G -dichotomies for 1-Borel sets

G0-dichotomies for ∞-Borel sets

Andres Eduardo Caicedo

Department of MathematicsBoise State University

XI Atelier International de theorie des ensemblesCIRM, Luminy, October 4-8, 2010

Caicedo G0-dichotomies for ∞-Borel sets

This is joint work with Richard Ketchersid.

I want to thank the organizers for the invitation, and the NSF forpartial support through grant DMS-0801189.


Introduction

The topic of this talk was motivated by Benjamin Miller’s recentresults.


One of the key aspects of Miller’s work is the use of “classical”arguments. By contrast, the results here use forcing, largecardinals, and consequences of determinacy.The other key aspect, that we definitely take advantage of, is thesoft deduction of many dichotomy theorems in classical descriptiveset theory from graph theoretic dichotomies via Baire categoryarguments.


As shown through Miller’s talks, these graph theoretic dichotomiescan be established in a vast generality, depending only in theexistence of Suslin representations for the relevant sets. Thus, theyhold under ADR of arbitrary graphs on R.


http://wwwmath.uni-muenster.de/u/ben.miller/Luminy.pdf

Ketchersid and I have shown that the appropriate versions of thesegraph theoretic dichotomies hold, for example, in natural models ofAD+, thus obtaining by soft arguments the other dichotomies aswell.Since in general models of AD+ not all sets of reals are Suslin, andthe dichotomies do not seem to reduce to the Suslin case by theusual reflection arguments, an approach different from Miller’s isneeded; we use arguments involving Vopenka-like forcing.I want to concentrate on the proof of the G0-dichotomy ofKechris-Solecki-Todorcevic in this context, to illustrate thetechnique.


Graphs and colorings

As in Miller’s framework, for us a graph G on a set X is a digraph,i.e., a subset of X2.Given such G, a Y -coloring of G is a function c : X → Y such that

G(x0, x1) =⇒ c(x0) 6= c(x1).

A set A ⊆ X is G-discrete, or independent iff A2 ∩G = ∅, soc : X → Y is a Y -coloring of G iff for all y ∈ Y , c−1[{y}] isG-discrete.We will be interested in coloring with ordinals.


G0

Fix s = (sn | n ∈ ω) dense in 2<ω with sn ∈ 2n.

Define the graph G0 on 2ω by:

G0(x0, x1) ⇐⇒∃n∃x∀i < 2 (xi = sn

_(i)_x).


Restriction on colorings of G0.

An immediate but key fact about G0 mentioned in Miller’s talks:

Fact

Any G0-discrete set A with the property of Baire must be meager.

The proof is a straightforward Baire category argument.Thus for any Baire-measurable coloring c : 2ω → Y of G0, c−1[{y}]is meager, hence, meager sets can not be closed under |Y |-sizedunions.This places limitations on definable colorings. For example, therecan not be a Baire measurable ω-coloring, or (under AD) anycolorings by ordinals.


Kechris-Solecki-Todorcevic

G0-dichotomy for analytic graphs:

For G an analytic graph on R exactly one of the following hold:

1 G is ω-colorable via a Borel measurable map.

2 There is a continuous map π : 2ω → R so that π is ahomomorphism of G0 into G.

The second possibility will be denoted G0 ≤c G.(We already knew that these possibilities are mutually exclusive.)


∞-Borel sets

To state our results, it is best to work in a fragment of AD+. Weneed the notion of ∞-Borel sets. Essentially, we generalize theiterative definition of the Borel hierarchy, by allowing well-orderableunions and intersections (of any length).Since we work without choice, rather than the sets themselves, weare more interested in their actual construction. Define the classbc<κ of < κ-Borel codes, for κ a cardinal, as the collection ofwell-founded trees on γ < κ describing how to build a set of realsby taking well-ordered unions and complements from basic sets.


More precisely, think of reals as subsets of ω. A code S can beseen as a formula φS in the propositional language L∞,0, where weallow the use of countably many propositional variables pi.The code S describes the set {x ∈ R | x |= φS}, where thesemantics are defined in the standard way, after setting x |= pi iffi ∈ x.A < κ-code is then a tree, and can be identified with a set ofordinals bounded below κ. An ∞-Borel code is a < κ-Borel codefor some κ.


