then φ(f(x)∆f(x′)) < 2ks < 2j+1 and φ(f(y)∆f(y′)) < 2kt <2j+1. Since limi φ((f(x)∆f(y))\i) > 2−j , this means that limi φ((f(x′)∆f(y′))\i) > 0 and ¬x′ E y′ as required.

Case 2. It is possible to thin out the sequence a so that (a)ω ⊂ C. In thiscase, we will thin out a further so that E � [a] ⊂ E0. The treatment will thenhook up into Case 1 of the equivalence relations reducible to EKσ . To perform


the thinning, by induction on i ∈ ω build a fusion sequence 〈ui, ci : i ∈ ω〉 so thatwhenever j ∈ ω and t, s ⊂ j+1 are two sets then for every x ∈ [

⋃i∈s ui∪uj , cj+1]

and every y ∈ [⋃i∈t ui, cj+1] it is the case that φ((f(x)∆f(y))\j) > 1

2ε(〈ui : i ∈t〉acj). Once such a construction is performed, writing c ∈ (ω)ω for its limit,it will satisfy E � [c] ⊂ E0. To see this, let x, y ∈ [c] be E0-unrelated points.One of them, say x, must have infinitely many elements that do not occur inthe other. Let d ≤ c be the sequence enumerating all j such that uj ⊂ xand note that ε(d) > 0 by the homogeneity assumption. Now there must beinfinitely many j ∈ ω such that uj ⊂ x and uj ∩ y = 0, and for each such j ∈ ω,writing s = {i < j : ui ⊂ x} and t = {i < j : ui ⊂ y}, the construction yieldsφ(f(x)∆f(y)) \ j) > ε(d). Therefore f(x)∆f(y) is not in the ideal J and x, yare not E-related.

Suppose that ui : i ∈ j and cj have been constructed. Thin out cj so thatfor every set s ⊂ j, for every x ∈ [

⋃i∈s ui, cj ] the set f(x) ∩ k is the same.

Enumerate the set P(j) × P(j) by 〈〈sk, tk〉 : k ∈ 22j〉, and by induction on kbuild finite sets ukj and Milliken cubes ckj so that

I1 u0j = 0 and the set uk+1

j is a union of ukj with some sets on the sequence

ckj ;

I2 cj = c0j ≥ c1j ≥ . . . ;

I3 for every k there is a number mk and sets wsk, wtk ⊂ mk so that for every

x ∈ [⋃i∈sk ui∪u

k+1j , ck+1

j ] and every y ∈ [⋃i∈tk ui, c

k+1j ] it is the case that

f(x) ∩mk = wsk, f(y) ∩ vk = wsk, and wsk∆wtk >12ε(〈ui : i ∈ t〉ackj ).

This is not difficult to do using the case assumption. Suppose that ckj , ukj have

been found. Let d = 〈ui : i ∈ s〉a〈ukj 〉ackj . The definition of ε(d) yields points

x0, x1 ∈ [⋃i∈sk∪uk+1

j ,ckj] such that φ(f(x0)∆f(x1)) > 1

2ε(d). Let mk ∈ ω be a

number such that φ((f(x0)∆f(x1)) ∩mk) > 12ε(d). Find a sequence ck+1

j ≤ ckjsuch that for every y ∈ [

⋃i∈tk ui, c

kj ] the set f(y)∩mk is the same, equal to some

W tk. Then either f(x0)(mk) or f(x1)(mk) is farther than j from f(y)(mk), say

that it is x0, and put wsk = f(x0) ∩mk. Use the continuity of the function f to

find a set uk+1j such that x0 ∈ [

⋃i∈s ui ∪ u

j+1i , ckj ] and for all points x in this

dented cube it is the case that f(x) ∩mk = wsk. This concludes the inductionstep.

After the induction on k has been performed, let uj , cj be the last set andthe last sequence arrived at in it. This completes the construction of the fusionsequence,

As a final remark in this section, canonization of objects more complicatedthan equivalence relations must fail badly in the case of weak Milliken cubesdue to the following simple

Proposition 5.10.8. The σ-ideal I has a coding function.

5.11. STRONG MILLIKEN CUBES 169

Proof. Let x, y ∈ X be two points, not equal modulo finite. Consider all setsu ⊂ x ∩ y which are an interval inside x and inside y, and are maximal withrespect to this property; list them in an increasing order as 〈un : n ∈ ω〉.Define f(x, y) ∈ 2ω by setting f(x, y)(n) = 1 if max((x ∪ y) \min(un) ∈ x andf(x, y)(n) = 0 otherwise.

Suppose now that B ⊂ X is an I-positive set; it contains a dented cube ofthe form [t, a] for some nonempty finite set t ⊂ ω and a ∈ (ω)ω. Let z ∈ 2ω be aprescribed value. Let x be the union of t the sets a(2n+1) for all n ∈ ω and thesets a(2n) for all n with z(2n) = 1. Let y be the union of t the sets a(2n + 1)for all n ∈ ω and the sets a(2n) for all n with z(2n) = 0. It is immediate thatf(x, y) = z.

5.11 Strong Milliken cubes

Definition 5.11.1. A strong Milliken cube is a set of the form {c ∈ (ω)ω : c ≤a} for a sequence a ∈ (ω)ω. A dented strong Milliken cube is a set of the form{tac : c ≤ a} where t is a finite increasing sequence of finite subsets of ω anda ∈ (ω)ω is such that max

⋃t < min(a(0)).

The σ-ideal on (ω)ω associated with the notion of strong Milliken cube isisolated in the same way as in the previous sections. Let P be the poset ofall pairs p = 〈tp, ap〉 such that tp is an increasing finite sequence of nonemptyfinite subsets of ω and ap ∈ (ω)ω is a sequence of sets above tp. The orderingis defined by q ≤ p if tp ⊂ tq, aq ≤ ap, and all sets on the sequence tq \ tpare unions of some sets on ap. For a condition p ∈ P write [p] to denote thedented cube obtained from tp, ap. Quite obviously, the poset P adds a genericsequence xgen ∈ (ω)ω which is the union of the first coordinates of conditionsin the generic filter. To define the associated σ-ideal, let I be the collection ofBorel subsets of (ω)ω which do not contain a dented strong Milliken cube. Thefollowing seems to be folklore knowledge:

Theorem 5.11.2. The poset P is proper. The collection I is a σ-ideal and themap p 7→ [p] is an isomorphism between P and a dense subset of PI .

Proof. The argument is best channeled through the following useful fact.

Claim 5.11.3. Whenever M is a countable elementary submodel of a largeenough structure, a ∈ (ω)ω is a P -generic sequence over M and b ≤ a, then bis a P -generic sequence over M as well.

Proof. We will show that for every condition p ∈ P and every open dense setD ⊂ P , there is a condition q ≤ p with tq = tp such that for every c ≤ taq aqthere is a number n such that 〈(c � n), aq \ n〉 ∈ D. The lemma will follow: ifD ∈ M then the sequence a must pass though a condition q ∈ M as above bygenericity, and then every sequence c ≤ a passes through a condition in D ∩Mby the properties of q.


To find the condition q ≤ p, assume for simplicity that tp = 0. Use a simplefusion argument to find a sequence a′p ≤ ap such that for every n ∈ ω and everysequence u ≤ a′p � n, either (a) 〈u, a′p \n〉 ∈ D or (b) there is no sequence c ≤ a′psuch that 〈u, b〉 ∈ D. Now let (a′p)

ω = B0 ∪ B1 be the partition into two Borelsets defined by c ∈ B0 if for some n ∈ ω, (a) above occurs for u = c � n. TheMilliken theorem shows that one of the sets B0, B1 contains a strong Millikencube. If aq ≤ a′p is such that (aq)

ω ∈ B0 then the condition q = 〈tp, aq〉 isas required. On the other hand, the set B1 cannot contain a strong Millikencube: if aq ≤ a′p is such that (aq)

ω ∈ B1 and r ≤ 〈tp, aq〉 is a condition in theopen dense set D, then (a) must have occurred for u = tr \ tp, contradicting theassumption that (aq)

ω ⊂ B1.

To finish the proof of the theorem, let J be the collection of those Borel setsB such that P xgen /∈ B. The collection J is a σ-ideal; we will show thatJ = I. It is clear that J ⊆ I: if B /∈ I then there is a condition p ∈ P suchthat [p] ⊂ B, thus p xgen ∈ B and B /∈ J . To show that I ⊆ J , look at

an arbitrary Borel set B /∈ J and a condition p ∈ P forcing xgen ∈ B. Let Mbe a countable elementary submodel of a large enough structure and let b be aP -generic sequence over M meeting the condition p. Let q ≤ p be the conditiondefined by tq = tp and aq = b \ tp; we will show that [q] ⊂ B and thereforeB /∈ I. Indeed, by the claim every sequence c ≤ b is generic over M , so everysequence c ∈ [q] is generic over M meeting the condition p, so M [c] |= c ∈ B bythe forcing theorem and c ∈ B by the analytic absoluteness between the modelsM [c] and V .

Thus, I is a σ-ideal; the map p 7→ [p] is a bijection between P and a densesubset of PI that is also an isomorphism of ≤ and ⊆. Finally, the propernessof the poset P also follows abstractly from the claim. If B ∈ PI is an I-positive Borel set and M is a countable elementary submodel of a large structurecontaining the set B, we must show that the set {x ∈ B : x is PI -generic overM} is I-positive. Find a condition p ∈M such that [p] ⊂ B, and find a sequencex ∈ (ω)ω generic over M which meets p. Let q = 〈tp, x \ tp〉. The claim showsthat [q] consists only of points generic over M and meeting the condition p. As[q] /∈ I, the proof is complete.

The canonization properties of strong Milliken cubes are much more com-plicated than those of weak Milliken cubes. We will just describe the part ofthe behavior which is parallel to the canonization properties of Ramsey cubes.Regarding the smooth equivalence relations, Klein and Spinas proved a paral-lel of the Promel-Voigt canonization theorem. They identified a well-organizedcollection Γ of continuous maps from (ω)ω to Vω+1 such that, writing Eγ for theequivalence relation {〈x, y〉 ∈ (ω)ω × (ω)ω : γ(x) = γ(y)}, the following holds:

Fact 5.11.4. [34] For every a ∈ (ω)ω and every smooth equivalence relation Eon (a)ω there is b ≤ a and γ ∈ Γ such that E � (b)ω = Eγ .

Among the countable equivalence relations on the space (ω)ω, the relationEtail stands out. It connects two sequences a, b if they have the same tail,

5.11. STRONG MILLIKEN CUBES 171

that is, if there is an integer n such that for all but finitely many m ∈ ω,a(m) = b(m + n). The first canonization theorem elaborates on the centralposition of Etail.

Theorem 5.11.5. For every a ∈ (ω)ω and every countable equivalence relationE on (a)ω there is b ≤ a such that E � (b)ω ⊆ Etail.

Proof. The whole argument follows closely the proof of the first sentence ofTheorem 5.9.5, with one additional partition trick. For b, c ∈ (ω)ω write c ≤∗ bif for some tail c′ of c, c′ ≤ b.

Claim 5.11.6. Whenever c ∈ (ω)ω and f : (c)ω → (ω)ω is a Borel functionsuch that for every e ≤ c, f(e) 6≤∗ e, then there is a sequence d ≤ c such thatfor every e ≤ d, f(e) 6≤∗ d.

Proof. First, divide (c)ω to two Borel classes: e ∈ B0 if⋃n f(e)(n) \

⋃n e(n) is

a finite set, and e ∈ B1 otherwise. The Milliken theorem divides the treatmentinto two cases.

Case 1. There is c′ ≤ c such that (c′)ω ⊂ B1. In this case, for every e ≤ c′,write eev for the sequence of even entries of e, and eod for the sequence of oddentries of e, and partition (c′)ω into two Borel classes: e ∈ C0 if

⋃n f(eev))(n)∩⋃

n eod(n) is finite and e ∈ C1 otherwise. The remainder of the argument isidentical with Claim 5.9.6; we will find d ≤ c′ such that for every e ≤ d it isthe case that

⋃n f(e)(n) \

⋃n d(n) is an infinite set for all e ≤ d. Therefore,

f(e) 6≤∗ d as required.

Case 2. There is c′ ≤ c such that (c′)ω ⊂ B0. In this case, for every e ≤ c′

write eco for the sequence 〈e(2n) ∪ e(2n + 1) : n ∈ ω〉. Since f(eco) 6≤∗ eco andCase 2 prevails, it must be the case that for infinitely many n, u = f(eco)(n)is a finite set that splits some v = eco(mn), (meaning that v has nonemptyintersection with both u and its complement) and this can happen in one offour ways. Writing v0 = e(2mn) and v1 = e(2mn + 1) (so that v = v0 ∪ v1) itmust be that either (a) u splits v0, or (b) u splits v1, (c) v0 ⊂ u and v1 ∩ u = 0,or (d) v0 ∩ u = 0 and v1 ⊂ u. one of these four cases will repeat infinitely manytimes, and we partition (c′)ω into corresponding four Borel pieces. The Millikentheorem divides the treatment in four corresponding subcases, of which we treatthe third, the others being similar.

Case 2c. There is d′ ≤ c′ such that for every e ≤ d′ there are infinitelymany n for which there exists mn such that e(2mn) ⊂ f(eco)(n) and e(2mn +1) ∩ f(eco)(n) = 0. Let d = d′co; we claim that d works as in the claim. Indeed,if e ≤ d, then write e ≤ d′ for the sequence defined by e(2m + 1) = d′(k) forthe largest k such that d′(k) ⊂ e(m), and e(2m) = e(m) \ e(2m + 1); notethat m must be odd as e ≤ d. By the homogeneity of the sequence d′ thereare infinitely many n for which there is mn such that e(2mn) ⊂ f(e)(n) ande(2mn+1)∩f(e)(n) = 0. A review of the definitions will show that then f(e)(n)splits the set d(m−1

2 ). Since this happens for infinitely many n, we conclude thatf(e) 6≤∗ d as required.


