TY - JOUR
T1 - Infinite games in the Cantor space and subsystems of second order arithmetic
AU - Nemoto, Takako
AU - MedSalem, Med Yahya Ould
AU - Tanaka, Kazuyuki
PY - 2007
Y1 - 2007
N2 - 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.
AB - 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.
KW - Determinacy
KW - Reverse mathematics
KW - Second order arithmetic
UR - http://www.scopus.com/inward/record.url?scp=34250369663&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=34250369663&partnerID=8YFLogxK
U2 - 10.1002/malq.200610041
DO - 10.1002/malq.200610041
M3 - Article
AN - SCOPUS:34250369663
SN - 0942-5616
VL - 53
SP - 226
EP - 236
JO - Mathematical Logic Quarterly
JF - Mathematical Logic Quarterly
IS - 3
ER -