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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
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