The rest of the proof is a literal repetition of the argument for Theorem 5.9.5.We just state the key claim and leave the rest to the reader.

Claim 5.11.7. Let M be a countable elementary submodel of a large structureand a ∈ (ω)ω a P -generic sequence over M . For any two sequences b, c ≤ a,M [b] = M [c]↔ b Etail c.

5.12 Products of superperfect sets

As another application of Milliken theorem, we consider the two-dimensionalproduct of Miller forcing. Otmar Spinas [56] computed the associated σ-idealand proved a canonization theorem for smooth equivalence relations. We providesimpler proofs of his results and show that E0 and EKσ are in the spectrum ofthe associated σ-ideal.

Let I be the σ-ideal on ωω σ-generated by compact sets, as in Section 5.1b.Let J be the collection of Borel subsets B of the space X = ωω × ωω such thatB contains no rectangle C ×D where C,D ⊂ ωω are Borel I-positive sets.

Theorem 5.12.1. [56], [24]

1. The collection J is a σ-ideal of Borel sets.

2. The quotient PJ is proper and adds a dominating real.

3. The σ-ideal J is not Π11 on Σ1

1.

4. If B ⊂ X is a Borel rectangle with I-positive sides and E a smooth equiv-alence relation on it, then there exists a Borel subrectangle C ⊂ B withI-positive sides such that E � C either is ee or vertical sections of C areE-unrelated, or horizontal sections of C are E-unrelated.

The last item is a weak form of canonization for smooth equivalence relations,and as such speaks about real degrees in the product extension. There, thegeneric is given by a pair 〈xlgen, xrgen〉 ∈ ωω × ωω. Each of the two points isMiller generic over the ground model, and therefore minimal over the groundmodel. The last item says that every real in the extension is either in the groundmodel or constructs one of the two coordinates in the generic pair. There areuncountably many other degrees above the minimal pair, and they have beenclassified in [56].

All the items in the theorem have been proved elsewhere. We will provideMilliken-style proofs that avoid the excessive length of the original arguments.The proof of (1) is due to Todorcevic.

Proof. Items (1) and (4) use the same trick recorded in the following claim.Define a function g : (ω)ω → ωω ×ωω by g(a) = 〈x, y〉 where x is the increasingenumeration of the set

⋃i a(2i) and y is the increasing enumeration of the set⋃

i a(2i+ 1). The following easy claim is the heart of our proofs:

5.12. PRODUCTS OF SUPERPERFECT SETS 173

Claim 5.12.2. Whenever B is a dented strong Milliken cube, then there aresuperperfect trees S, T ⊂ ω<ω such that

1. [S]× [T ] ⊂ g′′B;

2. for every n ∈ ω and every s, t ∈ S ∪ T with n ∈ rng(s) ∩ rng(t), both s, tcome from the same tree S or T , and they have a common initial segmentu such that n ∈ rng(u).

Proof. For simplicity of notation assume that B is a strong Milliken cube, B =(a)ω. Let ω =

⋃s∈ω<ω ds be a partition into infinite sets and consider the

superperfect tree U ⊂ ω<ω consisting of those sequences t such that t(0) = 0and t(i) ∈ dt�i for all i ∈ dom(t). For each branch x ∈ [U ] let g(x) ∈ ωω bethe increasing enumeration of the set

⋃i a(2t(i)) and find a superperfect tree S

such that [S] = g′′[U ]. For each branch x ∈ [U ] let h(x) ∈ ωω be the increasingenumeration of the set

⋃i a(2t(i) + 1) and find a superperfect tree T such that

[T ] = g′′[U ]. These two trees work.To see the first item, let y, z be some branches of S and T respectively.

By induction on j ∈ ω define sets b(2j) = (rng(y) \⋃i<j b(2i)) ∩min(rng(z) \⋃

i<j b(2i+1)), and b(2j+1) = (rng(z)\⋃i<j b(2i+1))∩min(rng(z)\

⋃i≤j b(2i)).

It is not difficult to see that the sequence b belongs to (a)ω and g(b) = y, z. Thesecond item is obvious from the construction.

The first item of the theorem is now easy. Let S, T be superperfect treesand [S]× [T ] =

⋃iBi be a partition into countably many Borel sets. We must

inscribe a product of superperfect trees into one of the Borel sets in the partition.Pulling back through the natural homeomorphisms of [S], [T ], and ωω, we mayassume that in fact S = T = ω<ω. Now use Theorem 5.11.2 to find a numberi ∈ ω and a dented strong Milliken cube C such that g′′C ⊂ Bi. It follows fromClaim 5.12.2 that Bi contains a product of superperfect trees as desired.

For the canonization, we will first introduce useful notation. (ω)<ω is the setof all finite sequences t of nonempty finite subsets of ω such that max(t(n)) <min(t(n + 1)) whenever n, n + 1 ∈ dom(t). If t ∈ (ω)<ω and a ∈ (ω)ω then[t, a] = {x ∈ (ω)ω : t is an initial segment of x, and the rest of x is below a inthe Milliken order}. [t, a]1 = {x ∈ (ω)ω : t � max(dom(t)) is an initial segmentof x, the next entry on x is the union of the last entry of t with some entries ona, and the rest of x is below a in the Milliken order}. If x ∈ (ω)ω and n ∈ ω thenx∩n ∈ (ω)<ω is the sequence described in the following way: if there is m suchthat x(m)∩n and x(m)\n are both nonempty sets, then t = x � ma〈x(m)∩n〉;if no such m exists, then t includes all entries of x which are subsets of n. Ift, s ∈ (ω)<ω and a ∈ (ω)ω, the the expression s ≤ [t, a] denotes the fact that t isan initial segment of s and the remaining entries of s are unions of some entriesof a.

Now let S, T ⊂ ω<ω be superperfect trees and E be a smooth equivalencerelation on [S] × [T ]. We must find superperfect subtrees s” ⊂ S and T ′ ⊂ Tsuch that E � ([S′] × [T ′]) is of the prescribed form. Pulling back the whole


context through the canonical homeomorphisms between ωω and [S], [T ], wemay assume that in fact S = T = ω<ω.

Let f : ωω × ωω → 2ω be a Borel function reducing the equivalence relationE to the identity. Repeating standard homogenization arguments, we can finda sequence a ∈ (ω)ω such that for every t ∈ (ω)<ω and every i ∈ ω there is atail b of a such that the bit f ◦ g(x)(i) is the same for all x ∈ [t, b], denotedas π0(t)(m), and the bit f ◦ g(x)(i) is the same for all x ∈ [t, b]1, denoted asπ1(t)(i). Thus, π0, π1 are both functions from (ω)<ω to 2ω.

For every t ∈ (ω)<ω consider the equivalence relation Et on [ω]<ℵ0 definedby u Et v if π(tau) = π(a v). Thinning out the cube a repeatedly, we mayassume that for every t ≤ a, Et has one of the following four forms:

• u Et v ↔ u = v–we will say that t is id-like;

• u Et v ↔ min(u) = min(v)–we will say that t is min-like;

• u Et v ↔ max(u) = max(v)–we will say that t is max-like;

• u Et v ↔ min(u) = min(v) and max(u) = max(v)–we will say that t ismin max-like;

• u Et v ↔ u = v–we will say that t is ee-like.

Define partitions qod and qev: qod(b) = 0 if there are infinitely many oddnumbers i such that b � i is not ee-like, and qod(b) = 1 otherwise; similarlyfor qev. Thinning out the sequence a if necessary, we may assume that a ishomogeneous for both partitions.

Case 1. The homogeneous color for one of the partitions qev, qod is 1; fordefiniteness assume that it is the partition qev. In this case, we will producesuperperfect trees S′ and T ′ such that vertical sections of [S′]× [T ′] are pairwiseE-unrelated. By induction on n ∈ ω, pick sets c(n) and numbers m(n) so that

I1 if t ≤ 〈c(i) : i ∈ n〉 and t is not ee-like then for every u ≤ 〈c(0), . . . c(n−1)the binary sequence π(tac(n)) is not equal to either π0(u) or π1(u);

I2 if tad ≤ 〈c(i) : i ∈ n〉 and t is not min or ee-like then same conclusion asin the previous item;

I3 the number m(n) is so large that the differences indicated in the previoustwo items appear already at entries smaller than m(n);

I4 for every sequence u ≤ 〈c(i) : i ≤ n〉, for every x ∈ [u, b] π0(u) � m(n) ⊂f ◦ g(x), and for every x ∈ [u, b]1 π1(u) � m(n) ⊂ f ◦ g(x).

This is not difficult to do. Consider the induction step to construct theset c(n) and the number m(n). There are only finitely many functions on thelist {π0(u), π1(u) : u ≤ 〈c(i) : i ∈ n〉}. On the other hand, the entries of thesequence a have pairwise distinct minima and maxima, so for every t as in thefirst (or similarly second) item, the functions π0(tau) : u ∈ rng(a) are pairwise

5.12. PRODUCTS OF SUPERPERFECT SETS 175

distinct, so there must be u ∈ rng(a) such that c(n) = a will satisfy I1 and I2.Any sufficiently large number m(n) will then work for I3. Lastly, find a smallenough tail of the sequence a so that I4 is satisfied.

Now find superperfect trees S′, T ′ as in Claim 5.12.2. These trees work asrequired. To verify this, let 〈x0, y0〉, 〈x1, y1〉 ∈ [S′] × [T ′] be two points in theproduct such that x0 6= x1; we must check that f(x0, y0) 6= f(x1, y1). Findsequences d0, d1 ≤ c such that 〈x0, y0〉 = g(e0) and 〈x1, y1〉 = g(e1). Sincex0 6= x1, the set

⋃i d0(2i) ∩

⋂i d1(i) is finite. The homogeneity property of

a implies that there is a finite sequence t, an initial segment of d0 of evenlength, which is not ee-like, and writing w for the set such that taw ⊂ d0,w ∩

⋃i d1(i) = 0. There are two distinct cases.

Case 1a. Suppose first that t is min-like. In such a case, let n0 < n1 <n2 < · · · < ni be an increasing sequence of numbers such that w =

⋃k≤i c(nk)

and look at the construction of the set c(n0). Let u = d1 � min(cn0). Then

f(x1, y1) begins either with π0(u) � m(n0) or π1(u) � m(n0) by I4 at n0. On theother hand, f(x0, y0) begins with π0(taw) � m(ni) by I4 at ni; as t is min-like,this is the same as π0(tac(n0)) � m(ni), which begins with π0(tac(n0)) � m(n0),and that sequence is distinct from both π0(u) � m(n0) and π1(u) � m(n0) byI2 at n0. Thus, f(x0, y0) 6= f(x1, y1), and the difference between these binarysequences appears already before m(n0). We are done in this case.

Case 1b. Suppose now that t is not min-like. Again, write w =⋃k≤i c(nk)

and look at the construction of the set c(ni). Let u = d1 � min(c(ni)). As inthe previous item, f(x1, y1) begins either with π0(u) � m(ni) or π1(u) � m(ni).On the other hand, f(x0, y0) begins with π0(taw) � m(ni) by I4 at ni, and thissequence is distinct from both π0(u) � m(ni) and π1(u) � m(ni) by I2 at ni.Thus, f(x0, y0) 6= f(x1, y1), and the difference between these binary sequencesappears already before m(ni). This completes the argument in Case 1.

Case 2. The homogeneous color of both partitions is 0. In this case, we willpresent a dented cube on which the function f ◦ g is constant. An applicationof Claim 5.12.2 will yield a product of superperfect trees on which E = ee,completing the proof.

An application of Theorem 5.11.2 shows that there is a dented cube [t, b]such that every s ≤ 〈t, b〉 is ee-like. To simplify the notation, assume t = 0.

The properness of the product was proved in ??? The dominating real is themaximum of the left generic and the right generic ???

The complexity estimate in the third item follows from [67, Proposition3.8.15]–if the quotient poset adds a dominating real then the σ-ideal cannot beΠ1

1 on Σ11.

Theorem 5.12.3. E0 is in the spectrum of J .

Proof. For any two increasing functions x, y ∈ ωω let f(x, y) = {n :there is anelement of rng(y) between x(n) and x(n + 1)} and let E be the equivalencerelation connecting 〈x0, y0〉 and 〈x1, y1〉 if their f -images have finite symmetricdifference. This is certainly a Borel equivalence relation reducible to E0; we willshow that it remains bireducible with E0 on every product of superperfect sets.


Let S, T ⊂ ω<ω be superperfect trees consisting of increasing functions. Wemust find a continuous reduction of E0 to E � [S]× [T ]. By induction on n ∈ ωbuild

• splitnodes sn of the tree S so that s0 ⊂ s1 ⊂ . . . ;

• splitnodes tv : v ∈ 2n of the tree T so that u ⊂ v → tu ⊂ tv;

• numbers i(n) and j(n) such that i(0) = max(dom(s0)) < j(0) < i(1) =max(dom(s1)) < j(1) < . . . , and for every n and every v ∈ 2n rng(tva0\tv)is a nonempty subset of the interval (sn+1(in), sn+1(in+1)), and rng(tva1\tv) is a nonempty subset of the interval (sn+1(jn), sn+1(jn + 1)).

