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Combinatorial properties of Hechler forcing

arXiv:math/9211202v1 [math.LO] 3 Nov 1992Combinatorial properties of Hechler forcingJ¨org Brendle1,∗, Haim Judah1,∗∗and Saharon Shelah2,∗∗1 Abraham Fraenkel Center for Mathematical Logic, Department of Mathematics, Bar–Ilan University,52900 Ramat–Gan, Israel2 Institute of Mathematics, The Hebrew University, Jerusalem, IsraelAbstractUsing a notion of rank for Hechler forcing we show: 1) assuming ωV1 = ωL1 , there is noreal in V [d] which is eventually different from the reals in L[d], where d is Hechler over V ;2) adding one Hechler real makes the invariants on the left-hand side of Cicho´n’s diagramequal ω1 and those on the right-hand side equal 2ω and produces a maximal almost disjointfamily of subsets of ω of size ω1; 3) there is no perfect set of random reals over V in V [r][d],where r is random over V and d Hechler over V [r], thus answering a question of the firstand second authors.1991 Mathematics subject classification. Primary 03E40, Secondary 03E15 28A05 54H05Key words and phrases.Hechler real, category, measure, random real, Cicho´n’s diagram, almostdisjoint family∗The first author would like to thank the MINERVA-foundation for supporting him∗∗The second and third authors would like to thank the Basic Research Foundation (the Israel Academyof Sciences and Humanities) for partially supporting them1

IntroductionIn this work we use a notion of rank first introduced by James Baumgartner and PeterDordal in [BD, § 2] and later developed independently by the third author in [GS, § 4] toshow that adding a Hechler real has strong combinatorial consequences. Recall that theHechler p. o.

D is defined as follows. (s, f) ∈D ⇐⇒s ∈ω<ω ∧f ∈ωω ∧s ⊆f ∧f strictly increasing(s, f) ≤(t, g) ⇐⇒s ⊇t ∧∀n ∈ω (f(n) ≥g(n))We note here that our definition differs from the usual one in that it generically addsa strictly increasing function from ω to ω.

This is, however, a minor point making thedefinition of the rank in section 1 easier. We indicate at the end of § 1 how it can bechanged to get the corresponding results in §§ 2 and 4 for classical Hechler forcing.The theorems of section 2 are all consequences of one technical result which is ex-pounded in 2.1.

We shall sketch how some changes in the latter’s argument prove thatadding one Hechler real produces a maximal almost disjoint family of subsets of ω of size ω1(2.2.). Recall that A, B ⊆ω are said to be almost disjoint (a. d. for short) iff|A ∩B| < ω;A ⊆[ω]ω is an a. d. family iffthe members of A are pairwise a. d.; and A is a m. a. d.family (maximal almost disjoint family) iffit is a. d. and maximal with this property.

—We shall then show that assuming ωV1 = ωL1 , there is no real in V [d] which is eventuallydifferent from the reals in L[d], where d is Hechler over V (2.4.). Here, we say that givenmodels M ⊆N of ZFC, a real f ∈ωω ∩N is eventually different from the reals in Miff∀g ∈ωω ∩M ∀∞n (g(n) ̸= f(n)), where ∀∞n abbreviates for all but finitely manyn.

(Similarly, ∃∞n will stand for there are infinitely any n.) — Next we will prove thatadding one Hechler real makes the invariants on the left-hand side of Cicho´n’s diagramequal ω1 and those on the right-hand side equal 2ω (2.5.). These invariants (which describecombinatorial properties of measure and category on the real line, and of the eventuallydominating order on ωω) will be defined, and the shape of Cicho´n’s diagram explained, inthe discussion preceding the result in § 2.

Theorem 2.5. should be seen as a continuation ofresearch started by Cicho´n and Pawlikowski in [CP] and [Pa]. They investigated the effectof adding a Cohen or a random real on the invariants in Cicho´n’s diagram.

— We closesection 2 with an application concerning absoluteness in the projective hierarchy (2.6. );2

namely we show that Σ14 −D-absoluteness (which means that V and V [d], where d is Hech-ler over V , satisfy the same Σ14-sentences with parameters in V ) implies that ωV1 > ωL[r]1for any real r; in particular ωV1 is inaccessible in L. So, for projective statements, Hechlerforcing is much stronger than Cohen or random forcing for Σ1n-Cohen-absoluteness (Σ1n-random-absoluteness) is true in any model gotten by adding ω1 Cohen (random) reals [Ju,§ 2].In § 3 we leave Hechler forcing for a while to deal with perfect sets of random realsinstead, and to continue a discussion initiated in [BaJ] and [BrJ]. Recall that given twomodels M ⊆N of ZFC, we say that g ∈ωω ∩N is a dominating real over M iff∀f ∈ωω ∩M ∀∞n (g(n) > f(n)); and r ∈2ω ∩N is random over M iffr avoids all Borel nullsets coded in M iffr is the real determined by some filter which is B-generic over M (whereB is the algebra of Borel sets of 2ω modulo the null sets (random algebra) – see [Je, section42] for details).

— A tree T ⊆2<ω is perfect iff∀t ∈T ∃s ⊇t (sˆ⟨0⟩∈T ∧sˆ⟨1⟩∈T). Fora perfect tree T we let [T] := {f ∈2ω; ∀n (f↾n ∈T)} denote the set of its branches.

Then[T] is a perfect set (in the topology of 2ω). Conversely, given a perfect set S ⊆2ω there isperfect tree T ⊆2<ω such that [T] = S. This allows us to confuse perfect sets and perfecttrees in the sequel; in particular, we shall use the symbol T for both the tree and the set ofits branches.

