Title: The level-by-level strength of Borel determinacy
Speaker: Sherwood Hachtman (University of Illinois- Chicago)
Abstract: All infinite games with Borel winning condition are determined. By celebrated results of D.A. Martin and H. Friedman, this fact is provable in ZFC, but the base theory ZFC cannot be substantially weakened: any proof of Borel determinacy requires an appeal to the axioms of Power Set and Replacement. This correspondence holds "level-by-level": proving determinacy for sets in the level n+4 of the Borel hierarchy requires (roughly) n+1 uncountable infinities. But what ambient set theory is strictly necessary? We present a sharpest-possible refinement of the Martin/Friedman results by isolating a family of weak reflection principles whose strength matches up level-by-level with that of Borel determinacy. This furnishes an analysis of the complexity of the simplest winning strategies for Borel games in terms of the level of L at which these strategies are born. Formulating these results in the setting of higher-order reverse mathematics, we separate open and clopen determinacy for games with moves of large type, extending a result of Schweber.