On some topics around the Wadge rank ${\omega }_2$

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.