**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 Zermeloâ€“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 Zermeloâ€“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.

Read more about Set Theory: History, Basic Concepts, Some Ontology, Axiomatic Set Theory, Applications, Objections To Set Theory As A Foundation For Mathematics

### Famous quotes containing the words set and/or theory:

“Hence anyone who seeks for the true cause of miracles, and strives to understand natural phenomena as an intelligent being, and not to gaze at them as a fool, is *set* down and denounced as a impious heretic by those, whom the masses adore as the interpreters of nature and the gods.”

—Baruch (Benedict)

“Everything to which we concede existence is a posit from the standpoint of a description of the *theory*-building process, and simultaneously real from the standpoint of the *theory* that is being built. Nor let us look down on the standpoint of the *theory* as make-believe; for we can never do better than occupy the standpoint of some *theory* or other, the best we can muster at the time.”

—Willard Van Orman Quine (b. 1908)