Every set has a least jump enumeration

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'. Remar kably, 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'(mu)(A), which is the least degree of the jumps of all sets X for which A is Sigma (0)(1)(X) . In addition this result provides an isomorphism invariant method for assi gning Turing degrees to certain torsion-free abelian groups.