Combinatorial principles in the core model for one Woodin cardinal

Abstract

We study the fine structure of the core model for one Woodin cardinal, building of the work of Mitchell and Steel on inner models of the form L[ E → ] . We generalize to L[ E → ] some combinatorial principles that were shown by Jensen to hold in L. We show that L[ E → ] satisfies the statement: “□ κ holds whenever κ ⩽ the least measurable cardinal λ of ◁ order λ ++”. We introduce a hierarchy of combinatorial principles □ κ, λ for 1 ⩽ λ ⩽ κ such that □ κ□ κ, 1 ⇒ □ κ, λ ⇒ □ κ, κ□ κ ∗ . We prove that if ( κ +) v = (κ +) L[ E → ] , then □ κ, c∝( κ) holds in V. As an application, we show that ZFC + PFA ⇒ Con(ZFC + “there is a Woodin cardinal”). We also obtain one Woodin cardinal as a lower bound on the consistency strength of stationary reflection at κ + for a singular, countably closed limit cardinal κ such that ( V κ + ) # exists; likewise for the failure of □ κ ∗ at such a κ.