체르멜로-프렝켈 집합론
An attempt to define the cardinality of a set as the equivalence class of all sets equinumerous to it is problematic in Zermelo–Fraenkel set theory, the standard form of axiomatic set theory, because the equivalence class of any non-empty set would be too large to be a set: it would be a proper class. Within the framework of Zermelo–Fraenkel set theory, relations are by definition restricted to sets (a binary relation on a set "A" is a subset of the Cartesian product ), and there is no set of all sets in Zermelo–Fraenkel set theory. In Zermelo–Fraenkel set theory, instead of defining the cardinality of a set as the equivalence class of all sets equinumerous to it one tries to assign a representative set to each equivalence class (cardinal assignment). In some other systems of axiomatic set theory, for example in Von Neumann–Bernays–Gödel set theory and Morse–Kelley set theory, relations are extended to classes.