Determinacy of Refinements to the Difference Hierarchy of Co-analytic Sets

TL;DR

This paper develops a technique to prove the determinacy of certain classes within the difference hierarchy of co-analytic sets, establishing bounds and bridging gaps in the understanding of their determinacy strength.

Contribution

It introduces a new method for proving determinacy of classes in the difference hierarchy of co-analytic sets from weak principles, extending previous results.

Findings

Established upper bounds for determinacy of classes ^2-\u03a0^1_1+\u03a3^0_lpha for all computable alpha

Proved determinacy for ^2-^1_1+\u2206^1_1 classes

Bridged the gap between known hypotheses for determinacy in this hierarchy.

Abstract

In this paper we develop a technique for proving determinacy of classes of the form $\omega^2-\Pi^1_1+\Gamma$ (a refinement of the difference hierarchy on the co-analytic sets lying between $\omega^2-\Pi^1_1$ and $(\omega^2+1)-\Pi^1_1$) from weak principles, establishing upper bounds for the determinacy-strength of the classes $\omega^2-\Pi^1_1+\Sigma^0_\alpha$ for all computable $\alpha$ and of $\omega^2-\Pi^1_1+\Delta^1_1$. This bridges the gap between previously known hypotheses implying determinacy in this region.