Bulletin of Symbolic Logic

The stable core

Sy-David Friedman

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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).

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Bull. Symbolic Logic, Volume 18, Issue 2 (2012), 261-267.

First available in Project Euclid: 4 April 2012

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Friedman, Sy-David. The stable core. Bull. Symbolic Logic 18 (2012), no. 2, 261--267. doi:10.2178/bsl/1333560807. https://projecteuclid.org/euclid.bsl/1333560807

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