Heike Mildenberger and Saharon Shelah- On Needed Reals

8/3/2019 Heike Mildenberger and Saharon Shelah- On Needed Reals

http://slidepdf.com/reader/full/heike-mildenberger-and-saharon-shelah-on-needed-reals 1/32

7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7

ON NEEDED REALS

HEIKE MILDENBERGER AND SAHARON SHELAH

Abstract. Given a binary relation R, we call a subset A of the range of RR-adequate if for every x in the domain there is some y ∈ A such that(x, y) ∈ R. Following Blass [4], we call a real η “needed” for R if in everyR-adequate set we find an element from which η is Turing computable.We show that every real needed for inclusion on the Lebesgue null sets,Cof ( N ), is hyperarithmetic. Replacing “R-adequate” by “R-adequate withminimal cardinality” we get the related notion of being “weakly needed”.We show that it is consistent that the two notions do not coincide for thereaping relation. (They coincide in many models.) We show that not allhyperarithmetic reals are needed for the reaping relation. This answerssome questions asked by Blass at the Oberwolfach conference in December

1999 and in [4].

0. Introduction

We consider some aspects of the following notions:

Definition 0.1. (1) (Needed reals). Suppose that we have a cardinal char-acteristic x of the reals of the following form: There are (in most cases:Borel) sets A−, A+ ⊆ R and there is a (in most cases: Borel) relation R ⊆ A− × A+ such that

x = ||R|| := min{|Y | : Y ⊆ A+ ∧ (∀x ∈ A−)(∃y ∈ Y )R(x, y)}.

We call ||R|| the norm of R. A set Y ⊆ A+ is called R-adequate if (∀x ∈ dom(R)) (∃y ∈ Y )R(x, y). We say that η ∈ ω2 is needed for R if for every R-adequate set Y there is some y ∈ Y such that η is Turing reducible to y, abbreviated η ≤T ur y.

If A+ ⊆ R but can be mapped continuously, injectively into R by a mapping c which is, as a function on the digits, computable in both directions, then we call the real η needed for R and c if for any R-adequate set Y ⊆ A+ there is some y ∈ Y such that η ≤Tur c(y). We

2000 Mathematics Subject Classification. 03E15, 03E17, 03E35, 03D65.The first author was supported by a Minerva fellowship and by grant 13983 of the Austrian

Fonds zur wissenschaftlichen Forderung.

The second author’s research was partially supported by the “Israel Science Foundation”,founded by the Israel Academy of Science and Humanities. This is the second author’s worknumber 725.

1



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7

2 HEIKE MILDENBERGER AND SAHARON SHELAH

call such a function c a coding. In this situation, a real η is called needed for R, if it is needed for R and c for any coding c.

(2) (Weakly needed Reals). We call a real η weakly needed for R if for any R-adequate set Y of minimal cardinality there is some y ∈ Y such that

η ≤Tur y.

Every needed real is weakly needed. Sections 4 to 7 will give some informationon the reverse direction. A very good motivation for the investigation of neededreals is given in [4].

In the rest of this introduction, we describe briefly what will be proved in thesections. In Section 1 we prove that only hyperarithmetic reals are needed forthe cofinality relation on the ideal of Lebesgue null sets. In the second sectionwe prove the analogous statement for the slalom relation. In the third sectionwe extract from these two results sufficient conditions for the property “everyneeded real for R is hyperarithmetic”. In the fourth section we construct aforcing extension such that all hyperarithmetic reals are weakly needed for thereaping relation in the extension. This quite difficult model is further used in thefifth section, where we build a composed relation for which there are more weaklyneeded reals than needed reals. In Section 6 we prove that all needed reals forthe reaping relation are in complexity less than 0ω = {x, y : x, y ∈ ω, x ∈ 0y},where 0y is the yth jump of the degree 0 of all recursive sets. Moreover, usingthe model of Section 4 once more, we get that it is consistent that for the reapingrelation weakly needed and needed do not coincide. In the final section we givea sufficient criterion for a relation R such that the two notions “needed for R”and “weakly needed for R” coincide. From the proof in Section 1, we deriveone example of a relation for which the criterion is true. The definitions of thementioned relations will be recalled at their first appearance.

1. Needed reals for Cof ( N )

In this section we answer affirmatively Blass’ question whether only hyper-arithmetic reals are needed for the cofinality relation on the ideal of Lebesguenull sets.

In this section we work with two particular relations on the reals: For functionsf, g : ω → ω we write f ≤∗ g and say g eventually dominates f if (∃n < ω)(∀k ≥n)(f (k) ≤ g(k)). The dominating relation is

D = {(f, g) : f, g ∈ ωω ∧ f ≤∗ g},

and the cofinality relation for the ideal of sets of Lebesgue measure zero is

Cof ( N ) = {(F, G) : F, G are Gδ-sets of Lebesgue measure 0 and F ⊆ G}.

We write cof ( N ) for ||Cof ( N )||.

Before stating our first theorem, we review some notation: For s ∈ω>

2 = {r :(∃m ∈ ω)(r : m → 2)}, we write lg(s) = dom(s). If r ∈ ω≥2 and s ∈ ω≥2, wewrite r s if r = s lg(r). Let r s denote that r s and r = s. A subset



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7

ON NEEDED REALS 3

T ⊆ ω>2 is called a tree if it is downward closed, i.e., if for all s ∈ T for all r s,we have that r ∈ T . An element r ∈ T is a leaf if there is no s ∈ T such thatr s. For a tree T ⊆ ω>2 and some n ∈ ω we set T n = T ∩ n>2. For t ⊆ n2set t = {f ∈ ω2 : f n ∈ t}. The set of infinite branches of T is denoted by

lim(T ) = {f ∈ ω2 : (∀n)(f n ∈ T )}. The same notation applies to trees onω>H for an arbitrary set H . We will consider trees whose nodes are not finitesequences but finite sets of finite sequences.

Leb denotes the Lebesgue measure on the measurable subsets of ω2, the productspace of ω copies of the space {0, 1} where each point has measure 1

2 .

We work with the Amoeba forcing in the representation of [2, 3.4B]:

Q =

p : p ⊆ ω>2, p is a tree without leaves and Leb(lim( p)) >

1

2

.

For trees p, q ∈ Q, q is a stronger forcing condition than p, abbreviated q ≥ p, if q ⊆ p. In addition to the Jerusalem convention, that stronger conditions are thegreater or equal ones, we also follow the alphabetical convention [6]: letters laterin the alphabet or carrying more primes or stars are used for stronger conditions.The weakest element in Q is ω>2, and we write 1 for it. We write G for someQ-generic filter and T =

G for the generic tree. For x ∈ V [G], let x

˜denote a

name of x.

Definition 1.1. Q1 is the set of conditions p ∈ Q that fulfill:

(1.1) for all n < ω, s ⊆ p ∩ n2, Leb(s ∩ lim( p)) = 12

.

Claim 1.2. Q1 is dense in Q.

Proof. Let p ∈ Q and Leb(lim( p)) = 12 + ε. Let sn, n < ω, be an enumeration

of i<ω P (i2). Now choose by induction on n p = p0 ⊇ p1 ⊇ p2 . . . in Q

such that Leb(lim( pn)) ≥ 12 + ε(1 −

j<n

12j+2 ). We set ε0 = ε. In step n,

we set εn := min({Leb(si ∩ lim( pn)) − 12

: i < n ∧ Leb(si ∩ lim( pn)) − 12

>

0} ∪ {εn−1}) and choose pn+1 ⊆ pn such that Leb(sn ∩ lim( pn+1)) = 12 and such

that Leb(lim( pn+1)) ≥ Leb(lim( pn)) − εn2n+2

. Then automatically also Leb(si ∩lim( pn+1)) = 1

2for i < n and once property (1.1) is true for a condition pn

and si, i < n, it holds also for all later pk because we chose the pk’s such thatthe differences in their measures are sufficiently small. Then by the choices,q =

n<ω pn ∈ Q1.

The following definition is crucial for building an algorithm that uses the oracleT already in V . For this purpose we require: incompatibility of a finite part of T with a condition p can be read off a finite part of p (this is (b)), that measure12 is forbidden in a preciser way than in equation (1.1) (this is (c)), and that the

convergence from above of | p∩k2|

2k: k ∈ ω to Leb(lim( p)) is sufficiently fast (this

is (d)).

Definition 1.3. We say p obeys g if the following holds:

(a) p ∈ Q1, g ∈ ωω, (∀n)(n < g (n)).



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7


(b) If n < ω, s ⊆ p∩n2 and Leb(s∩lim( p)) < 12

, then |{ρ∈g(n)2 : ρn∈s,ρ∈ p}|

2g(n)<

12 .

(c) If n < ω, s ⊆ p ∩ n2 and Leb(s ∩ lim( p)) ≥ 12

, then Leb(s ∩ lim( p)) ≥

12

1 + 1g(n)

.

(d) If n < ω, then Leb(lim( p)) ≥ | p∩ g(n)2|2g(n)

1 − 1

22n

.

The main part of the section will be the proof of

Theorem 1.4. Suppose that η ∈ V ∩ ωω, that M ∈ V is a Turing machine and that p obeys g and

(1.2) p Q “ M computes η from T ˜

”.

Then η is computable from g.

Proof. We fix such an η.Fix for a while j ∈ ω. Since the statement “for every j there is some j such

that M computes η( j) using T j ” is forced, there is some stronger condition

r that forces a value for j for the fixed j. So [r j] could serve as a conditionthat describes enough of the oracle T in order to give the right computation of η( j).

Now the assigment j → j (say the minimal one), is an element of V Q, andin general not in V . But since our computation is not allowed to use additionalinformation except for g, we will look, given j, at all possible r’s and j ’s simul-taneously. The procedure to give a computation in V will be built upon guessingfinite parts of conditions r and finite parts of T that are already determined bythe same finite part of r. But, such an approximation, starting with trials of size zero and successively increasing the size, could give a unique (and, of course,halting) computation that gives the same outcome on all possibilities within theguessed part and still be not the right guess because a too small part of r is used

and only a larger approximation would mirror correctly what happens in theforcing process. However, fortunately from some approximation size onwards,the outcome will not change any more. So we can remedy the problem of wrongguesses by first choosing a suitable n(∗) and then looking into densely many

forcing conditions above p ∩ n(∗)2 simultaneously, and search increasing in m foran aproximation of size m. Starting from some m, all larger approximations willgive the same result. The search will be based upon g. And, from the definitionof “ p obeys g” it follows that any eventually larger function could serve as well.

We assign some structure to the collection of finite initial segments of membersof Q1, that will allow us to work with finitely branching trees. These will be the

trees (Tn(∗) p,g ,) from Definition 1.7. The procedure that computes η relative to

g will first search for a sufficiently large finite approximation of r, and then argue

that this approximation already determines the run of M with oracle T on thegiven input j. The height of this approximation in Tn(∗) p,g will be an appropriate

measure for being a sufficiently large approximation of r.



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7

ON NEEDED REALS 5

Definition 1.5. A set t ⊆ m≥2 is a subtree of m≥2 iff t is not empty and closed under initial segments and (∀ν ∈ t ∩ m>2)(∃e ∈ 2)(ν e ∈ t).

