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The proper and semi-proper forcing axioms for forcing notions that preserve $ \aleph_2$ or $ \aleph_3$

Authors: Joel David Hamkins and Thomas A. Johnstone
Journal: Proc. Amer. Math. Soc. 137 (2009), 1823-1833
MSC (2000): Primary 03E55, 03E40
Published electronically: December 15, 2008
MathSciNet review: 2470843
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Abstract: We prove that the PFA lottery preparation of a strongly unfoldable cardinal $ \kappa$ under $ \neg 0^\sharp$ forces $ \operatorname{PFA}(\aleph_2$-preserving$ )$, $ \operatorname{PFA}(\aleph_3$-preserving$ )$ and $ \operatorname{PFA}_{\aleph_2}$, with $ 2^\omega=\kappa=\aleph_2$. The method adapts to semi-proper forcing, giving $ \operatorname{SPFA}(\aleph_2$-preserving$ )$, $ \operatorname{SPFA}(\aleph_3$-preserving$ )$ and $ \operatorname{SPFA}_{\aleph_2}$ from the same hypothesis. It follows by a result of Miyamoto that the existence of a strongly unfoldable cardinal is equiconsistent with the conjunction $ \operatorname{SPFA}(\aleph_2$-preserving$ ) +\operatorname{SPFA}(\aleph_3$-preserving$ )+ \operatorname{SPFA}_{\aleph_2}+2^\omega=\aleph_2$. Since unfoldable cardinals are relatively weak as large cardinal notions, our summary conclusion is that in order to extract significant strength from PFA or SPFA, one must collapse $ \aleph_3$ to $ \aleph_1$.

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Additional Information

Joel David Hamkins
Affiliation: Department of Mathematics, The Graduate Center of The City University of New York, 365 Fifth Avenue, New York, New York 10016 — and — Department of Mathematics, The College of Staten Island of The City University of New York, Staten Island, New York 10314

Thomas A. Johnstone
Affiliation: Department of Mathematics, New York City College of Technology of The City University of New York, 300 Jay Street, Brooklyn, New York 11201

Keywords: Forcing axiom, strongly unfoldable cardinal
Received by editor(s): November 20, 2007
Received by editor(s) in revised form: August 13, 2008
Published electronically: December 15, 2008
Additional Notes: The research of the first author has been supported in part by grants from the CUNY Research Foundation and the Netherlands Organization for Scientific Research (NWO), and he is grateful to the Institute of Logic, Language and Computation at Universiteit van Amsterdam for the support of a Visiting Professorship during his 2007 sabbatical there.
Parts of this article are adapted from the second author’s Ph.D. dissertation, The Graduate Center of The City University of New York, June 2007
Communicated by: Julia Knight
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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