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.
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∗Supported by RFBR, project no. 14-01-31278 mol-a.
Translated from Algebra i Logika, Vol. 58, No. 3, pp. 334-343, May-June, 2019.
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Dorzhieva, M.V. Computable Numberings of Families of Infinite Sets. Algebra Logic 58, 224–231 (2019). https://doi.org/10.1007/s10469-019-09540-4
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DOI: https://doi.org/10.1007/s10469-019-09540-4