— We will show in 3.1. that given models M ⊆N of ZFC such that thereis a perfect set of random reals in N over M, either there is a dominating real in N overM or µ(2ω ∩M) = 0 in N. This result is sharp and has some consequences concerning therelationship between cardinals related to measure and to the eventually dominating orderon ωω (cf [BrJ, 1.9] and the discussion preceding 3.2. for details).The argument for theorem 3.1. together with the techniques of § 1 yield the mainresult of section 4; namely, there is no perfect set of random reals over M in M[r][d], wherer is random over M, and d Hechler over M[r] (4.2.). This answers questions 2 and 2’ in[BrJ].Notation.

Our notation is fairly standard. We refer the reader to [Je] and [Ku] for settheory in general and forcing in particular.Given a finite sequence s (i.e.

either s ∈2<ω or s ∈ω<ω), we let lh(s) := dom(s)denote the length of s; for ℓ∈lh(s), s↾ℓis the restriction of s to ℓ. ˆ is used for concatena-tion of sequences; and ⟨⟩is the empty sequence.

Given a perfect tree T ⊆2<ω and s ∈T,we let Ts := {t ∈T; t ⊆s or s ⊆t}. — Given a p.o.

P ∈V , we shall denote P-names by3

symbols like τ, ˘f, ˘T, ... and their interpretation in V [G] (where G is P-generic over V ) byτ[G], ˘f[G], ˘T[G], ...Acknowledgement. We would like to thank Andrzej Ros lanowski for several helpfuldiscussions.§ 1.

Prelude — a notion of rank for Hechler forcing1.1. Main Definition (Shelah, see [GS, § 4] — cf also [BD, § 2]).

Given t ∈ω<ωstrictly increasing and A ⊆ω<ω, we define by induction when the rank rk(t, A) is α. (a) rk(t, A) = 0 ifft ∈A.

(b) rk(t, A) = α ifffor no β < α we have rk(t, A) = β, but there are m ∈ω and ⟨tk; k ∈ω⟩such that ∀k ∈ω: t ⊆tk, tk ∈ωm, tk(lh(t)) ≥k, and rk(tk, A) < α.Clearly, the rank is either < ω1 or undefined (in which case we say rk = ∞). Werepeat the proof of the following result for it is the main tool for §§ 2 and 4.1.2.

Main Lemma (Baumgartner–Dordal [BD, § 2] and Shelah [GS, § 4]). Let I ⊆Dbe dense.

Set A := {t; ∃f ∈ωω such that (t, f) ∈I}. Then rk(t∗, A) < ω1 for anyt∗∈ω<ω.Proof.Suppose rk(t∗, A) = ∞for some t∗∈ω<ω.Let S := {s ∈ω<ω strictlyincreasing; t∗⊆s and for all s∗with t∗⊆s∗and with ∀i ∈dom(s∗)\dom(t∗) (s∗(i) ≥s(i)),we have rk(s∗, A) = ∞}.

S ⊆ω<ω is a tree with stem t∗.Suppose S has an infinite branch ⟨si; i ∈ω⟩(i.e. s0 = t∗, lh(si) = lh(t∗) + i, andsi ⊆si+1).

Let g be the function defined by this branch: g = Si∈ω si. Then (t∗, g) ∈D.Choose (t, f) ≤(t∗, g) such that (t, f) ∈I.

Then t ∈A, i.e. rk(t, A) = 0; but also t ∈S,i.e.

rk(t, A) = ∞, a contradiction.So suppose S has no infinite branches, and let s∗be a maximal point in S. Thenwe have a sequence ⟨tk; k ∈ω⟩such that lh(tk) = lh(s∗) + 1, tk(lh(s∗)) ≥k, t∗⊆tk,∀i ∈dom(s∗)\dom(t∗) (tk(i) ≥s∗(i)), and rk(tk, A) < ∞. Now we can find a subset B ⊆ωand lh(t∗) ≤m ≤lh(s∗) and t ∈ωm such that ∀k ∈B (tk↾m = t) and k < ℓ, k, ℓ∈B,4

implies tk(lh(t)) < tℓ(lh(t)). Hence the sequence ⟨tk; k ∈B⟩witnesses rk(t, A) < ∞.

Onthe other hand t ∈S; i.e. rk(t, A) = ∞, again a contradiction.Usually Hechler forcing D′ is defined as follows.

(s, f) ∈D′ ⇐⇒s ∈ω<ω ∧f ∈ωω ∧s ⊆f(s, f) ≤(t, g) ⇐⇒s ⊇t ∧∀n ∈ω (f(n) ≥g(n))We sketch how to introduce a rank on D′ having the same consequences as the one on Ddefined above. Let Ω= {t; dom(t) ⊆ω ∧|t| < ω ∧rng(t) ⊆ω}.

Given t ∈ΩandA ⊆ω<ω we define by induction when the rank rk(t, A) is α. (a) rk(t, A) = 0 ifft ∈A.

(b) rk(t, A) = α ifffor no β < α we have rk(t, A) = β, but there are M ∈[ω]<ω and⟨tk; k ∈ω⟩such that dom(t) ⊂M and ∀k ∈ω: t ⊆tk, tk ∈ωM, rk(tk, A) < α and∀i ∈M \ dom(t) ∀k1 ̸= k2 (tk1(i) ̸= tk2(i)).We leave it to the reader to verify that the result corresponding to 1.2. is true for thisrank on D′, and that the theorems of §§ 2 and 4 can be proved for D′ in the same way asthey are proved for D.§ 2. Application I — the effect of adding one Hechler real on the invariantsin Cicho´n’s diagramBefore being able to state the main result of this section (the consequences of whichwill be 1) and 2) in the abstract) we have to set up some notation.Let A ⊆[ω]ω be an a. d. family.

