Some descriptive set theory and core models

AbstractWe analyse the trees given by sharps for Π12 sets via inner core models to give a canonical decomposition of such sets when a core model is Σ13 absolute. This is by way of analogy with Solovay's analysis of Π11 sets into ω1 Borel sets — Borel in codes for wellorders. We find that Π12 sets are also unions of ω1 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 Σ13 absolute thenTheorem. Every real is in K iff every Π13 set of reals contains a Π13 singleton