## 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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