The limits of determinacy in second order arithmetic: consistency and complexity strength

Abstract

We prove that determinacy for all Boolean combinations of ${F_{\sigma \delta }}$ (Π ⁰₃ ) sets implies the consistency of second-order arithmetic and more. Indeed, it is equivalent to the statement saying that for every set X and every number n, there exists a β-model of Π ¹ _n -comprehension containing X.

We prove this result by providing a careful level-by-level analysis of determinacy at the finite level of the difference hierarchy on ${F_{\sigma \delta }}$ (Π ⁰₃ ) sets in terms of both reverse mathematics, complexity and consistency strength. We show that, for n ≥ 1, determinacy for sets at the nth level in this difference hierarchy lies strictly between (in the reverse mathematical sense of logical implication) the existence of β-models of Π ¹ _n+2 -comprehension containing any given set X, and the existence of β-models of Δ ¹ _n+2 -comprehension containing any given set X. Thus the nth of these determinacy axioms lies strictly between Π ¹ _n+2 -comprehension and Δ ¹ _n+2 -comprehension in terms of consistency strength. The major new technical result on which these proof theoretic ones are based is a complexity theoretic one. The nth determinacy axiom implies closure under the operation taking a set X to the least Σ _n+1 admissible containing X (for n = 1; this is due to Welch [9]).