Equiconsistencies at subcompact cardinals
Tóm tắt
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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