An unpublished theorem of Solovay on OD partitions of reals into two non-OD parts, revisited

Abstract

A definable pair of disjoint non-OD sets of reals (hence, indiscernible sets) exists in the Sacks and 𝔼0-large generic extensions of the constructible universe L. More specifically, if [Formula: see text] is either Sacks generic or 𝔼0-large generic real over L, then it is true in L[a] that there is a lightface [Formula: see text] equivalence relation [Formula: see text] on the [Formula: see text] set [Formula: see text] with exactly two equivalence classes, and both those classes are non-OD sets.