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$G_{\delta \sigma }$ GAMES AND INDUCTION ON REALS

Part of: Set theory Computability and recursion theory General logic Game theory

Published online by Cambridge University Press:  13 September 2021

J. P. AGUILERA  and
P. D. WELCH
J. P. AGUILERA
Affiliation:
DEPARTMENT OF MATHEMATICS GHENT UNIVERSITY KRIJGSLAAN 281-S8 B9000GHENTBELGIUM and INSTITUTE OF DISCRETE MATHEMATICS AND GEOMETRY VIENNA UNIVERSITY OF TECHNOLOGY 1040 VIENNA, AUSTRIA E-mail: aguilera@logic.at
P. D. WELCH
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF BRISTOLCLIFTON, BRISTOL BS8 1UG, UKE-mail: p.welch@bristol.ac.uk
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Abstract

It is shown that the determinacy of $G_{\delta \sigma }$ games of length $\omega ^2$ is equivalent to the existence of a transitive model of ${\mathsf {KP}} + {\mathsf {AD}} + \Pi _1\textrm {-MI}_{\mathbb {R}}$ containing $\mathbb {R}$ . Here, $\Pi _1\textrm {-MI}_{\mathbb {R}}$ is the axiom asserting that every monotone $\Pi _1$ operator on the real numbers has an inductive fixpoint.

Keywords

determinacygames on realsinductive definitions

MSC classification

Primary: 03B30: Foundations of classical theories (including reverse mathematics)
Secondary: 03D60: Computability and recursion theory on ordinals, admissible sets, etc. 03D70: Inductive definability 03E60: Determinacy principles 91A44: Games involving topology or set theory
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 4 , December 2021 , pp. 1676 - 1690
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Association for Symbolic Logic

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