We will produce a set of D-names {τA; A ∈A}for functions in ωω as follows. For each A ∈A fix fA : A →ω onto with ∀n ∃∞m ∈A (fA(m) = n).

Now, if r ∈ωω is a real having the property that {n ∈ω; r(n) ∈A}is infinite, let gr : ω →ω be an enumeration of this set (i.e. gr(0) := the least n suchthat r(n) ∈A; gr(1) := the least n > gr(0) such that r(n) ∈A; etc.).

In this case we letτA(r) : ω →ω be defined as follows.τA(r)(n) := fA(r(gr(n))).5

As A is infinite, we have ∥−D”|rng( ˘d) ∩A| = ω”, where ˘d is the name for the Hechlerreal; in particular τA(d) will be defined in the generic extension. Thus we can think of⟨τA( ˘d); A ∈A⟩as a sequence of names in Hechler forcing for objects in ωω.2.1.

Main Theorem. Whenever A ⊆[ω]ω is an a. d. family in the ground model V ,d is Hechler over V , and f ∈ωω is any real in V [d], then {A ∈A; ∀∞n (f(n) ̸= τA(d)(n))}is at most countable (in V [d]).Remark.

Slight changes in the proof show that, in fact, {τA; A ∈A} is a Luzin set inV [d] for uncountable A. (Recall that an uncountable set of reals is called Luzin ifffor allmeager sets M, M ∩S is at most countable.)Proof.

The proof uses the main lemma (1.2.) as principal tool.

Let ˘f be a D-name fora real (for an element of ωω). Let In be the set of conditions deciding ˘f↾(n + 1) (n ∈ω).All In are dense.

Let Dn := {t; ∃f ∈ωω such that (t, f) ∈In} (cf the main lemma). Wewant to define when a set A ∈A is n-bad.For each t ∈ω<ω \ Dn strictly increasing we can find (according to the main lemmafor Dn) an m ∈ω and ⟨tk; k ∈ω⟩such that for all k ∈ω: tk is strictly increasing, t ⊆tk,tk ∈ωm, tk(lh(t)) ≥k, and rk(tk, Dn) < rk(t, Dn).

Let mt := m −lh(t). We define byinduction on i < mt when A ∈A is t −i −n-bad.

Along the way we also construct sets Bi(i < mt).i = 0. Let B0 = ω.

If there is A ∈A such that A ∩{tk(lh(t)); k ∈B0} is infinite,choose such an A0 and let A0 be t −0 −n-bad. Now let B1 = {k ∈ω; tk(lh(t)) ∈A0}.

Ifthere is no such A, let B1 = B0 = ω.i →i + 1 (i + 1 < mt). We assume that Bi+1 is defined and infinite.

If there is A ∈Asuch that A ∩{tk(lh(t) + i + 1); k ∈Bi+1} is infinite, choose such an Ai+1 and let Ai+1be t −(i + 1) −n-bad. Now let Bi+2 = {k ∈Bi+1; tk(lh(t) + i + 1) ∈Ai+1}.

If there is nosuch A, let Bi+2 = Bi+1.In the end, we set Bt := Bmt. We say that A ∈A is n-bad iffit is t −i −n-bad forsome strictly increasing t ∈ω<ω \ Dn and i < mt.

Finally A ∈A is bad iffit is n-bad forsome n ∈ω. Let A ˘f = {A ∈A; A bad }.

Since for n ∈ω, t ∈ω<ω and i < mt at mostone A ∈A is t −i −n-bad, A ˘f is countable.Claim. If A ∈A \ A ˘f, then ∥−D∃∞n ( ˘f(n) = τA( ˘d)(n)).Remark.

Clearly this claim finishes the proof of the main theorem.6

Proof. Suppose not, and choose (s, g) ∈D, k ∈ω, and A ∈A \ A ˘f such that(s, g) ∥−D∀n ≥k ( ˘f(n) ̸= τA( ˘d)(n)).Let ℓ≥k be such that |rng(s) ∩A| ≤ℓ; i.e.

s does not decide the value of τA( ˘d)(ℓ). Byincreasing s, if necessary, we can assume that |rng(s) ∩A| = ℓ.

Let Y := {t ∈ω<ω; tstrictly increasing, s ⊆t, ∀i ∈dom(t)\dom(s) (t(i) ≥g(i)), and |rng(t)∩A| = ℓ}. Chooset ∈Y such that rk(t, Dℓ) is minimal.Subclaim.

rk(t, Dℓ) = 0.Proof. Suppose not.

Then choose by the main lemma (1.2.) m ∈ω and ⟨tk; k ∈ω⟩(i.e.all tk are strictly increasing, t ⊆tk, tk ∈ωm, tk(lh(t)) ≥k, and rk(tk, Dℓ)

In fact, we require that m and ⟨tk; k ∈ω⟩are the same as the ones chosen for ℓ, tin the definition of ℓ-badness. Let mt = m−lh(t) as above, and look at Bt.

By construction(as A is not t−i−ℓ-bad for any i < mt) and almost-disjointness, A∩{tk(lh(t)+i); k ∈Bi+1}is finite for all i < mt.So there is k ∈Bt such that rng(tk) ∩A = rng(t) ∩A, i.e.|rng(tk) ∩A| = ℓ, and tk(i) ≥g(i) for all i ∈dom(tk) \ dom(s).Hence tk ∈Y andrk(tk, Dℓ) < rk(t, Dℓ), contradicting the minimality of rk(t, Dℓ). This proves the subclaim.Continuation of the proof of the claim.

As rk(t, Dℓ) = 0 we have an h ∈ωω suchthat (t, h) ∈Iℓ. Then (t, max(h, g)) ≤(s, g), and this condition decides the value of ˘f at ℓwithout deciding the value of τA( ˘d) at ℓ.

