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In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician, Bertrand Russell, in 1901. [ 1 ] [ 2 ] Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. [ 3 ]
This paradox is often incorrectly attributed to Bertrand Russell (e.g., by Martin Gardner in Aha!). It was suggested to Russell as an alternative form of Russell's paradox, [1] which Russell had devised to show that set theory as it was used by Georg Cantor and Gottlob Frege contained contradictions. However, Russell denied that the Barber's ...
Russell recognized that the statement x = x is true for every set, and thus the set of all sets is defined by {x | x = x}. In 1906 he constructed several paradox sets, the most famous of which is the set of all sets which do not contain themselves. Russell himself explained this abstract idea by means of some very concrete pictures.
Download as PDF; Printable version; ... Girard's paradox is the type-theoretic analogue of Russell's paradox in set theory. References
Type theory was created to avoid a paradox in a mathematical equation [b] based on naive set theory and formal logic. Russell's paradox (first described in Gottlob Frege's The Foundations of Arithmetic) is that, without proper axioms, it is possible to define the set of all sets that are not members of themselves; this set both contains itself and does not contain itself.
This paradox works in mainly the same way as the liar paradox. Grelling–Nelson paradox: Is the word "heterological", meaning "not applicable to itself", a heterological word? (A close relative of Russell's paradox.)
The Principles of Mathematics (PoM) is a 1903 book by Bertrand Russell, in which the author presented his famous paradox and argued his thesis that mathematics and logic are identical. [ 1 ] The book presents a view of the foundations of mathematics and Meinongianism and has become a classic reference.
The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are Russell's paradox and the Burali-Forti paradox. Axiomatic set theory was originally devised to rid set theory of such paradoxes.