On the strength of PFA I∗†

Building on the work of Schimmerling ([11]) and Steel ([17]), we show that the failure of square principle at a singular strong limit cardinal implies that there is a non-tame mouse. This is the first step towards getting a model of ADR + “Θ is regular” from PFA via the core model induction. One of the wholly grails of inner model program has been determining the exact consistency strength of forcing axioms such as Proper Forcing Axiom (PFA) and Martin’s Maximum (MM). As the consistency of PFA, MM and other similar axioms have been established relative to one supercompact cardinal, it is natural to conjecture that the exact consistency strength of such forcing axioms is one supercompact cardinal. Conjecture 0.1 (The PFA Conjecture) PFA is equiconsistent with one supercompact cardinal. Recently, in [21], Viale and Weiss showed that any of the known methods for forcing PFA requires a strongly compact cardinal and if the forcing used is proper then it requires a supercompact cardinal. This result suggest that The PFA Conjecture must indeed be true. ∗2000 Mathematics Subject Classifications: 03E15, 03E45, 03E60. †