Journal of Symbolic Logic

Weak systems of determinacy and arithmetical quasi-inductive definitions

P. D. Welch

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

Theorem 1. KP + Σ₂-Comprehension ⊢ ∃ α Lα ⊨“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.

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J. Symbolic Logic, Volume 76, Issue 2 (2011), 418-436.

First available in Project Euclid: 19 May 2011

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Welch, P. D. Weak systems of determinacy and arithmetical quasi-inductive definitions. J. Symbolic Logic 76 (2011), no. 2, 418--436. doi:10.2178/jsl/1305810756.

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