Enumerations in computable structure theory

Finally, we prove that for every computable successor ordinal α, there is a countable structure with isomorphic copies in just the Turing degrees of sets X such that Δα0relative toX is not Δα0. In particular, for every finite n, there is a structure with isomorphic copies in exactly the non-lown Turing degrees. This generalizes the result obtained by Wehner, and independently by Slaman, that there is a structure A with isomorphic copies in exactly the nonzero Turing degrees.