Infinite games in the Cantor space and subsystems of second order arithmetic

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Abstract

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. RCA0equation image-Det* ↔ equation image-Det* ↔ WKL0.

2. RCA0 ⊢ (equation image)2-Det* ↔ ACA0.

3. RCA0equation image-Det* ↔ equation image-Det* ↔ equation image-Det ↔ equation image-Det ↔ ATR0.

4. For 1 < k < ω, RCA0 ⊢ (equation image)k-Det* ↔ (equation image)k –1-Det.

5. RCA0equation image-Det* ↔ equation image-Det.

Here, Det* (respectively Det) stands for the determinacy of infinite games in the Cantor space (respectively the Baire space), and (equation image)k is the collection of formulas built from equation image formulas by applying the difference operator k – 1 times. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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