Weak systems of determinacy and arithmetical quasi-inductive definitions

Abstract

AbstractWe locate winning strategies for various -games in the L-hierarchy in order to prove the following:Theorem 1. KP + Σ2-Comprehension -Determinacy.”Alternatively: “there is a β-model of -Determinacy.” The implication is not reversible. (The antecedent here may be replaced with instances of Comprehension with only -lightface definable parameters—or even weaker theories.)Theorem 2. KP + Δ2-Comprehension + Σ2-Replacement + -Determinacy.(Here AQI is the assertion that every arithmetical quasi-inductive definition converges.) Alternatively: -Determinacy.Hence the theories: , and are in strictly descending order of strength.