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2016 Degrees That Are Not Degrees of Categoricity
Bernard Anderson, Barbara Csima
Notre Dame J. Formal Logic 57(3): 389-398 (2016). DOI: 10.1215/00294527-3496154

Abstract

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.

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Bernard Anderson. Barbara Csima. "Degrees That Are Not Degrees of Categoricity." Notre Dame J. Formal Logic 57 (3) 389 - 398, 2016. https://doi.org/10.1215/00294527-3496154

Information

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

zbMATH: 06621297
MathSciNet: MR3521488
Digital Object Identifier: 10.1215/00294527-3496154

Subjects:
Primary: 03D30

Rights: Copyright © 2016 University of Notre Dame

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