Given a < κ-Borel code S, write S(x) to mean “x is in the setcoded by S.” This is very absolute:

S(x) ⇐⇒ Lo(S,x)[S, x] |= φ(S, x),

where o(S, x) = ωCK1 (S, x) is the first admissible over S, x and φis an appropriate Σ1-formula.If it is important to distinguish S from the set it codes, we writeAS , BS , . . . for the latter.A < κ-Borel set is the interpretation of a < κ-Borel code. Denoteby B<κ the class of < κ-Borel sets. An ∞-Borel set is a < κ-Borelset for some κ.


The basic theory

Suslin sets have strong absoluteness properties. In an attempt togeneralize regularity results that hold in the Solovay model orunder AD+ about κ-Suslin sets, we weaken the assumption ofbeing Suslin to simply carrying an ∞-Borel code, and use Los’slemma on ultrapowers to replace the use of absoluteness. For this,it is convenient to work in the following theory:

Definition (BT)

ZF + DCR.

There is a fine σ-complete measure on Pω1(R).

BT holds, for example, in the Solovay model after Levy collapsinga measurable cardinal to ω1 and in models of Turing-determinacy,assuming DCR.


AD+

Our results apply to models of Woodin’s AD+:

Definition (AD+)

DCR.

< Θ-ordinal determinacy.

All sets of reals are ∞-Borel.

If AD+ holds in a model M , then it holds in L(P(R))M . We saythat a natural model of AD+ is one satisfying V = L(P(R)). It ison these models that we concentrate.


G0-dichotomy

Assuming BT:

G0-dichotomy for ∞-Borel graphs (C-Ketchersid)

Let µ be a fine σ-complete measure on Pω1(R). Suppose G is a< κ-Borel graph with code S. Then exactly one of the followingholds:

1 There is a B<κ∞S -measurable κ∞S -coloring.

2 G0 ≤c G.

Here κ∞S =∏τ κ

τS/µ where κτS is the first inaccessible of

HODL(S,τ)S .


The extent of the ∞-Borel sets

Under the assumption of BT, Woodin saw how to associate a code∃yS for a subset of Rm to a code S for a subset of Rn+m suchthat

(∃yS)(x) ⇐⇒ ∃yS(x, y)

for all x ∈ Rm. This easily yields that if S ⊆ ORD, then every setof reals in L(S,R) is ∞-Borel. From this we can easily prove (byadapting Solovay’s argument) that all sets of reals in L(S,R) havethe standard regularity properties, in particular, the Baire property.It follows that all functions f : R→ ORD are Baire measurable.(For our result, we actually need an explicit computation thatallows us to bound the size of ∃yS in terms of the size of S.)


Vopenka algebras

Given sets S1, . . . , Sk, we denote by HODS1,...,Sk the collection ofhereditarily ordinal definable sets with parameters fromORD ∪ {S1, . . . , Sk}.For σ ∈ Pω1(R) and S ⊂ ORD, let Hσ

S = (HODS)L(S,σ).Denote by bcσS the class (bc)H

σS of Borel codes in Hσ

S , and defineT ∼σS T ′ for T, T ′ ∈ bcσS iff (AT = AT ′)L(S,σ).Let Qσ

S be the Vopenka algebra bcσS/ ∼σS . Note that ∼σS is ordinaldefinable in L(S, σ) from S. The poset Qσ

S,<κ is defined similarly,

restricting to the set bcσS,<κ = (bc<κ)HσS .


By κσS we denote the first “true inaccessible” of L(S, σ). What wewant is to use κσS as a bound for the size of the antichains in Qσ

S .It is easy to see that it suffices to take κσS to be the least regular κsuch that for every α < κ there is no surjection from P(R× α)onto κ. It is straightforward to verify that κσS coincides with thefirst strongly inaccessible cardinal in L(S, σ)Coll(R,ω), which is amodel of choice.Fix a fine σ-complete measure µ on Pω1(R). Note that using µ wecan easily define a (normal) measure µω1 on ω1. It is easy to checkthat

HODS,x,µω1|= “ωV1 is measurable ”

for any real x. It follows that ωV1 is Mahlo in L[S, x] and thereforein L(S, σ) whenever σ ∈ Pω1(R) and x is a real coding the rangeof σ. In particular, R ∩ L(S, σ) is countable and, moreover, κσS iswell-defined, and its power set in L(S, σ) is countable in V .


