## Notre Dame Journal of Formal Logic

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

### PFA and Ideals on ${\omega}_{2}$ 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 $M{A}^{+Diag}(\Gamma )$.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 24 September 2012

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/00294527-1716793

**Mathematical Reviews number (MathSciNet)**

MR2981015

**Zentralblatt MATH identifier**

1253.03078

**Subjects**

Primary: 03E05: Other combinatorial set theory

Secondary: 03E35: Consistency and independence results 03E50: Continuum hypothesis and Martin's axiom [See also 03E57] 03E55: Large cardinals 03E57: Generic absoluteness and forcing axioms [See also 03E50]

**Keywords**

forcing axioms ideals duality theorem large cardinals proper forcing

#### 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