Archive for Mathematical Logic 44 (2):219-226 (2005)

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
Keywords Mathematics   Mathematics, general   Algebra   Mathematical Logic and Foundations
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Reprint years 2005
DOI 10.1007/s00153-004-0261-z
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References found in this work BETA

The Axiom of Choice.Thomas J. Jech - 1973 - Amsterdam, Netherlands: North-Holland.
Powers of Regular Cardinals.William B. Easton - 1970 - Annals of Mathematical Logic 1 (2):139.
Rank in Set Theory Without Foundation.M. Victoria Marshall & M. Gloria Schwarze - 1999 - Archive for Mathematical Logic 38 (6):387-393.

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