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Algebra and Logic

, Volume 58, Issue 3, pp 224–231 | Cite as

Computable Numberings of Families of Infinite Sets

  • M. V. DorzhievaEmail author
Article
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We state the following results: the family of all infinite computably enumerable sets has no computable numbering; the family of all infinite \( {\varPi}_1^1 \) sets has no \( {\varPi}_1^1 \) -computable numbering; the family of all infinite \( {\varSigma}_2^1 \) sets has no \( {\varSigma}_2^1 \) -computable numbering. For k > 2, the existence of a \( {\varSigma}_k^1 \) -computable numbering for the family of all infinite \( {\varSigma}_k^1 \) sets leads to the inconsistency of ZF.

Keywords

computability analytical hierarchy computable numberings Friedberg numbering Gödel’s axiom of constructibility 

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Novosibirsk State UniversityNovosibirskRussia

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