Russell's paradox

Russell's paradox, sometimes called Russell's antinomy, is a contradiction in set theory that was published by the philosopher and mathematician Bertrand Russell in 1901. The paradox demonstrates that any set theory built on an unrestricted comprehension principle is inconsistent. This principle states that for any well-defined property, there exists a set containing exactly those objects that have that property.

To see the problem, consider the set R, defined as the set of all sets that are not members of themselves. If R is not a member of itself, then by its definition it must be a member of itself. Conversely, if R is a member of itself, then it cannot be a member of itself, because it is defined as the set of sets that are not members of themselves. This contradiction is the paradox.

Russell also showed that a version of this paradox could be derived within the axiomatic system of Gottlob Frege, thereby undermining Frege's project of reducing mathematics to logic and challenging the logicist program. Two major solutions emerged in 1908: Russell's own type theory and Zermelo set theory. Zermelo's approach restricted the unrestricted comprehension principle through new axioms. With later contributions from Abraham Fraenkel, Zermelo set theory evolved into standard Zermelo–Fraenkel set theory (often called ZFC when including the axiom of choice). The key difference between the two solutions is that Zermelo modified the axioms of set theory while keeping a standard logical language, whereas Russell changed the logical language itself. The language of ZFC, with help from Thoralf Skolem, turned out to be first-order logic.

The paradox had already been discovered independently by Ernst Zermelo by 1902, and possibly as early as 1899. Zermelo did not publish it, and it remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen. Zermelo did not view the emergence of such paradoxes as a crisis; he believed they could be avoided if mathematicians confined themselves to a limited number of established axioms. By the end of the 1890s, Georg Cantor—the founder of modern set theory—had already realized that his theory would lead to a contradiction (related to Cantor's theorem), as he told Hilbert and Richard Dedekind in letters. Hilbert also formulated his own paradox, based on reasoning similar to Cantor's diagonal argu