### The stable core

Sy-David Friedman
Source: Bull. Symbolic Logic Volume 18, Issue 2 (2012), 261-267.

#### Abstract

Vopěnka [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 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).

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