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Equiconsistencies at subcompact cardinals

Equiconsistencies at subcompact cardinals We present equiconsistency results at the level of subcompact cardinals. Assuming SBH δ , a special case of the Strategic Branches Hypothesis, we prove that if δ is a Woodin cardinal and both □(δ) and □ δ fail, then δ is subcompact in a class inner model. If in addition □(δ +) fails, we prove that δ is $${\Pi_1^2}$$ Π 1 2 subcompact in a class inner model. These results are optimal, and lead to equiconsistencies. As a corollary we also see that assuming the existence of a Woodin cardinal δ so that SBH δ holds, the Proper Forcing Axiom implies the existence of a class inner model with a $${\Pi_1^2}$$ Π 1 2 subcompact cardinal. Our methods generalize to higher levels of the large cardinal hierarchy, that involve long extenders, and large cardinal axioms up to δ is δ +(n) supercompact for all n < ω. We state some results at this level, and indicate how they are proved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

Equiconsistencies at subcompact cardinals

Neeman, Itay; Steel, John
Archive for Mathematical Logic , Volume 55 (2) – Dec 22, 2015
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References (20)

Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
ISSN
0933-5846
eISSN
1432-0665
DOI
10.1007/s00153-015-0465-4
Publisher site
See Article on Publisher Site

Abstract

We present equiconsistency results at the level of subcompact cardinals. Assuming SBH δ , a special case of the Strategic Branches Hypothesis, we prove that if δ is a Woodin cardinal and both □(δ) and □ δ fail, then δ is subcompact in a class inner model. If in addition □(δ +) fails, we prove that δ is $${\Pi_1^2}$$ Π 1 2 subcompact in a class inner model. These results are optimal, and lead to equiconsistencies. As a corollary we also see that assuming the existence of a Woodin cardinal δ so that SBH δ holds, the Proper Forcing Axiom implies the existence of a class inner model with a $${\Pi_1^2}$$ Π 1 2 subcompact cardinal. Our methods generalize to higher levels of the large cardinal hierarchy, that involve long extenders, and large cardinal axioms up to δ is δ +(n) supercompact for all n < ω. We state some results at this level, and indicate how they are proved.

Journal

Archive for Mathematical Logic – Springer Journals

Published: Dec 22, 2015

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