Combinatorial principles in the core model for one Woodin cardinal Original Research Article

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 View the MathML source. We generalize to View the MathML sourcesome combinatorial principles that were shown by Jensen to hold in L. We show that View the MathML source 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 View the MathML source. We prove that if (View the MathML source, 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 (Vκ+)# exists; likewise for the failure of View the MathML source at such a κ.