Downward Transference of Mice and Universality of Local Core Models

Abstract

Abstract. If M is a proper class inner model of ZFC and ω M 2 = ω2, then every sound mouse projecting to ω and not past 0 ¶ belongs to M. In fact, under the assumption that 0 ¶ does not belong to M, K M ω2 is universal for all countable mice in V.Similarly, if M is a proper class inner model of ZFC, δ > ω1 is regular, (δ + ) M = δ + , and in V there is no proper class inner model with a Woodin cardinal, then K M δ is universal for all mice in V of cardinality less than δ.