Computable numberings of families of low sets and Turing jumps in the Ershov hierarchy

Abstract

If ν and μ are some Δcomputable numberings of families of sets of the naturals then P(x,y) ⇔ ν(x)′ ≠ μ(y) is a Σpredicate. Deriving corollaries from this result, we obtain a sufficient condition for existence of a Δcomputable numbering of the subfamily of all sets in a given family with the Turing jumps belonging to a fixed level of the Ershov hierarchy, and we deduce existence of a Σcomputable numbering of the family of all superlow sets. © 2010 Pleiades Publishing, Ltd.