The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal

Katetov’s problem

“…In this note we show the consistency of Katětov's metrization statement: Every compact space with T 5 square is metrizable. Our research in this area centers around a combinatorial analysis of forcing axioms compatible with the existence of a particular type of Souslin tree (related work appears in [1,15,16,12,13,21,27]). It turns out that for solving Katětov's problem it suffices to analyze the maximal amount of MA ω1 compatible with the existence of this type of tree.…”

Determinacy from strong reflection

“…This theorem, obtained in late 2000, builds on Woodin's proof of PD from the same hypotheses (9.85 of [25]), and represents the first proof of the consistency of the Axiom of Determinacy from Forcing Axioms. The first author subsequently obtained the same conclusion in [22] from a single failure of square (and hence from PFA) building on Woodin's theorem that PFA together with an inaccessible gives AD in the Solovay model.…”

“…As in any core model induction, given a critical ordinal β, the first step toward W * β+1 in V is to find a (hybrid) mouse operator J which codes up truth at the level of the first pointclass Σ ∼ J β (R) n having the scale property. In order to prove W Theorem 13 (Woodin,[25] [9.85]). (Woodin) Assume WRP (2) (ω 2 ) and that N S is saturated; then PD holds, and continues to hold in the universe after ω 2 is collapsed.…”

“…In [25], Woodin gives a proof that under WRP (2) (ω 2 ), closure of P (ω 1 ) under sharps entails closure of P (ω 2 ) under sharps. His proof of Theorem 10 involved analogous arguments for the M # n operation.…”

“…We digress briefly in order to prove the following theorem, due to Woodin and implicit in [25] (see Thm. 9.82 for example).…”