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Incompatible Ω-Complete Theories

Published online by Cambridge University Press:  12 March 2014

Peter Koellner and
W. Hugh Woodin
Peter Koellner
Affiliation:
Department of Philosophy, Harvard University, 25 Quincy Street, Cambridge, Ma 02138, USA, E-mail: koellner@fas.harvard.edu
W. Hugh Woodin
Affiliation:
Department of Mathematics, University of California, Berkeley, Berkeley, Ca 94720-3840, USA, E-mail: woodin@math.berkeley.edu
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Abstract

In 1985 the second author showed that if there is a proper class of measurable Woodin cardinals and and are generic extensions of V satisfying CH then and agree on all Σ12-statements. In terms of the strong logic Ω-logic this can be reformulated by saying that under the above large cardinal assumption ZFC + CH is Ω-complete for Σ12. Moreover, CH is the unique Σ12-statement with this feature in the sense that any other Σ12-statement with this feature is Ω-equivalent to CH over ZFC. It is natural to look for other strengthenings of ZFC that have an even greater degree of Ω-completeness. For example, one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for all of third-order arithmetic. Going further, for each specifiable segment Vλ of the universe of sets (for example, one might take Vλ to be the least level that satisfies there is a proper class of huge cardinals), one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for the theory of Vλ. If such theories exist, extend one another, and are unique in the sense that any other such theory B with the same level of Ω-completeness as A is actually Ω-equivalent to A over ZFC, then this would show that there is a unique Ω-complete picture of the successive fragments of the universe of sets and it would make for a very strong case for axioms complementing large cardinal axioms. In this paper we show that uniqueness must fail. In particular, we show that if there is one such theory that Ω-implies CH then there is another that Ω-implies ¬-CH.

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Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 4 , December 2009, pp. 1155 - 1170
Copyright
Copyright © Association for Symbolic Logic 2009

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