The Consistency Strength of Proper Forcing Axioms Relative to Supercompact Cardinals and Inner Model Theory

The determination of the exact consistency strength of the Proper Forcing Axiom (PFA) remains one of the central open problems in modern set theory. While the consistency of PFA was established by Baumgartner relative to a supercompact cardinal, recent developments in inner model theory and forcing iterations have significantly narrowed the gap between the known upper and lower bounds. This monograph analyzes the hierarchy of large cardinal axioms implicated by PFA, specifically examining the necessity of supercompactness versus strongly compact cardinals. We discuss the seminal results of Steel regarding the derivation of the Axiom of Determinacy in L(R) from PFA, and the subsequent work by Viale and Weiss which suggests that supercompactness is essential for proper forcing iterations. Furthermore, we explore the lower bounds established by Sargsyan via the core model induction, locating the strength of PFA strictly above a proper class of Woodin cardinals, specifically in the region of models of ADR + reg. We provide a rigorous synthesis of these results, establishing a correspondence between forcing axioms and the fine structure of inner models.