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 \({\Pi_1^2}\) 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 \({\Pi_1^2}\) 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.
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We dedicate this paper to Rich Laver, a brilliant mathematician and a kind and generous colleague.
This material is based upon work supported by the National Science Foundation under Grants Nos. DMS-1101204 (Neeman) and DMS-0855692 (Steel), and the Simons Foundation under Simons Fellowship No. 225854 (Neeman).
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Neeman, I., Steel, J. Equiconsistencies at subcompact cardinals. Arch. Math. Logic 55, 207–238 (2016). https://doi.org/10.1007/s00153-015-0465-4
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DOI: https://doi.org/10.1007/s00153-015-0465-4