Premium
Determinacy of Wadge classes and subsystems of second order arithmetic
Author(s) -
Nemoto Takako
Publication year - 2009
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200710081
Subject(s) - determinacy , mathematics , second order arithmetic , axiom , order (exchange) , discrete mathematics , exponentiation , arithmetic , binary number , pure mathematics , algebra over a field , combinatorics , mathematical analysis , peano axioms , finance , economics , geometry
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)