Notre Dame Journal of Formal Logic

Degrees of Categoricity and the Hyperarithmetic Hierarchy

Barbara F. Csima, Johanna N. Y. Franklin, and Richard A. Shore

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Abstract

We study arithmetic and hyperarithmetic degrees of categoricity. We extend a result of E. Fokina, I. Kalimullin, and R. Miller to show that for every computable ordinal α, 0(α) is the degree of categoricity of some computable structure A. We show additionally that for α a computable successor ordinal, every degree 2-c.e. in and above 0(α) is a degree of categoricity. We further prove that every degree of categoricity is hyperarithmetic and show that the index set of structures with degrees of categoricity is Π11-complete.

Article information

Source
Notre Dame J. Formal Logic, Volume 54, Number 2 (2013), 215-231.

Dates
First available in Project Euclid: 21 February 2013

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1361454975

Digital Object Identifier
doi:10.1215/00294527-1960479

Mathematical Reviews number (MathSciNet)
MR3028796

Zentralblatt MATH identifier
1311.03070

Subjects
Primary: 03D45: Theory of numerations, effectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55]

Keywords
computability theory computable structure theory Turing degrees isomorphisms

Citation

Csima, Barbara F.; Franklin, Johanna N. Y.; Shore, Richard A. Degrees of Categoricity and the Hyperarithmetic Hierarchy. Notre Dame J. Formal Logic 54 (2013), no. 2, 215--231. doi:10.1215/00294527-1960479. https://projecteuclid.org/euclid.ndjfl/1361454975


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References

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Notre Dame Journal of Formal Logic

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