Definition 1.6. We say that a subtree t of m≥2 obeys g if the following holds:

(a) t is a subtree of

m≥

2, g ∈ω

ω, (∀n)(n < g (n)).(b) If n ≤ m, g(n) ≤ m, s ⊆ t∩n2, and if Leb(s∩t) < 1

2, then |{ρ∈

g(n)2 : ρn∈s,ρ∈t}|2g(n)

<12

.

(c) If n ≤ m, g(n) ≤ m, s ⊆ t ∩ n2, and if Leb(s ∩ t) ≥ 12

, then Leb(s ∩ t) ≥12

1 + 1

g(n)

.

(d) If g(n) ≤ m, then

|t ∩ g(n)2|

2g(n)≥

|t ∩ m2|

2m≥

|t ∩ g(n)2|

2g(n)

1 −

1

22n

(note that the first inequality holds trivially).

The properties “t obeys g” and “ p obeys g” look similar. In order to decribe the

features of the similarities that will be useful later we make another definition:

Definition 1.7. Assume that p obeys g and n(∗) < ω and that m ≥ n(∗).

(1) N bn(∗) p,g = {q ∈ Q1 : q (n(∗) + 1) = p (n(∗) + 1) and q obeys g}.

(2) mTn(∗) p,g = {∅ = t ⊆ m≥2 : t is a subtree of m≥2 and t obeys g and t

(n(∗) + 1) = p (n(∗) + 1)}.

(3) We let Tn(∗) p,g =

0≤m<ω

mTn(∗) p,g .

(4) If t ∈ Tn(∗) p,g , then let m(t) be the minimal m such that t ⊆ m≥2.

(5) The partial order on Tn(∗) p,g is defined it as follows: s t iff t

(m(s) + 1) = s.

Our first claim about the trees Tn(∗) p,g and the neighborhoods Nbn(∗) p,g concernsthe easier inclusion: members of the neighborhoods can be seen as branches of the trees:

Claim 1.8. (1) Tn(∗) p,g is a tree whose m-th level is mT

n(∗) p,g . If 0 ≤ m ≤ m(t)

and t ∈ Tn(∗) p,g then t (m + 1) t and t (m + 1) ∈ mT

n(∗) p,g .

(2) If q ∈ N bn(∗) p,g and m < ω then q (m + 1) ∈ mT

n(∗) p,g .

In the next claim some easy and useful facts are listed.

Claim 1.9. (1) Every p ∈ Q1 obeys some g.

(2) q ∈ lim(Tn(∗) p,g ) iff (∀m)q (m + 1) ∈ mT

n(∗) p,g .

(3) If p ∈ Q1 obeys g and n(∗) ∈ ω then N bn(∗)

p,g

⊆ Q1.

(4) If g1 ≤∗ g2 then there is g12 recursive in g2 such that g1 ≤ g1

2, where ≤is the pointwise order.



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7


(5) If g1 ≤ g2 and p obeys g1, then p obeys g2.

(6) Fixing p, n(∗), the function m → mTn(∗) p,g is recursive in g.

The next claim will allow us to apply Konig’s Lemma at an important step in

the proof of Claim 1.13.

Claim 1.10. Suppose that

(1.3) (∃k)(g(k) ≤ n(∗) ∧ Leb(lim( p))

1 −

1

22k−1

>

1

2).

Then N bn(∗) p,g is the set of unions of ω-branches of the tree T

n(∗) p,g .

Proof. Suppose that for all m ∈ ω, q (m + 1) ∈ mTn(∗) p,g . We prove that

q ∈ Nbn(∗) p,g . We first prove that q ∈ Q. From the definition of “subtree of m≥2”

it follows that q has no leaves.The main point is: Why is Leb(lim(q)) > 1

2 ? For every m ≥ g(k), we have byclause (d) of Definition 1.6

|q ∩ g(k)2|

2g(k)≥

|q ∩ m2|

2m≥

|q ∩ g(k)2|

2g(k))

1 −

1

22k

.

But as g(k) ≤ n(∗) clearly q ∩ g(k)2 = p ∩ g(k)2. As p obeys g we have

Leb(lim( p)) ≥| p ∩ g(k)2|

2g(k)

1 −

1

22k

.

Since the quotients are approaching the measure from above, the right side is

greater than or equal to Leb(lim( p))

1 − 122k

. So |q∩m2|

2m≥ Leb(lim( p))

1 − 1

22k

and this holds for every m ∈ ω. Hence Leb(lim(q)) ≥ Leb(lim( p))

1 − 1

22k

and

the right hand side is strictly larger than 12 by equation (1.3).

Now we have to prove that q ∈ Q1. So let n ∈ ω and s ⊆ q ∩ n2. Suppose thatLeb(s ∩ lim(q)) = 1

2. Then for all m, Leb(s ∩ q ∩ m2) ≥ 1

2. So by 1.6(b) for all

m, Leb(s ∩ q ∩ m2) ≥ 12

1 + 1

g(n)

. Hence also the limit is greater than or equal

to 12

1 + 1

g(n)

.

Now we have to prove that q obeys g. This follows from Definition 1.6 and thenature of the limit process.

The reverse inclusion is Claim 1.8(2).

Definition 1.11. Assume that T is a Turing machine. We let ΓM,k be the set of finite partial characteristic functions h, dom(h) ⊂ ω>2 such that if M runs with the input k it uses only h as an oracle. So it answers h(ρ) =? for ρ ∈ dom(h)according to h and does not ask questions of the kind h(ρ) =? for ρ ∈ dom(h).We let eM,h(k) be the result of such a run.



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7

ON NEEDED REALS 7

Definition 1.12. For p ∈ Q set

( p) = {h : (∃m)(h : m≥2 → 2 ∧ p h ⊆ chT ˜

)}.

For t ∈ mTn(∗) p,g let

(t) =

h : (∃n)

h : n≥2 → 2 ∧ g(n) ≤ m ∧ h−1({1}) is a tree

∧ h−1{1} ⊆ t ∧|{ρ ∈ g(n)2 : h(ρ n) = 1 ∧ ρ ∈ t}|

2g(n)>

1

2

.

Claim 1.13. (1) For every p ∈ Q and finite u ⊆ ω>2 there is some h ∈ ( p)whose domain is a superset of u.

(2) If p ∈ Q1 obeys g and h : n≥2 → 2 then p ∩ g(n)2 determines the truth value of h ∈ ( p), in fact h ∈ ( p) iff h ∈ ( p (g(n) + 1)).

Proof. (2) By looking at the relation ≤Q we see: For h : n≥2 → 2, p h ⊆ chT ˜

iff (h−1({1}) ⊆ p or (h−1({1}) ⊆ p and Leb(h−1({1}) ∩ lim( p)) ≤ 12 )). But theright hand side of the iff-clause can be read off p (g(n) + 1) by clause (b) of theDefinition of “ p obeys g”.

Now we are ready to finish the proof of Theorem 1.4. So assume that p ∈ Q1,η ∈ ω2, p “M computes η from the oracle T

˜” and p obeys g. We show that η

is computable from g.Step a) Let k > 0 be such that

1

2< Leb(lim( p))

1 −

1

22k−1

.

Step b) Let n(∗) ≥ g(k).

Step c) Now we have for every q ∈ Nbn(∗) p,g and j < ω that (q)∩( p)∩ΓM,j = ∅.

Why?First, by the choice of n(∗), Leb(lim( p) ∩ lim(q)) > 1

2 . This follows from the

computation: lim( p) ⊆ p ∩ n(∗)2 ⊆ p ∩ g(k)2 and the same holds for q. Since both

obey g we have lim( p) > | p∩g(k)2|2g(k)

1 − 1

22k

and the same holds for q. Hence

lim( p) and lim(q) each are missing only a part of p ∩ g(k)2 of measure less than| p∩g(k)2|

2g(k)+2k . The missing part of lim( p) ∩ lim(q) in p ∩ g(k)2 is at most twice this.

Hence Leb(lim( p) ∩ lim(q)) ≥ | p∩g(k)2|2g(k)

1 − 1

22k−1

≥ Leb(lim( p))

1 − 1

22k−1

>

12

.So there is some r ∈ Q that is above both. As r ≥ p, it forces that M running

on j and oracle T ˜

gives η( j) and the run uses only h = T ˜

∩ n≥2 for some n.

Step d) If eM,h( j) is well-defined and h ∈ ( p), then eM,h( j) = η( j). This isbecause η ∈ V .



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7


Step e) For every j there is some m, such that for all t ∈ mTn(∗) p,g we have

that (t) ∩ ( p) ∩ ΓM,j = ∅. Why? This follows by Claim 1.10 and step c) and

Konig’s lemma applied to the finitely branching tree Tn(∗) p,g .

Step f) For every j there are m and t1 ∈

mTn(∗)

p,g such that for every t2 ∈

mTn(∗)

p,gwe have (t1)∩(t2)∩ΓM,j = ∅. This holds by e): We can take t1 = p (m+1).

Step g) For every j < ω and e ∈ 2 the following are equivalent

(i) η( j) = e.

(ii) For some m < ω and t1 ∈ mTn(∗) p,g for every t2 ∈ mT

n(∗) p,g there is h ∈

(t1) ∩ (t2) ∩ ΓM,j such that eM,h( j) = e.

(i) ⇒ (ii) is step f).(ii) ⇒ (i): Let m, t1 be as in (ii) Let t2 = p (m + 1). By (ii) there is some

h ∈ (t1) ∩ (t2) ∩ ΓM,j , eM,h( j) = e. By step b) h ∈ ( p). So by step d) weare done.

By Claim 1.9(6), the procedure indicated in (ii) of step g) is recursive.So finally Theorem 1.4 is proved.

Corollary 1.14. Suppose that p ∈ Q and p Q η is computable from T ˜

. Then ηis needed for the dominating relation.

Proof. Choose some q ≥ p, q ∈ Q1 and some machine M such that q QM computes η from T

˜. Then choose g such that q obeys g. By Theorem 1.4, η

is computable from g. Since q obeys also every g ≥ g and since finite changesin the oracle may only change the algorithm but not the fact whether a real iscomputable from the oracle, η is also computable from any g ≥∗ g. Hence η isneeded for the dominating relation.

The following equivalent formulation of neededness is useful to show that in ageneric extension that contains a real ξ such that for all reals η in the groundmodel ηRξ, all needed reals in the ground model can be computed from such aξ.

Fact 1.15. (Blass [3, following Definition 1]) An equivalent condition for “ η isneeded for R” is

(1.4) (∃x ∈ dom(R))(∀y ∈ range(R))(xRy → η ≤Tur y).

Proof. Suppose that η is needed for R and that there is no x as in (1). Then(∀x ∈ dom(R)) (∃y ∈ range(R))(xRy ∧ η ≤Tur y). So we can build a R-adequateset from all these y’s, that shows that η is not needed for R. For the otherimplication: Fix x as in (1). Every R-adequate set has to contain some y suchthat xRy and hence η ≤Tur y.

If R is a transitive relation and R is a given R-adequate set, then x with theproperty of equation (1.4) can be found in R.

Theorem 1.16. Every needed real for Cof ( N ) is needed for the dominating relation.



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7

ON NEEDED REALS 9

Proof. : Let {Ai : i < κ} be a Cof ( N )-adequate set, such that each Ai is a F σset. Let η ∈ ω2.

