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Degrees that are not degrees of categoricity. (English) Zbl 1436.03229
Summary: A computable structure \(\mathcal {A}\) is \(\mathbf {x}\)-computably categorical for some Turing degree \(\mathbf {x}\) if for every computable structure \(\mathcal {B}\cong\mathcal {A}\) there is an isomorphism \(f:\mathcal {B}\to\mathcal {A}\) with \(f\leq_{T}\mathbf {x}\). A degree \(\mathbf {x}\) is a degree of categoricity if there is a computable structure \(\mathcal {A}\) such that \(\mathcal {A}\) is \(\mathbf {x}\)-computably categorical, and for all \(\mathbf {y}\), if \(\mathcal {A}\) is \(\mathbf {y}\)-computably categorical, then \(\mathbf {x}\leq_{T}\mathbf {y}\).
We construct a \(\Sigma^{0}_{2}\) set whose degree is not a degree of categoricity. We also demonstrate a large class of degrees that are not degrees of categoricity by showing that every degree of a set which is 2-generic relative to some perfect tree is not a degree of categoricity. Finally, we prove that every noncomputable hyperimmune-free degree is not a degree of categoricity.

MSC:
03D45 Theory of numerations, effectively presented structures
03D30 Other degrees and reducibilities in computability and recursion theory
03C57 Computable structure theory, computable model theory
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References:
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