Some descriptive set theory and core models

Abstract

We analyse the trees given by sharps for Π 1 2 sets via inner core models to give a canonical decomposition of such sets when a core model is Σ 1 3 absolute. This is by way of analogy with Solovay's analysis of Π 1 1 sets into ω 1 Borel sets — Borel in codes for wellorders. We find that Π 1 2 sets are also unions of ω 1 Borel sets — but in codes for mice and wellorders. We give an application of this technique in showing that if a core model, K, is Σ 1 3 absolute then Theorem. Every real is in K iff every Π 1 3 set of reals contains a Π 1 3 singleton .