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 antinomies or paradoxes. This utility of set theory led to the article 'Mengenlehre', contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia. 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. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking, and culminated in Cantor's 1874 paper.Д¬antor's work initially polarized the mathematicians of his day. Modern understanding of infinity began in 1870вЂ“1874, and was motivated by Cantor's work in real analysis.

Especially notable is the work of Bernard Bolzano in the first half of the 19th century. 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. Set theory, however, was founded by a single paper in 1874 by Georg Cantor: ' On a Property of the Collection of All Real Algebraic Numbers'. Mathematical topics typically emerge and evolve through interactions among many researchers. 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. 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. Besides 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. Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of ZermeloвЂ“Fraenkel set theory with the axiom of choice. 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 twentieth century, of which ZermeloвЂ“Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied.

The non-formalized systems investigated during this early stage go under the name of naive set theory. In particular, Georg Cantor is commonly considered the founder of set theory. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s.

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. Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.

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