Suppose that (t, max(h, g)) ∥−D” ˘f(ℓ) = j”. Nowchoose i ≥max(h, g)(lh(t)) such that i ∈A and fA(i) = j (this exists by the choice of thefunction fA).

Then(tˆ⟨i⟩, max(h, g)) ∥−D ˘f(ℓ) = j = fA(i) = fA( ˘d(g ˘d(ℓ))) = τA( ˘d)(ℓ).This final contradiction ends the proof of the claim and of the main theorem.We will sketch how a modification of this argument gives the following result.2.2. Theorem.

After adding one Hechler real d to V , there is a maximal almostdisjoint family of subsets of ω of size ω1 in V [d].Sketch of proof. We start with an observation which will relate Luzin sets and maximalalmost disjoint families.Observation.

Let ⟨Nα; ω ≤α < ω1⟩, ⟨hα; ω ≤α < ω1⟩and ⟨rα; ω ≤α < ω1⟩besequences such that Nα ≺H(κ) is countable and Nα ≺Nβ for α < β, hα ∈αω ∩Nα is7

one-to-one and onto, rα ∈ωω is Cohen over Nα and ⟨rα; α < β⟩∈Nβ. Define recursivelysets Cα for α < ω1.

⟨Cn; n ∈ω⟩is a partition of ω into countable pieces lying in Nω. Forα ≥ω, Cα := {rα(n); n ∈ω ∧∀m < n (rα(n) ̸∈Chα(m))}.

Then {Cα; α ∈ω1} is an a.d. family.Proof. The construction gives almost–disjointness.

So it suffices to show that eachCα is infinite. But this follows from the fact that each rα is Cohen over Nα and that theunion of finitely many Cβ’s (for β < α) is coinfinite.Now let A = ⟨Aα; α < ω1⟩∈V be an a. d. family.

As ⟨τAα(d); α < ω1⟩is Luzinin V [d] (see the remark following the statement of theorem 2.1.) we can find a strictlyincreasing function φ : ω1 \ ω →ω1 and sequences ⟨Nα; ω ≤α < ω1⟩, ⟨hα; ω ≤α < ω1⟩such that for rα := τAφ(α)(d) the requirements of the above observation are satisfied.

Byccc–ness of D, we may assume that φ ∈V ; and hence, that φ = id, thinning A out ifnecessary. We want to show that the resulting family ⟨Cα; α < ω1⟩is a m. a. d. family.For suppose not.

Then there is a D-name ˘C such that∥−D∀α < ω1 (| ˘Cα ∩˘C| < ω).Let ˘f be the D-name for the strictly increasing enumeration of ˘C. As in the proof of 2.1.we let In be the set of conditions deciding ˘f↾(n + 1), Dn := {t; ∃f ∈ωω ((t, f) ∈In)},and we define when a set A ∈A is n-bad (so that at most countably many sets will ben-bad).Furthermore, for each α < ω1 we let σα be the D-name for a natural number suchthat∥−D ˘Cα ∩˘C ⊆σα.We let I′α be the set of conditions deciding σα, D′α := {t; ∃f ∈ωω ((t, f) ∈I′α)};analogously to the proof of theorem 2.1. we define when a set A ∈A is α-bad (so that atmost countably many sets will be α-bad).Next choose α < ω1 such that1) if Aβ is n-bad for some n, then β < α;2) if β < α and Aγ is β-bad, then γ < α.Claim.

∥−D| ˘Cα ∩˘C| = ω.Proof. Suppose not, and choose (s, g) ∈D and k ∈ω such that(s, g) ∥−D ˘Cα ∩˘C ⊆k.8

Let ℓ≥k be such that |rng(s) ∩Aα| ≤ℓ; without loss |rng(s) ∩Aα| = ℓ. Let Y := {t ∈ω<ω; t strictly increasing, s ⊆t, ∀i ∈dom(t)\dom(s) (t(i) ≥g(i)), and |rng(t)∩Aα| = ℓ}.By the argument of the subclaim in the proof of 2.1. there is a t ∈Y such that ∀m <ℓ(rk(t, D′hα(m)) = 0).

Hence there is an h ∈ωω such that (t, h) ∈Tm<ℓI′hα(m). Withoutloss h ≥g.

Then (t, h) ≤(s, g), and this condition decides the values of σhα(m) (m < ℓ);suppose that (t, h) ∥−D”∀m < ℓ(σhα(m) = sm)”. Choose ℓ′ larger that the maximum ofthe sm (m < ℓ) and k. Again using the argument of the subclaim (2.1.) find t′ ⊇t suchthat ∀i ∈dom(t′) \ dom(t) (t′(i) ≥h(i)), |rng(t′) ∩Aα| = ℓ, and rk(t′, Dℓ′) = 0.

Thusthere exists an h′ ∈ωω such that(t′, h′) ∥−D” ˘f(ℓ′) = j” for some j.Without loss h′ ≥h. Then (t′, h′) ≤(t, h).

As ∥−D” ˘f is strictly increasing”, j ≥ℓ′ ≥k;by construction we have in particular that (t′, h′) ∥−D”∀m < ℓ(j ̸∈˘Chα(m))”. Choosei ≥h′(lh(t′)) such that i ∈Aα and fAα(i) = j.

Then(t′ˆ⟨i⟩, h′) ∥−D ˘f(ℓ′) = j = fAα(i) = τAα( ˘d)(ℓ) = ˘rα(ℓ) ∈˘Cα.This final contradiction proves the claim, and the theorem as well.In our proof we constructed a m. a. d. family of size ω1 from a Luzin set in V [d]. Wedo not know whether this can be done in ZFC.2.3.

