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.