Incompatible Ω-Complete Theories

Journal of Symbolic Logic 74 (4):1155 - 1170 (2009)
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Abstract

In 1985 the second author showed that if there is a proper class of measurable Woodin cardinals and $V^{B1} $ and $V^{B2} $ are generic extensions of V satisfying CH then $V^{B1} $ and $V^{B2} $ agree on all $\Sigma _1^2 $ -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 $\Sigma _1^2 $ Moreover. CH is the unique $\Sigma _1^2 $ -statement with this feature in the sense that any other $\Sigma _1^2 $ -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 $V_\lambda $ of the universe of sets (for example, one might take $V_\lambda $ 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 $V_\lambda $ . 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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Author Profiles

Peter Koellner
Harvard University
W. Hugh Woodin
Harvard University

References found in this work

[Omnibus Review].Akihiro Kanamori - 1981 - Journal of Symbolic Logic 46 (4):864-866.
On the question of absolute undecidability.Peter Koellner - 2010 - In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Philosophia Mathematica. Association for Symbolic Logic. pp. 153-188.
A Δ22 well-order of the reals and incompactness of L.Uri Abraham & Saharon Shelah - 1993 - Annals of Pure and Applied Logic 59 (1):1-32.

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