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The Stable core

Published online by Cambridge University Press:  15 January 2014

Sy-david Friedman
Sy-david Friedman*
Affiliation:
Kurt Gödel Research Center, Währinger Strasse 22, A-1090 Vienna, AustriaE-mail: sdf@logic.univie.ac.at
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Abstract

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 =(L[S],S), where S is the Stability predicate. We refer to the inner model 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).

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 18 , Issue 2 , June 2012 , pp. 261 - 267
Copyright
Copyright © Association for Symbolic Logic 2012

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References

REFERENCES

[1] Friedman, S., Fine structure and class forcing , de Gruyter Series in Logic and its Applications, vol. 3, de Gruyter, 2000.CrossRefGoogle Scholar
[2] Vopěnka, P. and Hájek, P., The theory of semisets, Academia (Publishing house of the Czech Academy of Sciences), Prague, 1972.Google Scholar