Mice with Woodin cardinals from a Reinhardt

Abstract

Suppose there is a Reinhardt cardinal. Then (1) M_n(X) exists and is fully iterable (above X) for every transitive set X and every n < ω (here M_n(X) denotes the canonical minimal proper class inner model containing X and having n Woodin cardinals above the rank of X); and (2) Projective Determinacy holds in every set generic extension.