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Algebra Logika, 2020, Volume 59, Number 1, Pages 66–83 (Mi al935)  

Computable positive and Friedberg numberings in hyperarithmetic

I. Sh. Kalimullina, V. G. Puzarenkobc, M. Kh. Faizrakhmanova

a Kazan (Volga Region) Federal University
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
c Novosibirsk State University

Abstract: We point out an existence criterion for positive computable total $\Pi^1_1$-numberings of families of subsets of a given $\Pi^1_1$-set. In particular, it is stated that the family of all $\Pi^1_1$-sets has no positive computable total $\Pi^1_1$-numberings. Also we obtain a criterion of existence for computable Friedberg $\Sigma^1_1$-numberings of families of subsets of a given $\Sigma^1_1$-set, the consequence of which is the absence of a computable Friedberg $\Sigma^1_1$-numbering of the family of all $\Sigma^1_1$-sets. Questions concerning the existence of negative computable $\Pi^1_1$- and $\Sigma^1_1$-numberings of the families mentioned are considered.

Keywords: computable numbering, admissible set, analytical hierarchy, positive numbering, Friedberg numbering, negative numbering.

Funding Agency Grant Number
Russian Foundation for Basic Research 18-01-00574_а
18-01-00624_а
Ministry of Education and Science of the Russian Federation 1.451.2016/1.4
Siberian Branch of Russian Academy of Sciences I.1.1., проект № 0314-2019-0003
Russian Science Foundation 18-11-00028


DOI: https://doi.org/10.33048/alglog.2020.59.104

Full text: PDF file (274 kB)
First page: PDF file
References: PDF file   HTML file

UDC: 510.5
Received: 28.10.2018
Revised: 30.04.2018

Citation: I. Sh. Kalimullin, V. G. Puzarenko, M. Kh. Faizrakhmanov, “Computable positive and Friedberg numberings in hyperarithmetic”, Algebra Logika, 59:1 (2020), 66–83

Citation in format AMSBIB
\Bibitem{KalPuzFai20}
\by I.~Sh.~Kalimullin, V.~G.~Puzarenko, M.~Kh.~Faizrakhmanov
\paper Computable positive and Friedberg numberings in hyperarithmetic
\jour Algebra Logika
\yr 2020
\vol 59
\issue 1
\pages 66--83
\mathnet{http://mi.mathnet.ru/al935}
\crossref{https://doi.org/10.33048/alglog.2020.59.104}


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  • Алгебра и логика Algebra and Logic
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