On some topics around the Wadge rank $\omega_2$

TL;DR

This paper provides an alternative proof for the Wadge rank of the omega-th level of the decreasing difference hierarchy of coanalytic sets being omega_2, using hyperarithmetical processes with finite mind-changes, and explores related hierarchies and principles.

Contribution

It offers a new proof of the Kechris-Martin theorem and investigates the relationship between difference hierarchies, least number principles, and Weihrauch degrees.

Findings

Alternative proof of the Kechris-Martin theorem.

Characterization of the omega-th level as hyperarithmetical processes.

Analysis of the gap between increasing and decreasing hierarchies.

Abstract

Kechris and Martin showed that the Wadge rank of the $\omega$-th level of the decreasing difference hierarchy of coanalytic sets is $\omega_2$ under the axiom of determinacy. In this article, we give an alternative proof of the Kechris-Martin theorem, by understanding the $\omega$-th level of the decreasing difference hierarchy of coanalytic sets as the (relative) hyperarithmetical processes with finite mind-changes. Based on this viewpiont, we also examine the gap between the increasing and decreasing difference hierarchies of coanalytic sets by relating them to the $\Pi^1_1$- and $\Sigma^1_1$-least number principles, respectively. We also analyze Weihrauch degrees of related principles.