A countable definable set containing no definable elements

Abstract

We make use of a finite support product of Jensen forcing to define a model in which there is a countable non-empty Π 1 2 set X of reals containing no ordinal-definable real. 1 * Revised version. The revision includes an updated proof of Lemma 4.5 (the densitypreservation lemma for the product).† IITP RAS and MIIT, Moscow, Russia, kanovei@googlemail.com -contact author. Partial support of RFFI grants 13-01-00006 (of the 2014 version) and 17-01-00705 acknowledged.‡ IITP RAS, Moscow, Russia, lyubetsk@iitp.ru 1 The result was strengthened in [5], to the effect that the counterexample set X is a E0equivalence class, or a Vitali equivalence class (a shift of Q , the rationals), if the true reals of the real line R are considered.2 A question about ordinal definable real numbers. Mathoverflow, March 09, 2010.

A model of second-order arithmetic satisfying AC but not DC

“…Theorem 3.1 (Kanovei-Lyubetsky, [KL17]). In M , suppose thatṙ is a P <ω -name for a real such that for all n ∈ ω, 1l P <ωṙ =ẋ n .…”

“…Jensen used his forcing to show that it is consistent with ZFC that there is a Π 1 2 -definable non-constructible real singleton [Jen70]. Recently Lyubetsky and the third author extended the "uniqueness of generic filters" property of Jensen's forcing to finite-support products of P J [KL17]. They showed that in a forcing extension L[G] by the ω-length finite support-product of P J , the only L-generic reals for P J are the slices of the generic filter G. The result easily extends to ω 1 -length finite support-products as well.…”

A definable E 0 class containing no definable elements

“…6 Recall that [u] = {a ∈ 2 ω : u ⊂ a} is the Baire interval in 2 ω . 7 The code of a continuous f : 2 ω → 2 ω is the family of sets C t = {u ∈ 2 <ω : f "[u] ⊆ [t]} , t ∈ 2 <ω .…”

“…Following the mentioned conjecture, we proved in [7] that indeed, in a P <ω -generic extension of L, the set of all reals P-generic over L is a countable Π 1 2 set with no OD elements. The Π 1 2 definability is the best one can get in this context since it easily follows from the Π 1 1 uniformisation theorem that any non-empty Σ 1 2 set of reals definitely contains a Δ 1 2 element.…”

Countable OD sets of reals belong to the ground model

“…† IITP RAS, Moscow, Russia, lyubetsk@iitp.ru 1 The model presented in [11] was obtained via the countable product of Jensen's minimal ∆ 1 3 real forcing [6]. Such a product-forcing model was earlier considered by Enayat [4].…”