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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.

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
Full Text: DOI
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