Enumerations, countable structures and Turing degrees
Stephan Wehner · Proceedings of the American Mathematical Society · 1998
It is proven that there is a family of sets of natural numbers which has enumerations in every Turing degree except for the recursive degree. This implies that there is a countable structure which has representations in all but the recursive degree. Moreover, it is shown that there is such a structure which has a recursively represented elementary extension.