The stable core
Vopěnka  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).
Permanent link to this document: http://projecteuclid.org/euclid.bsl/1333560807
Digital Object Identifier: doi:10.2178/bsl/1333560807
Zentralblatt MATH identifier: 06048107
Mathematical Reviews number (MathSciNet): MR2931674