Enumerations in computable structure theory

Abstract We exploit properties of certain directed graphs, obtained from the families of sets with special effective enumeration properties, to generalize several results in computable model theory to higher levels of the hyperarithmetical hierarchy. Families of sets with such enumeration features were previously built by Selivanov, Goncharov, and Wehner. For a computable successor ordinal α , we transform a countable directed graph G into a structure G ∗ such that G has a Δ α 0 isomorphic copy if and only if G ∗ has a computable isomorphic copy. A computable structure A is Δ α 0 categorical (relatively Δ α 0 categorical, respectively) if for all computable (countable, respectively) isomorphic copies B of A , there is an isomorphism from A onto B , which is Δ α 0 ( Δ α 0 relative to the atomic diagram of B , respectively). We prove that for every computable successor ordinal α , there is a computable, Δ α 0 categorical structure, which is not relatively Δ α 0 categorical. This generalizes the result of Goncharov that there is a computable, computably categorical structure, which is not relatively computably categorical. An additional relation R on the domain of a computable structure A is intrinsically Σ α 0 (relatively intrinsically Σ α 0 , respectively) on A if in all computable (countable, respectively) isomorphic copies B of A , the image of R is Σ α 0 ( Σ α 0 relative to the atomic diagram of B , respectively). We prove that for every computable successor ordinal α , there is an intrinsically Σ α 0 relation on a computable structure, which not relatively intrinsically Σ α 0 . This generalizes the result of Manasse that there is an intrinsically computably enumerable relation on a computable structure, which is not relatively intrinsically computably enumerable. The Δ α 0 dimension of a structure A is the number of computable isomorphic copies, up to Δ α 0 isomorphisms. We prove that for every computable successor ordinal α and every n ≥ 1 , there is a computable structure with Δ α 0 dimension n . This generalizes the result of Goncharov that there is a structure of computable dimension n for every n ≥ 1 . 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 Δ α 0 relative to X is not Δ α 0 . In particular, for every finite n , there is a structure with isomorphic copies in exactly the non- low n 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.