<P> Cantor's work initially polarized the mathematicians of his day . While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a 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 . </P> <P> 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 . Bertrand Russell and Ernst Zermelo 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 The Principles of Mathematics . </P> <P> In 1906 English readers gained the book Theory of Sets of Points by husband and wife William Henry Young and Grace Chisholm Young, published by Cambridge University Press . </P> <P> The momentum of set theory was such that debate on the 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 ZFC, which became the most commonly used set of axioms for set theory . The work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory, which has since become woven into 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 . </P>

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