The complements of Lower cones of Degrees and the degree spectra of Structures.

We study Turing degrees a for which there is a countable structure A whose degree spectrum is the collection {x : x not less than or equal to a}. In particular, for degrees a from the interval [0', 0 "], such a structure exists if a' = 0 ", and there are no such structures if a " > 0 "'.