TY - GEN
T1 - Infinite games and transfinite recursion of multiple inductive definitions
AU - Yoshii, Keisuke
AU - Tanaka, Kazuyuki
N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.
PY - 2012
Y1 - 2012
N2 - The purpose of this research is to investigate the logical strength of weak determinacy of Gale-Stewart games from the standpoint of reverse mathematics. It is known that the determinacy of sets (open sets) is equivalent to system ATR 0 and that of Σ 2 0 corresponds to the axiom of Σ 1 1 inductive definitions. Recently, much effort has been made to characterize the determinacy of game classes above Σ 2 0 within second order arithmetic. In this paper, we show that for any k ε ω, the determinacy of Δ((Σ 2 0) k+1) sets is equivalent to the axiom of transfinite recursion of Σ 1 1 inductive definitions with k operators, denote [Σ 1 1] k -IDTR. Here, (Σ 2 0) k+1 is the difference class of k + 1 Σ 2 0 sets and Δ((Σ 2 0) k+1) is the conjunction of (Σ 2 0) k+1 and co-(Σ 2 0) k+1.
AB - The purpose of this research is to investigate the logical strength of weak determinacy of Gale-Stewart games from the standpoint of reverse mathematics. It is known that the determinacy of sets (open sets) is equivalent to system ATR 0 and that of Σ 2 0 corresponds to the axiom of Σ 1 1 inductive definitions. Recently, much effort has been made to characterize the determinacy of game classes above Σ 2 0 within second order arithmetic. In this paper, we show that for any k ε ω, the determinacy of Δ((Σ 2 0) k+1) sets is equivalent to the axiom of transfinite recursion of Σ 1 1 inductive definitions with k operators, denote [Σ 1 1] k -IDTR. Here, (Σ 2 0) k+1 is the difference class of k + 1 Σ 2 0 sets and Δ((Σ 2 0) k+1) is the conjunction of (Σ 2 0) k+1 and co-(Σ 2 0) k+1.
UR - http://www.scopus.com/inward/record.url?scp=84862239850&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84862239850&partnerID=8YFLogxK
U2 - 10.1007/978-3-642-30870-3_38
DO - 10.1007/978-3-642-30870-3_38
M3 - Conference contribution
AN - SCOPUS:84862239850
SN - 9783642308697
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 374
EP - 383
BT - How the World Computes - Turing Centenary Conference and 8th Conference on Computability in Europe, CiE 2012, Proceedings
T2 - Turing Centenary Conference and 8th Conference on Computability in Europe, CiE 2012
Y2 - 18 June 2012 through 23 June 2012
ER -