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Amoeba–absoluteness and projective measurability

arXiv:math/9209206v1 [math.LO] 15 Sep 1992Amoeba–absoluteness and projective measurabilityJ¨org Brendle∗Abraham Fraenkel Center for Mathematical Logic, Department of Mathematics, Bar–Ilan University, 52900Ramat–Gan, IsraelandMathematisches Institut der Universit¨at T¨ubingen, Auf der Morgenstelle 10, 7400 T¨ubingen, GermanyAbstract. We show that Σ14–Amoeba–absoluteness implies that ∀a ∈R (ωL[a]1< ωV1 ), andhence Σ13–measurability.

This answers a question of Haim Judah (private communication).IntroductionWe study the relationship between Amoeba forcing and projective measurability. Re-call that the Amoeba partial order A is defined as follows.A ∈A ⇐⇒A ⊆2ω ∧A open ∧µ(A) < 12A ≤B ⇐⇒B ⊆AAmoeba forcing generically adds a measure one set of random reals.

Its importance inthe investigation of measurability of projective sets stems from the classical result, due toSolovay, that(*)all Σ12–sets are measurable ⇐⇒∀a ∈R (µ(Ra(L[a])) = 1)∗The author would like to thank the MINERVA-foundation for supporting him1

(see, e.g., [JS 2, 0.1. and § 3]). Here Ra(M) denotes the set of reals random over a modelM of set theory.The connection between Amoeba forcing and projective measurability was made moreexplicit through Judah’s study of absoluteness between models V ⊆W of set theory suchthat W is a forcing extension of V [Ju].Definition (Judah [Ju, § 2]).

Let V be a universe of set theory. Given a forcing notionP ∈V we say that V is Σ1n −P–absolute ifffor every Σ1n–sentence φ with parameters in Vwe have V |= φ iffV P |= φ.

(So this is equivalent to saying that RV ≺Σ1n RV P . )Note that Shoenfield’s Absoluteness Lemma [Je, Theorem 98] says that V is always Σ12−P–absolute.

Furthermore, using (*), Judah showed [Ju, § 2](**)all Σ12–sets are measurable in V ⇐⇒V is Σ13 −A–absolute.Whereas there is no way of getting a characterization of Σ13–measurability analogous to(*), (**) suggests the investigation of the relation between Σ13–measurability and Σ14 −A–absoluteness.The main goal of this note is to establish one implication, namely thatΣ14 −A–absoluteness implies Σ13–measurability (Theorem 5 in § 2). Our tools for provingthis theorem are a partial earlier result of Judah’s, who showed Theorem 5 under theadditional assumption that ∀a ∈R (ωL[a]1< ωV1 ), and combinatorial ideas due to Cicho´nand Pawlikowski [CP], which will eventually yield that Judah’s additional assumption isin fact a consequence of Σ14 −A–absoluteness (§ 1 and Theorem 4 in § 2).Notation.

We shall mostly work with 2ω or ωω instead of R. L denotes the ideal ofLebesgue measure zero sets, and B is the ideal of meager sets. Σ1n(L) stands for all Σ1n–setsare Lebesgue measurable; and Σ1n(B) means all Σ1n–sets have the property of Baire.

For anon–trivial σ–ideal I ⊆P(2ω), let add(I) be the size of the smallest family of members inI whose union is not in I; cov(I) denotes the least κ such that 2ω can be covered by κ setsfrom I; unif(I) is the cardinality of he smallest subset of the reals which does not lie in I;and cof(I) is the size of the smallest F ⊆I such that every member of I is included in amember of F. We always have add(I) ≤cov(I) ≤cof(I) and add(I) ≤unif(I) ≤cof(I)(see, e.g., [CP] for details concerning these invariants in case I = L or B).Our forcing notation is rather standard (see [Je] for any notion left undefined here).We confuse to some extent Boolean–valued models V P and forcing extensions V [G], G2

P–generic over V . For p.o.s P, Q, P

For a sentence of the P–forcing language φ, ∥φ∥is the Boolean value of φ. P–namesfor objects in the forcing extension are denoted by symbols like ˘r. Finally, B will stand forthe random algebra, C for the Cohen algebra, and D for the Hechler p.o.

