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 , we transform a countable directed graph into a structure such that has a isomorphic copy if and only if has a computable isomorphic copy.
A computable structure is categorical (relatively categorical, respectively) if for all computable (countable, respectively) isomorphic copies of , there is an isomorphism from onto , which is ( relative to the atomic diagram of , respectively). We prove that for every computable successor ordinal , there is a computable, categorical structure, which is not relatively categorical. This generalizes the result of Goncharov that there is a computable, computably categorical structure, which is not relatively computably categorical.
An additional relation on the domain of a computable structure is intrinsically (relatively intrinsically , respectively) on if in all computable (countable, respectively) isomorphic copies of , the image of is ( relative to the atomic diagram of , respectively). We prove that for every computable successor ordinal , there is an intrinsically relation on a computable structure, which not relatively intrinsically . 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 dimension of a structure is the number of computable isomorphic copies, up to isomorphisms. We prove that for every computable successor ordinal and every , there is a computable structure with dimension . This generalizes the result of Goncharov that there is a structure of computable dimension for every .
Finally, we prove that for every computable successor ordinal , there is a countable structure with isomorphic copies in just the Turing degrees of sets such that relative to is not . In particular, for every finite , there is a structure with isomorphic copies in exactly the non- Turing degrees. This generalizes the result obtained by Wehner, and independently by Slaman, that there is a structure with isomorphic copies in exactly the nonzero Turing degrees.