# Class

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A term used in mathematics mainly as a synonym for the term "set" for denoting arbitrary collections of objects possessing some definite property of indication (for example, in algebra, equivalence classes with respect to a given equivalence relation). Sometimes the term class is preferred for collections whose elements are sets (for example, in recursion theory: denumerable classes). In some cases, influenced by axiomatic set theory (see 2)), the term "class" is used to emphasize the fact that the given collection is a proper class rather than a set in the narrow sense (for example, in algebra, the primitive classes of universal algebras, also called varieties). The set-theoretic operations on classes are defined in the same way as for sets.

A class in axiomatic set theory (more precisely, in the Gödel–Bernays axiomatic system) is one of the forms of primitive objects considered in these systems. The difference between sets and classes consists here in the fact that only sets, but not (proper) classes, are allowed to be elements of classes. The idea of introducing into set theory classes in the above sense is due to J. von Neumann and is based on his observation that the well-known contradictions in Cantor's set theory arise not as a result of allowing the formation of very large sets, but rather because such sets are allowed to be members of other sets. Apart from this restriction, all usual set-theoretic operations are allowed in the above-mentioned systems for classes, the result being a class and not a set. Furthermore, for each admissible (in some sense) predicate defined on sets, there exists a class consisting precisely of those sets that satisfy the predicate. It has been proved that the consistency of each of the systems of Gödel–Bernays and Zermelo–Fraenkel follows from the consistency of the other (which confirms the standpoint of von Neumann). See also Axiomatic set theory.

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How to Cite This Entry:
Class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Class&oldid=46348
This article was adapted from an original article by V.A. Dushskii, A.A. Borisenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article