PFA

From Cantor's Attic
Revision as of 19:21, 24 March 2014 by Jdh (Talk | contribs)

Jump to: navigation, search

A cardinal $\kappa$ is a PFA cardinal if $\kappa$ is not zero and the canonical forcing of the PFA of length $\kappa$, which is the countable support iteration that at each stage $\gamma$ forces with the lottery sum of all minimal-rank proper partial orders $\mathbb{Q}$ for which there is a family $\cal{D}$ of $\omega_1$ many dense sets in $\mathbb{Q}$ for which there is no filter in $\mathbb{Q}$ meeting them.

Every supercompact cardinal is a PFA cardinal. It is not yet clear whether the converse is true.