(see, e.g., [BJS]).Acknowledgments. I am very much indebted to both Haim Judah (for sharing withme his insight into projective measurability and motivating me to work in the area) andAndrzej Ros lanowski (for several stimulating discussions, concerning mainly the materialin § 1).§ 1.

The combinatorial componentWe start with a straightforward generalization of one version of the main result of[CP]. The proof is included for completeness’ sake.Theorem 1 (Cicho´n – Pawlikowski [CP, § 1]).

Assume that C ≤c P, and that for anyuncountable T ⊆P there is an s ∈C such that for all ℓ∈ω there exists F ⊆T of size ℓsuchthat any t extending s is compatible with T F ∈P. Then there is a family {Ax; x ∈ωω∩V }of Lebesgue measure zero sets in V C such that for all z ∈V P , {x ∈ωω ∩V ; z ̸∈Ax} is atmost countable.Proof.

Let {τn; n ∈ω} be a one–to–one enumeration of ω<ω; set code(τ) = n iffτ = τn for any τ ∈ω<ω. Let {Cn(i); i ∈ω} be an enumeration of all open intervals in theunit interval I = [0, 1] with rational endpoints of length 2−n, For x, y ∈ωω letBnx,y =Cn(τy(n)(code(x↾y(n + 1))))if code(x↾y(n + 1)) ∈dom(τy(n))∅if notLet Bx,y = TnSm>n Bmx,y.

Clearly µ(Bx,y) = 0. We claim that if c is Cohen over V ,Ax = Bx,c for x ∈ωω ∩V , then {Ax; x ∈ωω ∩V } is the required family.For suppose not.

Then there are a P–name ˘z, an uncountable set T ⊆ωω ∩V , T ∈V ,conditions px ∈P, and kx ∈ω (x ∈T) such that3

px ∥−P ∀n ≥kx (˘z ̸∈Bnx,˘c)(*).Choose T ′ ⊆T uncountable and k ∈ω such that ∀x ∈T ′ (kx = k). Fix s ∈C accordingto T ′.

Let ℓ≥k, ℓ≥lh(s), and choose F ⊆ωω of size 2ℓsuch that {px; x ∈F} satisfiesthe requirements of the Theorem. Next let n > ℓbe such that |{x↾n; x ∈F}| = 2ℓ.

LetF = {xi; i ∈2ℓ}, and choose i0, ..., i2ℓ−1 such that Cℓ(i0) ∪... ∪Cℓ(i2ℓ−1) = I. Let m ∈ωbe such that τm(code(x0↾n)) = i0, ..., τm(code(x2ℓ−1↾n)) = i2ℓ−1.

Let t ≤s be such thatt(ℓ) = m, t(ℓ+ 1) = n. Then Si∈2ℓCℓ(τt(ℓ)(code(xi↾t(l + 1)))) = I, i.e.t ∩\{px; x ∈F} ∥−P ˘z ∈[i∈2ℓCℓ(τ˘c(ℓ)(code(xi↾˘c(ℓ+ 1)))) =[i∈2ℓBℓxi,˘c,contradicting (*).As each open set in 2ω can be written as a countable disjoint union of sets of the form[σ] = {f ∈2ω; σ ⊆f}, where σ ∈2<ω, we can think of a condition A in the Amoebaalgebra A as a function φ : ω →Si∈ω P(2i) with φ(i) ∈P(2i) such that σ ∈φ(i) iffσ ∈2iand σ lies in the countable disjoint decomposition of A. We can furthermore assume thatφ has the property:(*)∀σ ∈2i \ φ(i) (µ(∪{[τ]; τ ⊇σ ∧∃j > i (τ ∈φ(j))}) < 2−i).

(Then φ is unique.) We define a p.o.

A′ as follows. (u, φ) ∈A′ ⇐⇒1) dom(φ) = ω ∧∀i ∈ω (φ(i) ∈P(2i)) ∧φ satisfies (∗)2) u ⊆φ (u is an initial segment of φ)3) µ(∪{[σ]; ∃i ∈ω (σ ∈φ(i))}) < 12(u, φ) ≤(v, ψ) ⇐⇒u ⊇v ∧∀i ∀σ ∈ψ(i) ∃j ≤i ∃τ ∈φ(j) (σ ⊇τ)Lemma 1.