For each i choose a countable elementary submodel N i of (H( 3), ∈) to whichη and Ai belong. We let Gi be a subset of QN i that is generic over N i and let

T i = T ˜ [Gi]. Now let A∗i be

A∗i = {ρ ∈ ω2 : no ρ ∈ ω2 which is almost equal to ρ

(i.e. ρ(n) = ρ(n) for every large enough n) belongs to lim(T i)}

A∗i is a null set: Since it is a tail set, by the zero-one law it can only have

measure zero or one. Since it is disjoint from the set lim(T i), that has measureone half, it is a null set.

By genericity of T i and because Ai ∈ N i and because Ai is a nullset in N iwe have that Ai ⊆ lim(T i)

c. The same argument shows that for all s ∈ ω>2 wehave that {s f : ∃s (|s| = |s| ∧ s f ∈ Ai)} is a subset of (lim(T i))c. Hence wehave that Ai ⊆ A∗

i . Therefore also {A∗i : i < κ} is a Cof ( N )-adequate set. We

choose i such that η is recursive in Ai and in all its supersets. Since Cof ( N ) istransitive, such an Ai exists by Fact 1.15 and the remark thereafter. Then η isalso recursive in A∗

i , because A∗i ⊇ Ai. If η is recursive in A∗

i it is also recursivein T i. Since this holds for arbitrary Gi, by Theorem 1.4, applied to some p ∈ Q1

that obeys some g, the real η is needed for dominating.

Fact 1.17. a) (Solovay [11]) Every real that is needed for the dominating relation is hyperarithmetic.

b) (Jockusch, [8]) Every hyperarithmetic real is needed for the dominating re-lation.

Blass [4, Theorem 6, Corollary 8] showed that every real that is needed for D

is also needed for Cof ( N ) and hence that all hyperarithmetic reals are neededfor Cof ( N ). So this gives the other inclusion in the following corollary:

Corollary 1.18. Exactly the hyperarithmetic reals are needed for the Cof ( N )-relation.

2. Needed reals for the slalom relation

In this section we deal with a forcing L which is closely related to the local-ization forcing from [2, page 106]. Theorem 2.4 is analogous to Theorem 1.4,but for the forcing L. Theorem 2.16 is analogous to Theorem 1.16 together with

Corollary 1.18.For u = u : ∈ ω, m < ω , let u n = u : < n.



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7


Definition 2.1.

L = { p : p = (n, u) = (n p, u p), u = u : ∈ ω, u ∈ [ω]≤,

|u| : ∈ ω is bounded },

p ≤ q ↔∈ω

u p ⊆ uq ∧ uq n p = u p n p ∧ n p ≤ nq

.

Again we denote the weakest element (0, ∅ : i < ω) of L by 1. For p ∈ L, wewrite b( p) = max{|u| : ∈ ω}.

Notation 2.2. Let G˜

be a name for an L-generic element. Let S = S G =

{u p n p : p = (n p, u p) ∈ G}. We write S

˜ for a L-name for S . We think of S as a

subset of ω × ω and have its characteristic function chS(, m) = 1 iff m ∈ u.

Definition 2.3. We say p = (n, u) ∈ L obeys (g, b) if (∀ < ω )( < g()), b ∈ ω,(∀)(u p ⊆ g()) and (∀)(|u p | ≤ b). Note that the condition 1 obeys every (g, b).

Theorem 2.4. Assume that M is a Turing machine and that η ∈ ω2. Suppose

that p ∈ L obeys (g, b) and

(2.1) p L M computes η from S ˜

.

Then η is computable from g.

Proof. As in the proof of Theorem 1.4, we use (2.1) and the fact that p obeys(g, b) in order to find densely many conditions above p and finite approximationsof chS and of the respective condition. We will keep the numbering of the claimsand of the definitions used in the proof of Theorem 2.4 parallel to the numberingin the proof of Theorem 1.4, though many of them are much easier for L and willbe almost obsolete or empty. But this procedure will help to establish a generalscheme.

Definition 2.5. A tuple (n, u : < m) is a finite part of a condition iff n ≤ m and for all < m, u is a non-empty finite set.

Definition 2.6. A finite part of a condition (n, u : < m) obeys (g, b) iff (∀ < m)(u ⊆ g() ∧ |u| ≤ b).

Now, in the following we do not only number analogously but also use similar

names Nbn(∗) p,g,b and mT

n(∗) p,g,b for the corresponding objects. We use g so that the

Tn(∗) p,g,b will be finitely branching, and we use b to get the boundedness clause in

the definition of a condition.

Definition 2.7. Let p = (n(∗), u p) be a condition that obeys (g, b).

(1) N bn(∗) p,g,b = {q = (nq, uq) ∈ L : nq ≥ n(∗) ∧ (∀i < n(∗))u pi = uqi ∧ (∀i)uqi ⊆

g(i) ∧ (∀i)|uq

i | ≤ b}. As the trees and neighborhoods in Definition 1.7 used only p n(∗), also here the part of u p above n(∗) is ignored. Thealgorithm will depend on g and on a finite part of p.



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7

ON NEEDED REALS 11

(2) If m ≥ n(∗), let mTn(∗) p,g,b = {(n, ui : i < m) : n ≥ n(∗), (∀i <

n(∗))(ui = u pi ) ∧ (∀i < m)(ui ⊆ g(i) ∧ |ui| ≤ b)}. If m < n(∗), let mT

n(∗) p,g,b = {(m, u p m)}.

(3) Tn(∗) p,g,b =

m<ω mT

n(∗) p,g,b.

(4) For (n, ui : i < m) ∈ mTn(∗) p,g,b we let m(n, ui : i < m) = m.

(5) For (n, u), (n, v) ∈ Tn(∗) p,g,b we write (n, u) (n, v) iff u is an initial

segment of v and u n = v n.

Claim 2.8. (1) Tn(∗) p,g,b is a tree whose m-th level is mT

n(∗) p,g,b. If 0 ≤ m ≤ m(t)

and t ∈ Tn(∗) p,g,b then t (m + 1) t and t (m + 1) ∈ mT

n(∗) p,g,b.

(2) If q = (n, u) ∈ N bn(∗) p,g,b and n ≤ m < ω then (n, u m) ∈ mT

n(∗) p,g,b.

Claim 2.9. (1) Every p ∈ L obeys some (g, b).

(2) q = (n, u) ∈ lim(T

n(∗)

p,g,b) iff (∀m)(min(m, n), u m) ∈m

T

n(∗)

p,g,b.

(3) If p = (n(∗), u) ∈ L obeys (g, b) and n(∗) ∈ ω then N bn(∗) p,g,b ⊆ L.

(4) If g1 ≤∗ g2 then there is g12 recursive in g2 such that g1 ≤ g1

2, where ≤is the pointwise order.

(5) If g1 ≤ g2 and p obeys (g1, b), then p obeys (g2, b).

(6) Fixing p, n(∗), the function m → mTn(∗) p,g,b is recursive in g.

Claim 2.10. Suppose that p ∈ L obeys (g, b). Then N bn(∗) p,g,b is compact as a subset

of P (ω>2), and is the set of unions of ω-branches of the tree Tn(∗) p,g,b.

Proof. This is obvious.

Definition 2.11. Assume that T is a Turing machine. We let ΓM,k be the set of finite partial characteristic functions h, dom(h) ⊂ ω×ω2 such that if M runswith the input k it uses only h as an oracle. So it does not ask questions of thekind h(ρ) =? for ρ ∈ dom(h). We let eM,h(k) be the result of such a run.

Definition 2.12. For p ∈ L set

( p) = {h : (∃m)(h : m × m → 2 ∧ p h ⊆ chS˜

)}.

For t = (n(∗), ui : i < m) ∈ mTn(∗) p,g let

(t) = {h : h : m × m → 2 ∧ (∀i < n(∗))h−1({1}) ∩ ({i} × ω) = {i} × ui

∧ (∀i ∈ [n(∗), m))h−1{1} ∩ ({i} × ω) ⊇ {i} × ui}.

Claim 2.13. (1) For every p ∈ L and finite z ⊆ ω × ω there is some h ∈( p) whose domain is a superset of z.



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7


(2) If p = (n(∗), u) ∈ L obeys (g, b), n(∗) ≤ m, and h : m × m → 2 then t =(n(∗), u m) determines the truth value of h ∈ ( p), in fact h ∈ ( p)iff h ∈ (t).

Proof. (2) By looking at ≤L we see: For h : m × m → 2, p

h ⊆ chS iff (h−1({1}) ⊇i<m{i} × ui or (∃i < n(∗))(h−1({1}) ∩ ({i} × ω) = {i} × ui)).

Now we finish the proof of Theorem 2.4: Assume p = (n(∗), ui : i ∈ ω) ∈ L,η ∈ ω2, p “M computes η from the oracle S

˜” and p obeys (g, b). We show that

η is recursive from g.

We name the steps in parallel to the steps in the end of the proof of Theo-rem 1.4. Since n(∗) is already given, the first two steps are empty.

Step c) Now we have that for every q ∈ Nbn(∗) p,g,b and j < ω that (q) ∩ ( p) ∩

ΓM,j = ∅. Why? It is easy to see that p and q are compatible in L. So thereis some r ∈ Q that is above both. As r ≥ p, it forces that M running on j andoracle S

˜gives η( j) and the run uses only h := chS ∩ m×m2 for some m.

Step d) If eM,h( j) is well-defined and h ∈ ( p), then eM,h( j) = η∗( j).

Step e) For every j there is some m, such that for all t ∈ mTn(∗) p,g,b we have such

that (t)∩ ( p)∩ ΓM,j = ∅. Why? this follows by c) and Konig’s lemma applied

to the finitely branching tree Tn(∗) p,g,b .

Step f) For every j there are m and t1 ∈ mTn(∗) p,g,b such that for every t2 ∈

mTn(∗) p,g we have (t1) ∩ (t2) ∩ ΓM,j = ∅. This holds by e): We can take

t1 = (n(∗), ui : i < m).

Step g) For every j < ω and e ∈ 2 the following are equivalent

(i) η( j) = e.

(ii) For some m < ω and t1 ∈ mTn(∗) p,g,b for every t2 ∈ mT

n(∗) p,g,b there is h ∈

(t1) ∩ (t2) ∩ ΓM,j such that eM,h( j) = e.

(i) ⇒ (ii) is step f).(ii) ⇒ (i): Let m, t1 be as in (ii) Let t2 = (n(∗), u m). By (ii) there is some

h ∈ (t1) ∩ (t2) ∩ ΓM,j , eM,h( j) = e. By step b) h ∈ ( p). So by step d) weare done.

By Claim 2.9(6), the procedure in (ii) of step g) is recursive.

Corollary 2.14. Suppose that p ∈ L and p Q “ η is computable from S ˜

”. Then η is needed for the dominating relation.

Proof. Choose some q ≥ p, q ∈ L and some machine M such that q Q “M computes η from S

˜”. Then choose (g, b) such that q obeys (g, b). By Theorem 2.4,

η is computable from g. Since q obeys also every (g, b) for g ≥ g and since finite

changes in the oracle may only change the algorithm but not the fact whether areal is computable from the oracle η is also computable from any g ≥∗ g. Henceη is needed for the dominating relation.



