# Topic Archive: forcing axioms

Set theory seminarFriday, December 11, 201510:00 amGC 3212

# Beyond the Continuum: The Search for Higher Analogues of Martin’s Axiom

University of Bonn

Martin’s axiom has been very successful in deciding numerous questions such as the Souslin hypothesis. Can we find higher analogues that, for example, decide the $\kappa$-Souslin hypothesis and allow us to meet $< 2^\kappa$ dense open sets? If one is careful enough and imposes closure conditions and stronger forms of the $\kappa$-cc, one obtains the principle BA$_\kappa$, independently discovered by Baumgartner, Laver and Shelah. BA$_\kappa$ shares many similarities with Martin’s axiom but decides the $\kappa$-Souslin hypothesis in the wrong way.

Set theory seminarFriday, March 21, 201410:00 amGC6417

# A proof of the relative consistency of PFA

The CUNY Graduate Center

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.

Set theory seminarFriday, March 7, 201410:00 amGC6417

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

The CUNY Graduate Center

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.