Russell's paradox

Russell's paradox
a paradox of set theory in which an object is defined in terms of a class of objects that contains the object being defined, resulting in a logical contradiction.
[1920-25; first proposed by Bertrand RUSSELL]

* * *

logic
      statement in set theory, devised by the English mathematician-philosopher Bertrand Russell (Russell, Bertrand), that demonstrated a flaw in earlier efforts to axiomatize the subject.

      Russell found the paradox in 1901 and communicated it in a letter to the German mathematician-logician Gottlob Frege (Frege, Gottlob) in 1902. Russell's letter demonstrated an inconsistency in Frege's axiomatic system of set theory by deriving a paradox within it. (The German mathematician Ernst Zermelo had found the same paradox independently; since it could not be produced in his own axiomatic system of set theory, he did not publish the paradox.)

      Frege had constructed a logical system employing an unrestricted comprehension principle. The comprehension principle is the statement that, given any condition expressible by a formula ϕ(x), it is possible to form the set of all sets x meeting that condition, denoted {x | ϕ(x)}. For example, the set of all sets—the universal set—would be {x | x = x}.

      It was noticed in the early days of set theory, however, that a completely unrestricted comprehension principle led to serious difficulties. In particular, Russell observed that it allowed the formation of {x | x ∉ x}, the set of all non-self-membered sets, by taking ϕ(x) to be the formula x ∉ x. Is this set—call it R—a member of itself? If it is a member of itself, then it must meet the condition of its not being a member of itself. But if it is not a member of itself, then it precisely meets the condition of being a member of itself. This impossible situation is called Russell's paradox.

      The significance of Russell's paradox is that it demonstrates in a simple and convincing way that one cannot both hold that there is meaningful totality of all sets and also allow an unfettered comprehension principle to construct sets that must then belong to that totality. (Russell spoke of this situation as a “vicious circle.”)

 Set theory avoids this paradox by imposing restrictions on the comprehension principle. The standard Zermelo-Fraenkel axiomatization (ZF; see the table—>) does not allow comprehension to form a set larger than previously constructed sets. (The role of constructing larger sets is given to the power-set operation.) This leads to a situation where there is no universal set—an acceptable set must not be as large as the universe of all sets.

      A very different way of avoiding Russell's paradox was proposed in 1937 by the American logician Willard Van Orman Quine (Quine, Willard Van Orman). In his paper “New Foundations for Mathematical Logic,” the comprehension principle allows formation of {x | ϕ(x)} only for formulas ϕ(x) that can be written in a certain form that excludes the “vicious circle” leading to the paradox. In this approach, there is a universal set.

Herbert Enderton
 

* * *


Universalium. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Russell's paradox — Part of 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 of Frege leads to a contradiction.It might be assumed that, for any formal… …   Wikipedia

  • russell's paradox — ˈrəsəlz noun Usage: usually capitalized R Etymology: after Bertrand Russell : a paradox that discloses itself in forming a class of all classes that are not members of themselves and in observing that the question of whether it is true or false… …   Useful english dictionary

  • Russell's paradox — The most famous of the paradoxes in the foundations of set theory, discovered by Russell in 1901. Some classes have themselves as members: the class of all abstract objects, for example, is an abstract object. Others do not: the class of donkeys… …   Philosophy dictionary

  • Russell's paradox — noun The following paradox: Let A be the set of all sets which do not contain themselves. Then does A contain itself? If it does, then by definition it does not; and if it does not, then by definition it does. See Also: Burali Forti paradox …   Wiktionary

  • Russell — is an English, Irish, or Scottish name derived from old French, the old French word for Red was rouse ; hence the carry over from French the English Russell, the name also derives from the animal, the fox. Its uses include:People*Arthur Russell… …   Wikipedia

  • Russell , Bertrand Arthur William — Russell , Bertrand Arthur William, third earl Russell (1872–1970) British philosopher and mathematician Russell, who was born at Trelleck, England, was orphaned at an early age and brought up in the home of his grandfather, the politician Lord… …   Scientists

  • Russell, Bertrand — ▪ British logician and philosopher in full  Bertrand Arthur William Russell, 3rd Earl Russell of Kingston Russell, Viscount Amberley of Amberley and of Ardsalla  born May 18, 1872, Trelleck, Monmouthshire, Wales died Feb. 2, 1970,… …   Universalium

  • Paradox — For other uses, see Paradox (disambiguation). Further information: List of paradoxes A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition. Typically,… …   Wikipedia

  • Russell's teapot — part of a series on Bertrand Russell …   Wikipedia

  • paradox — A paradox arises when a set of apparently incontrovertible premises gives unacceptable or contradictory conclusions. To solve a paradox will involve either showing that there is a hidden flaw in the premises, or that the reasoning is erroneous,… …   Philosophy dictionary

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”