Determinacy and monotone inductive definitions

Abstract

We prove the equivalence of the determinacy of $$\Sigma^0_3$$ (effectively Gδσ) games with the existence of a β-model satisfying the axiom of $$\Pi^2_1$$ monotone induction, answering a question of Montalban [8]. The proof is tripartite, consisting of (i) a direct and natural proof of $$\Sigma^0_3$$ determinacy using monotone inductive operators, including an isolation of the minimal complexity of winning strategies; (ii) an analysis of the convergence of such operators in levels of G¨odel’s L, culminating in the result that the nonstandard models isolated by Welch [18] satisfy $$\Pi^2_1$$ monotone induction; and (iii) a recasting of Welch’s [17] Friedman-style game to show that this determinacy yields the existence of one of Welch’s nonstandard models. Our analysis in (iii) furnishes a description of the degree of $$\Pi^2_1$$ -correctness of the minimal β-model of $$\Pi^2_1$$ monotone induction.