Set (mathematics)

A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no elements is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set.

This article is about what mathematicians call "intuitive" or "naive" set theory. For a more detailed account, see Naive set theory. For a rigorous modern axiomatic treatment of sets, see Set theory.

Sets are uniquely characterized by their elements; this means that two sets that have precisely the same elements are equal (they are the same set). This property is called extensionality. In particular, this implies that there is only one empty set.

Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century.

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