We present equiconsistency results at the level of subcompact cardinals. Assuming SBH ?, a special case of the Strategic Branches Hypothesis, we prove that if ? is a Woodin cardinal and both ?(?) and ?? fail, then ? is subcompact in a class inner model. If in addition ?(?+) fails, we prove that ? is $${\Pi_1^2}$$?12 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 ? so that SBH ? holds, the Proper Forcing Axiom implies the existence of a class inner model with a $${\Pi_1^2}$$?12 subcompact cardinal. Our methods generalize to higher levels of the large cardinal hierarchy, that involve long extenders, and large cardinal axioms up to ? is ?+(n) supercompact for all n < ?. We state some results at this level, and indicate how they are proved.
['Itay Neeman', 'John R. Steel']
Equiconsistencies at subcompact cardinals