Sets without Subsets of Higher Many-One Degree

Abstract

Previously, both Soare and Simpson considered sets without subsets of higher $\leq_T$, define the concept of a $\leq_r$-introimmune set. For the most common reducibilities $\leq_r$, a set does not contain subsets of higher $\leq_r$-degree if and only if it is $\leq_r$-introimmune. In this paper we consider $\leq_m$-introimmune and $\leq^P_m$-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 $\leq_r$-introimmune sets for $\leq_r\in\{\leq_m,\leq^P_m\}$? We answer the question by proving the existence of $\leq_m$-introimmune sets in the class $\Pi^0_1$, bi-$\leq_m$-introimmune sets in $\Delta^0_2$, and bi-$\leq^P_m$-introimmune sets in $\Delta^0_1$.