Some Pathological Sets in Special Model of Set Theory

Some Pathological Sets in Special Model of Set Theory Mariam Beriashvili, Ralf Schindler February 22, 2017 Abstract We Produce a model of ZF+DC in which there are Bernstein sets, Luzin sets, and Sierpinski sets, but there is no Vitali sets and hence no Hamel basis. Definition. • B ⊂ ω ω is called Bernstein iff B ∩ P 6= ∅ = 6 P \ B for all perfect P ⊂ ωω . • Letting E0 ⊂ (ω ω )2 be the Vitali equivalence relation defined by xE0 y iff ∃n0 ∀n ≥ n0 , x(n) = y(n), V ⊂ ω ω is called Vitali if V picks exactly one element from each E0 equvalence class, i.e. ∀x∃y∀z((zE0 x ∧ z ∈ V ) ↔ z = y) Building upon [1], [3] proves that ZF + ”there is a Bernstein set” does not yield a ”Vitali set” V rec in the redefined sense that V rec picks exactly one element from each Turing degree. We here show that a slight variation of the argument of [1] amd [3] show that ZF + ”there is a Bernstein set” does not yield a ”Vitali set” in the original sense as defined above. Theorem 1. ZF + ”there is a Bernstein set” does not prove ”there is a Vitali set”. Proof . Let G be a C(ω1 )-generic over L, where ω1 = ω1L and C(ω1 ) is the finite support product of ω1 Cohen forcing, cf. [2, p105]. For α < ω1 let β(α) be the least β > ω, β > α such that Lβ |= ”α ≤ ℵ0 ” and let eα : ω ↔ Lα be the Lβ(α) -least bijection. Let Eα ⊂ ω × ω be such that (ω; Eα ) ∼ =eα (Lα ; ∈) and let gα be the set of all n < ω such that there 1 are ξ < α, k, m < ω with eα (n) = (ξ, k, m) and p(ξ)(k) = m for some p ∈ G. I.e., Eα is a canonical code for Lα and gα codes Gα relative to Eα . We may define zα : ω → ω by zα (2l) = ( 1, iff (l)0 Eα (l)1 ; 0, iff (l)0 6 Eα (l)1 . and zα (2l + 1) = ( 1, iff l ∈ gα ; 0, iff l 6∈ gα (l)1 . . Here, (l)0 and (l)1 is the first and second component, resp., of l where construed as a pair of natural number, i.e., fixing a canonical e : ω ↔ ω × ω, e(l) = ((l)0 , (l)1 ). Claim 1 For every x ∈ ω ω ∩ L[G] there is some α < ω1 such that x ∈ L[zα ]. Proof: Given x, there is some α with x ∈ Lα [Gα]. But Lα [Gα] ∈ L[zα ] Claim 2 Let s ∈ C(ω1 ) be any condition. Let GS be he collection of all p ∈ C(ω1 ) for which there is a q ∈ G with dom(p(ξ)) = dom(q(ζ)) for all ξ < ω1 and p(ξ)(k) = ( s(ξ)(k), if k ∈ dom(s(ξ)); q(ξ)(k), o.w. Then Gs is C(ω1 )-generic over L, and L[Gs ] = L[G]. Also, s ∈ Gs . Proof: cf. [2] Claim 3. Let s ∈ C(ω1 ) be any condition. Let α < ω1 , let gαs = {n : ∃ξ < α, ∃k, m < ω[eα (n) = (ξ, k, m)∧p(ξ)(k) = m f or some p ∈ Gs ]} and let zαs : ω → ω be defined by zαs (2l) = ( 1, iff (l)0 Eα (l)1 ; 0, iff (l)0 6 Eα (l)1 . 