Russell's Paradox

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Russell's paradox

[′rəs·əlz ′par·ə‚däks]
The paradox concerning the concept of all sets which are not members of themselves which forces distinctions in set theory between sets and classes.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

Russell's Paradox

A logical contradiction in set theory discovered by Bertrand Russell. If R is the set of all sets which don't contain themselves, does R contain itself? If it does then it doesn't and vice versa.

The paradox stems from the acceptance of the following axiom: If P(x) is a property then

x : P

is a set. This is the Axiom of Comprehension (actually an axiom schema). By applying it in the case where P is the property "x is not an element of x", we generate the paradox, i.e. something clearly false. Thus any theory built on this axiom must be inconsistent.

In lambda-calculus Russell's Paradox can be formulated by representing each set by its characteristic function - the property which is true for members and false for non-members. The set R becomes a function r which is the negation of its argument applied to itself:

r = \ x . not (x x)

If we now apply r to itself,

r r = (\ x . not (x x)) (\ x . not (x x)) = not ((\ x . not (x x))(\ x . not (x x))) = not (r r)

So if (r r) is true then it is false and vice versa.

An alternative formulation is: "if the barber of Seville is a man who shaves all men in Seville who don't shave themselves, and only those men, who shaves the barber?" This can be taken simply as a proof that no such barber can exist whereas seemingly obvious axioms of set theory suggest the existence of the paradoxical set R.

Zermelo Fr?nkel set theory is one "solution" to this paradox. Another, type theory, restricts sets to contain only elements of a single type, (e.g. integers or sets of integers) and no type is allowed to refer to itself so no set can contain itself.

A message from Russell induced Frege to put a note in his life's work, just before it went to press, to the effect that he now knew it was inconsistent but he hoped it would be useful anyway.
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There is, however, reason to think that the "Borges" in "Borges and I" is a set, as my bizarre bastardization of Russell's paradox suggests.
One mayor issue around Frege's Theorem is why Frege did not choose (HP) as an axiom, even after knowing of Russell's Paradox. Frege derives (HP) using his definition of number.
This justification turns on Badiou's interpretation of Russell's paradox and the related paradoxes, which led Russell and subsequent logicians to seek devices to prevent the possibility of forming the problematic sets.
But to our analysis it is important that young Wittgenstein declares that he has solved Russell's paradox--'Herewith Russell's paradox vanishes' (1922, prop.
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Russell's paradox was presented in the context of set theory, which is basically the study of collections ranging from collections of concrete and familiar objects (such as the set of students in a class) to abstract objects like numbers.
Russell's Paradox, outlined in a letter to fellow mathematician Gottlob Frege, has an analogy in the statement by Epimenides, a Cretan, that "All Cretans are liars." Russell's mathematical statement of this paradox implied that there could be no truth in mathematics, since mathematical logic was flawed at a basic level.
He considered Russell's paradox (in its set version) and two variants: the barber who shaves those and only those men who do not shave themselves, and the library catalogue of those and only those catalogues which do not catalogue themselves.
Russell's Paradox), we substitute another less-problematic axiom, or introduce a set theory based on types, or whatever else is necessary to fix the problem.
Russell's paradox is a well-known example: is the set of all and only those sets that are not members of themselves a member of itself or not?
Arvai Wieschenberg and Peter Shenkin; "A Conversation about Russell's Paradox," by Paul E.
petitio principii); self-killing and self-generation of life; self-membership of classes (Russell's paradox); self-knowledge and self-reference (the Liar).

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