Determinacy of Wadge classes and subsystems of second order arithmetic
نویسنده
چکیده
In this paper we study the logical strength of the determinacy of infinite binary games in terms of second order arithmetic. We define new determinacy schemata inspired by the Wadge classes of Polish spaces and show the following equivalences over the system RCA0, which consists of the axioms of discrete ordered semi-ring with exponentiation, ∆1 comprehension and Π 0 0 induction, and which is known as a weaker system than the popular base theory RCA0: • Bisep(∆1,Σ1)-Det ↔ WKL0; • Bisep(∆1,Σ2)-Det ↔ ATR0 +Σ1 induction; • Bisep(Σ1,Σ2)-Det ↔ Sep(Σ1,Σ2)-Det ↔ Π1-CA0; • Bisep(∆2,Σ2)-Det ↔ Π1-TR0; where Det∗ stands for the determinacy of infinite games in the Cantor space.
منابع مشابه
Corrigendum of “Determinacy of Wadge classes and subsystems of second order arithmetic”
for a pair (θ0(x), θ1(x)) of Π0 formulae such that, for all f ∈ {0, 1}N, ∀n(∃m > n)θ0(f [m]) if and only if ¬∀n∃m > nθ1(f [m]). We may assume that {s ∈ {0, 1}<N : θ0(s) ∧ θ1(s)} = ∅ and θ0(⟨⟩) by replacing θ0(x) with θ′ 0(x) ≡ (θ0(x) ∧ ¬θ1(x)) ∨ (x = ⟨⟩) and θ1(x) with θ′ 1(x) ≡ θ1(x) ∧ ¬θ0(x) ∧ (x ̸= ⟨⟩) if necessary. The above tree Tθ0,θ1 is well-founded, for if F were an infinite path of Tθ0,...
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عنوان ژورنال:
- Math. Log. Q.
دوره 55 شماره
صفحات -
تاریخ انتشار 2009