Abstract.
Let Q be an equivalence relation whose equivalence classes, denoted Q[x], may be proper classes. A function L defined on Field(Q) is a labelling for Q if and only if for all x,L(x) is a set and
L is a labelling by subsets for Q if and only if
BG denotes Bernays-Gödel class-set theory with neither the axiom of foundation, AF, nor the class axiom of choice, E. The following are relatively consistent with BG.
(1) E is true but there is an equivalence relation with no labelling.
(2) E is true and every equivalence relation has a labelling, but there is an equivalence relation with no labelling by subsets.
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This research was partially supported by Fondecyt 1980855 and by Fondecyt 1040846
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Marshall, M., Schwarze, M. Labelling classes by sets. Arch. Math. Logic 44, 219–226 (2005). https://doi.org/10.1007/s00153-004-0261-z
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DOI: https://doi.org/10.1007/s00153-004-0261-z