A classification of low c.e. sets and the Ershov hierarchy

Abstract

In this paper, we prove several results about the Turing jumps of low c.e. sets. We show that only Δ-levels of the Ershov Hierarchy can properly contain the Turing jumps of c.e. sets and that there exists an arbitrarily large computable ordinal with a normal notation such that the corresponding Δ-level is proper for the Turing jump of some c.e. set. Next, we generalize the notion of jump traceability to the jump traceability with ${\Sigma }_{\alpha }^{-1}$- and ${\Delta }_{\alpha }^{-1}$-bound for every infinite computable ordinal α. It is known that jump traceability and superlowness coincide on the c.e. sets and we show that for every infinite computable ordinal α, jump traceability with ${\Sigma }_{\alpha }^{-1}$- or ${\Delta }_{\alpha }^{-1}$-bound of a c.e. set A is equivalent to the fact that $A^{\prime }\in {\Delta }_{\alpha }^{-1}$. Finally, we consider the generalized truth-table reducibilities ${⩽}_{gtt\left(\alpha \right)}$ and prove that for every (not necessarily the Turing jump of a c.e. set) set A and every limit computable ordinal α, $A\in {\Delta }_{\alpha }^{-1}$ iff $A{⩽}_{gtt\left(\alpha \right)}{⌀}^{\prime }$.