Weak systems of determinacy and arithmetical quasi-inductive definitions

Abstract

We locate winning strategies for various Σ⁰₃-games in the L-hierarchy in order to prove the following:

Theorem 1. KP + Σ₂-Comprehension ⊢ ∃ α L_α ⊨“Σ₂-KP + Σ⁰₃-Determinacy.”

Alternatively: Π¹₃-CA₀ ⊢“there is a β-model of Δ¹₃-CA₀ + Σ⁰₃-Determinacy.” The implication is not reversible. (The antecedent here may be replaced with Π¹₃(Π¹₃)-CA₀: Π¹₃ instances of Comprehension with only Π¹₃-lightface definable parameters—or even weaker theories.)

Theorem 2. KP + Δ₂-Comprehension + Σ₂-Replacement + AQI ⊬ Σ⁰₃-Determinacy.

(Here AQI is the assertion that every arithmetical quasi-inductive definition converges.) Alternatively:

Δ¹₃CA₀ + AQI ⊬ Σ⁰₃-Determinacy.

Hence the theories: Π¹₃-CA₀, Δ¹₃-CA₀+ Σ⁰₃-Det, Δ¹₃-CA₀+AQI, and Δ¹₃-CA₀ are in strictly descending order of strength.