Question (Fleissner, see [Mi, 4.7.]) Does the existence of a Luzin set imply theexistence of a m. a. d. family of size ω1?Remark.

It is consistent that there is a m. a. d. family of size ω1, but no Luzin set.This is known to be true in the model obtained by adding at least ω2 random reals to amodel of ZFC + CH.We next turn to consequences of theorem 2.1.2.4. Theorem.

Let V ⊆W be universes of set theory, ωV1 = ωW1 . Then no real inW[d] is eventually different from the reals in V [d], where d is Hechler over V .Remark.

Remember that Hechler forcing has an absolute definition. So d will beHechler over V as well.Proof.

Let A ⊆[ω]ω be an almost disjoint family in V of size ω1. Assume that thefunctions fA for A ∈A (defined at the beginning of this section) are also in V .

Then9

each real in W[d] can only be eventually different from countably many of the reals in{τA(d); A ∈A} ∈V [d], by the main theorem.To be able to explain our next corollary to the main theorem, we need to introduce afew cardinals. Given a σ-ideal I ⊆P(2ω), we letadd(I) := the least κ such that ∃F ∈[I]κ (S F ̸∈I);cov(I) := the least κ such that ∃F ∈[I]κ (S F = 2ω);unif(I) := the least κ such that [2ω]κ \ I ̸= ∅;cof(I) := the least κ such that ∃F ∈[I]κ ∀A ∈I ∃B ∈F (A ⊆B).We also defineb := the least κ such that ∃F ∈[ωω]κ ∀f ∈ωω ∃g ∈F ∃∞n (g(n) > f(n));d := the least κ such that ∃F ∈[ωω]κ ∀f ∈ωω ∃g ∈F ∀∞n (g(n) > f(n)).If M is the ideal of meager sets, and N is the ideal of null sets, then we can arrange thesecardinals in the following diagram (called Cicho´n’s diagram).cov(N )unif(M)cof(M)cof(N )2ωbdω1add(N )add(M)cov(M)unif(N )(Here, the invariants grow larger, as one moves up and to the right in the diagram.) Thedotted line says that add(M) = min{b, cov(M)} and cof(M) = max{d, unif(M)}.

Forthe results which determine the shape of this diagram, we refer the reader to [Fr].Asurvey on independence proofs showing that no other relations can be proved betweenthese cardinals can be found in [BJS]. We shall need the following characterizations of thecardinals unif(M) and cov(M), which are due to Bartoszy´nski [Ba].unif(M) = the least κ such that ∃F ∈[ωω]κ ∀g ∈ωω ∃f ∈F ∃∞n (f(n) = g(n));cov(M) = the least κ such that ∃F ∈[ωω]κ ∀g ∈ωω ∃f ∈F ∀∞n (f(n) ̸= g(n)).We are ready to give our next result, which says essentially that after adding one Hechlerreal, the invariants on the left-hand side of the above diagram all equal ω1, whereas thoseon the right-hand side are all equal to 2ω.10

2.5. Theorem.

After adding one Hechler real d to V , unif(M) = ω1 and cov(M) =2ω in V [d].Proof. (i) Let A ⊆[ω]ω be an a. d. family of size ω1 in V .

Then by the maintheorem no real is eventually different from {τA(d); A ∈A}, giving unif(M) = ω1 (byBartoszy´nski’s characterization). (ii) Let A ⊆[ω]ω be an a. d. family of size 2ω in V (such a family exists, see e.g.

[Ku,chapter II, theorem 1.3]). Suppose κ = cov(M) < 2ω, and let {gα; α < κ} be a familyof functions such that ∀g ∈V [d] ∩ωω ∃α < κ ∀∞n (g(n) ̸= gα(n)), using Bartoszy´nski’scharacterization.

As |A| = 2ω > κ, there is A′ ⊆A, |A′| ≥ω1, and α < κ such that∀A ∈A′ ∀∞n (τA(d)(n) ̸= gα(n)). This contradicts the main theorem.Remark.

Instead of Bartoszy´nski’s characterization we could have used the fact that{τA(d); A ∈A} is a Luzin set (see the remark after 2.1.). We leave it to the reader toverify that the existence of a Luzin set implies unif(M) = ω1; and that the existence ofa Luzin set of size 2ω implies cov(M) = 2ω.We close with an application concerning absoluteness in the projective hierarchy.

Wefirst recall a notion due to the second author [Ju, § 2]. Given a universe of set theory Vand a forcing notion P ∈V we say that V is Σ1n −P-absolute ifffor every Σ1n-sentence φwith parameters in V we have V |= φ iffV P |= φ.

So this is equivalent to saying thatRV ≺Σ1n RV P . Note that Shoenfield’s Absoluteness Lemma [Je, theorem 98] says that V isalway Σ12 −P-absolute.

Furthermore, Σ13 −D- absoluteness is equivalent to all Σ12-sets havethe property of Baire [Ju, § 2]. This is a consequence of Solovay’s classical characterizationof the latter statement which says that it is equivalent to: for all reals a, the set of realsCohen over L[a] is comeager.2.6.

Theorem. Σ14 −D-absoluteness implies that ω1 > ωL[r]1for any real r.Proof.

Suppose there is an a ∈R such that ωL[a]1= ωV1 . By Σ13 −D-absolutenesswe have that all Σ12-sets have the property of Baire (see above); i.e.

∀b ∈R (Co(L[b]) iscomeager) (Co(M) denotes the set of reals Cohen over some model M of ZFC). Notethat x ∈Co(L[b]) is equivalent to∀c (c ̸∈L[b] ∩BC ∨ˆc is not meager ∨x ̸∈ˆc),where BC is the set of Borel codes which is Π11 [Je, lemma 42.1], and for c ∈BC, ˆc is theset coded by c. As L[b] is Σ12 [Je, lemma 41.1], Co(L[b]) is a Π12-set.