This is not difficult to do. Let s0, t0 be arbitrary splitnodes of the respective treesS, T . Suppose that sn, tv : v ∈ 2n have been found. Let in = max(dom(sn)).For every v ∈ 2n find a splitnode tva0 in the tree T extending tv such that thenumber tva0(dom(tv)) is greater than all elements of rng(sn). Find a splitnodes′n of the tree S extending sn such that the number s′n(in + 1) is larger thanall elements of rng(tva0) for all v ∈ 2n. Let j(n) = max(dom(s′n)). For everyv ∈ 2n find a splitnode tva1 in the tree T extending tv such that the numbertva1(dom(tv)) is greater than all elements of rng(s′n). Find a splitnode sn+1

of the tree S extending s′n such that the number sn+1(j(n)) is larger than allelements of rng(tva0) for all v ∈ 2n. This completes the induction step.

Now for every z ∈ 2ω let h(z) = 〈⋃n sn,

⋃n tz�n〉. It is not difficult to

see that f ◦ h(z) = {in : z(n) = 0} ∪ {jn : z(n) = 1} modulo finitely manyexceptions caused by the interplay between s0 and t0. Thus, the continuousfunction z reduces E0 to E � [S]× [T ] as required.

Theorem 5.12.4. EKσ is in the spectrum of J .

5.13 Laver forcing

The Laver forcing P is the partial order of those infinite Laver trees T ⊂ ω<ω

such that there is a finite trunk and all nodes past the trunk split into infinitelymany immediate successors. The associated Laver ideal I on ωω is generatedby all sets Ag where g : ω<ω → ω is a function and Ag = {f ∈ ωω : ∃∞n f(n) ∈g(f � n)}.

Theorem 5.13.1. 1. Every analytic subset of ωω is either in I or it containsall branches of a Laver tree, and these two options are mutually exclusive;

2. the forcing P is < ω1-proper, preserves outer Lebesgue measure, and ad-joins a minimal generic extension;

3. I is a game ideal.

5.13. LAVER FORCING 177

Proof. The first item comes from [5], [67, Proposition 4.5.14]. The minimalityof Laver extension is due to Groszek [16, Theorem 7]; it can be also derivedfrom the general treatment in [67, Theorem 4.5.13]. The Laver game was in itssimplest form presented in [67, Section 4.5].

There are standard corollaries and remarks. The function π : P → PI definedby π(T ) = [T ], is a dense embedding of the poset P into PI . The study of theideal I is complicated by the fact that it is not Π1

1 on Σ11. Laver forcing adds

a dominating real; no ideal whose quotient forcing adds a dominating real canhave such a simple definition [67, Proposition 3.8.15]. It is well-known that theLaver forcing is nowhere c.c.c.: given a Laver tree T ⊂ ω<ω, for every splitnodet ∈ T write bt ⊂ ω for the infinite set {n ∈ ω : tan ∈ T}, fix an uncountablecollection {btα : α ∈ ω1} of pairwise almost disjoint infinite subsets of ω, and letAα = {x ∈ [T ]: for every n ∈ ω such that x � n is a splitnode of T , x(n) ∈ bx�nα }for every α ∈ ω1. The sets {Aα : α ∈ ω1} are easily checked to be I-positiveand their pairwise intersections are in the σ-ideal I. The computation of thespectrum of the Laver ideal is then facilitated by the general results such asTheorem 4.3.8 and Theorem 4.3.7 to yield

Corollary 5.13.2. The σ-ideal I has total canonization for equivalences clas-sifiable by countable structures.

Proposition 3.3.2 then yields

Corollary 5.13.3. Suppose that ω1 is inaccessible to the reals. Then the Laverideal has the Silver property for Borel equivalences classifiable by countable struc-tures.

Further canonization properties of the Laver ideal were discovered by MichalDoucha:

Fact 5.13.4. [10] The Laver ideal has total canonization for equivalence rela-tions Borel reducible to =J , where J is any Fσ P-ideal on ω. If ω1 is inaccessibleto the reals, then the Laver ideal has the Silver property for these equivalencerelations.

Question 5.13.5. Does the Laver ideal have total canonization for equivalencerelations Borel reducible to orbit equivalence relations of Polish group actions?

Further inquiry into the spectrum identifies a nontrivial feature:

Theorem 5.13.6. EKσ is in the spectrum of the Laver forcing.

Proof. Consider the equivalence relation E on the set of increasing functions inωω defined by x E y if and only if there is a number n ∈ ω such that for all m,x(m) < y(m + n) and x(m) < y(m + n). In other words, a finite shift of thegraph of x to the right is below the graph of y, and vice versa, a finite shift ofthe graph of x is below the graph of y.

It is not difficult to show that E ≤B E � B for every Borel E-positive set.Such positive set must contain all branches of some Laver tree T ; to simplify


the notation assume that T has empty trunk. Find a level-preserving injectionπ from the set of all finite increasing sequences of natural numbers into T sothat if the last number on s is less than the last number on t then also the lastnumber on π(s) is less than the last number on π(t). It is immediate that πextends to a continuous map which embeds E to E � B.

The last point is to show that EKσ is bireducible with E. To embed EKσin E, look at the two partitions ω =

⋃nKn =

⋃n Ln into consecutive intervals

such that |Kn| = n+ 1, |Ln| = 2(n+ 1) and for a point y ∈ dom(EKσ ) let f(y)be the function in ωω specified by f(y)(min(Kn) + i) = min(Ln) + i+ y(n) forevery n and i ∈ n. It is not difficult to see that f reduces EKσ to E. To embedE into EKσ , note that E =

⋃n Fn is a countable union of closed sets with

compact sections: just write Fn = {〈x, y〉: for every m ∈ ω, x(m) < y(m + n)and y(m) < x(m + n). Viewing ωω as a Gδ subset of a compact metric spaceY and consider the closures Fn of Fn in Y × Y . Since the vertical sections ofFn are compact, the closures Fn have the same sections corresponding to thepoints in ωω as the sets Fn. Thus, E, the relation on the space Y defined byy E z if either z, y /∈ ωω or there is n such that 〈y, z〉 ∈ Fn, is a Kσ equivalencerelation equal to E on ωω. By a result of Rosendal [44], it is Borel reducible toEKσ and the restriction of that Borel reduction reduces E to EKσ as desired.

Corollary 5.13.7. Whenever J is a σ-ideal on a Polish space X such that thequotient forcing PJ is proper and adds a dominating real, then EKσ is in thespectrum of the ideal J .

Proof. The assumptions imply that there is a Borel J-positive set B ⊂ X anda Borel function f : B → ωω such that for every point y ∈ ωω, the set {x ∈ B :f(x) does not modulo finite dominate y} is in the σ-ideal I. Let π : ω<ω → ωbe a bijection and let g : B → ωω be the Borel function defined inductively byg(x)(n) = f(x)(π(x � n)).

Observe that g-preimages of sets in the Laver σ-ideal I are in J . For, ifh : ω<ω → ω is any function and Ah ⊂ ωω is the associated generating setfor the Laver ideal, Ah = {y ∈ ωω : ∃∞n y(n) ∈ h(y � n)}, then the setBh = {x ∈ B : f(x) ◦ π modulo finite does not dominate h} is I-small by theproperties of the function h, and g−1Ah ⊂ Bh by the definitions.

Let F be the equivalence relation on B defined by x0 F x1 if and only ifg(x0) E g(x1), where E is the equivalence relation described in the previousproof. We will show that E witnesses the fact that EKσ is in the spectrum ofJ . It is clear that g reduces F to E, which is in turn reducible to EKσ . On theother hand, if C ⊂ B is a Borel J-positive set, then its g-image is an analyticI-positive set. A standard argument will then yield a Laver tree T ∈ P anda continuous function k : [T ] → C such that [T ] ⊂ G′′C and g ◦ k = id. Thecontinuous function k reduces E � [T ] to F � C, and since EKσ is reducibleto E � [T ] by the previous proof, we have that EKσ is reducible to F � C asdesired.

5.14. FAT TREE FORCINGS 179

Theorem 5.13.8. The Laver ideal has a coding function.

Proof. Consider the function f : (ωω)2 → 2ω defined by f(x, y)(n) = 1 ↔x(n) > y(n). This function works.

Every I-positive Borel set B ⊂ ωω contains all branches of some Laver treeT with trunk of length n, and then it is easy to find, for every binary sequencez ∈ ωω, two branches x, y ∈ [T ] so that f(x, y) = z on all entries past n. Sof ′′B2 contains a nonempty open set.

5.14 Fat tree forcings

The posets consisting of finitely branching trees with some condition enforc-ing very fast branching have been used in many independence results, such as[3]. While they are not well known outside of the forcing community, one cancalculate their associated ideals and prove canonization and anticanonizationresults.

Let Tini be a finitely branching tree without terminal nodes, and for everynode t ∈ Tini fix a submeasure φt on the set at of immediate successors of t inTini, such that limt φt(at) = ∞. Consider the fat tree forcing P of those treesT ⊂ Tini such that limt∈T φt(bt) =∞, where bt ⊂ at is the set of all immediatesuccessors of the node t in the tree T ; the ordering is that of inclusion.

This is a special case of the generalized Hausdorff measure forcings of [67,Section 4.4] or the liminf ∞ tree forcings of [?]. The computation of the asso-ciated ideal gives a complete information. The underlying space is X = [Tini]and the σ-ideal I on X is generated by all sets Af , where f is a function withdomain Tini associating with each node t ∈ Tini a set of its immediate suc-cessors of φt-mass ≤ nf for some number nf ∈ ω. The set Af is defined as{x ∈ X : ∃∞n x(n) ∈ f(x � n)}. The following sums up the work in [67]:

Theorem 5.14.1. Let φt : t ∈ Tini be submeasures such that limt φt(at) = ∞.Then

1. every analytic subset of X either is in the ideal I or contains all branchesof some tree T ∈ P , and these two options are mutually exclusive;

2. the ideal I is Π11 on Σ1

1;

3. I is a game ideal;

4. the forcing P is < ω1-proper, bounding, adds no independent reals, andadds a minimal forcing extension.

Proof. The first item comes from [67, Proposition 4.4.14], the second item comesfrom [67, Theorem 3.8.9]. The integer game associated with the ideal I is de-scribed in [67, Proposition 4.4.14]. The first three assertions in the last itemfollow from [67, Theorem 4.4.2, Theorem 4.4.8]. For the minimality of the exten-sion, the selection property and < ω1-properness imply that every intermediate


extension is c.c.c. by Theorem 3.2.3. Potential c.c.c. intermediate extensionswithout new reals are ruled out by Theorem 2.6.9, and intermediate c.c.c. ex-tensions with new reals are ruled out by Fact 2.6.6 as the poset P adds noindependent reals. Thus all intermediate forcing extensions must be trivial!

The usual corollaries apply. The map π : T 7→ [T ] is an isomorphism betweenthe poset P and a dense subset of the quotient poset PI , and Theorem 4.3.8and Proposition 3.3.2 yield

Corollary 5.14.2. The ideal I has the Silver property for Borel equivalencerelations classifiable by countable structures.

Further forcing and/or canonization properties very much depend on the choiceof the submeasures φt.

Theorem 5.14.3. There is a choice of submeasures such that the resulting idealI has the Silver property for all Borel equivalence relations.

Proof. For a submeasure φ on a finite set a let add(φ) be the minimum size ofa set B ⊂ P(a) such that φ(

⋃B) ≥ max{φ(b) : b ∈ B}+ 1. Submeasures with

large add can be obtained for example by starting with the counting measureand then inflicting on it the operation φ 7→ log(1 + φ) repeatedly many times.Now choose a tree Tini with the submeasures φt : t ∈ ω so that there is a linearordering of Tini in type ω in which add(φt) is much bigger than Πs|as| where sranges over all nodes of Tini preceding t in the linear order. We claim that thischoice of submeasures works.

In fact, we will show that the resulting σ-ideal I has the rectangular property.The Silver property then follows by the usual chain of implications: total can-onization for analytic equivalence relations is implied by Theorem 3.2.5 and theSilver property for Borel equivalence relations follows by Proposition 3.3.2. Analternative direct argument avoiding the use of Theorem 3.2.3 is also possible.

So let T, S ∈ P be fat trees and let [T ]× [S] =⋃nBn be a decomposition of

the rectangle into countably many Borel sets; we have to find fat trees T ′ ⊂ Tand S′ ⊂ S such that the product [T ′]× [S′] is wholly included in one piece ofthe partition. Thinning out the trees T and S if necessary we may assume thatthey have incomparable trunks. For every node t ∈ T ∪S beyond the trunks letbt be the set of immediate successors of t in T , equipped with the submeasureφt. Consider the space Y = Πt∈T bt and the continuous map f : Y → [S] × [T ]assigning to every point y ∈ Y the unique pair 〈x0, x1〉 ∈ X such that for all nbeyond the length of the trunk of the tree T , x0(n) = y(x0 � n), and similarlyon the x1-side. The creature forcing arguments of [50] show that the collectionJ of those Borel subsets of Y which do not contain a subset of the form Πtct,where the sets ct ⊂ bt are nonempty and the numbers φt(ct) tend to infinity, is aσ-ideal of Borel sets. Therefore, there are nonempty sets ct ⊂ bt such that theirmasses tend to infinity and the f -image of the product Πtct is wholly containedin some set Bn of the partition. Consider the trees S′ ⊂ S and T ′ ⊂ T thatat each of their nodes t branch into immediate successors from the set ct. A

5.14. FAT TREE FORCINGS 181

review of the definitions will show that S′, T ′ are fat trees and [S′]× [T ′] ⊂ Bnas desired.

Theorem 5.14.4. There is a choice of submeasures such that E2 and EKσ arein the spectrum of the associated σ-ideal.

