Weak systems of determinacy and arithmetical quasi-inductive definitions

TL;DR

This paper investigates the logical strength of Sigma^0_3-Determinacy by locating winning strategies within the L-hierarchy, revealing its position between known subsystems of second-order arithmetic.

Contribution

It establishes that Sigma^0_3-Determinacy is intermediate between Pi^1_3-CA_0 and Delta^1_3-CA_0 + AQI by analyzing strategies in the L-hierarchy.

Findings

Sigma^0_3-Determinacy is between Pi^1_3-CA_0 and Delta^1_3-CA_0 + AQI.

Winning strategies for Sigma^0_3-games are located in the L-hierarchy.

The result clarifies the logical strength of arithmetical quasi-inductive definitions.

Abstract

We locate winning strategies for various Sigma^0_3-games in the L-hierarchy in order to prove that Sigma^0_3 Determinacy is intermediate between Pi^1_3-CA_0 (even Pi^1_2-CA_0 (lightface) with Pi^1_3-lightface definable parameters allowed) and Delta^1_3-CA_0 + AQI. (Here "AQI" is the statement in second order number theory that every arithmeical quasi-inductive definition on any input stabilizes).