The Limits of Determinacy in Higher-Order Arithmetic

TL;DR

This paper establishes precise bounds on the strength of determinacy principles in higher-order arithmetic, extending existing theorems and connecting reflection principles with determinacy axioms.

Contribution

It extends the Montalbán-Shore theorem to higher levels of the Borel hierarchy and links reflection principles with determinacy axioms in higher-order arithmetic.

Findings

Level-by-level bounds on determinacy strength in hyperarithmetical hierarchy

Extension of Montalbán-Shore theorem to higher Borel levels

Equivalence between reflection principles and determinacy axioms

Abstract

We prove level-by-level upper and lower bounds on the strength of determinacy for finite differences of sets in the hyperarithmetical hierarchy in terms of subsystems of finite-and transfinite-order arithmetic, extending the Montalb\'an-Shore theorem to each of the levels of the Borel hierarchy beyond the one they treated. We also prove equivalences between reflection principles for higher-order arithmetic and quantified determinacy axioms, answering two questions of Pacheco and Yokoyama.