In fact, it will be obvious from the construction that there are very many com-plicated equivalence relations on the underlying space X. For every Borel equiv-alence relation E there will be a Borel equivalence relation F on X such thaton every Borel I-positive set B ⊂ X, E ≤ F � B holds. In order to see this,just note that the equivalence relations =K , where K is a Borel ideal on ω andx =K y if {n : x(n) 6= y(n)} ∈ K for x, y ∈ 2ω, are cofinal in the reducibilityordering of Borel equivalence relations by Fact 2.2.12, and use these Borel idealsin place of J in the proof below.

Proof. Use the concentration of measure as in Fact 2.7.2. Find a fast increasingsequence of numbers ni : i ∈ ω so that in the Hamming cube 〈2ni , µi, di〉, forevery subset of P(2ni) of cardinality 2Πj∈i2

nj , consisting of sets of µni-massat least 1/i, there is a ball of radius 2−i intersecting them all. Consider thetree Tini consisting of all finite sequences t such that ∀i ∈ dom(t) t(i) ∈ 2ni ,and for every t ∈ T of length i let φt be the normalized counting measure on2ni multiplied by i. We claim that the fat tree forcing given by this choice ofsubmeasures works.

To get EKσ in the spectrum, let J be an Fσ-ideal on 2ω such that =J

is bireducible with EKσ as in the construction after fact 2.2.12. Define anequivalence E on [Tini] by setting x E y if there is a set a ∈ J and a numberk ∈ ω such that for every i ∈ ω \ a, 2idni(x(i), y(i)) < k. This is clearly Kσ, soreducible to EKσ . We will show that =J is Borel reducible to E � B for everyBorel I-positive set B ⊂ [Tini].

So suppose B is such a Borel I-positive set, and inscribe a fat tree T intoit. Find a node t ∈ T such that past t, the nodes of the tree T split into aset of immediate successors of mass at least 1. Let j = dom(t). The choice ofthe numbers ni shows that for every i > j there is a point pi ∈ 2ni such thatits neigborhood of dni-radius 2−i intersects all sets of immediate successors ofnodes in T extending the node t of length i, and also their flips (a flip of a subsetof 2ni is obtained by interchanging 0’s and 1’s in all of its elements). Now it iseasy to find a continuous function h : 2ω → [T ] such that for every x ∈ 2ω andevery i > j, h(x)(i) is some point within dni-distance 2−i from pi if x(i) = 0,and h(x)(i) is some point within dni-distance 2−i from the flip of pi if x(i) = 1.Note that the dni-distance of pi from its flip is equal to 1. This clearly meansthat for x, y ∈ 2ω, x =J y if and only if h(x) E h(y), and therefore EKσ is inthe spectrum.

To get E2 in the spectrum, consider the equivalence relation F on Tini definedby x F y if Σ{1/j + 1 : for some i ∈ ω, ni ≤ j < 2ni and x(i)(j − ni) 6=y(i)(j − ni)} < ∞. This is clearly reducible to E2; we must show that E2 isBorel reducible to F � B for every I-positive Borel set B ⊂ [Tini].


A direct construction of the embedding is possible by a trick similar to theprevious paragraph. We note that if E2 was not reducible to F � B, thenF � B would be essentially countable by the E2 dichotomy, the ideal I hastotal canonization for essentially countable equivalences, and so there would bea Borel I-positive subset consisting of either pairwise inequivalent or pairwiseequivalent elements. So it is enough to show that every Borel I-positive setB contains a pair of equivalent elements, and a pair of inequivalent elements.Inscribe a fat tree T into B, choose the node t and the points pi ∈ 2ni : i >dom(t) as in the EKσ case. Note that u, v ∈ 2ni are sequences such that u is2−i-close to pi and v is 2−i-close to the flip of pi, then Σ{1/j + 1 : ni ≤ j < 2niand u(i)(j − ni) 6= v(i)(j − ni)} > 1/2− 4 · 2−i, and if u, v are both close to pior both close to the flip of pi, then this sum is smaller than 4 · 2−i. Thus, withthe continuous function h constructed in the EKσ argument, h(x) F h(y) if andonly if x E0 y; in particular, there are both F -related and F -unrelated pairs in[T ] ⊂ B.

5.15 The Lebesgue null ideal

Let λ be the usual Borel probability measure on 2ω, and let I = {B ⊂ 2ω :λ(B) = 0}. The quotient forcing PI is the random forcing.

Proposition 5.15.1. E0 is in the spectrum of I.

Proof. The classical Steinhaus theorem [3, Theorem 3.2.10] shows that in everyBorel set of positive mass there are two distinct E0-related points. Thus, Borelpartial E0-selectors must have Lebesgue mass zero, and since every Borel seton which E0 is smooth is a countable union of partial Borel E0 selectors, allsuch sets must have Lebesgue mass zero. It follows that E0 is non-smooth onevery Borel set B ⊂ 2ω of positive mass, and by the Glimm-Effros dichotomy,E0 ≤ E0 � B as desired.

Proposition 5.15.2. E2 is in the spectrum of I.

Proof. This follows from the treatment of the ideal J associated with E2 ofSection 5.6. As Claim 5.6.9 shows, J ⊂ I, and so E2 is Borel reducible toE2 � B for every Borel non-null set B ⊂ 2ω.

Theorem 5.15.3. If E ⊃ E2 is an equivalence relation classifiable by countablestructures, then E has an equivalence class of λ-mass 1.

Proof. We have no direct proof of this remarkable statement. Rather, we willfind a fat tree forcing as in Section 5.14 such that the associated σ-ideal Jon 2ω is a subset of the null ideal, and the equivalence relation E2 is ergodicon I. Then, the canonization of equivalence relations classifiable by countablestructures on I as in Corollary 5.14.2 will yield the theorem.

To see this abstract argument, let E ⊃ E2 be an equivalence relation classi-fiable by countable structures. Observe that E can have at most one J-positive

5.15. THE LEBESGUE NULL IDEAL 183

class. If there were two such classes, then as E2 is ergodic for the σ-ideal J , theywould have to contain E2-connected points. Since E2 ⊂ E, these two pointswould have to be also E-connected, which is impossible if they come from dis-tinct E-classes. Let B be the complement of the single J-positive equivalenceclass; we will argue that the set B has zero mass. First of all, it is a coanalyticset and therefore measurable. If B had nonzero mass, it would contain a Borelsubset B′ ⊂ B of nonzero mass. The set B′ is J-positive and its E-equivalenceclasses are all in J . Since I has total canonization for equivalence relationsclassifiable by countable structures by Corollary 5.14.2, it must be the case thatB′ contains an J-positive Borel subset B′′ on which E is the identity. Again,this is impossible by the ergodicity of the equivalence relation E2 ⊂ E: dividingB′′ into two disjoint Borel J-positive pieces, they should contain E2-connectedpoints, which would be also E-connected. This contradiction will complete theproof of the theorem.

For the construction of the σ-ideal J ⊂ I, let ω =⋃n Jn be a partition

of ω into successive finite intervals so long that for every n ∈ ω, 1/n2 ≥e−2−n−1/8kn+1 . Here, kn = Σj∈Jn1/(j+1)2. Such a partition exists since the sumΣj∈ω1/(j + 1)2 converges. Consider the set 2Jn equipped with the normalizedcounting measure µn and the distance dn(x, y) = Σx(j) 6=y(j)1/j + 1. Fact 2.7.4shows that every subset of 2J−n of µn-mass ≥ 1/n2 has 2−n-neighborhoodof µn-mass greater than 1/2. As in Corollary 2.7.3, if A,B ⊂ 2Jn are twosets of normalized counting measure at least 1/n2, then they contain pointsx ∈ A, y ∈ B of dn-distance at most 2−n: the 2−n−1-neighborhoods of thesets A and B have normalized counting measure greater than 1/2 and thereforethey intersect. Replacing the set A with the flips of its points and repeatingthis argument, we also find other points x ∈ A, y ∈ B of dn-distance at least(Σj∈Jn1/j + 1)− 2−n.

Consider the fat tree forcing on the initial tree Tini = Πn2Jn , and the sub-measure φn is simply the normalized counting measure on 2Jn multiplied by n2.Clearly, every branch through the tree Tini can be identified with the union ofthe labels on its nodes, which is an element of 2ω. With this identification, thegenerating sets of the associated σ-ideal I have µ-mass 0 by the Borel-Cantellilemma, since the sum of the numbers 1/n2 converges. And just as in the proofof Theorem 5.14.4, one can show that every two trees T, S ∈ P contain branchesx ∈ [T ], y ∈ [S] which are E2-related, and branches x ∈ [T ] and y ∈ [S] that areE2-unrelated.

Proposition 5.15.4. The spectrum of I is cofinal in the Borel equivalencerelations under the Borel reducibility order, and it includes EKσ .

Proof. Look at the space X = (2ω)ω with the product Borel probability measureinstead. Consider the ideal J associated with the product of Sacks forcing, asin Section 5.7. Then J ⊂ I, the ideal J is suitably homogeneous, and so thespectrum of J is included in the spectrum of I. This proves the proposition.

The equivalence F2 belongs to the spectrum of the null ideal, and there is an


ergodicity theorem parallel to Theorem 5.1.27. Consider the infinite product µof countably many copies of the measure λ; so µ is a Borel probability measureon the space X = (2ω)ω. Let I be the σ-ideal on the space X consisting of setsof µ-mass 0. The poset P = PI adds a generic point xgen ∈ X. Instead of theproduct forcings Pn for n ∈ ω we will utilize the posets P⊗n = PIn where In isthe σ-ideal on Xn consisting of the set of µn-mass zero, where µn is the productmeasure on Xn.

Definition 5.15.5. Let M be a transitive model of set theory and a ⊂ 2ω bea set. We will say that a is a random-symmetric set for M if no finite tuple ofdistinct elements of a belongs to any null set coded in M , and moreover, forevery finite set b ⊂ a, every m ∈ ω and every Borel set B ⊂ (2ω)m of nonzeromass coded in the model M [b], there is an m-tuple c consisting of elements ofa such that c ∈ B. A finite collection ai : i ∈ n of subsets of 2ω is mutuallyrandom-symmetric over M if no finite tuple of elements of

⋃i ai belongs to any

null set coded in M , and for every finite set b ⊂⋃i ai, every m ∈ ω, every Borel

set B ⊂ (2ω)m of nonzero mass coded in the model M [b] and every i ∈ n, thereis an m-tuple c consisting of elements of ai such that c ∈ B.

In the study of random symmetric sets, the poset Qn is helpful. It is the directlimit of the sequence 〈Qin : i ∈ ω〉, where Qin is the poset of Borel subsets of(2ω)n×i of nonzero mass, where the measure on (2ω)n×i is just the usual productmeasure of ni copies of λ. It is clear that Qn is a c.c.c. poset adding pointsxj,i : j ∈ n, i ∈ ω such that any finite tuple among them is random generic overthe ground model. It is important to distinguish between the posets P⊗n andQn.

Claim 5.15.6. Let M be a countable transitive model of set theory, and letai : i ∈ n be countable subsets of 2ω. The following are equivalent:

1. the sets ai : i ∈ n are mutually random symmetric over M ;

2. there is a sequence 〈xi : i ∈ n〉 of points in the space X which is Qn-genericover M and for every i ∈ n, rng(xi) = ai.

3. there is a sequence 〈xi : i ∈ n〉 of points in the space X which is P⊗n-generic over M and for every i ∈ n, rng(xi) = ai.

Moreover, if (1) or (2) hold, if yi : i ∈ n are any one-to-one enumerations ofthe sets ai and q ∈ Qn ∩ M is a condition, the set {~g ∈ Sn∞: the sequence〈yi ◦ gi : i ∈ n〉 is Qn-generic over M and meets q} is nonmeager in the productSn∞.

Corollary 5.15.7. In the poset P⊗n, the sentences of the forcing languagementioning only rng(xi) : i ∈ n and some elements of the ground model aredecided by the largest condition.

Corollary 5.15.8. If M is a transitive model of set theory and bi : i ∈ n arenonempty subsets of X such that every finite tuple of distinct elements of

⋃i bi is

5.15. THE LEBESGUE NULL IDEAL 185

P⊗n-generic over the model M , then the sets⋃x∈bi rng(x) : i ∈ n are mutually

random-symmetric.

Theorem 5.15.9. Let E be an analytic equivalence relation on X = (2ω)ω suchthat F2 ⊂ E. Either F2 ≤B E or else E has a co-null equivalence class.

Proof. The argument follows the lines of Theorem 5.1.27. There are cases 1through 4 just as in that proof. The first three are deal with in exactly thesame fashion, replacing product of Cohen forcings with random algebras P⊗n.

Case 4 (the negation of the first three cases) must be handled through thefollowing well known (?) claim:

Claim 5.15.10. For every countable elementary submodel M of a large enoughstructure, there is a perfect subset of X such that every finite tuple of its pairwisedistinct points is P⊗n-generic over M for the suitable n.

Proof. Let Q be the poset consisting of all Borel sets B such that for somen = n(B), B ⊂ X2n has nonzero product mass. Let C ≤ B if n(C) ≥ n(B) andfor every function π : 2n(B) → 2n(C) such that ∀s s ⊂ π(s), it is the case that∀x ∈ C π−1x ∈ B, where π−1x = y if y ∈ X2n is the unique element such thaty(s) = x(π(s)).