A and A′ are equivalent.Proof.We define Φ : A →A′ as follows.Φ(φ) = (u, φ), where u ⊆φ is suchthat dom(u) is maximal with the following property: for any extension ψ ⊇φ in A,ψ↾dom(u) = φ↾dom(u). We claim that Φ is a dense embedding.Clearly ψ ≤φ implies Φ(ψ) ≤Φ(φ), and ψ⊥φ implies Φ(ψ)⊥Φ(φ).

To check density,choose (u, φ) ∈A′. Let i := dom(u) −1; and set Sφ := {σ ∈2i; for no j ≤i does thereexist τ ∈u(j) such that σ ⊇τ}.

For σ ∈Sφ we have mσ := µ([σ] \ ∪{[τ]; τ ⊇σ ∧∃i ≥dom(u) (τ ∈φ(i))}) > 0. Let a := min{mσ; σ ∈Sφ}; and note that Pσ∈Sφ mσ > 12.4

Now define ψ satisfying (*) such that1) ∀i ∈dom(u) (ψ(i) = φ(i))2) ∀i ≥dom(u) ∀τ1 ∈φ(i) ∃j ≤i ∃τ2 ∈ψ(j) (τ2 ⊆τ1)3) 12 > µ(∪{[τ]; ∃i ∈ω (τ ∈ψ(i))}) > 12 −a24) for each σ ∈Sφ, µ([σ] \ ∪{[τ]; τ ⊇σ ∧∃i ≥n (τ ∈ψ(i))}) ≥a2This is clearly possible. By construction we have Φ(ψ) = (u, ψ) ≤(u, φ).Next define A′′ ⊆A′ by(u, φ) ∈A′′ ⇐⇒for some n ∈ω we have µ(∪{[σ]; ∃i ∈dom(u) (σ ∈u(i))}) > 12 −12n ,µ(∪{[σ]; ∃i ∈dom(u) −1 (σ ∈u(i))}) ≤12 −12n ,and µ(∪{[σ]; ∃i ≥dom(u) (σ ∈φ(i))}) <12n+7 .Clearly A′′ is dense in A′.

Finally we want to define h : A′′ →C giving rise to a completeembedding of C into A. To this end, let f : ω →ω be such that ∀n ∃∞i (f(i) = n).

For(u, φ) ∈A′′ and n ∈ω such that 12 −12n+1 ≥µ(∪{[σ]; ∃i ∈dom(u) (σ ∈u(i))}) > 12 −12nand each j ≤n choose ij minimal such that µ(∪{[σ]; ∃i ∈ij (σ ∈u(i))}) > 12 −12j , and leth((u, φ)) = ⟨f(i0)⟩ˆ...ˆ⟨f(in)⟩. We leave it to the reader to verify that h : A′′ →C is indeeda projection (in the forcing theoretic sense).

Furthermore, given T ⊆A′′ uncountable wecan find T ′ ⊆T uncountable and u such that all elements of T ′ are of the form (u, φ)for some φ. Then there is an s ∈C such that ∀(u, φ) ∈T ′ (h((u, φ)) = s).

Next, givenℓ∈ω, we can find F ⊆T ′ of size ℓsuch that ∩F ∈A′′. Clearly h(∩F) = s and so anyextension of s in C will be compatible with ∩F.

Hence we have proved that A′′ satisfiesthe requirements of Theorem 1. Using Lemma 1 we getTheorem 2.

There is a family {Ax; x ∈ωω ∩V } of Lebesgue measure zero sets inV A such that for all z ∈V A, {x ∈ωω ∩V ; z ̸∈Ax} is at most countable.Corollary 1. Let V ⊆W be models of ZFC such that ωV1 = ωW1 .

Then there is noreal random in W A over V A.Proof. Let {Ax; x ∈ωω ∩W} be as in Theorem 2 and note that ∀z ∈ωω ∩W A ∃x ∈ωω ∩V (z ∈Ax).

