Non-collapse of the effective Wadge hierarchy

Abstract

We study the recently suggested effective Wadge hierarchy in effective spaces, concentrating on the non-collapse property. Along with hierarchies of sets, we study hierarchies of k-partitions which are interesting on their own. In particular, we establish sufficient conditions for the non-collapse of the effective Wadge hierarchy and apply them to some concrete spaces, including the discrete space of natural numbers, the Baire space, and the Cantor space. While the latter two cases are obtained by easy effectivization of the corresponding topological result by T. Kihara and A. Montalban, the first case needs a different technique and is certainly the main technical result of this paper.