On some topics around the Wadge rank $ω_2$

Abstract

Kechris and Martin showed that the Wadge rank of the ω-th level of the decreasing difference hierarchy of coanalytic sets is ω 2 under the axiom of determinacy. In this article, we give an alternative proof of the Kechris-Martin theorem, by understanding the ω-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 Π 1 1 -and Σ 1 1 -least number principles, respectively. We also analyze Weihrauch degrees of related principles.