# Blog Archives

# Topic Archive: PFA

# wPFA

We isolate a new forcing axiom, ${\rm wPFA}$, which is strictly between ${\rm BPFA}$ and ${\rm PFA}$. ${\rm wPFA}$ is equiconsistent with a remarkable cardinal, it implies the failure of $\square_{\omega_1}$, but it is compatible with $\square_\kappa$ for all $\kappa \geq \omega_2$. This is part of joint work with J. Bagaria and V. Gitman.

# A proof of the relative consistency of PFA

I will use a supercompact cardinal to force the Proper Forcing Axiom (PFA). I will follow Baumgartner’s original argumet, but will use lottery sums instead of a Laver function.

# The consistency strength of PFA for posets preserving aleph_2 or aleph_3

While the consistency strength of PFA is quite high in the large cardinal hierarchy, it is reasonable to expect that tame fragments of PFA should require much weaker assumptions. I will present an argument of Hamkins and Johnstone (2008) which shows the consistency of PFA for posets preserving aleph_2 or aleph_3 from a strongly unfoldable cardinal, a much smaller large cardinal which is, roughly speaking, to strongness (or supercompactness) as weak compactness is to measurability.