Square in Core Models

Abstract

AbstractWe prove that in all Mitchell-Steel core models, □k holds for all k. (See Theorem 2.) From this we obtain new consistency strength lower bounds for the failure of □k if k is either singular and countably closed, weakly compact, or measurable. (Corollaries 5, 8, and 9.) Jensen introduced a large cardinal property that we call subcompactness; it lies between superstrength and supercompactness in the large cardinal hierarchy. We prove that in all Jensen core models, □k holds iff k is not subcompact. (See Theorem 15; the only if direction is essentially due to Jensen.)