It turns out that for µ-a.e σ, QσS = Qσ

S,<κσS. This allows us to

bound the complexity of the colorings we obtain.Our arguments use ultrapowers. Here,

∏S A

σ/µ is the version ofthe ultrapower of the sets Aσ, σ ∈ Pω1(R), where only S-invariantfunctions are used. A function f : Pω1(R)→ V is S-invariant ifff(σ) = f(τ) whenever L(S, σ) = L(S, τ).


The following observations are key features of QσS :

For all x ∈ σ, Gx = {T ∈ QσS :T (x)} is Hσ

S -generic.

HσS [x] = Hσ

S [Gx].For every T ∈ Qσ

S , there is x ∈ σ with T ∈ Gx.


The equality

QσS = bc

HσS∞ / ∼σS= bc

HσS

<κσS/ ∼σS

gives us that QσS is κσS-cc, so the forcing and all of its maximal

antichains are contained in HσS ∩ VκσS .

Define 〈H∞S ,Q∞S , κ∞S 〉 =∏S〈Hσ

S ,QσS , κ

σS〉/µ.

By Los’s lemma we have:

For all x ∈ R, Gx = {T ∈ Q∞S :T (x)} is H∞S -generic.

H∞S [x] = H∞S [Gx].T ∼∞S T ′ ⇐⇒ ∀x ∈ R[T (x) ⇐⇒ T ′(x)], so

Q∞S = bcH∞S<κ∞S

/ ∼∞S , and the corresponding interpretations are

in B<κ∞S .

For every T ∈ Q∞S , there is x ∈ R with T ∈ Gx.


The proof

Assume BT. We want to show a quantitative version of the factthat if G is an ∞-Borel graph on R, then either there is a coloringf : R→ ORD of G, or else there is a continuous homomorphism ofG0 to G.Let µ be a fine σ-complete measure on Pω1(R). We will workprimarily with Q∞S and so omit the superscript; when S is clearfrom context we omit it as well.


By Qn we denote the version of Q for subsets of Rn. This is notthe same as Q× · · · ×Q︸︷︷︸

n times

.

For p ∈ Qn, denote by p2 the code in Q2n such that

p2(x0, . . . , x2n−1) ⇐⇒ p(x0, . . . , xn−1) ∧ p(xn, . . . , x2n−1).


Fix an ∞-Borel code S for the graph G. Let

W∞ = {p ∈ Q :H∞ |= p2 Q2 ¬GS∞(x0, x1)},

where (x0, x1) is the standard name for the Q2-generic pair. If thisset is dense, we are done, since there is a natural well-ordering ofthe elements of Q, every pair of reals (in V ) is Q2-generic overH∞, and the map that assigns to each real r the least condition pin the generic corresponding to r, is clearly a coloring of G; thepoint is that GS∞ ∩ (R2)V = G, hence if ¬GS∞(y0, y1) then, infact, ¬G(y0, y1).


We are left with the task of building the homomorphism of G0 toG when W∞ fails to be dense. Pick a witness p, so for any p′ ≤ pthere is c ∈ Q2 with c ≤ p′2 and such that

H∞ |= c GS∞(x0, x1).

The construction is inductive. For n ∈ ω let An =

{(t0, t1) ∈ (2n)2 :∃m < n∃u ∈ 2<ω∀i < 2(ti = sm_(i)_u)}.

Note that A0 = ∅ and

An+1 = {(t0i, t1i) : (t0, t1) ∈ An, i ∈ 2} ∪ {(sn0, sn1)}.

(In Miller’s notation, An = G0(2n).)


We build a Lipschitz map from 2ω into 2ω, and use a forcingargument to guarantee that it works as the requiredhomomorphism. For this, we identify a σ so that the map can bein fact seen as an assignment of Qσ-generics over Hσ to reals, andfor this we build some Qσ-conditions (ensuring genericity) alongthe way.