Let G be a Q-generic filter over the model M . For every y ∈ 2ω, let Gybe the collection of those Borel sets B ⊂ X in M such that for some numbern ∈ ω, the condition {x ∈ X2n : x(y � n) ∈ B} belongs to the filter G. We mustargue that for distinct elements yk : k ∈ m of 2ω, the filters Gyk : k ∈ m areP⊗m-generic over M . Once this is done, the claim follows by considering theset of all points in X corresponding to some filter Gy : y ∈ 2ω.

Let B ∈ Q be a condition, m ∈ ω be a number smaller than n = n(B),and let D be a dense set in the random algebra on Xm. We will produce a setC ≤ B such that for every set a ⊂ 2n of size m and every map π : a → m,writing Ca,π = {x ∈ Xm : ∃y ∈ C∀s ∈ a y(s) = x(π(s))}, it will be the casethat Ca,π ∈ D. An obvious density argument will then finish the proof.

The construction of the set C ⊂ B is straightforward. List all the finitelymany possibilities for a, π as ai, πi : i ∈ j and by induction on i ∈ j + 1 builda decreasing chain of conditions Bi below B = B0 such that n(Bi) = n andBi+1ai,πi ∈ D. The condition C = Bj will then be as desired. To obtain the set

Bi+1 from Bi, let A = {x ∈ Xm: the set {y ∈ X2n\ai : y ∪ (x ◦ pii) ∈ Bi}has nonzero measure}. This is a Borel set by [33, Theorem 17.25]; the Fubinitheorem shows that it has nonzero mass. Let A′ ⊂ A be a set in D and letBi+1 = {y ∈ Bi : y ◦ πi ∈ A′}.

Now let M be a countable elementary submodel of a large structure, andlet P ⊂ X be a perfect set consisting of P -generic sequences over M . LetD ⊂ Pω ×X be the Borel set consisting of pairs 〈z, x〉 such that x enumeratesthe countable set

⋃n rng(z(n)). We will verify for any two pairs 〈z0, x0〉 and

〈z1, x1〉 in the set D, x0 E x1 if and only if rng(z0) = rng(z1). The proof is thencompleted by a reference to Corollary 5.1.30.


Now, if rng(z0) = rng(z1) then x0, x1 enumerate the same set, and sinceF2 ⊂ E, they must be E connected as desired. On the other hand, assume thatrng(z0) 6= rng(z1); there are several cases depending on the relationship of thetwo ranges. Assume for example that rng(z0) ∩ rng(z1), rng(z0) \ rng(z1), andrng(z1)\ rng(z0) are all nonempty sets. Then the sets a0, a1, a2 ⊂ 2ω defined bya0 =

⋃{rng(y) : y ∈ rng(z0) ∩ rng(z1)}, a1 =

⋃{rng(y) : y ∈ rng(z0) \ rng(z1)}

and a2 =⋃{rng(y) : y ∈ rng(z1) \ rng(z0)} are mutually random symmetric,

and since Case 2 fails, they are not E-equivalent as desired.

Proposition 5.15.11. F2 is in the spectrum of the null ideal.

Proof. Consider the product measure µ on the space X = (2ω)ω. Proceed as inthe fourth case of the above proof with E = F2. Clearly, the first three casesare impossible. Let B ⊂ X be a Borel nonnull set, and let M be a countableelementary submodel of a large structure containing B. Let f : X → X bea Borel function preserving F2 such that rng(f) consists of enumerations ofrandom symmetric sets for M . Let D ⊂ X ×X be the set {〈x, y〉 : y ∈ B andrng(f(x)) = rng(y)}. Corollary 5.1.30 applied to the Borel set D shows thatF2 ≤B F2 � B as required.

5.16 Finite product

???Needs a rewrite???

Fact 5.16.1. [67, Theorem 5.2.6] If the σ-ideals in the collection {In : n ∈ ω}are Π1

1 on Σ11, the quotient forcings are proper, bounding, and preserve the

Baire category, then the collection has the rectangular Ramsey property, and theproduct ideal inherits all mentioned properties as well.

The canonization properties of the product ideals may be difficult to identify.We will start with the finite products; the infinite products are much morechallenging, and they will be treated at the end of this section. Even in the caseof finite product of length n ∈ ω, there are obvious obstacles to canonization: ifFi : i ∈ n are equivalence relations on the coordinates of the product that resistfurther simplification, their product

∏i Fi will be impossible to simplify further

on the product. The natural hope is that there are no other obstacles. In thissection, we will exhibit three cases in which this hope comes true.

Theorem 5.16.2. Let n ∈ ω and {Ii : i ∈ n} be Π11 on Σ1

1 σ-ideals on therespective Polish spaces {Xi : i ∈ n} σ-generated by closed sets, with the coveringproperty. Let J denotes the product ideal on the product space. Then for everysequence 〈Bi : i ∈ n〉 of respectively Ii-positive Borel sets and every analyticequivalence E on

∏iBi there are Borel Ii-positive sets Ci ⊂ Bi such that E �∏

i Ci =∏i Fi where each Fi is either id or ee.

5.16. FINITE PRODUCT 187

Proof. First observe that the quotient posets are proper, preserve Baire cat-egory, and have the continuous reading of names by [67, Theorem 4.1.2]. Inthis context, the covering property is equivalent to the bounding property [67,Theorem 3.3.2], and so by the above Fact, the collection of σ-ideals has therectangular Ramsey property, and the product P is proper. Now, let a ⊂ nbe a set. The first step is to compute the reduced product associated with theequivalence Ea connecting two sequences if they have the same entries on theset a. Let Pa be the product of PIi : i ∈ a with PIi × PIi : i /∈ a. We willorganize the generic sequence into two, ~xlgen ∈ (2ω)n and ~xrgen ∈ (2ω)n whichuse the PIi-generic point for i ∈ a, and the left or right PIi-generic point fori /∈ a; thus, ~xlgen Ea ~xrgen.

It is not difficult (and not necessary for the proof of the theorem) to showthat Pa is naturally isomorphic to a dense subset of the reduced product P×EaPvia the map π : r 7→ 〈〈r(i) : i ∈ a, rleft(i) : i /∈ a〉, 〈r(i) : i ∈ a, rright(i) : i /∈ a〉〉.Note that the range of π is a subset of P ×Ea P and π preserves the ordering.To prove the density of the range of π, note that if 〈p, q〉 ∈ P ×Ea P is acondition (a pair of conditions in P such that some large forcing introduces aP -generic sequence to each which are Ea-related), then for every i ∈ a the setp(i) ∩ q(i) must be an Ii-positive Borel set, and so the sequence r given byr(i) = p(i) ∩ q(i) if i ∈ a and rleft(i) = p(i) and rright(i) = q(i) if i /∈ a is anelement of Pa mapped by π below the pair 〈p, q〉.

The next step is a version of the mutual genericity for the product.

Lemma 5.16.3. Let M be a countable elementary submodel of a large structure,and let {Bi : i ∈ n} be Ii-positive Borel sets in the model M . Let a ⊂ n be aset. There are Ii-positive Borel subsets {Ci ⊂ Bi : i ∈ n} such that the productΠi∈nCi consists of pairwise reduced product generic sequences for the model M .

This is to say that any two sequences ~x, ~y ∈ Πi∈nCi satisfying x(i) = y(i) ↔i /∈ a, are reduced product generic for the model M .

Proof. We must first introduce the game-theoretic characterization of the forc-ing properties of the quotient posets PI as described in [67, Section 3.10.10].The game GI associated with I between Players I and II starts with PlayerII indicating an initial I-positive Borel set Bini; then the game has ω manymegarounds, and each of the megarounds consists of finitely many rounds, ineach of which Player I plays an I-positive Borel set and Player II answers withits Borel I-positive subset. At some point of his choice, Player I decides to endthe megaround and pass to a next one. Given a play p of finite length at theend of which Player I ends its last megaround, write [p] for the intersection ofBini with all the sets obtained as unions of all Player II’s answers in a fixedmegaround of p; similar usage will prevail for infinite plays p. Player I wins aninfinite play p if he ended infinitely many megarounds and [p] /∈ I. In whatfollows, the letters p, q, . . . will denote either infinite plays of this game or finiteplays at the end of which Player I decided to end the current megaround.


Fact 5.16.4. [67, Theorem 3.10.24] Suppose that the σ-ideal I is Π11 on Σ1

1

and the quotient poset is proper. The following are equivalent:

1. PI preserves Baire category and is bounding;

2. Player I has a winning strategy in the game G.

We conclude that for every i ∈ n Player I has a winning strategy σi in the gameGIi . This provides ammunition for various fusion arguments like the followingclaim:

Claim 5.16.5. Let pi : i ∈ n be finite plays according to the respective strategiesσi, and let D be an open dense subset of

∏i PIi . Then there are extensions p′i

of pi such that the set∏i[p′i] is covered by finitely many elements of D.

Now, write n× ω<ω =⋃m Tm as an increasing union of finite sets Tm such

that for every m, i the set {t : 〈i, t〉 ∈ Tm} is closed under initial segments. Byinduction on m build functions fm with the domain Tm such that

I1 for every 〈i, t〉 ∈ Tm the value fm(i, t) is a finite play in the model Mrespecting the strategy σi, and fm(i, 0) starts with Bi;

I2 for m < m′ and 〈i, t〉 ∈ Tm, the play fm′(i, t) extends fm(i, t);

I3 for if 〈i, t〉 ∈ Tm and 〈i, taj〉 ∈ Tm+1 \ Tm and [fm(i, t)] has nonemptyintersection with the j-th open set in some fixed enumeration of a basisfor the space Xi, then [fm+1(i, taj)] is a subset of this intersection;

I4 whenever a ⊂ n is a set, and {〈i, ti〉 : i ∈ a} ∪ {〈i, t0i 〉, 〈i, t1i 〉 : i /∈ a} arenodes of Tm such that t0i 6= t1i for all i /∈ a, then the product

∏[fm(i, ti)]×∏

i/∈a[fm(i, t0i )]×∏i∈a[fm(i, t1i )] is covered by finitely many conditions in

them-th dense subset of Pa in the modelM under some fixed enumeration.

The induction process is trivial, using the claim repeatedly to assure that I4holds. In the end, for every i ∈ n and every y ∈ ωω look at the filter generatedby the conditions {[fm(i, y � k)] : m is the least such that 〈i, y � k〉 ∈ Tm}. Thisis an M -generic filter for PI and so there is a unique point gi(y) ∈ Bi in it. Wewill first argue that rng(gi) is an Ii-positive set.

Indeed, let {Ak : k ∈ ω} be closed sets in the σ-ideal Ii; we must producey ∈ ωω such that gi(y) /∈

⋃k Ak. By induction build an increasing sequence

of nodes tk ∈ ω<ω of length k such that [fm(i, tk)] ∩ Ak = 0, where m is thethe least such that 〈i, tk〉 ∈ Tm. The induction step is arranged by notingthat [

⋃m fm(i, tk)] is an Ii-positive set and therefore has Ii-positive intersection

with some basic open set disjoint from Ak (this is the only place where we usethe assumption that the generators of Ii are closed). If this basic open set is

enumerated by some j then the induction hypothesis at k + 1 with tk+1 = tak jwill hold. In the end, let y =

⋃k tk and observe that gi(y) /∈

⋃k Ak as desired.

Now let Ci be some Borel Ii-positive subset of the analytic Ii-positive setrng(gi). The induction hypothesis I4 above immediately implies that the setsCi have the required mutual genericity property.

5.17. THE COUNTABLE SUPPORT ITERATION IDEALS 189

The theorem immediately follows from the lemma. Let E be a Borel equiv-alence relation on the space

∏iXi, and find an inclusion minimal set a ⊂ n

such that some condition in Pa forces ~xlgen E ~xrgen. If a condition p =〈Bi : i ∈ (n \ a), Bi : i ∈ a,Ci : i ∈ a〉 in Pa forces this, then so doesq = 〈Bi : i ∈ (n \ a), Bi : i ∈ a,Bi : i ∈ a〉: whenever ~xlgen, ~xrgen are genericsequences meeting the condition q, in a further generic extension one can find acondition ~ygen such that the pairs ~xlgen, ~ygen and ~xrgen, ~ygen are generic sequencesmeeting the condition q, so these pairs consist of E-equivalent sequences by theforcing theorem, and consequently ~xlgen E ~xrgen by the transitivity of the re-lation E. Now let M be a countable elementary submodel of a large structureand use the claim to find sets Ci ⊂ Bi : i ∈ n whose product consists ofpairwise reduced product generic sequences for the model M . We claim thatE � Πi∈nCi = Ea.

To show this, suppose that ~x, ~y ∈ Πi∈nCi are sequences and let b = {i ∈ n :~x(i) = ~y(i)}. Find sequences ~x′, ~y′ in the product such that a = {i ∈ n : ~x(i) =~x′(i)} = {i ∈ n : ~y(i) = ~y′(i)} and a ∩ b = {i ∈ n : ~x′(i) = ~y′(i)}. The forcingtheorem applied to the reduced product Pa together with Borel absolutenessimplies that ~x E ~x′ and ~y E ~y′. Now if a ∩ b = a then the forcing theorem alsoimplies that ~x′ E ~y′ and so ~x E ~y. on the other hand, if a ∩ b 6= a then theminimal choice of the set a together with the forcing theorem applied to Pa∩bgive ¬~x′ E ~y′ and ¬~x E ~y!

Things get complicated very quickly beyond the relatively simple case ofthe previous theorem. We are still able to deal with the smooth σ-ideal ofSection 5.4.

5.17 The countable support iteration ideals

The countable support iteration is a central operation on forcings since thepioneering work of Shelah [49]. It corresponds to a certain operation on σ-ideals, the transfinite Fubini power, as shown in [67]. The natural question thenarises whether this operation preserves natural canonization properties of theσ-ideals in question.

