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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:
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