Abstract

Previously, both Soare and Simpson considered sets without subsets of higher -degree. Cintioli and Silvestri, for a reducibility , define the concept of a -introimmune set. For the most common reducibilities , a set does not contain subsets of higher -degree if and only if it is -introimmune. In this paper we consider -introimmune and -introimmune sets and examine how structurally easy such sets can be. In other words we ask, What is the smallest class of the Kleene's Hierarchy containing -introimmune sets for ? We answer the question by proving the existence of -introimmune sets in the class , bi--introimmune sets in , and bi--introimmune sets in