Notre Dame Journal of Formal Logic

Degrees That Are Not Degrees of Categoricity

Bernard Anderson and Barbara Csima

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A computable structure A is x-computably categorical for some Turing degree x if for every computable structure BA there is an isomorphism f:BA with fTx. A degree x is a degree of categoricity if there is a computable structure A such that A is x-computably categorical, and for all y, if A is y-computably categorical, then xTy.

We construct a Σ20 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

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

Received: 2 July 2013
Accepted: 6 February 2014
First available in Project Euclid: 7 April 2016

Permanent link to this document

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

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03D30: Other degrees and reducibilities

degree of categoricity computably categorical strong degree of categoricity CatSpec category spectrum computable structure


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.

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