The Maximality of the Core Model

Our main results are: 1) every countably certified extender that coheres with the core model K is on the extender sequence of K, 2) K computes successors of weakly compact cardinals correctly, 3) every model on the maximal 1-small construction is an iterate of K, 4) (joint with W. J. Mitchell) K‖κ is universal for mice of height ≤ κ whenever κ ≥ א2, 5) if there is a κ such that κ is either a singular countably closed cardinal or a weakly compact cardinal, and <ω κ fails, then there are inner models with Woodin cardinals, and 6) an ω-Erdös cardinal suffices to develop the basic theory of K.