Inner Model Theory for Supercompact Cardinals and the Consistency of the Proper Forcing Axiom

Mathematical Applications of Science Fiction We construct a canonical inner model L[E] for a super-compact cardinal, extending the hyper-Woodin hierarchythrough a fully iterable extender sequence E allowing forlong extenders. We establish the existence of a comparisonprocess for mice exhibiting λ-supercompactness, resolvingthe iterability problem for extenders overlapping Woodincardinals. Furthermore, we prove that the existence of sucha model Msc provides the exact consistency strength re-quired for the Proper Forcing Axiom (PFA). Specifically, wedemonstrate that if there exists a supercompact cardinal inthe core model K, then there is a forcing extension satisfyingPFA. Conversely, we show that PFA implies the existence ofan inner model with a supercompact cardinal, thereby estab-lishing the equiconsistency of ZFC + ∃κ(κ is supercompact)and ZFC + PFA via canonical inner model theory.