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Weak systems of determinacy and arithmetical quasi-inductive definitions

Published online by Cambridge University Press:  12 March 2014

P. D. Welch
P. D. Welch*
Affiliation:
School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK, E-mail: [email protected]
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Abstract

We 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 2 , June 2011 , pp. 418 - 436
Copyright
Copyright © Association for Symbolic Logic 2011

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