Forcing Axioms and the Consistency Strength of the Proper Forcing Conjecture via Supercompact Cardinals

The determination of the exact consistency strength of the Proper Forcing Axiom (PFA) and its strengthening, Martin's Maximum (MM), remains one of the central open problems in modern set theory. While the consistency of these axioms was established relative to a supercompact cardinal by Baumgartner and Foreman-Magidor-Shelah respectively, the question of whether this large cardinal hypothesis is optimal has driven significant research. This monograph analyzes the interplay between forcing axioms and large cardinal hierarchies, specifically focusing on the necessity of supercompactness. We examine the seminal consistency proofs involving countable support iterations and the preservation of properness. Furthermore, we discuss recent developments by Viale, Weiss, and others suggesting that the consistency strength of PFA is indeed a supercompact cardinal, utilizing principles such as the Infinite Strong Prediction (ISP) and covering properties. We conclude by evaluating the status of the Proper Forcing Conjecture in light of these lower-bound results.