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2012 PFA and Ideals on ω2 Whose Associated Forcings Are Proper
Sean Cox
Notre Dame J. Formal Logic 53(3): 397-412 (2012). DOI: 10.1215/00294527-1716793


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).


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Sean Cox. "PFA and Ideals on ω2 Whose Associated Forcings Are Proper." Notre Dame J. Formal Logic 53 (3) 397 - 412, 2012.


Published: 2012
First available in Project Euclid: 24 September 2012

zbMATH: 1253.03078
MathSciNet: MR2981015
Digital Object Identifier: 10.1215/00294527-1716793

Primary: 03E05
Secondary: 03E35 , 03E50 , 03E55 , 03E57

Keywords: duality theorem , Forcing axioms , Ideals , large cardinals , proper forcing

Rights: Copyright © 2012 University of Notre Dame

Vol.53 • No. 3 • 2012
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