Hence any real in W A lies in a measure zero set coded in V A.Using a similar argument as in [CP, § 3] we can proveCorollary 2. After adding one Amoeba real, cov(L) = add(L) = ω1 and unif(L) =cof(L) = 2ω.5

We note that in [BJS, § 2] results much stronger than Theorem 2 and the Corollarieswere proved for the Hechler p.o. D; e.g.

it was shown that after adding a Hechler real,add(B) = unif(B) = ω1 and cof(B) = cov(B) = 2ω [BJS, 2.5.]. Accordingly we ask:Question [BJS, 2.7.].

Is unif(B) = ω1 and cov(B) = 2ω after adding an Amoebareal?Before ending this section I wish to include a few comments, some of which are dueto Andrzej Ros lanowski.Definition (implicit in [Tr 2]). A p.o.

P is said to have (ω1, ω)–caliber ifffor anyuncountable T ⊆P of size ω1 there is a countable F ⊆T such that ∩F ∈P.This is equivalent to: any set of ordinals A in V P of size ≥ω1 has a countable subset B inV [Tr 2]. It is easy to see that if C ≤c P and P has (ω1, ω)–caliber, then the assumptionsof Theorem 1 are satisfied.

Furthermore the Amoeba algebra A has (ω1, ω)–caliber (theproof for this is similar to the corresponding proof for the random algebra B, given in [Tr2]). This gives an alternative argument to prove Theorem 2.

— Our reason for giving the(slightly more difficult) above argument involving A′ and A′′ is that along the same linesresults corresponding to Theorem 2 and the Corollary can be proved for p.o.s not having(ω1, ω)–caliber. We include two examples for such p.o.s:— the eventually different reals p.o.

E, due to Miller [Mi]:(s, G) ∈E ⇐⇒s ∈ω<ω ∧G ∈[ωω]<ω(s, G) ≤(t, H) ⇐⇒s ⊇t ∧G ⊇H ∧∀g ∈H ∀i (dom(t) ≤i < dom(s) →s(i) ̸= g(i))— the localization p.o. L (see, e.g., [Tr 3, § 2]):(σ, G) ∈L ⇐⇒σ ∈([ω]<ω)<ω ∧∀i ∈dom(σ) (|σ(i)| = i + 1) ∧G ∈[ωω]≤dom(σ)+1(σ, G) ≤(τ, H) ⇐⇒σ ⊇τ ∧G ⊇H ∧∀g ∈H ∀i (dom(τ) ≤i < dom(σ) →g(i) ∈σ(i))Let {fα; α < ω1} ⊆ωω be a family of pairwise eventually different reals (i.e.

α ̸= β →∃n ∀k ≥n (fα(k) ̸= fβ(k))). Then {(⟨⟩, {fα}); α < ω1} is an uncountable set of conditionsin E (and L) such that no countable subset has nontrivial intersection, thus witnessing thatE and L do not have (ω1, ω)–caliber.

We leave it to the reader to verify that both stillsatisfy the assumptions of Theorem 1, however (note that both have a definition similarto, but easier than, A′′). (The localization p.o.

L arose from Bartoszy´nski’s characterization of the cardinaladd(L) [Ba], and is closely related to the Amoeba algebra A. Truss [Tr 3, § 4] showed that6

A

The projective partWe first introduce a notion closely related to absoluteness, and discuss the relationshipbetween the two notions.Definition (Judah [Ju, § 2]). Let V be a universe of set theory.

Given a forcing notionP ∈V we say that V is Σ1n −P–correct ifffor every Σ1n–formula φ(x) with parameters inV and for every P–name τ for a real we have V [τ] |= φ(τ) iffV P |= φ(τ).Lemma 2. Suppose P

Then:(i) Σ1n −Q–correctness implies Σ1n −P–correctness. (ii) Σ1n+1 −Q–absoluteness + Σ1n −Q–correctness implies Σ1n+1 −P–absoluteness.Proof.

We prove both (i) and (ii) by induction on n.(i) n = 2 follows from Shoenfield’s Absoluteness Lemma. Suppose it is true for n ≥2and assume V is Σ1n+1 −Q–correct.

