PFA and guessing models

TL;DR

This paper investigates the consistency strength of PFA and related theories, establishing new lower bounds and showing their consistency relative to large cardinal assumptions, thus advancing understanding in set theory.

Contribution

It proves that the theory involving a variation of the Viale-Wei$ extss{ss}$ guessing hull principle is consistent relative to a supercompact cardinal, improving previous bounds.

Findings

The theory (T) is consistent relative to a supercompact cardinal.

The consistency of 'AD$_\mathbb{R}$ + $\Theta$ is regular' is established relative to (T) and PFA.

This work significantly raises the known lower bounds for the consistency strength of PFA and (T).

Abstract

This paper explores the consistency strength of The Proper Forcing Axiom ($\textsf{PFA}$) and the theory (T) which involves a variation of the Viale-Wei$\ss$ guessing hull principle. We show that (T) is consistent relative to a supercompact cardinal. The main result of the paper implies that the theory "$\sf{AD}$$_\mathbb{R} + \Theta$ is regular" is consistent relative to (T) and to $\textsf{PFA}$. This improves significantly the previous known best lower-bound for consistency strength for (T) and $\textsf{PFA}$, which is roughly "$\sf{AD}$$_\mathbb{R} + \textsf{DC}$".