In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction. The same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl and other members of the University of Göttingen.
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 qualifies as a member of itself, it would contradict its own definition as a set containing all sets that are not members of themselves. On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition. This contradiction is Russell's paradox. Symbolically:
In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and the Zermelo set theory, the first constructed axiomatic set theory. Zermelo's axioms went well beyond Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical Zermelo–Fraenkel set theory (ZF).
Read more about Russell's Paradox: Informal Presentation, Formal Presentation, Set-theoretic Responses, History, Applied Versions, Related Paradoxes
Famous quotes containing the words russell and/or paradox:
“A stranger came one night to Yussouf’s tent,
Saying, “Behold one outcast and in dread,
Against whose life the bow of power is bent,
Who flies, and hath not where to lay his head;
I come to thee for shelter and for food,
To Yussouf, called through all our tribes ‘he Good.’ “
“This tent is mine,” said Yussouf, “but no more
Than it is God’s; come in, and be at peace;”
—James Russell Lowell (1819–1891)
“... it is the desert’s grimness, its stillness and isolation, that bring us back to love. Here we discover the paradox of the contemplative life, that the desert of solitude can be the school where we learn to love others.”
—Kathleen Norris (b. 1947)