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Enumerations in computable structure theory. (English) Zbl 1081.03033
In the paper under review the authors show that for each computable successor ordinal \(\alpha\) there is a computable structure that is \(\Delta_{\alpha}^0\)-categorical, but not relatively \(\Delta_{\alpha}^0\)-categorical. Further, it is shown that for each computable successor ordinal \(\alpha\) and each natural number \(n\geq 1\) there is a computable structure with exactly \(n\) computable copies, up to \(\Delta_{\alpha}^0\)-isomorphism. Finally, the authors prove that for each computable successor ordinal \(\alpha\) there is a structure with copies in just the degrees of sets \(X\) such that \(\Delta_{\alpha}^0(X)\) is not \(\Delta_{\alpha}^0\). It follows, in particular, that for each natural number \(n\geq 1\) there is a structure with copies in just the non-low\(_n\) degrees.

MSC:
03C57 Computable structure theory, computable model theory
03D45 Theory of numerations, effectively presented structures
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