UCLA

We study the strength of determinacy hypotheses in levels of two hierarchies of subsets of Baire space: the standard Borel hierarchy, and the hierarchy of sets in the Borel sigma-algebra generated by coanalytic sets.

We begin with the third level of this hierarchy, the lowest level at which the strength of determinacy had not yet been characterized in terms of a natural theory. Building on work of Philip Welch, we show that this determinacy is equivalent to the existence of a wellfounded set model of an axiom of monotone induction.

For the fourth and higher levels, we prove best-possible refinements of old bounds due to Harvey Friedman and Donald A. Martin on the strength of determinacy in terms of iterations of the Power Set axiom. We introduce a novel family of reflection principles, and prove a level-by-level equivalence between determinacy at these levels and existence of a wellfounded model of the corresponding axiom. In the base case, we have the following concise result: determinacy at the fourth level of the Borel hierarchy is equivalent to the existence of a level of L satisfying that the set of real numbers exists, and every wellfounded tree on the reals is ranked.

We connect this result to work of Noah Schweber on higher order reverse mathematics. Schweber shows, using the method of forcing, that for games with real number moves, clopen determinacy does not imply open determinacy over a weak base theory of third order arithmetic. We show that the level of L mentioned above is a witness to this separation result, and furthermore, that it is (in the appropriate sense) the minimal (third-order) beta-model of a natural theory of projective transfinite recursion. We obtain that that determinacy for games on naturals with payoff in the fourth level of the Borel hierarchy falls strictly between the principles of open and clopen determinacy for games with real number moves in terms of beta-consistency strength.

Finally, we combine our methods with those of John Steel and Itay Neeman to characterize the strength of determinacy for sets in the pointclasses of the Borel hierarchy generated by analytic sets. Granted that the reals are closed under the sharp function, we show this determinacy is equivalent to the existence of an iterable mouse with a measurable cardinal of largest possible Mitchell order, conjuncted with an appropriate reflection principle.