Hence ∀b ∈R (Co(L[b])11

is comeager) which is equivalent to∀b∃c (c ∈BC ∧ˆc is meager ∧∀x (x ∈ˆc ∨x ∈Co(L[b])))is a Π14-sentence. So it is true in V D by Σ14 −D-absoluteness; in particular Co(L[a][d])is comeager in V [d] which implies that there is a dominating real in V [d] over L[a][d],contradicting theorem 2.4.2.7.

Question. Are there results similar to theorems 2.4., 2.5., and 2.6. for Amoebaforcing or Amoeba-meager forcing?We conjecture that the answer is yes because both the Amoeba algebra and theAmoeba-meager algebra contain D as a complete subalgebra (see [Tr, § 6]; a definitionof the algebras can also be found there).

But there doesn’t seem to be a way to introducea rank on these algebras (as in § 1).§ 3. Interlude — perfect sets of random reals3.1.

Theorem. Let V ⊆W be models of ZFC.

Suppose there is a perfect set ofrandom reals in W over V . Then either1) there is a dominating real in W over V ; or2) µ(2ω ∩V ) = 0 in W.Proof.

Suppose not, and let T ∈W be a perfect set of random reals. Define f ∈ωω∩Was follows.f(i) = min{k; ∀σ ∈T ∩2i (|Tσ ∩2k| > 4i)}Let g ∈ωω∩V be such that ∃∞i (g(i) ≥f(i)).

Let U be the family of all u ∈Qi∈ω P(2g(i))such that u(i) ⊆2g(i) and |u(i)|2g(i) = 2−i. U can be thought of as a measure space (namely,for u ⊆2g(i) with|u|2g(i) = 2−i let µi(u) =1(2g(i)2g(i)−i); and let µ be the product measure of theµi).Let N ≺⟨H(κ)W , ...⟩be countable with g, T ∈N.

As µ(2ω ∩V ) ̸= 0 in W, we cannothave that 2ω ∩V ⊆∪{B; µ(B) = 0, B ∈N, B Borel }; i.e. there are reals in V which are12

random over N. Let u∗∈U be such a real. Using u∗we can define a measure zero set Bin V as follows.B = {h ∈2ω; ∃∞i (h↾g(i) ∈u∗(i))}Let (for k ∈ω) Bk = {h ∈2ω; ∀i ≥k (h↾g(i) ̸∈u∗(i))}.

Clearly 2ω \ B = ∪k∈ωBk; andthe Bk form an increasing chain of perfect sets of positive measure.As all reals in T are random over V we must have T ⊆∪k∈ωBk. This gives us σ ∈Tand k ∈ω such that Tσ ⊆Bk (otherwise choose σ0 ∈T such that σ0 ̸∈B0, σ1 ∈Tσ0 suchthat σ1 ̸∈B1, etc.

This way we construct a branch in T which does not lie in ∪k∈ωBk, acontradiction).By construction, we know that for infinitely many i, we have |Tσ ∩2g(i)| > 4i andu∗(i) ∩(Tσ ∩2g(i)) = ∅. For each such i and u ⊆2g(i) with|u|2g(i) = 2−i, the probabilitythat u ∩(Tσ ∩2g(i)) = ∅(in the sense of the measure µi defined above) is≤(2g(i) −4i2g(i))2g(i)−i ≤(e−4i2g(i) )2g(i)−i = e−2i.So the probability that this happens infinitely often is zero.

But u∗is random over N, acontradiction.Corollary (Cicho´n [BaJ, § 2]). If r is random over V , then there is no perfect setof random reals in V [r] over V .Remark.

Theorem 3.1. is best possible in the following sense.1) It is consistent that there are V ⊆W and a perfect tree T of random reals in W overV and µ∗(2ω ∩V ) > 0 in W (µ∗denotes outer measure). To see this add a Laverreal ℓto V and then a random real r to V [ℓ]; set W = V [ℓ][r].

By [BaJ, theorem 2.7]there is a perfect tree of random reals in W over V ; and by [JS, § 1] µ∗(2ω ∩V ) > 0in V [ℓ] and hence in W.2) It is consistent that there are V ⊆W and a perfect tree T of random reals in W overV and no dominating real in W over V (see [BrJ, theorem 1]).Before being able to state some consequences of this result, we need to introduce twofurther cardinals.wcov(N ) := the least κ such that ∃F ∈[N ]κ (2ω \ S F does not contain a perfect set);wunif(N ) := the least κ such that there is a family F ∈[[2<ω]ω]κ of perfect sets with ∀N ∈N ∃T ∈F (N ∩T = ∅).13

We can arrange these cardinals and some of those of the preceding section in the followingdiagram.2ωcof(N )cov(N )dwunif(N )wcov(N )bunif(N )add(N )ω1(Here the invariants get larger as one moves up in the diagram.) The dotted line says thatwcov(N ) ≥min{cov(N ), b} (and dually, wunif(N ) ≤max{unif(N ), d}) (see [BaJ, § 2]or [BrJ, 1.9]).

Using the above result we get3.2. Theorem.

(i) wcov(N ) ≤max{b, unif(N )};(ii) wunif(N ) ≥min{d, cov(N )} — In fact, given V ⊆W models of ZFC such thatin W there is a real which is random over a real which is unbounded over V , there existsa null set N ∈W such that for all perfect sets T ∈V , T ∩N ̸= ∅.Proof. (i) follows immediately from theorem 3.1; and the first sentence of (ii) followsfrom the last sentence of (ii).

