On the consistency strength of the proper forcing axiom

By Matteo Viale, Christoph Weiß. 2010

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Abstract In recent work, the second author extended combinatorial principles due to Jech and Magidor that characterize certain large cardinal properties so that they can also hold true for small cardinals. For inaccessible cardinals, these modifications have no effect, and the resulting principles still give the same characterization of large cardinals. We prove that the proper forcing axiom PFA implies these principles hold for ω 2 . Using this, we argue to show that any of the known methods for forcing models of PFA from a large cardinal assumption requires a strongly compact cardinal. If one forces PFA using a proper forcing, then we get the optimal result that a supercompact cardinal is necessary.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a7e0404d7d7d7487b6fe7708cb194b519ab2651e