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Finite computable dimension and degrees of categoricity. (English) Zbl 06974479
Summary: We first give an example of a rigid structure of computable dimension 2 such that the unique isomorphism between two non-computably isomorphic computable copies has Turing degree strictly below $$0^{\prime\prime}$$, and not above $$0^\prime$$. This gives a first example of a computable structure with a degree of categoricity that does not belong to an interval of the form $$[0^{(\alpha)}, 0^{(\alpha + 1)}]$$ for any computable ordinal $$\alpha$$. We then extend the technique to produce a rigid structure of computable dimension 3 such that if $$\mathbf{d}_0$$, $$\mathbf{d}_1$$, and $$\mathbf{d}_2$$ are the degrees of isomorphisms between distinct representatives of the three computable equivalence classes, then each $$\mathbf{d}_i < \mathbf{d}_0 \oplus \mathbf{d}_1 \oplus \mathbf{d}_2$$. The resulting structure is an example of a structure that has a degree of categoricity, but not strongly.

##### MSC:
 03D45 Theory of numerations, effectively presented structures 03C57 Computable structure theory, computable model theory 03D80 Applications of computability and recursion theory 03D99 Computability and recursion theory
##### Keywords:
computability theory; computable structure theory
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##### References:
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