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Determinacy of Wadge classes and subsystems of second order arithmetic

✍ Scribed by Takako Nemoto

Publisher
John Wiley and Sons
Year
2009
Tongue
English
Weight
274 KB
Volume
55
Category
Article
ISSN
0044-3050

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✦ Synopsis

Abstract

In this paper we study the logical strength of the determinacy of infinite binary games in terms of second order arithmetic. We define new determinacy schemata inspired by the Wadge classes of Polish spaces and show the following equivalences over the system RCA~0~*, which consists of the axioms of discrete ordered semi‐rings with exponentiation, Δ~1~^0^ comprehension and Π~0~^0^ induction, and which is known as a weaker system than the popularbase theory RCA~0~:

  1. Bisep(Δ~1~^0^, Σ~1~^0^)‐Det* ↔ WKL~0~,

  2. Bisep(Δ~1~^0^, Σ~2~^0^)‐Det* ↔ ATR~0~ + Σ~1~^1^ induction,

  3. Bisep(Σ~1~^0^, Σ~2~^0^)‐Det* ↔ Sep(Σ~1~^0^, Σ~2~^0^)‐Det* ↔ Π~1~^1^‐CA~0~,

  4. Bisep(Δ~2~^0^, Σ~2~^0^)‐Det* ↔ Π~1~^1^‐TR~0~,
    where Det* stands for the determinacy of infinite games in the Cantor space (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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