Notre Dame Journal of Formal Logic

Degrees That Are Not Degrees of Categoricity

Abstract

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.

Article information

Source
Notre Dame J. Formal Logic, Volume 57, Number 3 (2016), 389-398.

Dates
Accepted: 6 February 2014
First available in Project Euclid: 7 April 2016

https://projecteuclid.org/euclid.ndjfl/1460032557

Digital Object Identifier
doi:DOI: 10.1215/00294527-3496154

Mathematical Reviews number (MathSciNet)
MR3521488

Zentralblatt MATH identifier
06621297

Subjects
Primary: 03D30: Other degrees and reducibilities

Citation

Anderson, Bernard; Csima, Barbara. Degrees That Are Not Degrees of Categoricity. Notre Dame J. Formal Logic 57 (2016), no. 3, 389--398. doi:DOI: 10.1215/00294527-3496154. https://projecteuclid.org/euclid.ndjfl/1460032557

References

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