Notre Dame Journal of Formal Logic

PFA and Ideals on ω2 Whose Associated Forcings Are Proper

Sean Cox


Given an ideal I, let PI denote the forcing with I-positive sets. We consider models of forcing axioms MA(Γ) which also have a normal ideal I with completeness ω2 such that PIΓ. Using a bit more than a superhuge cardinal, we produce a model of PFA (proper forcing axiom) which has many ideals on ω2 whose associated forcings are proper; a similar phenomenon is also observed in the standard model of MA+ω1(σ-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 (DRP(ICω1)) introduced by the author in earlier work is equivalent to a natural weakening of “there is an ideal I such that PI is proper”; and (2) for many natural classes Γ of posets, MA+ω1(Γ) is equivalent to an apparently stronger version which we call MA+Diag(Γ).

Article information

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

First available in Project Euclid: 24 September 2012

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Zentralblatt MATH identifier

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]

forcing axioms ideals duality theorem large cardinals proper forcing


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.

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