The stable core

Abstract

Vopenka [2] proved long ago that every set of ordinals is set-generic over HOD, Gödel's inner model of hereditarily ordinal-definable sets. Here we show that the entire universe V is class-generic over (HOD,S), and indeed over the even smaller inner model $\mathbb{S}=(L[S],S)$, where S is the Stability predicate. We refer to the inner model $\mathbb{S}$ as the Stable Core of V. The predicate S has a simple definition which is more absolute than any definition of HOD; in particular, it is possible to add reals which are not set-generic but preserve the Stable Core (this is not possible for HOD by Vopenka's theorem).