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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.

MSC:
03D45 Theory of numerations, effectively presented structures
03D28 Other Turing degree structures
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References:
[1] Ash, C. J., and J. Knight, Computable Structures and the Hyperarithmetical Hierarchy , vol. 144 of Studies in Logic and the Foundations of Mathematics , North-Holland, Amsterdam, 2000. · Zbl 0960.03001
[2] Fokina, E. B., I. Kalimullin, and R. Miller, “Degrees of categoricity of computable structures,” Archive for Mathematical Logic , vol. 49 (2010), pp. 51-67. · Zbl 1184.03026 · doi:10.1007/s00153-009-0160-4
[3] Harizanov, V. S., “Pure computable model theory,” pp. 3-114 in Handbook of Recursive Mathematics, Vol. 1 , vol. 138 of Studies in Logic and the Foundations of Mathematics , North-Holland, Amsterdam, 1998. · Zbl 0952.03037 · doi:10.1016/S0049-237X(98)80002-5
[4] Hirschfeldt, D. R., and W. M. White, “Realizing levels of the hyperarithmetic hierarchy as degree spectra of relations on computable structures,” Notre Dame Journal of Formal Logic , vol. 43 (2002), pp. 51-64. · Zbl 1048.03035 · doi:10.1305/ndjfl/1071505769
[5] Marker, D., “Non \(\Sigma_{n}\) axiomatizable almost strongly minimal theories,” Journal of Symbolic Logic , vol. 54 (1989), pp. 921-27. · Zbl 0698.03021 · doi:10.2307/2274752
[6] Rogers Jr., H., Theory of Recursive Functions and Effective Computability , McGraw-Hill Book Co., New York, 1967. · Zbl 0183.01401
[7] Sacks, G. E., Higher Recursion Theory , Perspectives in Mathematical Logic, Springer, Berlin, 1990. · Zbl 0716.03043
[8] Soare, R. I., Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets , Perspectives in Mathematical Logic, Springer, Berlin, 1987. · Zbl 0667.03030
[9] White, W. M., Characterizations for Computable Structures , Ph.D. dissertation, Cornell University, Ithaca, N.Y., 2000.
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