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All structures to be considered here have universe ω, and all languages come equipped with Gödel numberings. If is a structure, then D(), the open diagram of , can be thought of as a subset of ω, and it makes sense to talk about the Turing degree deg(D()). This depends on the presentation as well as the isomorphism type of .For example, consider the ordering = (ω, <). For any B ⊆ ω, it is possible to code B in a copy of as follows: Let π be the permutation of ω such that for each n ∈ ω, π leaves 2n and 2n + 1 fixed if n ∈ B and switches 2n with 2n + 1 if n ∉ B. Let be the copy of such that ≃π. Then n ∈ B iff the sentence 2n < 2n + 1 is in D(). In §4, this idea will be used to show that for any structure that is not completely trivial, {deg(D()): ≃ } is closed upwards.It would be satisfying to have a way of assigning Turing degrees to structures such that the degree assigned to a given structure measured the recursion-theoretic complexity of the isomorphism type and was independent of the presentation. Jockusch suggested the following.

ABSTRACT. It is proved that any O-minimal structure M (in which the underlying order is dense) is strongly O-minimal (namely, every N elementarily equivalent to M is O-minimal). It is simultaneously proved that if M is 0-minimal, then every definable set of n-tuples of M has finitely many "definably connected components."

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We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of F F -reducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all Σ 1 1 equivalence relations on hyperarithmetical subsets of ω.

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 ∆ 0 α isomorphic copy if and only if G * has a computable isomorphic copy.A computable structure A is ∆ 0 α categorical (relatively ∆ 0 α categorical, respectively) if for all computable (countable, respectively) isomorphic copies B of A, there is an isomorphism from A onto B, which is ∆ 0 α (∆ 0 α relative to the atomic diagram of B, respectively). We prove that for every computable successor ordinal α, there is a computable, ∆ 0 α categorical structure, which is not relatively ∆ 0 α categorical. This generalizes the result of Goncharov that there is a computable, computably categorical structure, which is not relatively computably categorical.

ABSTRACT. It is proved that any O-minimal structure M (in which the underlying order is dense) is strongly O-minimal (namely, every N elementarily equivalent to M is O-minimal). It is simultaneously proved that if M is 0-minimal, then every definable set of n-tuples of M has finitely many "definably connected components."

Makkai [10] produced an arithmetical structure of Scott rank ω1CK. In [9], Makkai's example is made computable. Here we show that there are computable trees of Scott rank ω1CK. We introduce a notion of “rank homogeneity”. In rank homogeneous trees, orbits of tuples can be understood relatively easily. By using these trees, we avoid the need to pass to the more complicated “group trees” of [10] and [9], Using the same kind of trees, we obtain one of rank ω1CK that is “strongly computably approximable”. We also develop some technology that may yield further results of this kind.