## Notre Dame Journal of Formal Logic

### Degrees of Categoricity and the Hyperarithmetic Hierarchy

#### 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 $\alpha$, $\mathbf{0}^{(\alpha)}$ is the degree of categoricity of some computable structure $\mathcal{A}$. We show additionally that for $\alpha$ a computable successor ordinal, every degree $2$-c.e. in and above $\mathbf{0}^{(\alpha)}$ 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 $\Pi_{1}^{1}$-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

https://projecteuclid.org/euclid.ndjfl/1361454975

Digital Object Identifier
doi:10.1215/00294527-1960479

Mathematical Reviews number (MathSciNet)
MR3028796

Zentralblatt MATH identifier
1311.03070

#### 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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