2 zαs (2l + 1) = ( 1, iff l ∈ gαs ; 0, iff l 6∈ gαs . (I.e., zαs is defined as zα above except for using gαs instead of gα ) Then zαs E0 zα . Proof: Immediate, as there are only finitely many pairs (ξ, k) such that there are p ∈ Gs and q ∈ G with p(ξ)(k) 6= q(ξ)(k). Definition. Let us write dα = {z : zE0 zα } for the E0 -equivalence class of zα . By claim 3, {zαs : s ∈ C(ω1 )} ⊂ dα and by Claim 1: Claim1′ . For every x ∈ ω ω ∩ L[G] there is some α < ω1 s.t. x ∈ L[Z] for all z ∈ dα . Let us now consider the model L[G] N = HODωω ∩L[G]∪{(dα :α<ω1 } i.e. the class of all X ∈ L[G] which inside L[G] are hereditarily ordinal definable from parameters in (ω ω ∩ L[G]) ∪ {(dα : α < ω1 )}, cf. [Sch. p. 86]. N |= ZF . Claim 4. N |= ¬AC, i.e., the axiom of choice fails in N , in fact: There is no well-ordering of the reals in N. Proof: Suppose L[G] has a well-ordering of its reals which is definable from α ~ ∈ OR, ~y ∈ ω ω ∩ L[G] and (dα : α < ω1 ). Let α < ω1 be such that ~y ∈ Lα [Gα]. Let α̃ > α,α̃ < ω1 . Then zα̃ must be definable in L[G] from α ~ , γ, ~y , and (dα : α < ω1 ) for some ordinal γ. Let us assume w.l.o.g. that ~y ∈ ω ω ∩ L; the argument in the general case is just a simple variant of the argument that is to come. There is a formula φ such that for all k, m < ω, z α̃ (k) = m) iff L[G] |= φ(k, m, α ~ , γ, ~y , (dα : α < ω1 )). 3 Let the formula ψ define (dα : α < ω1 ) from G over L[G], i.e., ~ d~ = (dα : α < ω1 ) ↔ ψ(d, ~ Ġ)). Hence zα̃ (k) = (m) iff L[G] |= ∀d( ~ L[G] |= ”(k, m, α ~ , γ, ~y , d), iff C(ω1 ) ∃p ∈ GL where ~ ”φ(ǩ, m̌, α ~ˇ , γ̌, ~yˇ, d), ~ Ġ)” ψ(d, where ~ Ġ)” ψ(d, C(ω ) ~ Suppose s ∈ C(ω1 ) is any condition such that sL 1 ”¬φ(ǩ, m̌, α ~ˇ , γ̌, ~yˇ, d), ~ Ġ). Using Claim 2, Gs is C(ω1 )-generic over L, s ∈ Gs , and where ψ(d, s L[G ] = L[G]. This gives that ~ L[Gs ] = L[G] |= ”¬φ(k, m, α ~ , γ, ~y , d), where ~ Gs )” ψ(d, ~ Gs ) holds true in L[G], then in fact . However, Claim 3 beys us that if ψ(d, d~ = (dα : α < ω) and therefore L[G] |= ¬φ(k, m, α ~ , γ, ~y , (dα : α < ω1 )). Contradiction! We have shown that zα̃ (k) = m iff C(ω1 ) ∃p ∈ GpL iff ~ whereψ(d, ~ Ġ)” ”φ(ǩ, m̌, α ~ˇ , γ̌, ~yˇ, d)”, C(ω ) ~ 1C(ω11 ) L ”φ(ǩ, m̌, α ~ˇ , γ̌, ~yˇ, d)” ~ Ġ). But then zα̃ ∈ L, cf. [Sch., p. 118]. However, L[zα̃ ] contains where ψ(d, a Cohen real over L. Contradiction! We have verified Claim 4. Claim 5. N |= ”There is no Vitali Set”. Proof. Suppose there is some V ∈ N such that V ∩ dα is singleton for each α < ω1 . There is then a sequence (zα∗ : α < ω1 ) in N such that zα∗ ∈ dα for every α < ω1 . Let <α be the canonical well-ordering of L[zα∗ ] as being defined inside L[zα∗ ]. For x ∈ ω ω ∩ L[G] let α(x) be the least α < ω1 such that x ∈ L[zα∗ ]. By claim 1̀, α(x) is always well-defined. We may then define a well-ordere < of ω ω ∩ N inside N as follows x < y iff α(x) < α(y) or α(x) = α(y.) Contradiction with Claim 4. Claim 6. N |= ”There is a Bernstein Set”. 