The question may be natural, but it seems to be also very hard, even foriterations of length two in the general case. We will prove a rather restrictivepreservation result–Theorem 5.17.12. For Π1

1 on Σ11 σ-ideals I whose quotient

forcing PI has a rather strong preservation property (the weak Sacks property),the equivalence relations reducible to EKσ in the countable support iterationsimplify down to an unavoidable and fairly simple collection of obstacles to can-onization. The weak Sacks property may look quite restrictive, but in practice agood number of forcings satisfy it, it is easy to check and some forcings with thisproperty have nontrivial features in the spectrum–the E0, E2, or Silver forcingsof previous sections do satisfy the weak Sacks property. We will also show that


in a certain more general case, very complex equivalence relations arise alreadyat the first infinite stage of the iteration; on the other hand, in the countablesupport iterations of Sacks forcing a much stronger canonization result holds.

5.17a Iteration preliminaries

Let us first recall the analysis of the countable support iteration of definableproper forcing as introduced in [67, Section 5.1]:

Definition 5.17.1. [67, Definition 5.1.1] Let X be a Polish space and I a σ-idealon X. Let α ∈ ω1 be a countable ordinal. The space Xα of all α-sequences ofelements of X is equipped with the product topology. The α-th Fubini iterationIα of I is the σ-ideal on the space Xα generated by sets Af , where f : X<α → Iis an arbitrary function and Af = {~x ∈ Xα : ∃β ∈ α ~x(β) ∈ f(~x � β)}.

Definition 5.17.2. Let I be a σ-ideal on a Polish space X and J a σ-ideal ona Polish space Y . The symbol I ∗ J stands for the Fubini product of I and J :I ∗ J = {B ⊂ X × Y : {x ∈ X : {y ∈ Y : 〈x, y〉 ∈ B} /∈ J} ∈ I}}.

If I, J are Π11 on Σ1

1 σ-ideals such that the quotient forcings are properand every analytic positive set contains a Borel positive subset, then the two-step iteration of posets PI and PJ is naturally equivalent to the quotient PI∗J .This pattern persists to transfinite iterations. We will need a canonical form ofIα-positive Borel set:

Definition 5.17.3. [67, Definition 5.1.5] An I, α-tree is a tree p ⊂ X<α suchthat each node extends into an I-positive set of immediate successors and thetree is closed under unions of inclusion-increasing sequences of nodes whoseheight is bounded below α.

Fact 5.17.4. [67, Section 5.1.3] Suppose that

1. the σ-ideal I is Π11 on Σ1

1 and every analytic I-positive set contains anI-positive Borel subset;

2. the quotient forcing PI is provably proper.

Then

1. the σ-ideal Iα is Π11 on Σ1

1, and every analytic Iα-positive set contains anIα-positive Borel subset;

2. the coutable support iteration PαI of PI of length α is in the forcing senseequivalent to the quotient PIα ;

3. every Iα-positive analytic set contains all branches of a Borel I, α-tree.

If suitable large cardinals exist, the above fact holds even for σ-ideals with def-inition more complicated than Π1

1 on Σ11, obviously except for the part of the

conclusion that evaluates the complexity of the Fubini iteration ideal. This


makes it possible to apply the analysis to forcings such as Laver or Mathias,whose associated ideals are more complicated. Large cardinals or other assump-tions transcending ZFC are necessary for the argument to go through in the caseof Laver forcing. The countable support iterations in which the iterands varycan be handled in a quite similar way, and we will not bother to explicitly statethe trivial adjustments necessary.

Let I be a σ-ideal on a Polish space X and let α ∈ ω1 be a countableordinal. There are obvious obstacles to canonization of analytic equivalencerelations in the iterated Fubini power Iα on the product Polish space Xα. Letβ ≤ α be an ordinal; define idβ to be the identity on Xβ and idβ × ee to be theequivalence relation on Xα that relates the sequences with identical initial β-segments. These are smooth equvalence relations, and their associated modelscorrespond to the initial segment of the iteration of length β. Also, if E isan analytic equivalence relation on the space X which cannot be simplifiedconsiderably on a Borel I-positive set, we must consider the equivalences idβ ×E × ee for each β ∈ α. These are defined as ~x (idβ × E × ee) ~y if ~x � β = ~y � βand ~x(β) E ~y(β). Other obstacles are not immediately apparent, but theyseem to be difficult to exclude even already in the case of two step iteration!The main result of this section, Theorem 5.17.12, asserts that in a certain fairlyspecial case, there are no other obstacles to canonization of equivalence relationsreducible to EKσ .

5.17b Weak Sacks property

In order to state and prove our main result, we must define a rather commonpreservation property, provide its strategic characterizations, and apply themto prove common preservation theorems.

Definition 5.17.5. A forcing P has the weak Sacks property if for every func-tion f ∈ ωω in the extension there is a ground model infinite set a ⊂ ω and aground model function g with domain a such that for every n ∈ a, |g(n)| ≤ 2n

and f(n) ∈ g(n).

This is the obvious weakening of the Sacks property that requires the in-finite set a to be actually equal to ω [3, Definition 6.3.37]. It is not difficultto show that a great number of posets have the weak Sacks property, such asevery quotient PI if I is σ-generated by a σ-compact family of compact sets ofthe ambient space, the limsup creature forcings of [?] , the E0 forcing, the E2

forcing, or the Silver forcing. The weak Sacks property implies the boundingproperty and preservation of Baire category, which provides a host of nonexam-ple posets. The poset of σ-splitting sets [58] is proper, bounding, and preservesBaire category, and still does not have the weak Sacks property. It follows from[68] that if I is a Π1

1 on Σ11 σ-ideal such that the quotient PI is bounding, then

PI preserves P-points if and only if PI adds no independent reals and has theweak Sacks property.

In order to state and prove our main result, we cannot avoid a closer studyof the weak Sacks property. Our central reference for definable forcing [67]


does not mention it, since this is a new development; thus, we will have toprovide the basic proofs. As is the case with other standard forcing preservationproperties [67, Section 3.10], the treatment with the weak Sacks property hingeson its strategic characterizations. We will introduce three different games whichcharacterize the weak Sacks property in definable forcing. The first one speaksdirectly about posets and names for natural numbers, the last two use a quotientform of the poset in question.

Definition 5.17.6. Let P be a partial order. The game G1(P ) between playersI and II proceeds in infinitely many rounds in the following way. Player II firstindicates an initial condition pini ∈ P . After that, in round n Player I indicatesa natural number in ∈ ω, Player II plays a P -name τn for a natural number,and Player I answers with a set an ⊂ ω of size in. In the end, Player I wins ifthere is a condition q ≤ pini forcing τn ∈ an for all n ∈ ω.

Definition 5.17.7. Let I be a σ-ideal on a Polish space X. The game G2(I)between Players I and II proceeds in infinitely many rounds. In the beginning,Player II indicates an I-positive Borel set Kini ⊂ X. At round n, Player I playsa finite set Kn ⊂ Kini and Player II answers with an open set On ⊂ X. Theplayers must observe the rules Kn ⊂ On and Kn+1 ⊂ On. Player I wins if theset Kini ∩

⋂nOn is I-positive.

Definition 5.17.8. Let I be a σ-ideal on a Polish space X. The game G3(I)between Players I and II proceeds in the same way as G2(I), except in roundn Player II first chooses a set An ∈ I, Player I answers with a finite set Kn ⊂Kini \An and finally Player II indicates an open set On ⊂ X.

In the last two games, Player II will certainly not hurt himself by playingsmaller open sets, so we may assume that his open sets come from some fixedcountable basis closed under finite intersections, and the closure of On+1 is asubset of On, and each point of On is within 2−n-distance of some point of Kini.That way, the result of the play, the set

⋂nOn, is a compact subset of Kini.

Theorem 5.17.9. Suppose that I is a σ-ideal on a Polish space X such thatthe quotient forcing PI is proper and bounding. The following are equivalent forl = 1, 2, 3:

1. PI has the weak Sacks property;

2. Player II has no winning strategy in the game Gl(I).

Proof. Start with l = 1. The nonexistence of a winning strategy for Player IIin G1(I) easily leads to the weak Sacks property; given a name f for a functionin ωω and a condition p ∈ PI , consider the strategy of Player II consistingof playing pini = p and f(m) whenever Player I plays the natural number m.This is not a winning strategy, and the resulting condition of any counterplaywinning for Player I will force the function f into a tunnel of the required sort.On the other hand, the weak Sacks property implies that Player II cannot have


a winning strategy in G1(PI). If σ were such a putative winning strategy withan initial move pini ∈ P , consider the name f which assigns each m the finitesequence of all names the strategy σ answers to all finite plays of length ≤ min which all natural number moves of Player I were of size ≤ m and all of hisset moves were subsets of m. Use the weak Sacks property to find a conditionq ≤ pini, an infinite set a ⊂ ω and a ground model function h with domain asuch that for every m ∈ a, the size of h(m) is at most 2m and q f(m) ∈ h(m)(as a finite sequence of natural numbers). Now by induction build counterplayst0 ⊂ t1 ⊂ . . . against the strategy σ such that ti ends at round i with movesmi, bi of Player I and τi of Player II, and q τi τi ∈ bi. This way, PlayerII must win as witnessed by the condition q. The induction step is simple:once ti has been constructed, find m ∈ a larger than the maximum of all setsand numbers played along ti, let Player I extend ti by mi+1 = m, and whenthe strategy σ answers with τi+1, let bi+1 be the set of all numbers that thefunction h permits as possible values of the name τi+1. The set τi+1 is of sizeat most 2m, and the induction can proceed.

The case l = 2 is strictly easier than l = 3, and the game G2 is mentionedhere only because it also appeared in the pivotal paper [69]. The game G3 seemsto be much more difficult for Player I than G2; essentially we are saying thatin game G2 Player I has very many winning moves in each round. Suppose forcontradiction that PI has the weak Sacks property and Player II has a winningstrategy σ in the game G3, beginning with an initial move Kini. Let M bea countable elementary submodel of a large structure containing all objectsmentioned so far, and let L ⊂ Kini be a compact I-positive set consisting ofM -generic points only, such that all sets in M are relatively clopen in L. Such aset exists by the properness and boundedness assumption [67, Theorem 3.3.2].

Consider a natural number m ∈ ω and a finite partial play t ∈ M in whichPlayer II follows the winning strategy, ending with some open set On−1 and anI-small set An. Since An ∈ M , any m-tuple from the compact set L ∩ On−1

is a legal move of Player I after t. A compactness argument with the space(L ∩ On−1)m shows that there is a finite collection R(t,m) consisting of basicopen subsets of X such that every 2m-tuple of L∩On−1 is contained in one of thesets in R(t,m) and each of the sets in R(t,m) can be provoked as the next moveof the strategy σ after some m-tuple of points in L∩On−1 is played as the nextmove of Player I. Let S(t,m) be the set of open sets with nonempty intersectionwith L such that each set in R(t,m) is either disjoint from them or a supersetof them. Observe that whenever {Pi : i ∈ 2m} are sets in S(t,m) then there isa move Kn of Player I which provokes the strategy σ to answer with a set Onwhich is a superset of

⋃i Pi: just pick a point xi ∈ Li ∩Pi for each i ∈ 2m, note

that there must be an open set O ∈ R(t,m) with {xi : i ∈ 2m} ⊂ O, and since∀i ∈ 2m O ∩ Pi 6= 0, it must be the case that ∀i ∈ 2m Pi ⊂ O,

⋃i Pi ⊂ O, and

O = On will work. By the elementarity of the model M , there must be a playin the model M extending t by one round such that its last move covers

⋃i Pi.

Enumerate all finite plays in the model M as 〈tk : k ∈ ω〉, and let f bethe name from ω to basic open subsets of X defined by f(m) =some sequence


〈Pk : k ∈ m〉 such that ∀k Pk ∈ S(tk,m) ∧ xgen ∈ Pk. By the weak Sacksproperty, there is a compact I-positive set L′ ⊂ L, a ground model infinite seta ⊂ ω and a ground model function h such that for every m ∈ a, |h(m)| = 2m

and L′ f(m) ∈ h(m). We will show that every finite play t ∈ M followingthe strategy σ whose last open set move covers L′ can be extended by one moreround to a play in M following σ such that the last open set move still coversL′. This will yield an infinite play against the strategy σ whose resulting setcovers L′, so is I-positive, contradicting the assumption that σ was a winningstrategy. So, find a number m ∈ a larger than some k such that t = tk. Thecollection A = {O ∈ S(tk,m) : O is k-th element of some sequence in h(m)} hassize at most 2m, and L′ ⊂

⋃A. As the previous paragraph shows, there is a

move in the model M that, after t has been played, provokes the strategy σ torespond with a superset of

⋃A, and that is our required one round extension

of the play t.

Theorem 5.17.10. Suppose that I is a Π11 on Σ1

1 σ-ideal on a Polish spaceX such that the quotient forcing PI is proper and bounding. Then the gamesG1(PI), G2(I), G3(I) are all determined.

Proof. This is just a standard unraveling argument as those in [67, Section3.10].

Thus, we can conclude that the weak Sacks property in suitably definableforcing is equivalent to Player I having a winning strategy in any (or all) gamesdefined above. The first game leads to a rather standard corollary: the weakSacks property is preserved under the countable support iteration of definableforcing. The third game will be used to provide a strong canonization result forsuch iterations.


1 σ-ideal on a Polish space X such thatthe quotient poset is provably proper and has the weak Sacks property. Thenthe countable support iterations of PI of arbitrary length has the weak Sacksproperty as well.

