Determinacy on the edge of second‐order arithmetic, I

Abstract This is the first of two articles on the strength of ‐‐determinacy for , the strongest theories of determinacy contained in Hilbert's second‐order arithmetic . In this article, we refute two natural conjectures on the strength of these principles in terms of inductive definability, strengthening the well‐known Montalbán–Shore theorem. Our results show that ‐‐determinacy does not coincide in strength with any subsystem of previously considered in the literature. In the process, we establish connections between these determinacy principles, the complexity of sets definable by analytical inductive definitions, and nonstandard admissible sets.