The latter is proved by an argument which closely followsthe lines of the proof of theorem 3.1, and is therefore left to the reader.The most interesting question concerning the relationship of the cardinals in the abovediagram is the following (question 3’ of [BrJ]).3.3. Question.

Is it consistent that wcov(N ) > d? Dually, is it consistent thatwunif(N ) < b?14

§ 4. Application II — adding a Hechler real over a random real does notproduce a perfect set of random reals4.1.

Theorem. Let V ⊆W be models of ZFC such that1) there is no dominating real in W over V ;2) 2ω ∩V is non-measurable in W.Then there is no perfect set of random reals in W[d], where d is Hechler over W.Remark.

This result clearly contains theorem 3.1. as a special case; still we decided tobring the latter as a separate result because it has consequences for the cardinals involved(see above, 3.2.). Also, the proof of theorem 4.1. can be seen as a combination of theargument for 3.1. and the techniques developed in § 1.4.2.

Corollary. There is no perfect set of random reals in V [r][d], where r is randomover V , and d is Hechler over W = V [r].Proof of theorem 4.1.

We work in W. Let ˘T be a D-name for a perfect tree. We wantto show that T = ˘T[G] (G D-generic over W) contains reals which are not random over V .We say that A ⊆ω<ω is large iff∀(s, f) ∈D ∃s′ ∈A with (s′, f) ≤(s, f) (By (s′, f) wemean here and in the sequel the condition (s′, f ′) where f ′↾dom(s′) = s′ and f ′(n) = f(n)for n ≥dom(s′)).Claim.

The following set A is large: s ∈A ⇐⇒for some k < ω and ⟨tℓ, f 1ℓ, f 2ℓ; ℓ∈ω⟩we have s ⊆tℓ, tℓ∈ωk, tℓ(lh(s)) ≥ℓ, f 1ℓ̸= f 2ℓ∈2ω, f 1ℓ↾ℓ= f 2ℓ↾ℓ, and ∀f ∈ωω (withtℓ⊆f) ∀m ∈ω ∀i ∈{1, 2} ((tℓ, f) ̸ ∥−f iℓ↾m ̸∈˘T).Proof. Let sp ˘T be the D-name for the subset of ω which describes the levels at whichthere is a splitting node in ˘T.

By thinning out T (in the generic extension) if necessary,we may assume that∥−D the j-th member of sp ˘T (denoted by τj) is > ˘d(j),where ˘d is (as always) the D-name for the Hechler real. Let (s∗, f ∗) ∈D, lh(s∗) = j∗.

So(s∗, f ∗) forces no bound on τj∗— even no (s∗, f ′) does (*). We assume there is no s ∈Awith (s, f ∗) ≤(s∗, f ∗) and reach a contradiction.Let I be the dense set of conditions forcing a value to τj∗; and let B = {s ∈ω<ω; ∃f ∈ωω ((s, f) ∈I)}.

By the main lemma 1.2. we have rk(s∗, B) < ω1. We prove by induction15

on the ordinal β < ω1(**) if s ∈ω<ω is such that (s, f ∗) ≤(s∗, f ∗) and rk(s, B) = β, then ∃m < ω ∀f ∈ωω(with s ⊆f) ((s, f) ̸ ∥−τj∗̸= m).If we succeed for s = s∗then we get a contradiction to (*).β = 0. So s ∈B.

Thus for some f ′ ≥f ∗, (s, f ′) forces a value to τj∗: (s, f ′) ∥−τj∗= m,for some m ∈ω, giving (**).β > 0. By the definition of rank there are k ∈ω, tℓ∈ωk (ℓ∈ω) such that s ⊆tℓ,tℓ(lh(s)) ≥ℓ, and rk(tℓ, B) = βℓ< β.

(We consider only ℓwith ℓ≥max(rng(f ∗↾k)).) Byinduction hypothesis there are mℓ∈ω such that ∀f ∈ωω (with tℓ⊆f) ((tℓ, f) ̸ ∥−τj∗̸=mℓ).

We consider two subcases.Case 1. For some m we have infinitely many ℓsuch that mℓ= m. Then we can usethis m for s and get (**).Case 2.

⟨mℓ; ℓ∈ω⟩converges to ∞. Replacing it by a subsequence, if necessary,we may assume that it is strictly increasing.

We show that ⟨tℓ; ℓ∈ω⟩witnesses s ∈A,contradicting our initial assumption.For each ℓlet Tℓ= {ρ ∈2<ω; for no f ∈ωω does (tℓ, f) ∥−ρ ̸∈˘T}. Clearly Tℓ⊆2<ω,⟨⟩∈Tℓ, and Tℓis closed under initial segments.

Also we have that ρ ∈Tℓimplies eitherρˆ⟨0⟩∈Tℓor ρˆ⟨1⟩∈Tℓ(otherwise we can find f0, f1 ∈ωω such that (tℓ, f0) ∥−ρˆ⟨0⟩̸∈˘Tand (tℓ, f1) ∥−ρˆ⟨1⟩̸∈˘T; let f = max{f0, f1}; choose p ≤(tℓ, f) such that p ∥−ρ ∈˘T(by assumption on ρ); but then there exists q ≤p such that either q ∥−ρˆ⟨0⟩∈˘T orq ∥−ρˆ⟨1⟩∈˘T, a contradiction).Finally, Tℓhas a splitting node at level mℓ; i.e. for some ρ = ρℓ∈Tℓ∩2mℓ, we haveρˆ⟨0⟩∈Tℓand ρˆ⟨1⟩∈Tℓ(if not, for each ρ ∈2mℓ∃fρ ∈ωω such that (tℓ, fρ) ∥−”ρˆ⟨0⟩̸∈˘Tor ρˆ⟨1⟩̸∈˘T”; let f = max{fρ; ρ ∈2mℓ}.

