On the consistency strength of the proper forcing axiom
Copyright policy )On the consistency strength of the proper forcing axiom
Recently the second author introduced combinatorial principles that characterize supercompactness for inaccessible cardinals but can also hold true for small 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.
- University of California, Irvine United States
- University of California United States
- University of California United States
- UNIVERSITY OF CALIFORNIA United States
- University of California United States
Generic absoluteness and forcing axioms, Mathematics(all), Large cardinals, strongly compact cardinal, Slender, PFA, 03E05, 03E35, 03E55, 03E57, 03E65, Strongly compact, Mathematics - Logic, Other combinatorial set theory, Supercompact, supercompact cardinal, guessing, Guessing, ineffable cardinal, Ineffable, standard iteration, FOS: Mathematics, Standard iteration, Consistency and independence results, Thin, Logic (math.LO)
Generic absoluteness and forcing axioms, Mathematics(all), Large cardinals, strongly compact cardinal, Slender, PFA, 03E05, 03E35, 03E55, 03E57, 03E65, Strongly compact, Mathematics - Logic, Other combinatorial set theory, Supercompact, supercompact cardinal, guessing, Guessing, ineffable cardinal, Ineffable, standard iteration, FOS: Mathematics, Standard iteration, Consistency and independence results, Thin, Logic (math.LO)
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