Set Theory

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 in mathematics as a whole.

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. The non-formalized systems investigated during this early stage go under the name of naive set theory. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox, and Burali-Forti paradox) various axiomatic systems were proposed in the early 20th century, of which Zermelo-Fraenkel set theory (with or without axiom of choice) is still the best-known and most studied.

Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo-Fraenkel set theory with the axiom of choice. Beside its foundational role, set theory also provides the framework to develop a mathematical theory of infinity, and has various applications in computer science (such as in the theory of relational algebra), philosophy, and formal semantics. Its foundational appeal, together with its paradoxes, 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. 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.

Mathematical topics typically emerge and evolve through interactions among many researchers. Set theory, however, was founded by a single paper in 1874 by Georg Cantor entitled "On a Property of the Collection of All Real Algebraic Numbers."

Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1870-1874, and was motivated by Cantor's work in real analysis. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking, and culminated in Cantor's 1874 paper.

Cantor's initial work polarized the mathematics of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as the founder of mathematical constructivism, did not. Cantorian set theory eventually became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, and the "infinity of infinities" ("Cantor's paradise") resulting from the power set operation. This utility of set theory led to the article "Mengenlehre," contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia.

The next wave of excitement in set theory came around 1900, when it was discovered that some interpretations of Cantorian set theory gave rise to several contradictions, called antimonies or paradoxes. Bertrand Russell and Ernst Zemelo independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves," which leads to a contradiction since it must be a member of itself and not a member of itself. In 1899, Cantor had himself posed the question: "What is the cardinal number of the set of all sets?" and obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his Principles of Mathematics. Rather than the term set, Russell used the term Class, which has subsequently been used more technically.

In 1906, the term set appeared in the book, "The Theory of Sets of Points," by husband and wife William Henry Young and Grace Chisholm Young, published by Cambridge University Press.

The momentum of set theory was such that debate on paradoxes did not lead to its abandonment. The work of Zermelo in 1908 and the work of Abraham Fraenkel and Thoralf Skolem in 1922 resulted in the set of axioms for set theory. The work of analysts, such as that of Henri Lebesgue, demonstrated the great mathematical utility of set theory, which has since become woven in the fabric of modern mathematics. Set theory is commonly used as a foundational system, although in some areas--such as algebraic geometry and algebraic topology--category theory is thought to be a preferred foundation.

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