Degrees of Categoricity and the Hyperarithmetic Hierarchy

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

doi:10.1215/00294527-1960479

Cited by 60 publications

(44 citation statements)

References 6 publications

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“…We note that Theorem 5.4 can be improved significantly by exploiting the structures introduced by Csima, Franklin, and Shore [5]. Indeed, the result remains true for any degree d that is d.c.e.…”

Section: Constructionmentioning

confidence: 76%

Computable categoricity versus relative computable categoricity

et al. 2013

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Abstract. We study the notion of computable categoricity of computable structures, comparing it especially to the notion of relative computable categoricity and its relativizations. We show that every 1-decidable computably categorical structure is relatively ∆ 0 2 -categorical. We study the complexity of various index sets associated with computable categoricity and relative computable categoricity. We also introduce and study a variation of relative computable categoricity, comparing it to both computable categoricity and relative computable categoricity and its relativizations.

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Section: Constructionmentioning

confidence: 76%

Computable categoricity versus relative computable categoricity

et al. 2013

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“…This fills in a gap that was missing from [CFS13] above limit ordinals, making further progress towards Question 5.1 of that paper. We have not yet explained what a strong degree of categoricity is.…”

Section: Introductionmentioning

confidence: 76%

Degrees of categoricity above limit ordinals

Csima

Deveau

Harrison-Trainor

et al. 2020

COM

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A computable structure A has degree of categoricity d if d is exactly the degree of difficulty of computing isomorphisms between isomorphic computable copies of A. Fokina, Kalimullin, and Miller showed that every degree d.c.e. in and above 0 (n) , for any n < ω, and also the degree 0 (ω) , are degrees of categoricity. Later, Csima, Franklin, and Shore showed that every degree 0 (α) for any computable ordinal α, and every degree d.c.e. in and above 0 (α) for any successor ordinal α, is a degree of categoricity. We show that every degree c.e. in and above 0 (α) , for α a limit ordinal, is a degree of categoricity. We also show that every degree c.e. in and above 0 (ω) is the degree of categoricity of a prime model, making progress towards a question of Bazhenov and Marchuk.

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“…Already for n = 2 and even for algebraically very well understood classes, 1 Here 0 (n+1) stands for the (n + 1)th iterate of the Halting problem. We note that there are variations of Definition 1.1 such as the notion of relative Δ 0 n -categoricity [3], and also related notions of categoricity spectra [20] and degrees of categoricity [19,10].…”

Section: Non-computable Isomorphisms Between Computable Structuresmentioning

confidence: 99%