An unpublished theorem of Solovay on OD partitions of reals into two non-OD parts, revisited

摘要

A definable pair of disjoint non-OD sets of reals (hence, indiscernible sets) exists in the Sacks and E-0-large generic extensions of the constructible universe L. More specifically, if a is an element of 2(omega) is either Sacks generic or E-0-large generic real over L, then it is true in La that there is a lightface Pi(1)(2) equivalence relation Q on the Pi(1)(2) set U = 2(omega) L with exactly two equivalence classes, and both those classes are non-OD sets.