Geodesic
Computable numberings of families of infinite sets
Algebra i logika, Tome 58 (2019) no. 3, pp. 334-343

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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 $\Pi^{1}_{1}$ 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
Mots-clés : Gödel's axiom of constructibility. 
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     author = {M. V. Dorzhieva},
     title = {Computable numberings of families of infinite sets},
     journal = {Algebra i logika},
     pages = {334--343},
     publisher = {mathdoc},
     volume = {58},
     number = {3},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2019_58_3_a2/}
}
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M. V. Dorzhieva. Computable numberings of families of infinite sets. Algebra i logika, Tome 58 (2019) no. 3, pp. 334-343. http://geodesic.mathdoc.fr/item/AL_2019_58_3_a2/