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 ∆ 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.

Spectra of structures and relations

“…This complements a recent result in [7], which proves the existence, for each n ∈ ω, of a structure whose spectrum contains exactly the non-low n degrees, i.e., those degrees c with c (n) > T 0 (n) . Moreover, we can use an arbitrary degree d in place of 0 in the following proof, thereby building structures with spectrum {c : c ≥ T d}.…”

Spectra of theories and structures

“…In other words, there is a theory spectrum S such that S −(α) = {d | d (α) ∈ S} is not a theory spectrum. This stands in contrast to Lemma 2.8, which implies α-jump inversion for theory spectra and finite α, and to Lemma 5.5 of [9], which gives α-jump inversion for structure spectra and computable successor ordinals α.…”

Preserving Categoricity and Complexity of Relations

“…THEOREM 1.4 [6]. For each computable successor ordinal α, there is a structure A that is Δ 0 α categorical but not relatively Δ 0 α categorical.…”

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Keywords: Δ 0 α categorical structure, structure that is not relatively Δ 0 α categorical, field.In [3,6,7], it was proved that for each computable ordinal α, there is a structure that is Δ 0 α categorical but not relatively Δ 0 α categorical. The original examples were not familiar algebraic kinds of structures.

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