A quasi-lower bound on the consistency strength of PFA

Friedman, Sy-David; Holy, Peter

doi:10.1090/S0002-9947-2014-05955-2

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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