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In set theory and its applications throughout mathematics, a 'class' is a collection of sets (or sometimes other mathematical objects) 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).
A class that is not a set is called a 'proper class'.
Many objects in mathematics are too big for sets and need to be described with classes. Examples include large categories or the class-field of surreal numbers. The usual construction to show that a given "thing" is a proper class is to show that such a "thing" has at least as many elements as there are ordinal numbers. For an example of such a proof, see free lattices.
A proper class cannot be an 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 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 which do not contain themselves is proper, and the Burali-Forti paradox becomes a proof that the class of all ordinal numbers is proper.
The standard Zermelo-Fraenkel 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 Neumann-Bernays-Gödel 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.
In other set theories, such as New Foundations or the theory of semisets, the concept of "proper class" still makes sense (not all collections are sets) but the criterion of sethood is not size. For example, any set theory with a universal set has proper classes which are subclasses of sets.
The word "class" is sometimes used synonymously with "set," most notably in the term "equivalence class."
This usage dates from a historical period where classes and sets were not distinguished as they are in modern terminology.
Many discussions of "classes" in the 19th century and earlier are really referring to sets, or perhaps to a more ambiguous concept.