# Frank Tall: Completely Baire spaces, Menger spaces, projective sets, Hurewicz’ theorems, and an application to Model Theory

Place: Fields Institute (Room 210)

Date: October 13, 2017 (13:30-15:00)

Speaker: Frank Tall

Title:Completely Baire spaces, Menger spaces, projective sets, Hurewicz’ theorems,  and an application to Model Theory

Abstract: We prove the following are equiconsistent:
(1) There is an inaccessible cardinal.
(2) Every projective Menger set of reals is σ-compact.
(2’) Every co-analytic Menger set of reals is σ-compact.
(3) Every projective set of reals with every closed subset Baire is Polish.
(3’) Every analytic set of reals with every closed subset Baire is Polish.
(1), (2), (2’) are from Tall-Todorcevic-Tokg ̈z 2017; (1), (3), (3’) are from Tall-Zdomskyy, in preparation.

Researchers previously derived (2), (3) from the Axiom of Projective Determinacy, and negations of (2’) and (3’) from V = L. We substitute a perfect set version of Todorcevic’s Open Graph Axiom for PD and the L[a] existence of an a ⊆ ω such that $\omega_1=\omega_1$ for V = L.

We (Tall-Zdomskyy) also construct in ZFC a separable metric space X such that every closed subset of X ω is Baire, but X includes no dense completely metrizable subspace. Such a space was previously constructed by Eagle-Tall (2017) from a non-meager P-filter, which is not known to exist in ZFC. Such a space can be used to construct an abstract logic in which the Omitting Types Theorem holds but a stronger, game-theoretic version of the OTT does not.