## Notre Dame Journal of Formal Logic

### PFA and Ideals on $\omega_{2}$ Whose Associated Forcings Are Proper

Sean Cox

#### 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 $MA^{+\omega_{1}}(\sigma\mbox{-closed})$ 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 ($\mathit{DRP}(IC_{\omega_{1}})$) 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, $MA^{+\omega_{1}}(\Gamma)$ is equivalent to an apparently stronger version which we call $MA^{+\operatorname{Diag}}(\Gamma)$.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 53, Number 3 (2012), 397-412.

Dates
First available in Project Euclid: 24 September 2012

https://projecteuclid.org/euclid.ndjfl/1348524118

Digital Object Identifier
doi:10.1215/00294527-1716793

Mathematical Reviews number (MathSciNet)
MR2981015

Zentralblatt MATH identifier
1253.03078

#### Citation

Cox, Sean. PFA and Ideals on $\omega_{2}$ Whose Associated Forcings Are Proper. Notre Dame J. Formal Logic 53 (2012), no. 3, 397--412. doi:10.1215/00294527-1716793. https://projecteuclid.org/euclid.ndjfl/1348524118

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