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7

ON NEEDED REALS 13

Definition 2.15. S ∈ ω([ω]<ω) is called a slalom iff for all n, |S (n)| ≤ n.

Theorem 2.16. Exactly the hyperarithmetic reals are needed for the slalom re-lation

SL = {(f, S ) : f ∈ω

ω ∧ S is a slalom and (∀n ∈ ω)(f (n) ∈ S (n))}.Proof. First we show that only hyperarithmetic reals are needed for SL: Let{S i : i < κ} be an SL-adequate set, η ∈ ω2 ∩ V be needed for SL. We takeN i (H ( 3), ∈) such that η, S i ∈ N i. Then we let Gi be L-generic over N i. Nowwe set S ∗i = S Gi

=

{u p n p : p ∈ Gi}. Then we have that for all but finitelymany n, S i(n) ⊆ S ∗i (n). Let η be computable from S i. Then for all wider slalomsS i than S i there is a (possibly even) wider slalom from which η is computable aswell, because that collection of slaloms wider than S i is SL-adequate. Then bydensity η ≤Tur S Gi

for all generic Gi. Hence we may choose p and M and (g, b)such that Theorem 2.4 applies.

All hyperarithmetic reals are needed for SL, because all of then are neededfor D. Suppose that {S in : n ∈ ω : i < κ} is SL-adequate and that η ∈ ω2

is hyperarithmetic. Then D = {max S in : n ∈ ω : i < κ} is D-adequate andhence there is some element max S in : n ∈ ω ∈ D from which η is computable.But then of course η is also computable in S in : n ∈ ω.

3. A general connection

In this section, we collect sufficient conditions and give a general scheme for theproofs of “every real needed for R is hyperarithmetic”. As in Theorems 1.4 and2.4 we use a forcing that adds an R-dominating real ρ. The first step is to provethat “being computable in ρ and being in V implies being hyperarithmetic”. Aform of this step will be given in Theorem 3.1. The second step is to show thatevery needed real for R is computable from any generic ρ. We write a generalversion of this step in Theorem 3.6.

Now we take (Q, R) instead of our two examples (Q, Cof ( N )) and (L, SL). Ris a Borel binary relation on the reals, and Q is a notion of forcing adding someelement in the range of the extension of R. Since R is Borel, we can use its codeand thus get a unique extension of R to a larger model of ZFC. From the workin the previous two sections we collect the following scheme:

Theorem 3.1. Assume that

(a) There is a notion “ p obeys g” such that if g ≥ g and p obeys g then pobeys g. For a dense subset Q of Q we have (∀ p ∈ Q)(∃g)( p obeys g).

(b) T p,g is a recursive finitely branching tree whose nodes are finite functions.

(c) Q is a forcing notion, p ∈ Q, p obeys g, ρ is a Q-name, and

p Q ρ ∈ lim(T p,g).

(d) For a dense set of p0 ∈ Q there is some p ≥ p0 such that the following conditions are fulfilled:



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7


( α) Let ( p) = {ν ∈ T p,g : p ν ⊆ ρ}. This is a subtree of T p,g.

( β ) Let S ∗ p,g =

t : for some leafless subtree T of T p,g and some k,

t = {ν ∈ T : level Tp,g (ν ) ≤ k}, and order S ∗ p,g naturally.

(γ ) S p,g is a recursive subtree of S ∗ p,g such that

(i) T p,g is the union of an ω-branch of S p,g,

(ii) for every branch t = t : ∈ ω of S p,g there is q ∈ Qsuch that q is compatible with p and Tq,g =

∈ω t.

(d) η ∈ ω2 or ωω

Then the following holds: if p Q “ η is recursive in ρ˜

” then η is hyperarith-metic.

Proof. For some p as in (c) and Turing machine M

p Q “M computes η from ρ˜

”.

Now we prove some intermediate facts, and the proof of 3.1 will be finished with3.4.

Fact 3.2. For every ω-branch tk : k ∈ ω of S p,g and j ∈ ω for some (equiv-alently every) large enough m ∈ ω for every ν ∈ tm ∩ level k(T p,g) if M runs on input j and oracle ν it finishes (so we do not ask oracle questions outside thedomain) and gives the result η( j) = k.

Proof. There is q such that Tq,g ⊆n∈ω tn . Let r ≥ q, and let G ⊆ Q be

generic with r ∈ G. If M runs with ρ˜

[G] m it gives η( j), so for some ν ∈ T p,g ,ν ⊆ ρ

˜[G]. Now we proceed as in 1.4. If there is some ρ of height k that gives

another computation result, then it is incompatible with r. But then this iswitnessed by some initial segment of r. Take m larger than all these initialsegments.

Fact 3.3. For j ∈ ω, for every large enough m, for every t ∈ l evelm(S p,g) thereis ν ∈ t ∩ l evelm(T p,g) such that if M runs with ν as an oracle then it computesη( j).

Proof. By the previous fact and Konig’s lemma applied to S p,g.

Crucial Fact 3.4. For j ∈ ω, k ∈ 2, the following are equivalent:

(i) η( j) = k.

(ii) There are some m and some t0 ∈ l evelm(S p,g) such that for every t1 ∈l evelm(S p,g) there is ν ∈ t0 ∩ t1 such that if we let M run with input j and oracle ν then the run finishes and there are no questions to theoracle that do not have an answer, and it gives answer k.

Proof. Analogous to the end of the proof of Theorem 1.4.

So we have proved 3.1.



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7

ON NEEDED REALS 15

Remark 3.5. Usually, S p,g depends only on a finite part of p, so that we havethat Q =

k∈ω Qk, and for all k ∈ ω we have S p,g as above being the same for

each p ∈ Qk.

Theorem 3.6. Suppose Q is a notion of forcing and ρ is a Q-name and 1 (∀x)(xRρ). Then 1 “every real in V that is needed for R is recursive in ρ

˜”.

Proof. Let p ∈ Q and η ∈ V . Since η is needed for R, by Fact 1.15 there is somex in dom(R) that for any y such that xRy, η ≤Tur y. Now if p xRρ, then p η ≤Tur ρ.

4. Weakly needed reals for the reaping relation

In this section we show that for any ground model V there is a forcing extensionV [G] such that all hyperarithmetic reals from V are weakly needed in V [G] for thereaping relation. The extension is necessarily a model where weakly needed and

needed are different and the CH fails, because of the following: In Section 6 weshall prove in ZFC that not all hyperarithmetic reals are needed for the reapingrelation, answering another question from Blass’ work [4]. In a model of CH,the notions “needed real” and “weakly needed real” coincide, and thus in such amodel not all hyperarithmetic reals in any submodel are weakly needed for thereaping relation. If we take a ground model V with CH then from the coincidenceof needed and weakly needed and from the fact that there are so few needed reals,we see that there are hyperarithmetic reals in V that are weakly needed in V [G]but not weakly needed in V . So the model of this section, together with theresult from Section 6, gives an example for the fact that in contrast to the notionof “being needed”, the notion of “being weakly needed” is not absolute.

Theorem 4.1. For any ground model V there is a generic extension V [G] by some c.c.c. forcing such that in V [G] every hyperarithmetic real in V is weakly needed for the reaping relation.

The proof of this theorem will occupy the whole section. First we recall thedefinition of the reaping relation:

Definition 4.2. The relation

R = {(f, X ) : f ∈ ω2, X ∈ ω[ω] ∧ f X is constant }

is called the reaping or the refining or the unsplitting relation. We say “ X refinesf ” if f X is constant. We say “ R refines f ” if there is some X ∈ R that refines

f . Finally we say “ R refines F ” if for every f ∈ F we have that R refines f .The norm of this relation is called r, the reaping number or the refining number or the unsplitting number.



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7


In this section we often use (finite) boolean combinations. For any finite set uand η ∈ u2 and Ai, i ∈ u, we set

Ai =

Ai, if = 1,ω \ Ai, if = 0;

and

A[η] =i∈u

Aη(i)i .

Definition 4.3. Let g ∈ ωω be strictly increasing and g(n) > n.

(1) We say A ∈ [ω]ω is g-slow if (∃∞n)|A ∩ g(n)| ≥ n.

(2)

F g = {f : dom(f ) ∈ [ω]ω, for i ∈ dom(f ) we have that f (i) = (f 1(i), f 2(i))

and f 2(i) ∈ [g(f 1(i))]≥f 1(i) and limsupf 1(i) : i ∈ dom(f ) = ω}.

(3) We say that a sequence A = Ai : i < κ of infinite subsets of ω is(g, κ)-o.k. if

if k < ω, f 0, . . . , f k−1 ∈ F g,∈k

dom(f ) = B ∈ [ω]ω

and limsupmin{f 1 (i) : ∈ k} : i ∈ B = ω,

then for some α = α(f : < k)we have that:

For every u ∈ [κ \ α]<ω and η ∈ u2 the set

{n ∈ B : (∀ < k)(f 2 (n) ∩ A[η] = ∅)} is infinite.

Remarks: f ∈ F g implies thati∈dom(f ) f 2(i) is g-slow. If g ≤ g then F g ⊆

F g .

Claim 4.4. We get an equivalent notion to “ A is (g, κ)-o.k.”, if we modify theDefinition 4.3(3) as in (a) and/or as in (b), where

(a) We demand 4.3(3) only for f ∈ F g that additionally satisfy dom(f 0) =· · · = dom(f k−1) = ω.

(b) We demand 4.3(3) only for f 0, . . . , f k−1 ∈ F g such that min{f 1 (i) : i <k} : i < B is strictly increasing (we can even demand, increasing faster than any given h), and for i ∈ B, max{f 1 (i) : < k} < min{f 1 (i +1) : < k}.

Proof. (a) Suppose the f 0, . . . , f k−1 ∈ F g in the original sense, and that we haverequired the analogue of 4.3(3) only for F g in the restricted sense. We supposethat

<k dom(f ) = B and take a strictly increasing enumeration {br : r ∈ ω}

of B. Then we take˜f : ω → [ω]

<ω

,˜

f (r) = f (br) for r ∈ ω. The analogue of 4.3(3) for the F g in the restricted sense gives α ∈ κ and infinite intersections in

4.3(3) for the f . The intersections are also infinite for the original f .



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7

ON NEEDED REALS 17

(b) Suppose that k < ω, f 0, . . . , f k−1 ∈ F g,∈ω dom(f ) = B ∈ [ω]ω and

lim supmin{f 1 (i) : ∈ k} : i ∈ B = ω. Then we can thin out the domain B tosome infinite B, inductively on i such that the f B fulfil all the requirementsfrom 4.4(b).

The following lemma describes the combinatorics that is used in the finalmodel:

Lemma 4.5. Let g ∈ ωω. If r < κ = cf(κ) and if there is some A that is (g, κ)-o.k., then every real that is computable in every function g ≥∗ g is weakly needed for the refining relation.

Proof. Let R = {Bα : α < |R|} be a refining family of size r < κ. Since thefamily A is refined by R, for every i < κ there are some αi < |R| and ν (i) ∈ {0, 1}

such that Bαi ⊆ Aν(i)i . Since κ is regular and since r < κ, there are some < 2

and some β < |R| such that

Y = {i < κ : ν (i) = ∧ αi = β }

is unbounded. So Bβ

⊆ i∈Y

A

i. We claim that B

βis not g-slow. Why?

