论文标题
二阶算术中的确定性和反思原理
Determinacy and reflection principles in second-order arithmetic
论文作者
论文摘要
众所周知，确定性公理的几种变体在逆向数学研究中起着重要作用，以及决定性和理解的层次结构之间的关系由田中，Nemoto，Montalbán，Shore和其他人揭示。我们证明了Kołodziejczyk和Michalewski的结果的变化，这些变化是$σ^0_2 $ sets和二阶算术中的任意布尔组合的确定性。具体而言，我们证明：超过$ \ MathSf {aca} _0 $，$π^1_2 $ - $ \ $ \ MATHSF {REF}（\ Mathsf {aca} _0）$等于$ \ forall n。 $π^1_3 $ - $ \ MATHSF {REF}（π^1_1 $ - $ \ MATHSF {Ca} _0）$等于$ \ forall n。（σ^0_1）_n $ - $ - $ \ $ \ MATHSF {det} $;和$π^1_3 $ - $ \ MATHSF {REF}（π^1_2 $ - $ \ MATHSF {Ca} _0）$等于$ \ forall n。我们还重述了Montalbán和Shore的结果，以表明$π^1_3 $ - $ \ MATHSF {REF}（\ MATHSF {Z} _2）$等于$ \ forall n。
It is known that several variations of the axiom of determinacy play important roles in the study of reverse mathematics, and the relation between the hierarchy of determinacy and comprehension are revealed by Tanaka, Nemoto, Montalbán, Shore, and others. We prove variations of a result by Kołodziejczyk and Michalewski relating determinacy of arbitrary boolean combinations of $Σ^0_2$ sets and reflection in second-order arithmetic. Specifically, we prove that: over $\mathsf{ACA}_0$, $Π^1_2$-$\mathsf{Ref}(\mathsf{ACA}_0)$ is equivalent to $\forall n.(Σ^0_1)_n$-$\mathsf{Det}^*_0$; $Π^1_3$-$\mathsf{Ref}(Π^1_1$-$\mathsf{CA}_0)$ is equivalent to $\forall n.(Σ^0_1)_n$-$\mathsf{Det}$; and $Π^1_3$-$\mathsf{Ref}(Π^1_2$-$\mathsf{CA}_0)$ is equivalent to $\forall n.(Σ^0_2)_n$-$\mathsf{Det}$. We also restate results by Montalbán and Shore to show that $Π^1_3$-$\mathsf{Ref}(\mathsf{Z}_2)$ is equivalent to $\forall n.(Σ^0_3)_n$-$\mathsf{Det}$ over $\mathsf{ACA}_0$.
