PFA and guessing models

Abstract

This paper explores the consistency strength of The Proper Forcing Axiom ($\text{PFA}$) 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 "$\mathsf{AD}$${}_{\mathbb{R}}+\Theta$ is regular" is consistent relative to (T) and to $\text{PFA}$. This improves significantly the previous known best lower-bound for consistency strength for (T) and $\text{PFA}$, which is roughly "$\mathsf{AD}$${}_{\mathbb{R}}+\text{DC}$".