Otherwise we have C = {n < ω : |Bβ ∩ g(n)| ≥ n} ∈ [ω]ω. We take a partialfunction f = (f 1, f 2) with C = dom(f ), f 1(n) = n and f 2(n) = Bβ ∩ g(n).Then f ∈ F g. Now let α ∈ κ be given. Then we take u0 such that u0 = {γ },γ ∈ Y , γ > α and η0 = {(γ, 0)} and η0 = {(γ, 1)}. Then we do not have(∃∞n)f 2(n) ∩ A0

γ = ∅ and (∃∞n)f 2(n) ∩ A1γ = ∅ at the same time, because Bβ is

refining Aγ . So A is not (g, κ)-o.k., in contrast to our assumption.

But now we can compute recursively from Bβ some g ≥∗ g, for example wemay take g(n) =(the nth element of Bβ) +1. Hence every real that is computablein every function g ≥∗ g is recursive in Bβ.

Now we show that there is a version of Lemma 4.5 that works simultaneouslyfor all hyperarithmetic reals in V .

Lemma 4.6. There is some g : ω → ω such that every hyperarithmetic real iscomputable in any g ≥ g.

Proof. For any number e ∈ ω for a Turing machine take a real re and a lowerbound ge ∈ ωω such that for all g ≥ ge, e computes re with the oracle g, if thereare such re, ge. Now take g eventually dominating all the ge, e ∈ ω.

We will find A that is (g, κ)-o.k. in a forcing extension. However, the construc-tion works only for g ∈ V . So the constellation in which we use Lemmata 4.5and 4.6 is as follows:

Corollary 4.7. Let g ∈ V be as in Lemma 4.6 in V . If in V [G], r < κ = cf(κ)and there is some A that is (g, κ)-o.k., then every hyperarithmetic real in V is in V [G] weakly needed for the refining relation.

So, how do we get a c.c.c. forcing extension in which r < κ = cf(κ) and inwhich there is some A that is (g, κ)-o.k.? The rest of this section will be devotedto this issue. We consider the case κ = cf(κ) > ℵ1 and intend to show that for



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7


every g it is consistent that “r = ℵ1 and there is some A that is (g, κ)-o.k.” Theconstruction works for any fixed g ∈ V . It is open whether a statement like “forall g ∈ V [G], there is some Ag that is (g, κ)-o.k. and r < κ = cf(κ)” is consistent.

We give a sketch of the construction in the consistency proof. We first add κ

Cohen reals to some ground model where there are at most κ reals. We show thatfrom these we get some A that is (g, κ)-o.k. for all g simultaneously. The nextstep is to extend further, in ℵ1 steps, so that along this iteration a refining familyof size ℵ1 is added. The lengthy work is to show that we can find an extensionsuch that A stays (g, κ)-o.k. for one chosen g. This is not trivial because F g isenlarged.

Definition 4.8. (1) K g = K = {(P, A˜

) : P is a ccc forcing and P “ A˜

is(g, κ)-o.k.”}. For a fixed g, we often leave out the subscript.

(2) (P 1, A1˜

) ≤K (P 2, A2˜

) iff P 1 P 2 and A1˜

= A2˜

.

We really mean the same names Ai˜

not just the same interpretations. Indeedwe think of a finite support iteration P α, Qβ : β < ℵ1, α ≤ ℵ1 giving the P α’s.

But we formulated 4.8 a bit more general, because also in the next claim theQβ’s do not appear.

Claim 4.9. (1) We have that K = ∅. In fact, if P is the forcing adding κCohen reals and A

˜is the enumeration of the κ Cohen reals, then (P, A

˜) ∈

K g for any function g ∈ V .

(2) If (P α, A˜

) ∈ K g for α < δ, δ a limit cardinal, and P α : α < δ isincreasing and continuous w.r.t. the complete embedding relation, and P =

α<δ P α, and P has the c.c.c., then (P, A

˜) ∈ K g and α < δ ⇒

(P α, A˜

) ≤K (P, A˜

).

Proof. (1) Suppose that f 0, . . . , f k−1 ∈ V [Gκ] are injective functions. We take αsuch that f 0, . . . , f k−1 ∈ V [Gα] where Gα is a generic filter for the first α Cohenreals. Suppose that η ∈ u2, u ⊆ κ \ α. Now a density argument gives thatthese A[η] “flip for infinitely many n ∈ B” to 0 or to 1 within f 2 (n) for every < k.

(2) Now we show that P “A˜

is (g, κ)-o.k.”. Only the case of cf(δ) = ω isnot so easy. We suppose that δ =

n∈ω α(n), 0 < α(n) < α(n + 1). Towards a

contradiction we assume that p∗ ∈ P δ, and

p∗ P “B˜

, f ˜

: < k form a counterexample to A˜

being (g, κ)-o.k.”

For each n ∈ ω we find qn,i : i ∈ ω such that

(α) qn,i ∈ P ,

(β ) qn,0 = p∗,

(γ ) P |= qn,i ≤ qn,i+1,

(δ) for some bn,i˜ , f 1

n,,i˜ , f

2

n,,i˜ P α(n)-names we have

qn,i “bn,i˜

is the i-th member of B˜

, f ˜

(bn,i˜

) = (f 1n,,i˜

, f 2n,,i˜

)”,



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7

ON NEEDED REALS 19

(ε) qn,i α(n) = qn,0 α(n) = p∗ α(n).

How do we choose these? Let n and α(n) be given. Then we choose qn,iincreasing in i such that qn,i ∈ P and bn,i, (f 1)n,i, (f 2)n,,i in V and

qn,i <k

the ith element of B˜

= ˇbn,i ∧ f ˜

( ˇbn,i) = ( ˇ(f 1)n,,i, ˇ(f 2)n,,i).

Then we take

bn,i˜

= (bn,i, qn,i P α(n)),

f 1n,,i˜

= ((f 1)n,,i, qn,i P α(n)),

f 2n,,i˜

= ((f 2)n,,i, qn,i P α(n)),

qn,i = p∗ α(n) ∪ qn,i [α(n), δ).

Here, the restriction α is any reduction function witnessing P αP (see [1]), andin the general case, if P α is not the initial segment of length α of some iteration,

the term q

n,i [α(n), δ) has to be interpreted as some element from a quotientforcing algebra.

Now for every n we define P α(n)-names

Bn

˜= {bn,i

˜: i < ω},

f ,n˜

: Bn

˜→ V,

f ,n˜

(bn,i˜

) = (f 1,n˜

(bn,i˜

), f 2,n˜

(bn,i˜

)) = (f 1,n,i˜

, f 2,n,i˜

).

Now we have that

p∗ “Bn

˜∈ [ω]ℵ0 , f ,n

˜is a function with domain B

n˜

and

limsupf 1,n˜

(b) : b ∈ Bn

˜ = ω and

f 2,n,i˜

when defined is a subset of [0, g(f 1,n,i˜

)) of cardinality ≥ f 1,n,i˜

”.

As (P α(n), A˜

) is in K we have for every n

p∗ α(n) P α(n) “ for some β ˜

< κ for every u ⊆ [κ \ β ˜

]ℵ0 for every η ∈ u2b ∈ B

n˜

:<k

f 2,n(b)˜

∩ A˜

[η] = ∅

is infinite.”

Let β n˜

< κ be such a P α(n)-name. Since P α(n) has the ccc, there is some β ∗n < κsuch that P α(n) β n

˜< β ∗n < κ. Since κ is regular we have that β ∗ =

n∈ω β ∗n < κ.

It suffices to prove that

p∗ “β ∗ is as required in the definition of (g, κ)-o.k.”

If not, then there are counterexamples u ∈ [κ \ β ∗]<ℵ0 , η ∈ u2, q and b∗ suchthat



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7


p∗ ≤ q ∈ P = P δ

q “b ∈ B˜

: (∀ < k)(f 2

˜

(b) ∩ A[η]

˜= ∅) ⊆ [0, b∗]”.

(4.1)

For some n(∗) < ω we have that q ∈ P α(n(∗)). Let G ⊆ P be generic over V ,and let q ∈ Gα(n(∗)). So by the choice of β n(∗) < β ∗ we have that

q P α(n(∗)) C = {b ∈ Bn(∗)

˜: (∀ < k)(f 2,n(∗)

˜(b) ∩ A[η]

˜= ∅)} is infinite”.

Recall that Bn(∗)

˜

and f ,n(∗)

˜(b) are P α(n(∗))-names and that A[η]

˜is a P 0-name.

Now Bn(∗)

˜

= {bn(∗),i

˜: i < ω}, so for some i we have that bn(∗),i

˜[G] > b∗. So

qn(∗),i ∈ G ∩ P α(n(∗)) forces “the i-th member of B˜

is bn(∗),i

ãnd f

˜(bn(∗),i

˜) =

f ,n(∗)

˜(bn(∗),i

˜) = (f 1,n(∗),i

˜

, f 2,n(∗),i

˜

). Note that qn(∗),i α(n(∗)) = p∗ α(n(∗))

according to ε), and hence qn(∗),i ⊥ q. So there is some r ≥ q and r ≥ qn(∗),i.Such an r forces the contrary of the property forced in (4.1), and finally wereached a contradiction.

The conclusion of the next claim is a strengthening of 4.3(3). Let D be anultrafilter on ω. Instead of “infinite” we require “being in D”. Since ultrafiltersare closed under finite intersections we need to mention only one function in F g.

Claims 4.10 and 4.11 are like [10]. For h : ω → ω we write limDh(i) : i ∈ω = ω if for all m < ω we have that {i : h(i) > m} ∈ D.

Claim 4.10. Assume that in V :

(a) A is (g, κ)-o.k.

(b) κ = 2ℵ0 is regular.

Then there is an ultrafilter D on ω such that

if f ∈ F g and dom(f ) ∈ D and limDf 1(n) : n ∈ dom(f ) = ω

then for some αf < κ for every u ∈ [κ \ αf ]<ℵ0 and η ∈ u2

we have that {n ∈ dom(f ) : f 2(n) ∩ A[η] = ∅} ∈ D.

(4.2)

Proof. The following is a mock forcing argument. We work with the partial orderAP , which is < κ-closed. We have to meet only κ dense sets. So, by taking oneunion over κ conditions in the end, we find a generic in V . Let F g = {f j : j < κ}.Let AP be the set of tuples (D,i,α) such that

(i) D is a filter on ω containing the co-finite subsets, ∅ ∈ D, i,α < κ,

(ii) D is generated by < κ members,

(iii) if k < ω and for < k, j < i, and dom(f j) ∈ D and limDf 1j(n) : n ∈

dom(f j) = ω and u ∈ [κ \ α]<ℵ0 , η ∈ u2, then

n ∈

<k

dom(f j) :<k

f 2j(n) ∩ A[η] = ∅

= ∅ mod D.



