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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A quasi-lower bound on the consistency strength of PFA


Authors: Sy-David Friedman and Peter Holy
Journal: Trans. Amer. Math. Soc. 366 (2014), 4021-4065
MSC (2010): Primary 03E35, 03E55, 03E57
Published electronically: March 24, 2014
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Abstract: A long-standing open question is whether supercompactness provides a lower bound on the consistency strength of the Proper Forcing Axiom (PFA). In this article we establish a quasi-lower bound by showing that there is a model with a proper class of subcompact cardinals such that PFA for $ (2^{\aleph _0})^+$-linked forcings fails in all of its proper forcing extensions. Neeman obtained such a result assuming the existence of ``fine structural'' models containing very large cardinals, however the existence of such models remains open. We show that Neeman's arguments go through for a similar notion of an ``L-like'' model and establish the existence of L-like models containing very large cardinals. The main technical result needed is the compatibility of Local Club Condensation with Acceptability in the presence of very large cardinals, a result which constitutes further progress in the outer model programme.


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Additional Information

Sy-David Friedman
Affiliation: Kurt Gödel Research Center for Mathematical Logic, Universität Wien, Währinger Strasse 25, 1090 Wien, Austria
Email: sdf@logic.univie.ac.at

Peter Holy
Affiliation: Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom
Email: maxph@bristol.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-05955-2
PII: S 0002-9947(2014)05955-2
Keywords: PFA, outer model programme, Local Club Condensation, Acceptability, large cardinals
Received by editor(s): July 5, 2012
Published electronically: March 24, 2014
Additional Notes: The authors wish to thank the Austrian Research Fund (FWF) for its generous support of this research through Project P22430–N13. The second author also wishes to thank the EPSRC for its generous support through Project EP/J005630/1.
Article copyright: © Copyright 2014 American Mathematical Society