Infinite games and transfinite recursion of multiple inductive definitions

Abstract

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 20 corresponds to the axiom of Σ 11 inductive definitions. Recently, much effort has been made to characterize the determinacy of game classes above Σ 20 within second order arithmetic. In this paper, we show that for any k ε ω, the determinacy of Δ((Σ 20) k+1) sets is equivalent to the axiom of transfinite recursion of Σ 11 inductive definitions with k operators, denote [Σ 11] k -IDTR. Here, (Σ 20) k+1 is the difference class of k + 1 Σ 20 sets and Δ((Σ 20) k+1) is the conjunction of (Σ 20) k+1 and co-(Σ 20) k+1. © 2012 Springer-Verlag.