Let φ(x) be a Σ1n+1–formula, φ(x) = ∃yψ(y, x) whereψ is Π1n. Suppose first that V [τ] |= φ(τ).

Then V [τ] |= ∃xψ(x, τ). So there is a P-name σsuch that V [τ] = V [σ, τ] |= ψ(σ, τ).

By induction V P |= ψ(σ, τ); thus V P |= φ(τ).Assume now that V P |= φ(τ). Hence V P |= ∃xψ(x, τ); and we can again find a P–name σ such that V P |= ψ(σ, τ).

By induction V [σ, τ] |= ψ(σ, τ). So Σ1n −Q–correctnessimplies V Q |= ψ(σ, τ); thus V Q |= φ(τ).

Hence by Σ1n+1 −Q–correctness V [τ] |= φ(τ). (ii) n = 1 follows from Shoenfield’s Absoluteness Lemma.

Suppose (ii) is true forn ≥1 and assume V is Σ1n+2 −Q–absolute and Σ1n+1 −Q–correct.By (i) V is alsoΣ1n+1 −P–correct. Let φ be a Σ1n+2–sentence, φ = ∃xψ(x), where ψ is Π1n+1.

Suppose firstthat V |= φ; i.e. V |= ψ(a) for some a ∈V .

By induction V P |= ψ(a); thus V P |= φ.Assume now that V P |= φ; i.e.V P |= ψ(τ) for some P–name τ.By Σ1n+1 −P–correctness V [τ] |= ψ(τ). Hence Σ1n+1 −Q–correctness implies V Q |= φ.

Thus V |= φ byΣ1n+2 −Q–absoluteness.Lemma 3 (Truss [Tr 1, 6.5]). D

Definition (Judah – Shelah [JS 1, § 0]). A ccc notion of forcing (P, ≤) is called Sousliniffit can be thought of as a Σ11–subset of the reals R with both ≤and ⊥(incompatibility)being Σ11–relations (in the plane R2).Note that all p.o.s discussed in this paper are Souslin.Theorem 3 (Judah [Ju, § 2]).

Assume that ∀a ∈R (ωL[a]1< ωV1 ), and P ∈V is aSouslin forcing. Then V is Σ13 −P–correct.Theorem 4.

Σ14 −A–absoluteness implies that ∀a ∈R (ωL[a]1< ωV1 ).Corollary 3. Σ14−A–absoluteness implies Σ13−A–correctness, Σ14 −D–absoluteness,and Σ13 −D–correctness.Theorem 5.

Σ14 −A–absoluteness implies Σ13(L) and Σ13(B).The proof of Theorem 4 follows the lines of the proof of 2.6 in [BJS]. Theorem 5is a consequence of Theorem 4 and a result in [Ju, § 2].We give the proof here forcompleteness’ sake.

— Note that Σ13 −D–absoluteness is equivalent to Σ12(B) [Ju, § 2].Thus the implication Σ13 −A–absoluteness =⇒Σ13 −D–absoluteness (immediate fromLemmata 2 and 3) is just another version of the Raisonnier–Stern Theorem; and Corollary3 may be thought of as the corresponding result for Σ14.Proof of Theorem 4. Suppose there is an a ∈R such that ωL[a]1= ωV1 .

By Σ13 −A–absoluteness we have Σ12(L); i.e. ∀b ∈R (µ(Ra(L[b])) = 1) (see the beginning of thissection).

Note that x ∈Ra(L[b]) is equivalent to∀c (c ̸∈L[b] ∩BC ∨ˆc is not null ∨x ̸∈ˆc),where BC is the set of Borel codes which is Π11 [Je, Lemma 42.1], and for c ∈BC, ˆcis the set coded by c.As L[b] is Σ12 [Je, Lemma 41.1], Ra(L[b]) is a Π12–set.Hence∀b ∈R (µ(Ra(L[b])) = 1) which is equivalent to∀b∃c (c ∈BC ∧ˆc is null ∧∀x (x ∈ˆc ∨x ∈Ra(L[b])))is a Π14–sentence. So it is true in V A by Σ14–absoluteness; in particular Ra(L[a][r]) (wherer is Amoeba over V ) has measure one in V [r] which implies that there is a random real inV [r] over L[a][r], contradicting Corollary 1 in § 1.Proof of Corollary 3.

