In the foundations of mathematics, Russell’s paradox (also known as Russell’s antinomy), discovered by Bertrand Russell in 1901, showed that some attempted formalizations of the naïve set theory created by Georg Cantor led to a contradiction. The same paradox had been discovered in 1899 by Ernst Zermelo but he did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other members of the University of Göttingen. At the end of the 1890s Cantor himself had already realized that his definition would lead to a contradiction, which he told Hilbert and Richard Dedekind by letter.
According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves.