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Degrees of categoricity and the hyperarithmetic hierarchy. (English) Zbl 1311.03070
Let \(\mathbf d\) be a Turing degree. A computable structure is \(\mathbf d\)-computably categorical if any of its computable isomorphic copies is isomorphic to it via a \(\mathbf d\)-computable isomorphism. If there is a least degree with this property, then this degree is called the degree of categoricity of this structure. A degree is called a degree of categoricity if it is the degree of categoricity for some computable structure. If \(\mathbf d\) is a degree of categoricity with the property that there are isomorphic computable structures \({\mathcal A}_0\) and \({\mathcal A}_1\) for which \(\mathbf d\) is the degree of categoricity and every isomorphism from \({\mathcal A}_0\) onto \({\mathcal A}_1\) computes \(\mathbf d\), then \(\mathbf d\) is called strong degree of categoricity.
The authors prove the following results:
1) for any computable ordinal \(\alpha\), \({\mathbf 0}^{(\alpha)}\) is the strong degree of categoricity for some computable structure;
2) if in addition \(\alpha\) is a successor ordinal, then any degree \(2\)-c.e. in and above \({\mathbf 0}^{(\alpha)}\) is a strong degree of categoricity;
3) every degree of categoricity is hyperarithmetic;
4) the set of codes of all structures having a degree of categoricity is \(\Pi_1^1\)-complete.

03D45 Theory of numerations, effectively presented structures
03D28 Other Turing degree structures
Full Text: DOI Euclid
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