Degrees of categoricity and the hyperarithmetic hierarchy.

*(English)*Zbl 1311.03070Let \(\mathbf d\) 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.

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.

Reviewer: Andrei S. Morozov (Novosibirsk)

##### MSC:

03D45 | Theory of numerations, effectively presented structures |

03D28 | Other Turing degree structures |

##### Keywords:

computability theory; computable structure theory; Turing degrees; isomorphisms; relative categoricity; degree of categoricity
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\textit{B. F. Csima} et al., Notre Dame J. Formal Logic 54, No. 2, 215--231 (2013; Zbl 1311.03070)

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