PFA and guessing models

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}$$_\mathbb{R} + Θ$ is regular" is consistent relative to (T) and to $\textsf{PFA}$. This improves significantly the previous known best lower-bound for consistency strength for (T) and $\textsf{PFA}$, which is roughly "$\sf{AD}$$_\mathbb{R} + \textsf{DC}$".

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