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7

ON NEEDED REALS 21

Let (D1, i1, α1) ≤AP (D2, i2, α2) if both tuples are in AP and

(α) D1 ⊆ D2, i1 ≤ i2, α1 ≤ α2, and

(β ) if k < ω and { j0, . . . , jk−1} ⊆ i1, dom(f j) ∈ D2 and limD2f 1j(i) : i ∈

dom(f j) = ω and u ⊆ [α1, α2) is finite and η ∈

u

2 thenn ∈

<k

dom(f j) :<k

f 2j(n) ∩ A[η] = ∅

∈ D2.

Now we have that (AP , ≤AP) is a non-empty partial order. Take i = α = 0 andD the filter of all cofinite subsets of ω. In (AP , ≤AP) every increasing sequenceof length < κ has an upper bound, namely, take the filter generated by the unionin the first coordinate and take the supremum in the second and in the thirdcoordinate.

Now we come to the first kind of sets we want to meet: If B ⊆ ω and (D,i,α) ∈AP then there are some D, i, α such that (D, i, α) ≥AP (D,i,α) and thatB ∈ D or that ω \ B ∈ D. Why? Try D = the filter generated by D ∪ {B}

and the same i and α. If this fails then we can find k < ω, such that for < kwe have j < i, such that dom(f j) ∈ D and limDf 1j(i) : i ∈ dom(f j) = ω,

u ∈ [κ \ α]<ℵ0 , η ∈ u2 and such thatn ∈

<k

dom(f j) : f 2j(n) ∩ A[η] = ∅

∩ B = ∅ mod D.

Let α < κ be such that α ≤ α and<k u ⊆ α. Let D be the filter generated

by

D ∪

n ∈

<k

dom(f j) : f 2j(n) ∩ A[η] = ∅

:

k < ω, j < i, u ∈ [α

\ α]<ℵ0

, η ∈u

2

.

Then ω \ B ∈ D, and (D, i , α) ∈ AP .

Finally, there is a second useful kind of dense sets: If (D,i,α) ∈ AP then forsome D, α we have that (D, i + 1, α) ∈ AP .

Proof. Let M (H (χ), ∈) such that M ∩ κ ∈ κ, (D,i,α) ∈ M , F g ∈ M , and|M | < κ. Suppose that dom(f i) ∈ D and that limDf 1i (k) : k ∈ dom(f i) = ω.Let α = M ∩κ. Let D1 be the filter in the boolean algebra in P (ω)∩M generatedby

(D ∩ M ) ∪

n ∈

<kdom(f j) : f 2j(n) ∩ A[η] = ∅

:

k < ω, j ≤ i, u ∈ [α \ α]<ℵ0 , η ∈ u2

.



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7


Since in M , A is (g, κ)-o.k., this has the infinite intersection property. Let D2

be an ultrafilter in M extending D1. Let D be the filter on ω in V that D2

generates.

Now we take an increasing chain (Dj , ij, αj) : j < κ in the partial order

(AP , ≤AP) such that ij is unbounded in κ and D :=j<κ Dj is an ultrafilter.

Then D fulfills (4.2).

Now we use equation (4.2) of 4.7, which implies that A is (g, κ)-o.k., to con-struct an extension in which A is still (g, κ)-o.k. The following preservationtheorem is a bit more general: it works also when the Dη’s do not coicide. Inour application, however, they will coincide.

Claim 4.11. Assume that

(a) A is (g, κ)-o.k.

(b) D = Dη : η ∈ <ωω, Dη = D, D is an ultrafilter on ω as in 4.7.

(c) QD = {T : T ⊆ <ωω is a subtree with exactly one -minimal element,and for some η ∈ T, η ν ∈ T ⇒ {k : ν k ∈ T } ∈ Dν}, ordered by inverse inclusion. (The -minimal η of this sort is called the trunk of T , tr(T ).)

Then QD“ A is (g, κ)-o.k.”.

Proof. We use the fact [10] that QD has the pure decision property: Let ϕi,i ∈ ω, be countably many sentences of the QD-forcing language. We think

of names f ˜

, < k, for some elements of F g and ϕi = “

the i-th element of

B˜

=<k dom(f

˜)

= bi and<k f

˜(bi) = ( ˇf 1,i,

ˇf 2,i)”. The pure decision

property says:

∀ p ∈ QD ∃q ≥tr p ∀r ≥ q ∀i r ϕi → (∃si ∈ r)q[si] ϕi,

where we write ≥tr for the pure extension: q ≤tr r if r ⊆ q and tr(q) = tr(r),and q[si] = {η ∈ q : si η}.

Towards a contradiction we assume that there is a counterexample. By Claim 4.4(first (b) and then (a)) we may assume that it is of the following form

p∗ “f ˜

: < k are functions from ω to ω

and for i ∈ ω, max{f 1˜

(i) : < k} < min{f 1˜

(i + 1) : < k}

and there is no α < κ such that the statement

Definition 4.3(3) holds.”

(4.3)

We find q such that

(α) q ∈ P

(β ) q ≥tr p∗,



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7

ON NEEDED REALS 23

(γ ) for all i ∈ ω for all f 1,i ∈ ω, f 2,i ⊆ [0, g(f 1,i)) of size bigger than f 1,i wehave that

if r ≥ q, r “f ˜

(i) = ( ˇf 1,i,ˇf 2,i)”,

then also for some si ∈ r, the condition q[si]

forces the same.”

We fix such a q.Now we set for ν ∈ q and < k

B1ν, = {i ∈ ω : some pure extension of q[ν] decides f (i)

˜}.

We say (ν, ) is 1-good if B1ν, ∈ D. Let for i ∈ B1

ν,, hν,(i) = (h1ν , h2

ν,) the

value of f (i)˜

that is given by the pure decision. This is well-defined because anytwo pure extensions are compatible. Of course, by the requirements we had puton the counterexample, we have that limDh1

ν,(i) : i ∈ B1ν, = ω.

We say that (ν, ) ∈ q × k is 2-good, if it is not 1-good and we have for allm ∈ ω that

M ν,,m = { j ∈ ω : (∃i ∈ ω)(hν j,(i)) is well-defined,and h1

ν j,(i) > m)

∈ D.

So, for 2-good but not 1-good (ν, ) we may define for j ∈ M ν,,m,

gν,( j) =hν j,(iν j,),

where iν j, is such that hν j,(iν j,) is defined in h1ν j,(iν j,) > m

and if there is a maximal such i, then take this as iν j,.

We show that there is M ν,,m ∈ D, M ν,,m ⊇ M ν,,m such that for j ∈ M ν,,mthere is a maximal such i: If hν j,(i) is defined and i < i then there is somepure extension deciding hν j,(i) since there are only finitely many possibilitiesfor its values, by the third line of (4.3). Hence some pure extension decides the

value. Hence also hν j,(i

) is defined. If hν j,(i) is defined for all i, then (ν j,)is 1-good. Hence , if (ν j,) is 2-good but not 1-good, then there is a maximal iwitnessing j ∈ M ν,,m. If { j : (ν j,) is 1-good } ∈ D, then by gluing together

suitable pure extensions rj of q[ν j] together we get a pure extension of q[ν] thatshows that (ν, ) is 1-good. Hence X = { j : (ν j is 2-good and not 1-good } ∈ D.So we may take M ν,,m = M ν,,m ∩ X . In order to simplify notation, we assume

that M ν,,m = M ν,,m.

Also from the third line of equation (4.3) we get that for every ν ∈ q either forall < k, (ν, ) is 1-good or no (ν, ) is 1-good. In the latter case there is someiν , such that for all < k, dom(hν,) = iν or dom(hν,) = iν + 1. Moreover, alsoby (4.3) we get that if for some < k, for all m, M ν,,m ∈ D, then for all < k,for all m, M ν,,m ∈ D. So if (ν, ) is 2-good, then all (ν, ) are 2-good. We call ν

i-good if there is some such that (ν, ) is i-good. We set M ν,m =<k M ν,,m.We fix some pseudo-intersection M ν of M ν,m : m ∈ ω, such that limiν j :

j ∈ M ν = ω.



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7


Then we also have that limDmin{g1ν,( j) : < k} : j ∈ M ν = ω, because for

each z < ω, { j : min{g1ν,( j) : < k} < z} is a finite set. Hence gν, ∈ F g. By

combining with an enumeration of M ν , we may assume that dom(gν,) = ω ∈ D.We will not write this enumeration, in order to prevent too clumsy notation, but

we shall later apply that D is as in 4.7 for F g, and therefore we need that thedomains are in D.

Now we take χ sufficiently large and N (H (χ), ∈) such that f ˜ : < k ∈

N , B1ν,, hν,, gν, : ν ∈ q, < k ∈ N , q, D ∈ N . We take α∗ = sup(N ∩ κ). We

claim that q forces that α∗ is as in the Definition 4.3(3).

If not, then there are counterexamples u ∈ [κ \ α∗]<ℵ0 and η ∈ u2 andr ∈ QD, r ≥ q, and b∗ such that

r ≥ q, and

r QD“

<kdom(f

˜) = ω and

(∀i ∈ ω)max{f 1˜

(i) : < k} < min{f 1˜

(i + 1) : < k}

and

b ∈ ω : (∀ < k)(f 2˜

(b) ∩ A[η]

˜= ∅)

⊆ [0, b∗]”.

(4.4)

First case: There is some ν ∈ r with tr(r) ν such that all (ν, ) are 1-good.

Now we take for each t ∈ ω, some pure extension of q[ν]t of r[ν] such that it forces

<k(hν, t = f ˜ t). Since A is (g, κ)-o.k., and since all is reflected to N , and

by the choice of α∗, we have that I = {n ∈ ω : (∀ < k)(h2ν,(n) ∩ A[η] = ∅} is

infinite. So we take t ∈ I such that t > b∗. Now q[ν]t contradicts (4.4).

Second case. There is some ν ∈ r such that all ν , < k are 2-good but not1-good. We set gν,( j) = hν j,(iν j,) as purely decided above q[ν j]. Fact:

Now gν, : < k is as required in the definition of ¯

A being (g, κ)-o.k., becauseω = limDg1(iν j) : j ∈ ω.

Now we take for each t ∈ ω, some pure extension q[ν j]t of r[ν j] such that it

determines<k gν, t. Since A is (g, κ)-o.k., and since all is reflected to N ,

and by the choice of α∗, we have that J = {n ∈ ω : (∀ < k)(g2ν,(n) ∩ A[η] = ∅}

is infinite. Then also J = {iν n : n ∈ J } is infinite. So we take t > b∗, t ∈ J .

Now the gluing together of q[ν j]t , j ∈

<k M ν,,t, contradicts (4.4) because we

have gν,( j) = hν j,(iν j,) = f (iν j), if q[ν]t ∈ G. Here we write f for f

˜[G].

Third case: All ν ∈ r are neither 1-good nor 2-good. We shall prove somethingstronger:

An end-segment of the generic {η : there is some element q ∈ G

with trunk η} can be thinned out (so that still infinitely manypoints are left) and injected into an infinite subset of {n ∈ ω :<k f 2

˜[G](n) ∩ A[η] = ∅}.



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7

ON NEEDED REALS 25

This is more than enough.Let iν, = max(B1

ν,) < ω, because (ν, ) is neither 1-good nor 2-good. Let

i∗ν = dom(hν,) such that i∗ν = i∗ν, or i∗ν = i∗ν, + 1. By the premise (4.3),there are such i∗ν . There is r ≥ q with no ν ∈ r being 1-good or 2-good in N .

