Math. Log. Quart. 53, No. 3, 226 – 236 (2007) / DOI 10.1002/malq.200610041 Infinite games in the Cantor space and subsystems of second order arithmetic Takako Nemoto ∗ , MedYahya Ould MedSalem ∗∗ , and Kazuyuki Tanaka ∗∗∗ Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan Received 30 June 2006, accepted 12 December 2006 Published online 15 May 2007 Key words Reverse mathematics, determinacy, second order arithmetic. MSC (2000) 03B30, 03E60, 03F35 In this paper we study the determinacy strength of infinite games in the Cantor space and compare them with their counterparts in the Baire space. We show the following theorems: 1. RCA0 ⊢ Δ 0 1 -Det ∗ ↔ Σ 0 1 -Det ∗ ↔ WKL0. 2. RCA0 ⊢ (Σ 0 1 )2-Det ∗ ↔ ACA0. 3. RCA0 ⊢ Δ 0 2 -Det ∗ ↔ Σ 0 2 -Det ∗ ↔ Δ 0 1 -Det ↔ Σ 0 1 -Det ↔ ATR0. 4. For 1 <k<ω, RCA0 ⊢ (Σ 0 2 ) k -Det ∗ ↔ (Σ 0 2 ) k−1 -Det. 5. RCA0 ⊢ Δ 0 3 -Det ∗ ↔ Δ 0 3 -Det. Here, Det ∗ (respectively Det) stands for the determinacy of infinite games in the Cantor space (respectively the Baire space), and (Σ 0 n ) k is the collection of formulas built from Σ 0 n formulas by applying the difference operator k − 1 times. c  2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction In [5] and [6], we have investigated the logical strength of the determinacy of infinite games in the Baire space N N up to ∆ 0 3 . In this paper we study a particular case of infinite games, where the players play 0 or 1 at each turn. Our investigation is motivated by the following question of reverse mathematics [7]: What set existence axiom is needed to prove the determinacy of a given class of formulas? The base theory in this paper is RCA 0 , a system that consists of basic arithmetic, recursive comprehension axiom, and Σ 0 1 induction. We use other subsystems of second order arithmetic, such as WKL 0 and ATR 0 , which are stronger than RCA 0 , to measure the logical strength of different classes of games in the Cantor space. Let C -Det ∗ (respectively Det) denote the determinacy of games in a class C of games in which the players play 0 or 1 (respectively a natural number) at each turn. Here, we briefly describe the contents of this paper. In Section 2 we define some basic concepts such as the language and subsystems of second order arithmetic. For those who are familiar with reverse mathema- tics, this section can be skipped. In Section 3, we first show that ∆ 0 1 -Det ∗ , Σ 0 1 -Det ∗ , and WKL 0 are pairwise equivalent over RCA 0 . We also prove the equivalence between (Σ 0 1 ) 2 -Det ∗ and ACA 0 . In Section 4, we show that ∆ 0 2 -Det ∗ , Σ 0 2 -Det ∗ , and ATR 0 are pairwise equivalent over RCA 0 . In the last section, we study ∆ 0 3 -Det ∗ as well as (Σ 0 2 ) k -Det ∗ . Finally, we should remark that Steel [8] has also mentioned some reverse-mathematical results on the binary games. However, his claims are made under the assumption of full induction, and no proofs are given there. ∗ e-mail: sa4m20@math.tohoku.ac.jp ∗∗ e-mail: dah@lri.fr ∗∗∗ Corresponding author: e-mail: tanaka@math.tohoku.ac.jp c  2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim