Mathematical class

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In mathematics, a class is a collection of sets that can be unambiguously defined by a property that all its members share. Some classes are sets, for instance the class of all integers that are even, but others are not, for instance the class of all ordinal numbers or the class of all sets. Classes that are not sets are called proper classes.

A proper class cannot be element of a set or a class and is not subject to the Zermelo-Fraenkel axioms of set theory; thereby a number of paradoxes of naive set theory, such as Russells paradox, are avoided.

The standard Zermelo-Fraenkel set theory axioms do not talk about classes and classes are defined afterwards as equivalence classes of logical formulas. Another approach is taken by the von Neumann- Bernays-Gödel set theory: classes are the basic objects in this theory and sets are then defined to be those classes which are elements of other classes. The proper classes then are those classes that are not element of any other class.

Several objects in mathematics are too big for sets and need to be described with classes, for instance categories or the surreal numbers.

The word "class" is sometimes used synonymous with "set", for instance in the term equivalence class.

For other meaning of word class, see Class.