Woodin Cardinals and the Consistency Strength of the Proper Forcing Axiom relative to Supercompactness

The Proper Forcing Axiom (PFA) represents a significant strengthening of Martin's Axiom, asserting the existence of generic filters for a broad class of proper partial orders. This monograph investigates the consistency strength of PFA, situating it within the hierarchy of large cardinals. We analyze the seminal results establishing that the consistency of PFA follows from the existence of a supercompact cardinal. Conversely, we explore the lower bounds of its consistency strength, detailing the derivation of inner models with Woodin cardinals from PFA. Specifically, we examine Steel's theorem that PFA implies the Axiom of Determinacy in L(R), thereby enforcing the existence of infinitely many Woodin cardinals in the core model. Furthermore, we discuss recent developments by Viale and Weiss, which suggest that PFA implies combinatorial principles on 2 characteristic of supercompactness, hinting at an equiconsistency between PFA and a supercompact cardinal.