Determinacy and reflection principles in second-order arithmetic

Abstract

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, Montalb\'an, Shore, and others. We prove variations of a result by Ko{\l}odziejczyk and Michalewski relating determinacy of arbitrary boolean combinations of $\Sigma^0_2$ sets and reflection in second-order arithmetic. Specifically, we prove that: over $\mathsf{ACA}_0$, $\Pi^1_2$-$\mathsf{Ref}(\mathsf{ACA}_0)$ is equivalent to $\forall n.(\Sigma^0_1)_n$-$\mathsf{Det}^*_0$; $\Pi^1_3$-$\mathsf{Ref}(\Pi^1_1$-$\mathsf{CA}_0)$ is equivalent to $\forall n.(\Sigma^0_1)_n$-$\mathsf{Det}$; and $\Pi^1_3$-$\mathsf{Ref}(\Pi^1_2$-$\mathsf{CA}_0)$ is equivalent to $\forall n.(\Sigma^0_2)_n$-$\mathsf{Det}$. We also restate results by Montalb\'an and Shore to show that $\Pi^1_3$-$\mathsf{Ref}(\mathsf{Z}_2)$ is equivalent to $\forall n.(\Sigma^0_3)_n$-$\mathsf{Det}$ over $\mathsf{ACA}_0$.