Exact correspondence between large cardinals and forcing axioms

Determine the exact correspondence between large cardinal axioms and forcing axioms (including the Proper Forcing Axiom and Martin's Maximum) by identifying, in terms of consistency strength, the precise large cardinal hypotheses that are necessary and sufficient to obtain models of each forcing axiom and, conversely, the exact large cardinal consequences implied by each forcing axiom.

Background

The paper surveys three major lines of axioms extending ZFC—determinacy axioms, large cardinal axioms, and forcing axioms—and discusses how they interact. It notes known implications, such as that a supercompact cardinal yields models of Martin’s Maximum or the Proper Forcing Axiom, and that the Proper Forcing Axiom yields models with certain large cardinals below supercompact.

Despite these partial results, the paper emphasizes that a complete, exact understanding of the relationship between large cardinal hypotheses and forcing axioms—capturing both necessary and sufficient conditions across the hierarchy—remains unresolved and is identified as a central open problem motivating substantial research.