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Determinacy and monotone inductive definitions

  • Sherwood Hachtman
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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 \(\Pi^1_2\) monotone induction, answering a question of Montalbán [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 \(\Pi^1_2\) 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 \(\Pi^1_2\)-correctness of the minimal β-model of \(\Pi^1_2\) monotone induction.

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© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA

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