Proof. This is a simple strategy diagonalization argument completely parallelto the proofs in [67, Section 6.3.5] and we omit it.

5.17c The preservation result

It is now time to state the main result of this section.

Theorem 5.17.12. Let I be a σ-ideal on a compact metric space X such that Iis Π1

1 on Σ11, every I-positive analytic set has a Borel I-positive subset, and the

quotient poset PI is provably proper and has the weak Sacks property. Let α ∈ ω1

be a countable ordinal. Then, for every Borel Iα-positive Borel set B and everyequivalence relation E on B reducible to EKσ there is an ordinal β ≤ α andan equivalence F on X reducible to EKσ and a Borel Iα-positive subset C ⊂ Bsuch that on C, E = Eβ × F .


Proof. The proof of the theorem goes by induction on α, and starts with theinvestigation of the successor step, where in fact most of the difficulty lies.

Claim 5.17.13. Let I, J be Π11 on Σ1

1 σ-ideals on Polish spaces X,Y such thatthe quotients PI , PJ are proper and have the weak Sacks property. Write K forthe Fubini iteration of I followed with J . Let E be an equivalence relation onX × Y reducible to EKσ , and let B ⊂ X × Y be a K-positive Borel set. Thereis a Borel K-positive Borel set C ⊂ B such that

1. either there is an equivalence relation F on X, reducible to EKσ , such thatE � C = F × ee;

2. or there is an equivalence relation G on Y , reducible to EKσ , such thatE � C = id×G.

In the second case, no two points of C with distinct X coordinates are E-equivalent.

Proof. To simplify the notation, assume that both σ-ideals I, J live on theCantor space 2ω. Let B ⊂ 2ω × 2ω be a Borel I ∗ J-positive set, and let Ebe a Borel equivalence relation on B reducible to EKσ ; let h : B → ωω be theBorel reducing functio;. To establish a suitable terminology for the equivalencerelation EKσ , define the notion of distance between elements y, z ∈ ωω as thesupremum of {|y(n) − z(n)| : n ∈ ω}. The distance can be infinite, and EKσconnects those points in ωω with finite distance.

Since the iteration PI ∗ PJ is a bounding forcing, we can thin out B ifnecessary to a compact I ∗J-positive set on which the reduction h is continuous.There are two distinct cases.

Case 1. Suppose first that the set A = {x ∈ 2ω : ∃y ∈ 2ω φ(x, y)} is I-positive, where φ(x, y) = 〈x, y〉 ∈ B ∧{z ∈ 2ω : 〈x, z〉 ∈ B ∧ 〈x, y〉 E 〈x, z〉} /∈ J ;Note that the set A is analytic since the σ-ideal J is Π1

1 on Σ11. If A is I-positive,

then it has a Borel I-positive subset A′. Now for every x ∈ A there is y ∈ 2ω suchthat φ(x, y) holds, this is a Π1

2 formula and by Shoenfield’s absoluteness 2.4.1A′ forces in PI that there is y ∈ 2ω such that φ(xgen, y); let τ be a PI -namefor such a y. Let M be a countable elementary submodel of a large structurecontaining A′ and τ , and find an I-positive compact set A′′ ⊂ A′ consisting ofM -generic points only such that the function g : x 7→ τ/x is continuous on it;there is such a set since the poset PI is bounding. Note that for every x ∈ A′′,M [x] |= φ(x, g(x)) by the forcing theorem, and therefore φ(x, g(x)) holds in V bythe analytic absoluteness. Consider the set C = {〈x, y〉 ∈ B : x ∈ A′′ ∧ 〈x, y〉 E〈x, g(x)〉}. This is clearly a Borel K-positive set and if 〈x0, y0〉, 〈x1, y1〉 are twopoints in C then they are E-related just in case 〈x0, g(x0)〉, 〈x1, g(x1)〉 are. Item(1) of the claim holds with the equivalence relation F connecting points x0, x1

if 〈x0, g(x0)〉 E 〈x1, g(x1)〉 .Case 2. Suppose now that Case 1 fails. Let B1 ⊂ 2ω be a Borel I-positive

set such that for every x ∈ B1, Bx /∈ J . Since J is a Π11 on Σ1

1 σ-ideal, this is aΠ1

2 statement about the set B and therefore absolute for all forcing extensions.


In particular, B1 as a condition in PI forces the PI -generic xgen into B1 andso also forces Bxgen

/∈ J . A similar absoluteness argument also shows thatB1 forces that the Borel equivalence relation Exgen

on Bxgen, connecting points

y0, y1 ∈ Bxgenif 〈xgen, y0〉 E 〈xgen, y1〉, has J-small classes.

Let τ be a PI -name for a winning strategy for Player I in the unraveledversion of the game G3(J,Bxgen). Note that we may assume that the strategyproduces a sequence of finite sets 〈Lm : m ∈ ω〉 in Bxgen

such that whenevern 6= m ∈ ω are distinct natural numbers then no point in Lm is Exgen

-equivalentto any point in Ln. At any rate, Player II can force the strategy τ to play inthis way at no cost to himself, just adding the J-small set [

⋃m∈n Ln]Exgen to

his move at round n.Let M be a countable elementary submodel of a large structure containing

τ and let B2 ⊂ B1 be a Borel I-positive set consisting of M -generic points only;we can also B2 to be compact and such that all sets in M are relatively clopenin B2 by the bounding property of the forcing PI . For every point x ∈ B2, lookinto the model M [x], at the strategy τ /x. Observe that the nonexistence of awinning counterplay for Player II against τ /x is a wellfoundedness statement inthe model M [x], and since the model M [x] is wellfounded, it holds even in V .Therefore, every counterplay by Player II against τ /x in which Player II usesonly moves from the model M [x], will result in a J-positive compact subset ofBx. We will find a Borel I-positive set B3 ⊂ B2 and for every point x ∈ B3

a counterplay qx against the strategy τ /x such that the x0 6= x1 ∈ B3 impliesthat no point of the result of qx0 is equivalent to no point of the result of qx1 .This will complete the proof.

Fix a winning strategy σ for Player I in the game G2(I,B2). By inductionon n ∈ ω build

• finite counterplays p0 ⊂ p1 ⊂ . . . against σ. The play pn will last nrounds and its last move will be played of Player II. There will be a finiteset an ⊂ 2<ω such that every t ∈ an has exactly one continuation in thelast move of Player I in the play pn, and vice versa, every point in the lastmove of Player I will have exactly one intial segment in an. Also, everysequence in an−1 has an extension in an and every sequence in an has aninitial segment in an−1.

• maps fn with domain an such that for every t ∈ an, the value fn(t) isa name for a finite couterplay against τ . The name fn(t) will be in themodel M . It will be the case that, writing s for the unique initial segmentof t in an−1, B2 ∩ [t] forces fn−1(s) to be an initial segment of fn(t).

• There will also be maps gn with domain an such that for every t ∈ an,g(t) is a clopen subset of 2ω, B2 ∩ [t] forces fn(t) to have gn(t) as the lastmove of Player II, and if t0 6= t1 ∈ an then the h-images of points in theset B ∩ ([t0] × gn(t0)) have distance at least n from the h-images of thepoints in the set B ∩ ([t1]× gn(t1)).

Suppose the induction succeeds. Let B3 be the result of the play⋃n pn, so

B3 = {x ∈ 2ω : for every n, x has an initial segment in an}. For every point


x ∈ B3, consider the play qx =⋃{fn(t)/x : t is the unique initial segment of x

in an, n ∈ ω}. It follows from the second item that this is a play against thestrategy τ/x whose initial segments belong to the model M [x], and as explainedearlier, its result (equal to

⋂{gn(t) : t is the unique initial segment of x in an,

n ∈ ω}) is a J-positive compact subset of Bx. The last item then implies thatif x0 6= x1 ∈ B3 are distinct then no point of the result of qx0 is equivalent to nopoint of the result of qx1 as desired. It should be noted that there is no reasonto believe that the play qx actually belongs to the model M [x].

The induction is elementary except for arranging the last item. Supposepn−1, an−1, fn−1, gn−1 have been constructed. To extend the play pn−1 by onemore round, let first Player II play an empty set, and the strategy σ respondswith a finite set Kn. For every x ∈ Kn, look into the model M [x] and finda tentative infinite extension rx ∈ M of the play fn−1(s)/x (where s is theunique initial segment of x in an−1) played in accordance with the strategyσ. Denote the finite sets produced by the strategy in these various plays byLm,x : m ∈ ω, x ∈ Kn. Enumerate the set Kn as {xi : i ∈ j} and by inductionon i ∈ j find increasing numbers mi larger than the length of fn−1(s)/xi suchthat for every i0 ∈ i1 ∈ j, no point in {xi0} × Lmi is E-related to any point in{xi1}×Lm for anym ≥ mi0+1. This is certainly possible since the set {xi0}×Lmiis finite, while the sets {xi1} × Lm : m ∈ ω are finite and E-unrelated by thechoice of the (name for the) strategy τ . Thus, the sets {xi} × Lmi : i ∈ j arepairwise E-unrelated, and by the continuity of the reduction h, there are clopensets Oi, Pi ⊂ 2ω for i ∈ j such that xi ∈ Oi, Lmi ⊂ Pi, and whenever i0 ∈ i1 ∈ jare numbers then the h-images of points in B ∩Oi0 ×Pi0 have distance at leastn from the h-images of points in B ∩Oi1 × Pi1 . Note that Player II can legallyuse a subset of Oi as an answer after the play rxi � mi. By the forcing theoremapplied to the model M [xi], there must be a name zi, a clopen set O′i ⊂ Oi anda condition Di ∈ M such that xi ∈ Di and Di zi is an extension of the playg(s) (where s is the unique initial segment of xi in s) in which Player I uses O′i ashis last move. The set Di∩Pi ∈M is relatively clopen in B3, contains the pointxi, and by the choice of the set B3 there is an initial segment ti ⊂ xi such thatB3 ∩ [ti] ⊂ Di ∩ Pi. Extend the initial segments ti : i ∈ j if necessary to makethem pairwise incompatible, let an = {ti : i ∈ j}, fn(ti) = zi, gn(ti) = O′i andbow to the applause of the audience for the successfully completed inductionstep!

The theorem is now fairly easy to prove by induction on α. The successorstep is immediately handled by the claim. For the limit step, we will considerthe iterations of length ω, and the rest will follow by the usual bootstrappingargument. So assume that B ⊂ Xω is a Borel Iω-positive set and E is anequivalence relation on it reducible to EKσ . For every n ∈ ω, let An = {~x ∈ Xn :there is an Iβ\n-positive set D ⊂ Xω\n such that the product {~x}×D is a subsetof the set B and of a single E-equivalence class}. This is an analytic set. If thereis n ∈ ω such that the set An ⊂ Xn is In-positive, then similarly to the firstcase of the claim, there is a Borel set B′ ⊂ B and an equivalence relation F on


Xn such that E � B′ is equal to F ×ee. Then, it is possible to use the inductionhypothesis for the iteration of length n to get the desired simplification of theequivalence relation E on some Iω-positive set C ⊂ B.

If, on the other hand, the sets {An : n ∈ ω} are In-small for every n ∈ ω,then the set A =

⋃n(An × Xω\n) ⊂ Xω is analytic and Iω-small. The first

reflection theorem implies that A has an Iω-small Borel superset A′. Look atthe set B′ = B \ A′; it is Iω-positive, and so contains an Iω-positive compactsubset B′′ ⊂ B′. We will find a Borel I-positive Borel set C ⊂ B′′ such thatE � C = id.

Choose a condition 〈pn : n ∈ ω〉 in the iteration forcing the generic to belongto the set B′′, let n ∈ ω, and consider what happens after the first n stages ofthe iteration. The remainder of the iteration is forced to be equal to the two stepiteration of PI and PIω\n+1 under some condition, and that condition forces thetail of the generic sequence to belong to some compact subset ofX×Xω\n+1 suchthat the E-equivalence classes in its vertical sections are Iω\n+1-small (recallthe definitions of the sets An here). By the Claim applied in this situation to theiteration of PI and PIω\n+1 (note that the reminder of the iteration has the weakSacks property as well), the compact subset can be thinned out so that pointsin its distinct vertical sections are E-unrelated. By induction on n ∈ ω, one canthen strengthen the condition 〈pn : n ∈ ω〉 so that for every n ∈ ω, p � n forcesthat there is a compact subset of X × Xω\n+1 such that points in its distinctvertical sections are E-unrelated, and the remainder of the condition forces thetail of the generic sequence to belong to this compact set. Now let M be acountable elementary submodel of a large structure containing the condition〈pn : n ∈ ω〉, and let C ⊂ Xω be the Borel Iω-positive set of all M -genericsequences meeting this condition. By an absoluteness argument, C ⊂ B′′. Also,for any two distinct sequences ~x, ~y ∈ C, if n is the largest number such that~x � n = ~y � n, the forcing theorem shows that there is a compact subset ofX ×Xω\n+1 in the model M [~x � n] in which distinct vertical sections containno E-related points, and the tails of the sequences belong to this compact set,hnece ~x, ~y must be E-unrelated. Thus, E � C = id as required.

Corollary 5.17.14. Under the assumption of the theorem, if in addition PIadds a minimal real degree, then the real degrees in the extension by the iterationof length α are well-ordered in ordertype α.

Corollary 5.17.15. Under the assumptions of the theorem, if F,G ≤B EKσand ≤B F →I≤B G, then the same thing holds for Iα.

In particular, E0 is not in the spectrum of the countable support iteration ofE2 forcing. E2 is not in the spectrum of the countable support iteration of Silverforcing or the products of Sacks forcing, and in the countable support iterationof E0 forcing, every equivalence relation ≤B EKσ simplifies to one reducible toE0.


