June 2011 Weak systems of determinacy and arithmetical quasi-inductive definitions
P. D. Welch
J. Symbolic Logic 76(2): 418-436 (June 2011). DOI: 10.2178/jsl/1305810756

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 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.

Citation

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P. D. Welch. "Weak systems of determinacy and arithmetical quasi-inductive definitions." J. Symbolic Logic 76 (2) 418 - 436, June 2011. https://doi.org/10.2178/jsl/1305810756

Information

Published: June 2011
First available in Project Euclid: 19 May 2011

zbMATH: 1225.03082
MathSciNet: MR2830409
Digital Object Identifier: 10.2178/jsl/1305810756

Rights: Copyright © 2011 Association for Symbolic Logic

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Vol.76 • No. 2 • June 2011
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