A Groszek‐Laver pair of undistinguishable ‐classes

Abstract

A generic extension L[x, y] of the constructible universe L by reals x, y is defined, in which the union of E 0 -classes of x and y is a lightface 1 2 set, but neither of these two E 0 -classes is separately ordinal-definable.

On Russell typicality in Set Theory

“…of a non-OD generic real a ∈ 2 ω , introduced in [14] and also applied in [7,21,20]. This is done by a forcing notion P having the following key properties, see [14].…”

Models of set theory in which separation theorem fails

“…The idea to use finite-support products of Jensen's forcing in order to obtain models with different definability effects belongs to Enayat [5]. It was exploited to obtain generic models with: countable non-empty Π 1 2 sets (even E 0 -classes) with no OD elements [16,18], a countable Π 1 2 Groszek -Laver pair [6], planar Π 1 2 sets with countable cross-sections and OD-non-uniformizable [17,20], and also a model where the separation theorem fails for both Σ 1 3 and Π 1 3 [17]. The latter result corresponds to the case n = 3 of Theorem 1.2, in which case (III) is immediately true by Shoenfield.…”

“…The latter result corresponds to the case n = 3 of Theorem 1.2, in which case (III) is immediately true by Shoenfield. On the other hand, conditions similar to (I), (II) for n = 3, are involved in the forcing constructions in [6,16,17,18,20], and in [12] itself, where a CCC forcing J ∈ L is defined to add a real a ∈ 2 ω so that a is the only J-generic real in L[a], and "being a J-generic real" is Π 1 2 . These properties are implied by a special construction of J = α<ω 1 J α in L from countable sets J α of perfect trees.…”

“…Therefore there exists a -decreasing sequence Φ = n j , ϕ j j<ω of pairs n j , ϕ j ∈ ω × Sys(π), M + -generic in the sense that it intersects every set D ∈ M, D ⊆ ω × Sys(π), open dense in ω × Sys(π). 6 Let us fix any such a M + -generic sequence Φ.…”

Definable Minimal Collapse Functions at Arbitrary Projective Levels

“…Several variations of Jensen's forcing are known. In particular, a model in [15] in which, for a given n ≥ 3 there exists a minimal nonconstructible 1 n singleton but all 1 n sets x ⊆ are constructible, an 2 -long iteration of Jensen's forcing in [1], a model in [17] in which there is an equivalence class of the equivalence relation E 0 1 (a E 0 -class, for brevity), which is a lightface 1 2 set in , not containing OD elements, an extension of this result to arbitrary projective classes in [19], related models in [9] ( 1 2 Groszek-Laver pairs of E 0 -classes), [18,20,21] (nonuniformizable projective sets with countable vertical cross-sections), and a very recent [8] (a model in which the axiom of dependent choices 1 2 -DC fails, but the countable AC holds for sets of reals).…”