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Algebra Logika, 2019, Volume 58, Number 3, Pages 334–343 (Mi al898)  

Computable numberings of families of infinite sets

M. V. Dorzhieva

Novosibirsk State University

Abstract: We state the following results: the family of all infinite computably enumerable sets has no computable numbering; the family of all infinite $\Pi^{1}_{1}$ sets has no $\Pi^{1}_{1}$-computable numbering; the family of all infinite $\Sigma^{1}_{2}$ sets has no $\Sigma^{1}_{2}$-computable numbering. For $k>2$, the existence of a $\Sigma^{1}_{k}$-computable numbering for the family of all infinite $\Sigma^{1}_{k}$ sets leads to the inconsistency of $ZF$.

Keywords: computability, analytical hierarchy, computable numberings, Friedberg numbering, Gödel's axiom of constructibility.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-31278__
∗Supported by RFBR, project no. 14-01-31278 mol-a.


DOI: https://doi.org/10.33048/alglog.2019.58.303

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English version:
Algebra and Logic, 2019, 58:3, 224–231

Bibliographic databases:

UDC: 510.5
Received: 27.01.2018
Revised: 24.09.2019

Citation: M. V. Dorzhieva, “Computable numberings of families of infinite sets”, Algebra Logika, 58:3 (2019), 334–343; Algebra and Logic, 58:3 (2019), 224–231

Citation in format AMSBIB
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\issue 3
\pages 334--343
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