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Set ○꠹｜Definition｜1st｜20251119205401-00-⌔
Set (mathematics)
In mathematics, a set is a collection of different things1234; the things are called elements or members of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, functions, or even other sets.56
Mathematics typically does not define precisely what constitutes a “set” or “collection”, because such a definition would have to be in terms of something else previously defined. Instead, sets serve as foundational objects whose behavior is described by axioms modeled on intuition about collections,7 and then essentially all other mathematical objects are rigorously defined in terms of sets.8
Set theory studies possible axiom systems and their consequences. Since the first half of the 20th century, ZFC (Zermelo–Fraenkel set theory with the axiom of choice) has been the axiom system most commonly used.
Printed 2026-06-28.
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Link to original Footnotes
Cantor, Georg; Jourdain, Philip E.B. (Translator) (1915). Contributions to the founding of the theory of transfinite numbers. New York Dover Publications (1954 English translation). By an ‘aggregate’ (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Ganzen) M of definite and separate objects m of our intuition or our thought. Here: p.85 ↩
P. K. Jain; Khalil Ahmad; Om P. Ahuja (1995). Functional Analysis. New Age International. p. 1. ISBN 978-81-224-0801-0. ↩
Samuel Goldberg (1 January 1986). Probability: An Introduction. Courier Corporation. p. 2. ISBN 978-0-486-65252-8. ↩
Thomas H. Cormen; Charles E Leiserson; Ronald L Rivest; Clifford Stein (2001). Introduction To Algorithms. MIT Press. p. 1070. ISBN 978-0-262-03293-3. ↩
Halmos 1960, p. 1. ↩
Maddocks, J. R. (2004). Lerner, K. Lee; Lerner, Brenda Wilmoth (eds.). The Gale Encyclopedia of Science. Gale. pp. 3587–3589. ISBN 0-7876-7559-8. ↩
This is analogous to the role of points and lines in Euclidean geometry: Euclid never gives a meaningful definition of “point”. Instead, Euclid gives axioms modeled on our intuition on how points and lines behave. ↩
For example, the ordered pair (x, y) may be formally defined as the set { { x }, { x, y } }, from which and can be recovered, in order. ↩
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