Without loss of generality, we take q like that. Now we try to shrink q purely.Let ν 0 = tr(q).

First: We have that f ˜ i∗ν is decided by q. The range of i∗ν j : ν j ∈ q is

bounded modulo D because ν is not 2-good. Hence we may assume that thereis just one value i∗∗ν . So say (after shrinking q) that it is constant with valuei∗∗ν ≥ i∗ν .

Second we have that ν 0 ν ∈ q implies that q[ν] decides f ˜ i∗∗ν .

Third we have that if i ∈ [i∗ν , i∗∗ν ] then limDf 1ν j,(i) : j ∈ ω = ω by the

definition of i∗ν and i∗∗ν . So define gν,,i by gν,,i( j) = hν j,(i). So gν,,i ∈ N is afunction of the right form.

We have by the definition of α∗, for all i ∈ [i∗ν , i∗∗ν ) for all ν ∈ q for all u, ηthat

A := {b : (∀ < k)g2ν,,i(b) ∩ A[η] = ∅} ∈ D.

Since the range of

{η : there is some element q ∈ G with trunk η} =: ηω iseventually contained in every set in D, we now find the following infinite set: Wetake ηn : n ∈ ω such that ηn ∈ range(ηω) ∩ A and such that i∗∗ηn < i∗ηn+1 . We

set ξn = ηn |ηn − 1|. Then we have for almost all n such that ξn ∈ A and hencefor all i ∈ [i∗ξn , i∗∗ξn): gξn,,i(ηn(|ηn − 1|)) = hξn ηn(|ηn−1|),(i) = hηn,(i) = f (i).

Son∈ω[i∗ξn , i∗∗ξn ] ⊆∗ {b : (∀ < k)f 2 (b) ∩ A[η] = ∅} is infinite.

Claim 4.12. Let κ = cf(κ) > ω1. Let V |= 2ω ≤ κ and let P 0 = Cκ be the forcing adding κ Cohen reals. We fix some function g ∈ V such that every hyperarithmetic function is computable in every g ≥ g. We let G0 be P 0-generic

over V and set V 1 = V [G0]. Let in V 1, A be the enumeration of the κ Cohen reals.

(1) In V 1, there is (P, A) ∈ K g such that P “ r = ℵ1”.

(2) For (P, A) as in (1), we have that in V 1, P “every hyperarithmetic real in V is weakly needed for the reaping relation”.

Proof. (1) By 4.7 we have that A is (g, κ)-o.k. in V 1. According to 4.9, we maychoose in V 1 a -increasing sequence such that (P i, A) ∈ K , P i+1 = P i ∗ QDi

ãnd

P j , Qi : i < ℵ1, j ≤ ℵ1 is a finite support iteration and Di

˜= Di

η

˜: η ∈ <ωω

Diη = Di ∈ V P i enjoy the properties required there.Then P forces that r = ℵ1: P consecutively adds (“shoots”) ℵ1 reals through

the ultrafilters Dα

in the intermediate models V[Gα], α < ω1. Let f ∈ V P

bea real. Then, by the countable chain condition, there is some α < ω1 such thatf ∈ V P α . Then f −1[{0}] ∈ Dα or f −1[{1}] ∈ Dα. Since QDα adds a real rα that



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7


is almost a subset of every member of Dα, we have that rα refines f . Hence inV P , {rα : α < ω1} witnesses r = ℵ1.

(2) Now by part (1) and by Lemma 4.6 the proof of (2) follows.

Finally, taking P = P 0 ∗ P

˜

and G P -generic over V and A as in 4.11, statement

(2) yields that in V [G] every hyperarithmetic real from V in weakly needed forthe reaping relation, and thus the proof of Theorem 4.1 is finished.

5. There may be more weakly needed reals than needed reals

Under CH, or if ||R|| = 2ℵ0 , the notions “needed for R” and “weakly neededfor R” coincide. In this section, we show that there is some quite simply definedrelation R and that there is some model of ZFC in which there are more weaklyneeded reals for R than needed reals for R. The idea is to use the forcing modelfrom the previous section.

Claim 5.1. There is a simply defined relation R for which it is consistent that the notions “weakly needed” and “needed” do not coincide. In fact, in the forcing

models V , V [G] from the previous section, in V [G] every needed real for R isrecursive (hence in V ), and all the hyperarithmetic (and possibly more) reals areweakly needed for R.

Proof. Let R0 be the ordinary reaping relation, which we write for functions onω2 × ω2:

ηR0ν ⇔ η, ν ∈ ω2 ∧ (∃∞n)ν (n) = 1 ∧ η ν −1{1} is almost constant.

Let R1 be as follows:

ηR1ν ⇔ η, ν ∈ ω2 ∧ (∃∞n)η(n) = 1 ∧ (∃∞n)ν (n) = 1 ∧|ν −1{1} ∩ η−1{1} ∩ n|

|η−1{1} ∩ n|: n ∈ ω

converges to

1

2.

We set R = R0 ∪ R1 and use V P from the previous section. There we havethat P = P 0 ∗ Q

˜, P 0 is the forcing adding κ Cohen reals, and A

˜is an enumeration

of the names of these Cohen reals, and Q is the iteration described in 4.9. Thenin V P we have that ||R|| ≤ ||R0|| = ℵ1.

We first show that every hyperarithmetic real η ∈ V is weakly needed for R inthis model. We take some R-adequate set R in V P of power ℵ1. We let

Y = {i < κ : (∃x ∈ R)(AiRx)}.

So, by the definition of adequate we have that Y 0 ∪ Y 1 = κ. If |Y 0| = κ, then bythe proof of 4.5, we get some x ∈ R whose enumeration f with f (n) = m if m isthe nth element of x is so large in the eventual domination order that the real ηis computable from it.

We now show that |Y 1| < κ. Then it follows that |Y 0| = κ. Towards acontradiction, we assume that |Y 1| = κ. In the model from the previous sectionwe have that P =

i<ω1

P i, P 0 adds κ Cohen reals Aα, α < κ, P i increasing and



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7

ON NEEDED REALS 27

continuous, P i+1 = P i ∗ QDi

ãs there, P = P 0 ∗ Q

˜. We work in V P 0 . We have

that for some p∗ ∈ Q and some Q-names ν i˜

, i < ω1,

p∗ Q/P 0 |Y 1˜

| = κ ∧ R˜

= {ν i˜

: i < ω1}.

Let Y ∗

= {α : ∃ pα ≥ p∗

, pα Q/P 0 α ∈ Y 1˜ }. By the ccc of Q/P 0, we have thatY ∗ ∈ [κ]κ, and for α ∈ Y ∗ we choose pα such that p∗ ≤ pα Q/P 0 “α ∈ Y 1˜

”. So forα ∈ κ we have that AαR1ν i(α) and hence for a large enough n∗ for κ many α ∈ Y ∗

(without loss of generality, for all α ∈ Y ∗) we have (∀n ≥ n∗)(|ν−1i(α)

{1}∩Aα∩n|

|Aα∩n|∈

14

, 34

). Moreover, there is a ∆-system for the dom pα ∈ [κ \ {0}]<ω whose root

is u∗, and we assume that for α ∈ Y ∗, i(α) = i∗.So we may assume that for j ∈ u∗ we have that pα( j) is an object with trunk

ρj and not just a P 0-name. By pure decidability for some ν ∗ ∈ V P 0 we have:For every α ∈ Y ∗ and m for some pure extension q of pα with the same domainq ν i∗

˜ m = ν ∗ m. By the choice of n∗ we get an easy contradiction: Suppose

p ∈ P 0 and

p P 0 “∀α ∈ Y ∗

∀n ≥ n∗

∃qα ≥tr pα,

qα Q/P 0 “|ν ∗

˜−1{1} ∩ Aα ∩ n(∗)|

|Aα ∩ n(∗)|∈

1

4,

3

4

” ”.

This is impossible, because we may assume that ν ∗ ∈ V (it needs only countablymany of the κ Cohen reals) and we may arrange all other Aα’s so that the quotientwill be arbitrary. The forcing P/P 0 does not change the value of the quotient.

Now we show that if a real is not recursive then it is not needed for R. If η is not recursive and x ∈ ω2, let {x, η} ∈ N (H (χ), ∈), N countable. Letν = ν (x, η) be random over N , and we claim

(5.1) η ≤Tur ν.

Proof of (5.1): Otherwise we would have that η is recursive in the ground model.

This is proved in [5] and in [4, Proposition 14]. Since the proof is short, we repeatit here: Suppose

(5.2) p Random “M computes η from the oracle ν ˜

”.

Then by the Lebesgue density theorem we find s ∈ <ω2 such that above s, phas Lebesgue measure > 99

100 · Leb({ρ : s ρ}. The we set

Bn = {ν ∈ ω2 : s ν and from ν M computes η(n) correctly}.

From (5.2) we get that Leb(Bn) ≥ 99100

· Leb({ρ : s ρ}. So for every sufficientlylarge m ∈ ω we have that

(5.3) 2m−lg(s) ≤ |{ν ∈ m2 : s ν and from ν M computes η(n) correctly}|.

So we can run a machine, that has s as a fixed ingredient, and which, given input

n, increases m successively, and then computes η(n) with all possible oraclesabove s of length m ≥ lg(s) and decides with (5.3), when it is true for m (andhence for all later m), which is the right value. So equation (5.1) is proved.



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7


Thus the collection {ν (x, η) : x ∈ ω2} is an R1-adequate family and hence anR-adequate family. So, if η is needed for R, there is some ν such that η ≤Tur ν ,and hence by equation (5.1) η is recursive.

6. Needed reals for the reaping relation

In this section we prove in ZFC that for any submodel not all hyperarithmeticreals in it are needed for the reaping relation. Since in the model V [G] fromSection 4 all hyperarithmetic reals in V are weakly needed for the reaping relation,the model V [G] shows that also for the reaping relation it is consistent that weaklyneeded and needed do not coincide. In contrast to the result on the relation Rfrom the previous section, we do not prove that only recursive reals are neededfor the reaping relation. It is open whether our result here is sharp.

Theorem 6.1. If η is needed for the reaping relation, then η is arithmetical.

So, applied to the pairs of models from Section 4, we get:

Conclusion 6.2. For any V , V [G], not all hyperarithmetic reals are needed for the reaping relation in V [G].

Since there are non-arihmetical hyperarithmetic reals, 6.2 follows from 6.1.

Proof of 6.1: Suppose that η is needed for the reaping relation. Then byFact 1.15 there is some B∗ ⊆ ω such that:

(6.1) For all X , if X ⊆ B∗ = B1∗ or X ⊆ ω \ B∗ = B2

∗ then η is recursive in X .

For all X that refine B∗, we have that η is recursive in X . Note that equa-tion (6.1) is similar to η being hyperarithmetic: the difference is that η is com-putable also in every infinite subset of the complement of B∗. Unless η is recur-sive, we have that B∗ in equation (6.1) is infinite and coinfinite.

Notation 6.3. Let (M n1 , M n2 , an1 , an2 ) : n < ω be a recursive list of the quadru-

ples (M 1, M 2, a1, a2) such that (i) M 1, M 2 are Turing machines (with reference to an oracle),

(2) a1, a2 are finite disjoint sets.

