Computable isomorphisms, degree spectra of relations, and Scott families

AbstractThe spectrum of a relation R on a computable structure is the set of Turing degrees of the image of R under all isomorphisms between A and any other computable structure B. The relation R is intrinsically computably enumerable (c.e.) if its image under all such isomorphisms is c.e. We prove that any computable partially ordered set is isomorphic to the spectrum of an intrinsically c.e. relation on a computable structure. Moreover, the isomorphism can be constructed in such a way that the image of the minimum element (if it exists) of the partially ordered set is computable. This solves the spectrum problem. The theorem and modifications of its proof produce computably categorical structures whose expansions by finite number of constants are not computably categorical and, indeed, ones whose expansions can have any finite number of computable isomorphism types. They also provide examples of computably categorical structures that remain computably categorical under expansions by constants but have no Scott family