The stable core

Abstract

Vopěnka [2] proved long ago that every set of ordinals is set-generic over HOD, Godel’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 S = (L[S], S), where S is the Stability predicate. We refer to the inner model 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 Vopěnka’s theorem). For an infinite cardinal α, H(α) consists of those sets whose transitive closure has size less than α. Let C denote the closed unbounded class of all infinite cardinals β such that H(α) has cardinality less than β whenever α is an infinite cardinal less than β. Definition 1 For a finite n > 0, we say that α is n-Stable in β iff α < β, α and β are limit points of C and (H(α), C ∩ α) is Σn elementary in (H(β), C ∩ β). The Stability predicate S places the Stability notion into a single predicate. S consists of all triples (α, β, n) such that α is n-Stable in β. The ∆2 definable predicate S describes the “core” of V , in the following sense. ∗The author wishes to thank the Austrian Science Fund (FWF) for its generous support through Project P 22430-N13.