We know that (tℓ, f) ̸ ∥−mℓ̸= τj∗; so there isp ≤(tℓ, f) such that p ∥−mℓ= τj∗; i.e. p ∥−mℓ∈sp ˘T; we now get a contradiction asbefore).Hence we can find f 1ℓ, f 2ℓ∈[Tℓ] such that f 1ℓ↾(mℓ+ 1) = ρℓˆ⟨0⟩and f 2ℓ↾(mℓ+ 1) =ρℓˆ⟨1⟩.

Thus ⟨tℓ; ℓ∈ω⟩, ⟨f 1ℓ, f 2ℓ; ℓ∈ω⟩witness s ∈A. This final contradiction proves theclaim.Continuation of the proof of the theorem.

We assume that ∥−D ˘T = {τj; j ∈ω}; i.e.τj[G] (j ∈ω) will enumerate the tree T = ˘T[G] in the generic extension. We also let ˘Tjbe the name for the tree Tτj[G]; i.e.

∥−D ˘Tj = {ν ∈˘T; ν ⊆τj or τj ⊆ν}. For each j ∈ω16

there is — according to the claim for ˘Tj instead of ˘T — a large set Aj ⊆ω<ω; and fors ∈Aj there is a sequence ⟨ts,jℓ, f 1,s,jℓ, f 2,s,jℓ; ℓ∈ω⟩that witnesses s ∈Aj. For everyj ∈ω, s ∈Aj and m ∈ω we define Sj,s,m = {f i,s,jℓ↾k; k ∈ω, i ∈{1, 2}, m ≤ℓ∈ω}.

Byconstruction the function fj,s,m defined by fj,s,m(k) = |Sj,s,m ∩2k| converges to ∞. Byassumption 1) we can choose g ∈ωω ∩M such that ∀j, s, m ∃∞i (|Sj,s,m ∩2g(i)| > 4i).Now let U be as in the proof of theorem 3.1.; and choose u∗∈U as there (i.e.

u∗is random over a countable model N containing g and all Sj,s,m — using assumption 2)).We also define B and Bk (k ∈ω) as in the proof of theorem 3.1.We assume that ∥−D” ˘T is a perfect set of reals random over V ”; in particular ∥−D ˘T ⊆Sk∈ω Bk. So there are (s∗, f ∗) ∈D, j ∈ω and k ∈ω such that(s∗, f ∗) ∥−D ˘Tj ⊆Bk(cf the corresponding argument in the proof of theorem 3.1.

).Without loss s∗∈Aj(otherwise increase the condition using the claim). Let m > max(rng(f ∗↾kj,s∗)) wherekj,s∗is such that for all ℓ∈ω, ts∗,jℓ∈ωkj,s∗.

Then ∀ℓ≥m, (ts∗,jℓ, f ∗) is an extension of(s∗, f ∗). So we must have Sj,s∗,m ⊆Bk (because for any element of the former set we havean extension of (s∗, f ∗) forcing this element into ˘Tj).The rest of the proof is again as in the proof of theorem 3.1.

For infinitely many i wehave |Sj,s∗,m ∩2g(i)| > 4i; for each such i, the probability that u∗(i) ∩(Sj,s∗,m ∩2g(i)) = ∅is ≤e−2i; the probability that this happens infinitely often is zero, contradicting the factthat u∗is random over N.References[Ba] T. Bartoszy´nski, Combinatorial aspects of measure and category, FundamentaMathematicae, vol. 127 (1987), pp.

225-239. [BaJ] T. Bartoszy´nski and H. Judah, Jumping with random reals, Annals of Pureand Applied Logic, vol.

48 (1990), pp. 197-213.

[BJS] T. Bartoszy´nski, H. Judah and S. Shelah, The Cicho´n diagram, submitted17

to Journal of Symbolic Logic. [BD] J. Baumgartner and P. Dordal, Adjoining dominating functions, Journal ofSymbolic Logic, vol.

50 (1985), pp. 94-101.

[BrJ] J. Brendle and H. Judah, Perfect sets of random reals, submitted to IsraelJournal of Mathematics. [CP] J. Cicho´n and J. Pawlikowski, On ideals of subsets of the plane and on Cohenreals, Journal of Symbolic Logic, vol.

51 (1986), pp. 560-569.

[Fr] D. Fremlin, Cicho´n’s diagram, S´eminaire Initiation `a l’Analyse (G. Choquet, M.Rogalski, J. Saint Raymond), Publications Math´ematiques de l’Universit´e Pierreet Marie Curie, Paris, 1984, pp. 5-01 - 5-13.

[GS] M. Gitik and S. Shelah, More on ideals with simple forcing notions, to appearin Annals of Pure and Applied Logic. [Je] T. Jech, Set theory, Academic Press, San Diego, 1978.

[Ju] H. Judah, Absoluteness for projective sets, to appear in Logic Colloquium 1990. [JS] H. Judah and S. Shelah, The Kunen-Miller chart (Lebesgue measure, the Baireproperty, Laver reals and preservation theorems for forcing), Journal of SymbolicLogic, vol.

55 (1990), pp. 909-927.

[Ku] K. Kunen, Set theory, North-Holland, Amsterdam, 1980. [Mi] A. Miller, Arnie Miller’s problem list, to appear in Proccedings of the Bar-Ilanconference on set theory of the reals, 1991.

[Pa] J. Pawlikowski, Why Solovay real produces Cohen real, Journal of SymbolicLogic, vol. 51 (1986), pp.

957-968. [Tr] J. Truss, Sets having calibre ℵ1, Logic Colloquium 76, North-Holland, Amster-dam, 1977, pp.

595-612.18

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