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

Takako Nemoto, Med Yahya Ould MedSalem, Kazuyuki Tanaka

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)

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. RCA0 ⊢ Δ1 0* ↔ ∑10-Det* ↔ WKL0. 2. RCA0 ⊢ (∑10) 2-Det* ↔ ACA0. 3. RCA0 ⊢ Δ20* ↔ ∑2 0-Det* ↔ Δ10-Det ↔ ∑10-Det ↔ ATR0. 4. For 1 < k < w, RCA0 ⊢ (∑20)k- Det* ↔ (∑20)k-1-Det. 5. RCA 0 ⊢ Δ30* ↔ Δ30-Det. Here, Det* (respectively Det) stands for the determinacy of infinite games in the Cantor space (respectively the Baire space), and (∑n0)k is the collection of formulas built from ∑n0 formulas by applying the difference operator k - 1 times.

Original languageEnglish
Pages (from-to)226-236
Number of pages11
JournalMathematical Logic Quarterly
Volume53
Issue number3
DOIs
Publication statusPublished - 2007 Jun 20

Keywords

  • Determinacy
  • Reverse mathematics
  • Second order arithmetic

ASJC Scopus subject areas

  • Logic

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