A proper class cannot be an element of a set or a class and is not subject to the ZermeloFraenkel axioms of set theory; thereby a number of paradoxes of naive set theory are avoided. Instead, these paradoxes become proofs that a certain class is proper. For example, Russell's paradox becomes a proof that the class of all sets is proper, and the BuraliForti paradox becomes a proof that the class of all ordinal numbers is proper.
The standard ZermeloFraenkel set theory axioms do not talk about classes; classes exist only in the metalanguage[?] as equivalence classes of logical formulas. Another approach is taken by the von NeumannBernaysGödel axioms[?]; classes are the basic objects in this theory, and a set is then defined to be a class that is an element of some other class. The proper classes, then, are those classes that are not elements of any other class.
Several objects in mathematics are too big for sets and need to be described with classes, for instance large categories or the classfield of surreal numbers.
The word "class" is sometimes used synonymously with "set", for instance in the term "equivalence class".
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