Every Set has a Least Jump Enumeration

Abstract

Given a computably enumerable set W, there is a Turing degree which is the least jump of any set in which W is computably enumerable, namely 0′. Remarkably, this is not a phenomenon of computably enumerable sets. It is shown that for every subset A of N, there is a Turing degree, c′μ(A), which is the least degree of the jumps of all sets X for which A is ∑ 1 0 ( X ) . In addition this result provides an isomorphism invariant method for assigning Turing degrees to certain torsion-free abelian groups.