The maximality of the core model

Abstract

Our main results are: 1) every countably certified extender that coheres with the core model K K is on the extender sequence of K K , 2) K K computes successors of weakly compact cardinals correctly, 3) every model on the maximal 1-small construction is an iterate of K K , 4) (joint with W. J. Mitchell) K ‖ κ K\|\kappa is universal for mice of height ≤ κ \le \kappa whenever κ ≥ ℵ 2 \kappa \geq \aleph _2 , 5) if there is a κ \kappa such that κ \kappa is either a singular countably closed cardinal or a weakly compact cardinal, and ◻ κ > ω \square _\kappa ^{>\omega } fails, then there are inner models with Woodin cardinals, and 6) an ω \omega -Erdös cardinal suffices to develop the basic theory of K K .