Russell's paradox
A paradox showing unrestricted set comprehension leads to contradiction.
Russell's paradox, also called Russell's antinomy, is a contradiction in set theory that Bertrand Russell published in 1901. It shows that if a set theory allows an unrestricted comprehension principle—the idea that for any well-defined property there exists a set of all objects with that property—then contradictions arise.
Consider the set \( R \) of all sets that are not members of themselves. If \( R \) is not a member of itself, then by definition it must be a member of itself. But if it 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 found that a version of this paradox could be derived in Gottlob Frege's axiomatic system, undermining Frege's attempt to reduce mathematics to logic and challenging the logicist programme. Two major solutions appeared in 1908: Russell's own type theory and Zermelo set theory. Zermelo's approach restricted the unrestricted comprehension principle, while Russell modified the logical language itself. Zermelo's system, with contributions from Abraham Fraenkel, became the standard Zermelo–Fraenkel set theory (ZFC when including the axiom of choice). 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, but he did not publish it. It remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen. Zermelo did not see the paradoxes as a crisis, believing they could be avoided if mathematicians stuck to a limited set of axioms. By the end of the 1890s, Georg Cantor had already realized his theory would lead to a contradiction (related to Cantor's theorem), as he told Hilbert and Richard Dedekind by letter. Hilbert also formulated his own paradox, based on reasoning similar to Cantor's diagonal argument, which he said prompted Zermelo's version of Russell's paradox.
Most common sets are not members of themselves. Call a set "normal" if it is not a member of itself, and "abnormal" if it is. For instance, the set of all squares in a plane is not a square, so it is normal. The complementary set of everything that is not a square is itself not a square, so it is abnormal. Now consider the set \( R \) of all
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- Mathematical logic, set theory
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- Russell's paradox
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Lore & Background
Russell's paradox arises from considering the set R of all sets that are not members of themselves. If R is not a member of itself, then its definition entails that it is a member of itself; yet if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. This contradiction is expressed symbolically as R ∈ R ⟺ R ∉ R.
Russell also showed that a version of the paradox could be derived in the axiomatic system constructed by Gottlob Frege, undermining Frege's attempt to reduce mathematics to logic. The paradox had already been discovered independently by Ernst Zermelo by 1902, and possibly as early as 1899, though Zermelo did not publish it. Georg Cantor had also realized that his theory would lead to a contradiction, as he told David Hilbert and Richard Dedekind by letter.
Two influential ways of avoiding the paradox were both proposed in 1908: Russell's own type theory and Zermelo set theory. Zermelo's axioms restricted the unlimited comprehension principle, while Russell modified the logical language itself. Zermelo set theory, with additional contributions by Abraham Fraenkel, developed into Zermelo–Fraenkel set theory (ZFC when including the axiom of choice).
Reader's Guide
Russell's paradox is significant because it exposed a fatal flaw in naive set theory, which assumed that for any well-defined property there exists a set of all objects having that property. The paradox forced mathematicians to reexamine the foundations of set theory and led to the development of axiomatic systems designed to avoid such contradictions. Zermelo's solution, which became the basis of ZFC, restricted set comprehension by allowing only subsets of existing sets to be formed, preventing the construction of the Russell set. Russell's own solution, type theory, instead modified the logical language by introducing a hierarchy of types. The paradox also contributed to the decline of the logicist programme, as it showed that Frege's system was inconsistent. Today, ZFC remains the canonical axiomatic set theory, and the paradox is a standard example of the need for careful axiomatic foundations in mathematics.
Did You Know?
- Russell's paradox was published by Bertrand Russell in 1901.
- The paradox had already been discovered independently by Ernst Zermelo by 1902, and possibly as early as 1899.
- Two influential ways of avoiding the paradox were both proposed in 1908: Russell's own type theory and Zermelo set theory.
- Georg Cantor had realized that his theory would lead to a contradiction, as he told Hilbert and Dedekind by letter.
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