PFA and Ideals on ω2 Whose Associated Forcings Are Proper

Abstract

Given an ideal $I$, let ${\mathbb{P}}_I$ denote the forcing with $I$-positive sets. We consider models of forcing axioms $MA(\Gamma )$ which also have a normal ideal $I$ with completeness ${\omega }_2$ such that ${\mathbb{P}}_I\in \Gamma$. Using a bit more than a superhuge cardinal, we produce a model of PFA (proper forcing axiom) which has many ideals on ${\omega }_2$ whose associated forcings are proper; a similar phenomenon is also observed in the standard model of $M{A}^{+{\omega }_1}\left(\sigma \text{-closed}\right)$ obtained from a supercompact cardinal. Our model of PFA also exhibits weaker versions of ideal properties, which were shown by Foreman and Magidor to be inconsistent with PFA.

Along the way, we also show (1) the diagonal reflection principle for internally club sets ($\mathit{DRP}\left(I{C}_{{\omega }_1}\right)$) introduced by the author in earlier work is equivalent to a natural weakening of “there is an ideal $I$ such that ${\mathbb{P}}_I$ is proper”; and (2) for many natural classes $\Gamma$ of posets, $M{A}^{+{\omega }_1}(\Gamma )$ is equivalent to an apparently stronger version which we call $M{A}^{+Diag}(\Gamma )$.