Follows from Theorems 3 and 4 and Lemmata 2 and 3.Proof of Theorem 5 (Judah). Let φ(x) be a Σ13–formula and A = {x; φ(x)}.

Weshall show that A is measurable in V . First note that the sentence A has measure zero is8

equivalent to∃c (c ∈BC ∧µ(ˆc) = 0 ∧∀x (¬φ(x) ∨x ∈ˆc)),which is Σ14. So by Σ14 −A–absoluteness, if A is null in V A, it is also null in V .Hence assume that A is not null in V A.

As µ(Ra(V )) = 1 in V A, there is r ∈Ra(V )∩Ain V A; i.e. V A |= φ(r).

By Σ13 −A–correctness V [r] |= φ(r). Now let φ(x) = ∃yψ(x, y),where ψ is Π12.Then there is an s ∈V [r] such that V [r] |= ψ(r, s).

If a ∈V codesthe parameters of ψ and of the name of s, we have by Shoenfield’s Absoluteness LemmaL[a][r] |= ψ(r, s). Let ˘r be the B–name for the random real r and s(˘r) a B–name for s.Then the Boolean value ∥ψ(˘r, s(˘r))∥is non–zero.

Furthermore, if r′ ∈∥ψ(˘r, s(˘r))∥∩V israndom over L[a], then L[a][r′] |= ψ(r′, s(r′)) and — by absoluteness — V |= ψ(r′, s(r′));in particular V |= φ(r′).By Σ13 −A–absoluteness we have that µ(Ra(L[a])) = 1 in V (cf Introduction). Andthe previous discussion gives us that Ra(L[a]) ∩∥ψ(˘r, s(˘r))∥⊆A.

This shows that anynon–null Σ13–set has positive inner measure; and it is easy to conclude from this that anyΣ13–set is indeed measurable.Finally, Σ13(B) follows along the same lines because A adds a comeager set of Cohenreals.Questions. 1) Does Σ13(L) imply Σ14 −A–absoluteness?2) Does Σ14–Amoeba–meager–absoluteness (or Σ14 −D–absoluteness) imply Σ13(B)?

(cf[Tr 1, § 5] for Amoeba–meager forcing — the problem here is whether Σ14–Amoeba–meager–absoluteness implies ∀a ∈R (ωL[a]1< ωV1 ); cf [BJS, § 2] for D — the problem here is thatD does not add a comeager set of Cohen reals)3) Does ∀n (V is Σ1n −A–absolute ) imply projective measurability?4) (Judah) Does Σ13(L) imply Σ13(B)? (cf Corollary 3)References[Ba] T. Bartoszy´nski, Combinatorial aspects of measure and category, FundamentaMathematicae, vol.

127 (1987), pp. 225-239.9

[BJS] J. Brendle, H. Judah and S. Shelah, Combinatorial properties of Hechlerforcing, preprint. [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.

[Je] T. Jech, Set theory, Academic Press, San Diego, 1978. [Ju] H. Judah, Absoluteness for projective sets, to appear in Logic Colloquium 1990.

[JS 1] H. Judah and S. Shelah, Souslin forcing, Journal of Symbolic Logic, vol. 53(1988), pp.

1188-1207. [JS 2] H. Judah and S. Shelah, ∆12–sets of reals, Annals of Pure and Applied Logic,vol.

42 (1989), pp. 207-223.

[Mi] A. Miller, Some properties of measure and category, Transactions of the Amer-ican Mathematical Society, vol. 266 (1981), pp.

93-114. [Tr 1] J. Truss, Sets having calibre ℵ1, Logic Colloquium 76, North-Holland, Amster-dam, 1977, pp.

595-612. [Tr 2] J. Truss, The noncommutativity of random and generic extensions, Journal ofSymbolic Logic, vol.

48 (1983), pp. 1008-1012.

[Tr 3] J. Truss, Connections between different Amoeba algebras, Fundamenta Mathe-maticae, vol. 130 (1988), pp.

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