Non-tame mouse from the failure of square at a singular

Building on the work of Schimmerling ([8]) and Steel ([14]), we show that the failure of square principle at a singular strong limit cardinal implies that there is a non-tame mouse. The proof presented is the first inductive step beyond L(R) of the core model induction that is aimed at getting a model of ADR + “Θ is regular” from the failure of square at a singular strong limit cardinal or PFA. One of the holly grails of inner model program has been to determine 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. Most of the partial results have been obtained by assuming a weakening of PFA, namely the failure of square at various cardinals. For instance, in [14], Steel showed that the failure of square at a singular strong limit cardinal implies that AD holds in L(R). We extend this result further by showing the following theorem. ∗2000 Mathematics Subject Classifications: 03E15, 03E45, 03E60. †