4 Proof: Let B = {b ∈ ω ω : ∃ even α[b ∈ L[z] for all/some z ∈ dα+1 ∧ b 6∈ L[z] for all/some z ∈ dα ]} and ′ = {b ∈ ω ω : ∃oddα[b ∈ L[z] for all/some z ∈ dα+1 ∧ b ∈ / L[Z]f orall/somez ∈ dα }, as being defined in N . Obviously, ′ B ∩ B = ∅. Let P ⊂ ω ω be a perfect set in N , say P = [T ] for some perfect tree T , T ∈ L[z], z ∈ dα , α even. We work in N . Pick z ∗ ∈ dα+1 . We may easily find some b ∈ ω ω such that L[T, b] = L[z ∗ ]. In particular, b ∈ L[z ∗ ]. If b ∈ L[z], then L[z ∗ ] = L[T, b] ⊂ L[z], which condracts z ∗ ∈ dα+1 and z ∈ dα . Hence n ∈ / L[z ′ ] for any z ′ ∈ dα . We have shown that B ∩ P 6= ∅. Virtually the same argument shows that B ′ ∩ P 6= ∅. But then B is Bernstein. We may verify that there are Luzin and Sierpinski sets in N . Definition • L ⊂ ω ω is called Luzini iff L is uncountable and L ∩ M ≤ ℵ0 for every meager set. • S ⊂ ω ω is called Sierpinski iff S is uncountable and S ∩ N ≤ ℵ0 for all null set. In what follows, we shall feel free using the above introduced notions. For each α < ω1 , let κ(α) < ω1 be the least κ such that Lκ [z] |= ZF C − for all/some z ∈ dα . Lemma 1. N |= ”There is a Sierpinski set”. Proof: Let us define a normal function f : ω1 → ω1 as follows, working entirely inside N . For α < ω1 , let Hα be the collection of all Gδ null sets which have a reale code in Lκ(α)[z] for some/all z ∈ dα . Notice that H is S countable, so that H is a null set for all α < ω1 . Given α < ω1 , let f (α) be the least β > α such that there is some x ∈ ω ω such that for all/some z ∈ dβ and for all/some z̄ ∈ dα : x ∈ Lκ(β) \ (Lκ(α) ∪ [ H) For limit λ, let f (λ) = supα<λ f (α). We then let S be the collection of reals x such that for some α < ω1 with β being f (α), i.e. S = {x ∈ ω ω : ∃α < ω1 [x ∈ Lκ(f (α))[z] \ (Lκ(α)[z̄∪S H) for some/all z ∈ df (α) and z̄ ∈ dα ])}. It is easy to see that S is a Sierpinski set. Virtually the same proof shows: Lemma 2. N |= ” There is a Luzini Set”. 5 Proof: As the proof of the previous lemma, replacing H with the collection of all meager sets which have a real code in Lκ(α)[z] , for some/all z ∈ dα . For the record, let us also state: Lemma 3. N |= ”There is no Hamel basis”. This immediately follows from above mentioned results about Bernstein sets and Vitali sets together with the following. Definition. Recall that a Hamel basis is a basis for R construed as a vector space over Q. Lemma (Folklore). In ZF C, if there is a Hamel basis, then there is a Vitali Set. Proof: Fix a Hamel basis B. For each x, there is a unique finite bx ⊂ B of leas size such that [x]E0 ⊂< bx >. Using a well-ordering of the finite sequences of rational, we may then for each x ∈ ω ω pick y ∈ [x]E0 such that if X y= ~r bx , ~r ∈<ω Q, P then ~r is the least r~′ such that r~′ ∈ [x]E0 . this gives a Vitali set. We showed there are Bernstein, Luzini and Sierpinski sets in N , but no Vitali sets and no Hamel basis. References 1. Nies, A., dl./dropboxusercontent.com/n/310127/Blog/Blog2012.pdf, p.48 2. Schindler, R., Set Theory. Exploring independence and truth, SpringerVerlag, 2012 3. Wang, W., Wu, L., Yu, L., Cofinal maximal chains in the Turing degrees 6