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This article is about the branch of mathematics. For musical set theory, see Set theory (music).
A Venn diagram illustrating the intersection of two sets.
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.
The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.
Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
Contents [hide] * 1 History * 2 Basic concepts * 3 Some ontology * 4 Axiomatic set theory * 5 Applications * 6 Areas of study * 6.1 Combinatorial set theory * 6.2 Descriptive set theory * 6.3 Fuzzy set theory * 6.4 Inner model theory * 6.5 Large cardinals * 6.6 Determinacy * 6.7 Forcing * 6.8 Cardinal invariants * 6.9 Set-theoretic topology * 7 Objections to set theory as a foundation for mathematics * 8 See also * 9 Notes * 10 Further reading * 11 External links |
Mathematical topics typically emerge and evolve through interactions among many researchers. Set...