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}$".
Categories: math.LO