Equiconsistencies at subcompact cardinals

Archive for Mathematical Logic 55 (1-2):207-238 (2016)
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Abstract

We present equiconsistency results at the level of subcompact cardinals. Assuming SBHδ, a special case of the Strategic Branches Hypothesis, we prove that if δ is a Woodin cardinal and both □ and □δ fail, then δ is subcompact in a class inner model. If in addition □ fails, we prove that δ is Π12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_1^2}$$\end{document} 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 SBHδ holds, the Proper Forcing Axiom implies the existence of a class inner model with a Π12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_1^2}$$\end{document} 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+ supercompact for all n < ω. We state some results at this level, and indicate how they are proved.

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Jack Steel
University of Edinburgh

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References found in this work

Suitable extender models I.W. Hugh Woodin - 2010 - Journal of Mathematical Logic 10 (1):101-339.
Suitable extender models II: Beyond ω-huge.W. Hugh Woodin - 2011 - Journal of Mathematical Logic 11 (2):115-436.
Square in core models.Ernest Schimmerling & Martin Zeman - 2001 - Bulletin of Symbolic Logic 7 (3):305-314.
Characterization of □κin core models.Ernest Schimmerling & Martin Zeman - 2004 - Journal of Mathematical Logic 4 (01):1-72.
Iteration Trees.D. A. Martin & J. R. Steel - 2002 - Bulletin of Symbolic Logic 8 (4):545-546.

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