The key fact is the following preliminary lemma:

Lemma

Suppose that q ∈ Q2n is such that q is below the 2n-fold sum p2n .For any s ∈ 2n, we have q2 ∧ JGS∞(xs0, xs1)KQ2n+1 > 0.


Using Los’s lemma, fix a countable σ for which the correspondingversion of the key lemma holds. By fixing an appropriate sequence(Dn :n < ω) so that Dn is dense in Qn,σ, we can ensure thefollowing:Suppose that a sequence of conditions

(qn :n < ω)

is such that qn ∈ Dn and qn+1 ≤ q2n ∈ Q2n+1. Then this sequence

generates Qm,σ-generics over Hσ in a natural way: Consider anypairwise different s0, . . . , sm ∈ 2ω, m < ω. Given n, let

{s0, . . . , sm} � n = {s0 � n, . . . , sm � n}.

Let k be such that {s0, . . . , sm} � k has size m. Then thesequence of conditions

{πn(qn) :n ≥ k}

is Qm,σ-generic over Hσ, where πn is an appropriate projection onthe set of coordinates si � n for i ≤ m.


Assume we have defined an approximation σn : 2n → 2n, and wewant to build σn+1. We have also identified some conditionqn ∈ Q2n,σ with the property that the appropriate projections meetthe first n sets Di, and if (xs)s∈2n is the standard name for thegeneric real, then qn decides each xs � n, and σn(s) is preciselythis sequence. Moreover, if (t0, t1) ∈ An then qn GS (xt0 , xt1).To extend, we find qn+1 ∈ Q2n+1,σ below q2n guaranteeing theabove for n+ 1 instead of n, and the map σn+1 is given byσn+1(s) = t iff

qn+1 xs � n+ 1 = t

for s, t ∈ 2n+1.


To do this extension, simply note that the “σ-version” of the keylemma applies to qn. Extend q2n ∧ JGS (xsn0, xsn1)KQ2n+1,σ so it

meets Dn+1 and decides each xs � n+ 1 for s ∈ 2n+1.As observed above, An+1 is obtained by taking 2 copies of An andadding the tuple (sn0, sn1). Of this last tuple we took careexplicitly through the key lemma. The other tuples are alreadytaken care of, since the corresponding reals are in one of the 2“copies” of qn.This completes the inductive construction of the Lipschitz mapΣ =

⋃n σn.


Finally, we argue that Σ is indeed a homomorphism of G0 to G.For if G0(r0, r1), then there is some m and some t ∈ 2ω such thatri = sm

_(i)_t for all i ∈ 2.Carrying out the construction above, we have that (Σ(r0),Σ(r1))is Q2,σ-generic over Hσ, and (in particular) GS(Σ(r0),Σ(r1)), bygenericity.This completes the proof.


Applications

We obtain appropriate versions of many other dichotomy theoremsfrom the G0-dichotomy and its extensions. In particular, we have:

Theorem (C-Ketchersid)

Under BT, R/E0 is a successor of R.

This was previously known under ADR but not in general (forexample, in L(R) under AD).


We can also establish appropriate versions of the Glimm-Effrosdichotomy, and recover the following trichotomy result, that wehad established in earlier work:

Theorem (C-Ketchersid)

In models of BT of the form V = L(S,R), or in natural models ofAD+, for every set X, exactly one of the following holds:

1 X is well-orderable.

2 X is linearly orderable, but not well-orderable. In this case,|R| = |2ω| ≤ X ≤ |2κ| for some (well-ordered) κ.

3 |R/E0| ≤ |X|.


These results are the first steps towards understanding “small”cardinalities of not necessarily well-orderable sets in natural modelsof AD+.

There are no infinite Dedekind-finite sets.

Any infinite well-ordered κ has exactly two successors: κ andκ+ R.

R has at least two successors: R + ω1 and R/E0. There aresets larger than R that do not embed either of these. They areall linearly orderable, but do not embed ω1. Some of them aresuccessors, but at the moment there is no full classification.


The end.


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G -dichotomies for 1-Borel sets