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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A quasi-lower bound on the consistency strength of PFA
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by Sy-David Friedman and Peter Holy PDF
Trans. Amer. Math. Soc. 366 (2014), 4021-4065 Request permission

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
  • MR Author ID: 191285
  • 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
  • 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.
  • © Copyright 2014 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 366 (2014), 4021-4065
  • MSC (2010): Primary 03E35, 03E55, 03E57
  • DOI: https://doi.org/10.1090/S0002-9947-2014-05955-2
  • MathSciNet review: 3206451