Weak systems of determinacy and arithmetical quasi-inductive definitions

We locate winning strategies for various Σ3-games in the L-hierarchy in order to prove the following: THEOREM 1. KP +Σ2-Comprehension ` ∃αLα |=“Σ2-KP +Σ3-Determinacy .” Alternatively: Π3-CA0 `“there is a β-model of ∆ 1 3-CA0 + Σ 0 3Determinacy.” The implication is not reversible. (The antecedent here may be replaced with Π3(Π 1 3)-CA0: Π 1 3 instances of Comprehension with only Π3-lightface definable parameters or even weaker theories.) THEOREM 2. KP +∆2-Comprehension + Σ2-Replacement +AQI 6 `Σ3-Determinacy . (Here AQI is the assertion that every arithmetical quasi-inductive definition converges.) Alternatively: ∆3CA0 + AQI 6 ` Σ 0 3-Determinacy. Hence the theories: Π3-CA0,∆ 1 3-CA0 + Σ 0 3-Det,∆ 1 3-CA0 + AQI, and ∆ 1 3-CA0 are in strictly descending order of strength. §