Some Pathological Sets in Special Model of Set Theory

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 ⊂ ωω is called Bernstein iff B ∩ P 6= ∅ 6 = P \ B for all perfect P ⊂ ωω. • Letting E0 ⊂ (ωω)2 be the Vitali equivalence relation defined by xE0y iff ∃n0∀n ≥ n0, x(n) = y(n), V ⊂ ωω is called Vitali if V picks exactly one element from each E0 equvalence class, i.e. ∀x∃y∀z((zE0x ∧ 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 = ω L 1 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