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Set Theory 𓆩⚪𓆪｜Definition｜1st｜20260324120834-00-⌔

Set theory - Wikipedia

Set theory

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Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole.12

The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory.3 The non-formalized systems investigated during this early stage go under the name of naive set theory.4 After the discovery of paradoxes within naive set theory (e.g. Russell’s paradox,5 Cantor’s paradox,6 and the Burali-Forti paradox,7 et al.), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied.8

Set theory is commonly employed as a foundational system for the whole of mathematics,9 particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice.10 Besides its foundational role, set theory also provides the framework to develop a mathematical theory of infinity,11 and has various applications in computer science (such as in the theory of relational algebra),1213 philosophy,1 formal semantics,14 and evolutionary dynamics.15 Its foundational appeal, together with its paradoxes, and its implications for the concept of infinity and its multiple applications have made set theory an area of major interest for logicians and philosophers of mathematics.164 Contemporary research into set theory covers a vast array of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.1

Printed 2026-06-28.

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Footnotes

  1. Bagaria, Joan (January 31, 2023) [October 8, 2014], Zalta, Edward N.; Nodelman, Uri (eds.), “Set Theory”, Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University, retrieved May 9, 2026 ↩ ↩2 ↩3

  2. Ferreirós, José (December 3, 2025) [April 10, 2007], Zalta, Edward N.; Nodelman, Uri (eds.), “The Early Development of Set Theory”, Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University, retrieved May 9, 2026 ↩

  3. Dauben, Joseph Warren (1990), Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton: Princeton University Press ↩

  4. Kanamori, Akihiro (1996), “The Mathematical Development of Set Theory from Cantor to Cohen”, Bulletin of Symbolic Logic, 2 (1): 1–71, doi:10.2307/421046 ↩ ↩2

  5. Rang, B.; Thomas, W. (1981), “Zermelo’s discovery of the”Russell Paradox"", Historia Mathematica, 8 (1): 15–22, doi:10.1016/0315-0860(81)90002-1 ↩

  6. Rowe, David E. (2024), “On the Origins of Cantor’s Paradox”, The Mathematical Intelligencer, 46 (2): 102–116, doi:10.1007/s00283-022-10259-x ↩

  7. Moore, Gregory H.; Garciadiego, Alejandro (1981), “Burali-Forti’s paradox: A reappraisal of its origins”, Historia Mathematica, 8 (3): 319–350, doi:10.1016/0315-0860(81)90070-7 ↩

  8. Kanamori, Akihiro (2004), “Zermelo and set theory”, Bulletin of Symbolic Logic, 10 (4): 487–553, doi:10.2178/bsl/1102083759 ↩

  9. Džamonja, Mirna (2017), “Set Theory and its Place in the Foundations of Mathematics: A New Look at an Old Question”, Journal of Indian Council of Philosophical Research, 34 (2): 415–424, doi:10.1007/s40961-016-0082-6 ↩

  10. Feferman, Solomon; Friedman, Harvey M.; Maddy, Penelope; Steel, John R. (2000), “Does Mathematics Need New Axioms?”, Bulletin of Symbolic Logic, 6 (4): 401–446, doi:10.2307/420965 ↩

  11. Ferreirós, José (1995), ""What Fermented in Me for Years”: Cantor’s discovery of transfinite numbers”, Historia Mathematica, 22 (1): 33–42, doi:10.1006/hmat.1995.1003 ↩

  12. Codd, E. F. (1970), “A Relational Model of Data for Large Shared Data Banks”, Communications of the ACM, 13 (6): 377–387, doi:10.1145/362384.362685 ↩

  13. Codd, E. F. (1972), “Relational Completeness of Data Base Sublanguages”, in Rustin, Randall (ed.), Data Base Systems: Courant Computer Science Symposia Series 6, Prentice-Hall, pp. 65–98 ↩

  14. Montague, Richard (1970), “Universal Grammar”, Theoria, 36 (3): 373–398, doi:10.1111/j.1755-2567.1970.tb00434.x ↩

  15. Tarnita, Corina E.; Antal, Tibor; Ohtsuki, Hisashi; Nowak, Martin A. (2009), “Evolutionary dynamics in set structured populations”, Proceedings of the National Academy of Sciences, 106 (21): 8601–8604, doi:10.1073/pnas.0903019106 ↩

  16. Maddy, Penelope (1988), “Believing the Axioms. (2 parts)”, The Journal of Symbolic Logic, 53 (2, 3): 481–511, 736–764, doi:10.2307/2274520 ↩

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