5.17d A few special cases

In a number of situations, much better canonization theorems can be provedabout the iteration σ-ideals. Here, we will show that in the very specific case ofthe iteration of Sacks forcing, the canonization obtained in Theorem 5.17.12 infact holds for all Borel equivalence relations as opposed to just those reducibleto EKσ .

Theorem 5.17.16. Let I be the σ-ideal of countable subsets of 2ω. Let α ∈ ω1

be a countable ordinal. Then analytic→Iα {idβ × ee : β ≤ α}.

Proof. To simplify the notation, write X = 2ω, Xα = (2ω)α, Pα to be the posetof Iα-positive analytic subsets of Xα ordered by inclusion (note that compactsets are dense in Pα as the iteration is bounding), and for a condition p ∈ Pαand an ordinal β ∈ α write p � β = {~x ∈ Xβ : {~y ∈ Xα\β : ~xa~y ∈ p} /∈ Iα\β}.Since the σ-ideal Iα\β is Π1

1 on Σ11, this is an Iβ-positive analytic set and as

such a condition in the poset Pβ . For an ordinal β ∈ α, consider the reducedproduct Pαβ consisting of pairs 〈p, q〉 ∈ Pα × Pα for which p � β = q � β. Thereduced product adds sequences ~xlgen, ~xrgen ∈ (2ω)α which coincide on theirfirst β many coordinates. If M is a countable elementary submodel of a largestructure, we call sequences ~x, ~y ∈ (2ω)α reduced product generic over M if theset β = {γ ∈ α : ~x(γ) = ~y(γ)} is an ordinal and the sequences are Pαβ -genericfor the model M . As in the product case, there is a key lemma:

Lemma 5.17.17. Let M be a countable elementary submodel of a large enoughstructure, and B ∈ PIα ∩M is a condition. There is a Borel Iα-positive setC ⊂ B consisting of pairwise reduced product generic sequences over M .

To prove the lemma, consider a fusion type poset Q for PIα . A conditionp ∈ Q is a triple p = 〈ap, np, rp〉 where ap ⊂ α is a finite set, np ∈ ω, and rp isa function from (2np)ap to PIα � B such that for every s, t ∈ dom(rp) and everyordinal β ∈ ap, if s � β = t � β then rp(t) � β = rp(s) � β. If M is a countableelementary submodel of a large structure and G ⊂ Q is a filter generic over M ,then one can form the set C ⊂ (2ω)α =

⋂p∈G

⋃rng(rp). This is the desired set.

Checking its required properties is easy and left to the reader; we only state thecentral claim.

Claim 5.17.18. Let β ∈ α, p ∈ Q and let D ⊂ Pαβ be an open dense set. Thereis a condition q ≤ p such that β ∈ aq and for every pair s, t ∈ dom(rq) suchthat β is the smallest ordinal on which they differ, 〈rq(s), rq(t)〉 ∈ D.

Granted the lemma, the proof is completed as follows. Let E be an analyticequivalence relation on a Borel Iα-positive set B ⊂ (2ω)α. Let β be the leastordinal such that there is a condition 〈B0, C0〉 ∈ Pαβ such that 〈B0, C0〉 ~xlgen E~xrgen. Observe that then 〈B0, B0〉 ~xlgen E ~xrgen: given a Pαβ -generic pair〈~x0, ~x1〉 below the condition 〈B0, B0〉, a simple mutual genericity argument willproduce a sequence ~x2 such that both pairs 〈~x0, ~x2〉 and 〈~x1, ~x2〉 are Pαβ -genericbelow the condition 〈B0, C0〉. By the forcing theorem then, ~x0 E ~x2 E ~x1, andby the transitivity of the relation E, ~x0 E ~x1.


Now let M be a countable elementary submodel of a large structure con-taining E and B0 and use the lemma to produce a Borel Iα-positive set C ⊂ B0

consisting of pairwise reduced product generic sequences. For any two sequences~x, ~y ∈ C write γ(~x, ~y) to denote the set of all ordinals on which their values agree.If γ < β then M [~x, ~y] |= ¬~x E ~y by the minimality of the ordinal β and theforcing theorem; the analytic absoluteness between M [~x, ~y] and V then showsthat ¬~x E ~y holds in V . If γ = β then by a similar argument ~x E ~y. If γ > βthen we must use the transitivity of the equivalence relation E again. Use amutual genericity argument to produce a sequence ~z ∈ B such that both pairs〈~x, ~z〉 and 〈~y, ~z〉 are Pαβ -generic over M , use the forcing theorem and absolute-ness to argue that ~x E ~z E ~y, and by transitivity ~x E ~y again. In conclusion,E � C = idβ × ee as desired.

On the other end of the complexity spectrum, if the σ-ideal I to be iteratedcontains an arbitrary ergodic equivalence relation, the canonization situationalready at the first infinite stage of the iteration is so complex as to allow onlynegative results.

Proposition 5.17.19. Suppose that I is a Π11 on Σ1

1 equivalence relation ona Polish space X such that every analytic I-positive set has a Borel I-positivesubset and the quotient PI is provably proper. Suppose that E is a Borel equiv-alence relation on X with all equivalence classes in I, ergodic with respect to I.Then the iteration ideal Iω has a coding function, and its spectrum is cofinalamong the analytic equivalence relations.

Proof. Let f(~x, ~y)(n) = 1 if ~x(n) E ~y(n). We claim that this is a coding functionfor Iω. Suppose that B ⊂ Xω is a Borel Iω-positive set. By Fact 5.17.4, itcontains all branches of a Borel I-branching tree T ⊂ X<ω. Let z ∈ 2ω be anarbitrary point. By induction on n ∈ ω, build nodes sn, tn ∈ T of length n suchthat sn+1(n) E tn+1(n) iff z(n) = 1. The induction step uses the ergodicity ofthe equivalence relation E: the sets {x ∈ X : san x ∈ T} and {x ∈ X : tan x ∈ T}are I-positive, and the ergodicity of E implies the existence of both an E-relatedand E-unrelated pair whose components come from the respective two sets. Inthe end, the sequences ~x =

⋃n sn and ~y =

⋃n yn satisfy f(~x, ~y) = z as required.

For the spectrum, suppose that J is an analytic ideal on ω and let EJ be theequivalence relation on Xω defined by ~x EJ ~y if the set {n ∈ ω : ¬~x(n) E ~y(n)}belongs to the ideal J . It is not difficult to prove that =J is reducible toEJ restricted to any Borel I-positive subset of Xω. Thus the complexity ofequivalence relations in the iteration grows to encompass a cofinal set of Borelequivalence relations as well as a complete analytic equivalence relation.

We do not know anything tangible about the canonization properties ofiterations of posets that add unbounded reals. The following theorem showsthat the mutual genericity arguments from the iteration of Sacks forcings abovecannot work: there is a coding function on pairs. The argument is a slightupgrade of the proof of Velickovic-Woodin coding on triples [62].


Theorem 5.17.20. Let I be the σ-ideal of σ-compact subsets of ωω. There isa Borel function f : ωω × (ωω × ωω) → 2ω such that for every I-positive Borelset B ⊂ ωω and every I ∗ I-positive Borel set C ⊂ ωω × ωω the image f ′′B ×Ccontains a nonempty open set.

Corollary 5.17.21. Let J be a Π11 on Σ1

1 σ-ideal on a Polish space X, suchthat every analytic J-positive set contains a Borel J-positive subset. If the posetPJ is proper and adds an unbounded real then the two step iteration ideal J hasa coding function.

To prove the corollary from the theorem, just find a Borel J-positive setB ⊂ X and a Borel function g : B → ωω such that preimages of I-small sets areJ-small, and consider the function h : B×B → 2ω given by h(〈y0, y1〉, 〈x0, x1〉) =f(g(y0), g(x0), g(x1)). This is a coding function on the J ∗J-positive set B×B.

Proof. To define the functional value f(y, x0, x1), let 〈ni : i ∈ ω〉 enumeratethose numbers n such that x0(m) > y(m) for some m between x1(n) and x1(n+1). Let f(y, x0, x1)(i) = 0 if in the interval [x1(ni), x1(ni + 1)) the values of x, yoscilate fewer than four times, and f(y, x0, x1)(i) = 1 otherwise. If some of thenumbers ni are undefined, let f(y, x0, x1) =trash. This function works.

Suppose that B ⊂ ωω and C ⊂ ωω × ωω are I-positive and I ∗ I-positiveBorel sets respectively. Let T be a superperfect tree such that [T ] ⊂ C. Forsimplicity, assume that the trunk of T is 0; we will show that f ′′B × C = 2ω.Let z ∈ 2ω be a prescribed value; we must find points y ∈ B and 〈x0, x1〉 ∈ Csuch that f(y, x0, x1) = z.

For the iteration of Miller forcing of length 2, write x0gen for the first genericpoint and x1gen for the second. Find a condition p in the iteration which forces

the pair 〈x0gen, x1gen〉 into C. Find a countable elementary submodel M of alarge enough structure and by induction on i ∈ ω build finite integer sequencessi, ti, ui, numbers ni and conditions pi in the iteration such that

I1 the sequences si extend as i increases, the same for ti and ui. The length|si| is smaller than the last entry on the sequence ui, the length of ti islarger than the last entry of ui, and ni = |ui| − 1;

I2 si is a splitnode of the tree S, and pi is a condition of the form 〈Ti, Ui〉such that ti is a splitnode of the tree Ti and pi forces ui to be a splitnodeof the tree Ui;

I3 f(si, ti, ui) = z � i;

I4 the conditions pi decrease as i increases and pi ∈ M meets the i-th opendense subset of the iteration in the model M in some fixed enumeration.

If this induction succeeds, then in the end consider the points y =⋃i si, x0 =⋃

i ti and x1 =⋃i ui. Clearly, f(y, x0, x1) = z by item I3. The point y ∈ ωω is a

branch through the tree S and therefore an element of the set B. Writing g forthe filter generated by the conditions {pi : i ∈ ω} in the two step iteration, it is


obvious that g is M -generic and x0 = x0gen/g, x1 = x1gen/g and so 〈x0, x1〉 ∈ Cby the forcing theorem. This will conclude the proof.

To start the induction, just find a condition p0 ≤ p in the model M suchthat p0 = 〈T0, U0〉, t0 is the trunk of T0 and T0 forces some specific sequence u0

to be the trunk of U0; moreover, let s0 = 0. Now suppose si, ti, ui are given, andassume that z(i) = 0 (the case z(i) = 1 is similar). Use the superperfect propertyof the trees S, Ti to choose splitnodes si+1 ∈ S and t′ ∈ Ti such that |si+1| < |t′|,si ⊂ si+1, ti ⊂ t′, and si+1(|si|) > ti(ni), and moreover, the sequences si+1, t

′

oscilate only twice on dom(si+1) \ ni. Find some condition T ′ ≤ Ti � t′ in theMiller forcing in the model M indicating a number m > dom(si+1) such that

u′ = uai m ∈ Ui. Find some condition pi+1 ≤ 〈T ′, Ui � u′〉 in the model M in thei+1-th open dense set and strengthen the first coordinate if necessary to decidethe first splitnode of Ui+1 to be some definite ui+1. Further, we can strengthenthe first coordinate if necessary to make sure that its trunk ti+1 is longer thanthe last entry on ui+1. This concludes the induction step.

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Index

E0-box, 116Π1

1 on Σ11, 8, 11, 40, 41, 92, 105, 116,125, 136, 142, 161, 194

Axiom of Determinacy, 33

capacity, 32, 40, 83coding function, 7, 94, 99, 134, 140,

168, 179, 201concentration of measure, 26, 94, 181,

183cube

s-cube, 124dented strong Milliken, 169dented weak Milliken, 160Hamming, 26, 94, 181Ramsey, 154reverse, 124, 130Silver, 76, 148strong Milliken, 169weak Milliken, 160

equivalenceE0, 164E1, 86, 124, 142E2, 101, 136, 144, 181, 182F2, 103, 186EKσ , 153, 158, 161, 177, 181, 183Etail, 170classifiable by countable structures,

13, 79, 136, 144, 151, 161, 177countable, 74, 171essentially countable, 5, 13, 78, 122hypersmooth, 13, 86, 124, 129orbit, 85smooth, 13, 73, 118, 130

forcing

< ω1-proper, 37E0, 115E2, 136PEI , 65, 136, 142, 151, 157Coll(ω, κ), 14, 47, 54Cohen, 100Gandy-Harrington, 12, 125, 126Laver, 176proper, 5, 15, 23, 161reduced product, 69, 113

fusion sequence, 130, 143, 149, 161

gameboolean, 106symmetric category, 106

generic ultrapower, 21

idealσ-continuity, 40, 104σ-finite Hausdorff mass, 3, 32, 40,

83analytic P-ideal, 161c.c.c., 24calibrated, 92game, 29, 105generated by closed sets, 104Laver, 3, 30, 83, 176meager, 100polar, 32porosity, 104, 112

inaccessible, 34, 177

kernel, 66

modelV [[x]]E , 54, 136, 142V [x]E , 47, 68, 136, 142, 151, 153,

164

208

INDEX 209

propertycovering, 92, 94, 186free set, 6, 92, 97, 99, 113Fubini, 24multiple Silver, 43mutual generics, 114rectangular Ramsey, 24, 25, 40,

112, 113selection, 36, 89, 115, 161, 179separation, 81, 144Silver, 6, 41, 92, 97, 99, 180weak Laver, 97weak Sacks, 191

scrambling tool, 29spectrum, 7, 101, 136, 142, 164, 177,

181

total canonization, 5, 41

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