Without loss of generality, an1 ∪ an2 ⊆ n and each quadruple appears infinitely often. (This will be used in 6.10.)

Definition 6.4. We say E = E n : n ∈ ω is special if

(i) E n is an equivalence relation on ω \ n, and

(ii) for m < n, E n refines E m (ω \ n), i.e., every E m-class is the union of some E n-classes plus some subset of n,

(iii) if A is an E n-equivalence class, then A \ (n + 1) is divided by E n+1 in at most two equivalence classes, and E 0 has finitely many classes,

(iv) if

( α) A is an E n-equivalence class and



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7

ON NEEDED REALS 29

( β ) there is a partition X 1, X 2 of A \ (n + 1) such that for all j < ω ,Y i ⊆ ω, i = 1, 2, (if ani ⊆ Y i ⊆ X i ∪ ani , hi < ω, and if themachine M ni running with input j and oracle Y i finishes its run giving hi, then h1 = h2),

then E n+1 induces such a partition of A. (But note: it need not be thesame partition.)

Lemma 6.5. There is a special E that has as a three place relation {n,x,y :xE ny} Turing degree ≤ 0ω and such that if A is an E n-equivalence class then A ≤Tur 0n+1.

Proof. We choose E n by induction on n.n = 0. If for every m there is a partition (c0, c1) of m such that for i ∈ {1, 2}

for every bi ⊆ ci and j < n, if M 0i running with input j and oracle bi m andgives the results ki then k0 = k1, then we choose among these pairs (cm1 , cm2 )such that cm1 is minimal in the lexicographical order. If (cm1 , cm2 ) are defined forevery m, then we have that m1 ≤ m2 ≤ m3 ⇒ cm2

1 ∩ m1 ≤lex cm3

1 ∩ m1. So

cm1 : m ∈ ω converges to some c1. Now we define E 0, having two classes: c1

and ω \ c1. The relation E 0 is computable in 01.In the step from n to n + 1, the relation E n+1 is defined similarly, with the

modification that we use the description of E n as a parameter and take partitions(c0, c1) of (m \ n) ∩ C for each E n-class C and oracles bi ∪ ani . Clearly using 0n+1

we can define E n+1 and it is in the degree 0n+2.Note different successful computations have the same outcome. 6.5

Lemma 6.6. If η is needed for reaping and E that is special, then there aren ∈ ω and an E n-class A such that η is recursive in A.

Proof. We assume the contrary. Based on this assumption, forcing with QE,B∗

and absoluteness we will lead to a contradiction. The proof will be finished with

Claim 6.10.Definition 6.7. For a special E we define Q = QE,B∗

as the following notion of forcing:

(1) p ∈ Q has the form p = (n,A,b1, b2) = (n p, A p, b p1, b p2) such that

(i) n < ω,

(ii) A is an E n-equivalence class,

(iii) A is infinite,

(iv) b1, b2 are disjoint subsets of n,

(v) b1 ⊆ B∗, b2 ⊆ ω \ B∗.

(2) p ≤ q iff

(i) n p ≤ nq, A p ⊇ Aq, b pi ⊆ bqi , for i = 1, 2,

(ii) (bq1 ∪ bq2) \ (b p1 ∪ b p2) ⊆ A p.



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7


(3) Bi˜

=

{b pi : p ∈ GQ

˜} is a Q-name of a subset Bi ∈ V [G] of B∗

i for i = 1, 2.

Q is equivalent to Cohen forcing and independent of E and B∗. Nevertheless

we keep the complicated conditions, because they fit better to the investigationof the η’s needed for the reaping relation. We shall show: η is computable in thegeneric (hence it is recursive) or it is computable in some E n-class A. So, for thefollowing three claims we assume that η is not computable in any E n-class A.

Claim 6.8. For i = 1, 2 we have

(1) Q “ bi˜

is an infinite subset of B∗i ”.

(2) For densely many p∗, p∗ Q “ i=1,2(M n

p∗

i computes η with the oracle

bi˜

)”.

Proof. (1) It is enough, to find for a given p ∈ Q some q ≥ p, q ∈ Q such that fori = 1, 2, b pi = bqi . Now A p ∩ B∗

i is infinite, because of the hypothesis on B∗ and

because η is not recursive in A p by the assumption. We may choose h ∈ A p ∩ B∗i ,h ≥ max(max(b p1), max(b p2)) + 2 and an infinite E h-class A ⊆ A p, which existsbecause A p is infinite and because E h has finitely many equivalence classes. Wedefine q as nq = h + 2, Aq = A, bq1 = b p1 ∪ {h}, bq3−i = b p3−i ∪ {h + 1}.

(2) The statement made in equation (6.1) on B∗ and on η is Π11 and holds

in V ; hence it holds in V [G] as well by Shoenfield’s absoluteness theorem [7,

Theorem 98, p. 530]. We arrange by possibly increasing n p∗

that the machines

witnessing the recursiveness are M np∗

i . Now we apply it in V [G] to part (1) of this claim.

We fix p∗, M np∗

1 , M np∗

2 as in part (2) of Claim 6.7.

Fact 6.9. There is some q ≥ p∗

such that for i = 1, 2, M nq

i = M np

∗

i and such that bqi = anq

i .

Proof. For some nq ≥ n p∗

the quadruple (M nq

1 , M nq

2 , anq

1 , anq

2 ) is equal to

(M np∗

1 , M np∗

2 , b p∗

1 , b p∗

2 ). Let A be an infinite E np∗ -class which is a subset of A p∗ .

So we take q = (n∗, A , anq

1 , anq

2 ).

Claim 6.10. For n∗, A the demands (α) + (β ) of clause (iv) of 6.4 hold, hencethe conclusion.

Proof. We work first in V [G]. There, by 6.8, X i = A∩B∗i and A exemplify 6.4(iv).

But 6.4(iv) is a Σ12-statement of the parameters (A, an1 , an2 ), and therefore it holds

in V as well by Shoenfield’s absoluteness theorem.

Finally we finish the proof of Theorem 6.1: Let A1 = A2 be the E n∗

+1-equivalence classes which are ⊆ A, with Ai for M i as in 6.4(iv). So by 6.4(iv), ηis computable in E n∗+1.



7 2 5

r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7

ON NEEDED REALS 31

7. Coincidence

In this section we give a condition on a relation R under which the notions“needed for R” and “weakly needed for R” coincide and show that the conditionis fulfilled for the relation R

randomdefined below.

Definition 7.1. The domain of the relation Rrandom is {T ⊆ <ω2 with no leavesand Leb(lim(T )) > 1

2 }, i.e., the domain of the notion of forcing from Section 1.The range of Rrandom is ω2. We set T Rrandomν iff ν ∈ AT := {ρ ∈ ω2 : for some ρ ∈ lim(T ) we have that ρ =∗ ρ}.

Definition 7.2. R is boring if

(a) R is a 2-place Borel relation on ω2 and (∀x ∈ ω2)(∃y ∈ ω2)(xRy) , and

(b) for every x1, x2 ∈ ω2, if x2 is not recursive, there is x ∈ 2ω such that

(∀ν )

xRν → (x1Rν ∧ ¬(x2 ≤Tur ν ))

.

Claim 7.3. (1) Assume that R is boring. Then the notions of being needed

for R and being weakly needed for R coincide and coincide with being recursive.

(2) The relation Rrandom is boring.

Proof. (1) We have show that every weakly needed real for R is recursive. Sinceevery recursive real is needed for R, and since weakly needed reals are needed,this will complete the cycle of implications.

Suppose that x∗ ∈ ω2 is not recursive. We show that x∗ is not weakly neededfor R. Let Y be an R-adequate set of minimal cardinality. Let Y ∗ = {ν ∈ Y :¬x∗ ≤Tur ν }. Y ∗ ⊆ Y , and hence |Y ∗| ≤ |Y | = ||R||. We show that Y ∗ is alsoR-adequate. Then, by the definition of Y ∗, x∗ is not needed for R, and the proof is finished.

Let x1 ∈

ω

2 be given. We take x2 = x

∗

, and apply (b) of the definition of “boring”. So we get x as there. Since Y is R-adequate we find some ν ∈ Y suchthat xRν . Hence by R’s boringness we have that x1Rν ∧ x2 ≤Tur ν . So ν ∈ Y ∗

and x1Rν .(2) Let x1, x2 be given. We take N (H ( 3), ∈) such that x1, x2 ∈ N . Let T

be Amoeba-generic over N . Then T = x is as claimed in Definition 7.2(b): Letν ∈ 2ω be such that ν ∈ AT . The closed set T is a subset of x1 by the Amoebagenericity of T . Hence x1Rrandomν . The set {ν : x2 ≤Tur ν } is a tail set andhence has measure zero or one. Since every real recursive in a generic for randomforcing is recursive (see the proof or equation (5.1) or [4, Proposition 14] or [5])and since x2 is not recursive, every generic real for the random forcing avoids theset. Hence it has measure zero and is disjoint from AT , and therefore for ν ∈ AT ,¬(x2 ≤Tur ν ).

Conclusion 7.4. Needed reals for Rrandom and weakly needed reals for Rrandom

coincide and are just all the recursive reals.



r e

v i s i o n : 2 0 0 3 - 0 5 - 2 7

m o d i f i e d : 2 0 0 3 - 0 5 - 2 7


References

[1] Uri Abraham. Proper forcing. In Matthew Foreman, Akihiro Kanamori, and MenachemMagidor, editors, Handbook of Set Theory . Kluwer, To appear.

[2] Tomek Bartoszynski and Haim Judah. Set Theory, On the Structure of the Real Line . A

K Peters, Wellesley, Massachusetts, 1995.[3] Andreas Blass. Needed reals. Talk at Oberwolfach December 1999 .[4] Andreas Blass. Needed reals and recursion in generic reals. Ann. Pure Appl. Logic, 109:77–

88, 2001.[5] K. de Leeuw, E.F. Moore, C.E. Shannon, and N. Shapiro. Computability by probabilistic

machines. In J. McCarthy and C.E. Shannon, editors, Automata Studies, pages 183 –212.Princeton University Press, 1956.

[6] Martin Goldstern. A taste of proper forcing. In Carlos Augusto di Prisco, Jean A. Larson,Joan Bagaria, and A.R.D. Mathias, editors, Set Theory , pages 71–82. Kluwer AcademicPublishers, 1998.

[7] Thomas Jech. Set Theory . Addison Wesley, 1978.[8] Carl G. Jockusch Jr. Uniformly introreducible sets. J. Symbolic Logic, 33:521–536, 1968.[9] John Oxtoby. Measure and Category . Springer, second edition, 1980.

[10] Saharon Shelah. Long Iterations for the Continuum. Preprint [Sh:707] , 2000.[11] Robert Solovay. Hyperarithmetically computable sets. Trans. Amer. Math. Soc., 239:99–

122, 1978.

Current address: Heike Mildenberger, Institut fur formale Logik, Universitat

Wien, Wahringer Str. 25, 1090 Wien, Austria

Saharon Shelah, Institute of Mathematics, The Hebrew University of Jerusalem,

Givat Ram, 91904 Jerusalem, Israel

E-mail address: [email protected]

E-mail address: [email protected]

Documents

Heike Mildenberger and Saharon Shelah- On Needed Reals