Determinacy and reflection principles in second-order arithmetic

TL;DR

This paper explores the connections between determinacy principles and reflection in second-order arithmetic, establishing equivalences between various reflection schemas and determinacy assertions for complex sets.

Contribution

It extends prior work by proving new equivalences between reflection principles and determinacy of Boolean combinations of sets in second-order arithmetic.

Findings

Over ACA_0, Pi^1_2-Ref(ACA_0) is equivalent to certain determinacy principles.

Pi^1_3-Ref(Pi^1_1-CA_0) corresponds to determinacy for all n of (Sigma^0_1)_n sets.

Pi^1_3-Ref(Pi^1_2-CA_0) is equivalent to determinacy for all n of (Sigma^0_2)_n sets.

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 $\mathsf{ACA}_0$.