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equipotence, equipotent, equicardinality, equipollence, equipollent, equinumerosity,
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03E10 no label found


This entry defines cardinality, as a concept, only for sets. Can it be extended to deal with proper classes? If so, do they vary in cardinality?

The cardinality |X| of a set X is most usefully taken to be the cardinal number with which X is equinumerous. The arithmetic of cardinal numbers then provides a means of calculating the cardinalities of sets constructed via set operations from the cardinalities of their constituents. Note, for example, that |{0, 1}| = 2, and |A|^|B| = |{f | f:B -> A}|. The set {f | f:X -> {0, 1}} is just the set of characteristic functions of subsets of X. Thus, the power set of X has cardinality 2^|X|.
There are no proper classes in ZFC, so questions about them simply don't arise. In the class theories NBG and MK all proper classes are equinumerous with the class of ordinal numbers so, by abusing conventional usage, all proper classes could be said to have the same cardinality. But why bother?

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