An urelement contains no elements, belongs to some set, and is not identical with the empty set (Moore 1982, p. 3; Rubin 1967, p. 23). "Ur" is a German prefix which is difficult to translate literally, but has a meaning close to "primeval." Urelements are also called "atoms" (Rubin 1967, Moore 1982) or "individuals" (Moore 1982).

In "pure" set theory, all elements are sets and there are no urelements. Often, the axioms of set theory are modified to allow the presence of urelements for ease in representing something. In fact, before Paul Cohen developed the method of forcing, some of the independence theorems in set theory were shown if urelements were allowed.

See also

Empty Set, Set Theory

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Moore, G. H. Zermelo's Axiom of Choice: Its Origin, Development, and Influence. New York: Springer-Verlag, 1982.Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967.

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Cite this as:

Weisstein, Eric W. "Urelement." From MathWorld--A Wolfram Web Resource.

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