# Set Theory

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.

### Other articles related to "set theory, set, sets, theory":

Minimal Model (set Theory)
... In set theory, a minimal model is a minimal standard model of ZFC ... If there is a set W in V which is a standard model of ZF, and the ordinal κ is the set of ordinals which occur in W, then Lκ is the class of constructible sets of W ... If there is a set which is a standard model of ZF, then the smallest such set is such a Lκ ...
Axiom Of Reducibility - Criticism of The Axiom of Reducibility - Von Neumann 1925
... In Russell, all of mathematics and set theory seems to rest upon the highly problematic "axiom of reducibility", while Weyl and Brouwer systematically reject the ... to be formulated in such a way that all the desired theorems of Cantor's set theory follow from them, but not the antinomies ... We must, however, avoid forming sets by collecting or separating elements, and so on, as well as eschew the unclear principle of "definiteness" that can still be ...
Scott–Potter Set Theory
... foundations of mathematics that is of relatively recent origin, Scott–Potter set theory is a collection of nested axiomatic set theories set out by the philosopher Michael Potter, building on earlier work by ... Scott (1974), and showed how the resulting axiomatic set theory can do what is expected of such theory, namely grounding the cardinal and ordinal numbers, Peano arithmetic and ...
Objections To Set Theory As A Foundation For Mathematics
... From set theory's inception, some mathematicians have objected to it as a foundation for mathematics ... The most common objection to set theory, one Kronecker voiced in set theory's earliest years, starts from the constructivist view that mathematics is loosely related to computation ... view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in ...
Azriel Levy - Selected Works
... Levy A hierarchy of formulas in set theory, Memoirs of the American Mathematical Society, 57, 1965 ... not imply the axiom of choice, Axiomatic Set Theory, Symposia Pure Math ... Levy Basic Set Theory, Springer-Verlag, Berlin, 1979, 391 pages ...

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

every subjective phenomenon is essentially connected with a single point of view, and it seems inevitable that an objective, physical theory will abandon that point of view.
Thomas Nagel (b. 1938)

O, he’s drunk, Sir Toby, an hour agone; his eyes were set at eight i’ th’ morning.
William Shakespeare (1564–1616)