Russell's paradox

Part of the foundation 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.The assumption that sets can be freely defined by any criterion is disproved by a set containing exactly the sets that are not members of themselves. If such a set qualifies as a member of itself, it would contradict its own definition as a set containing 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.
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<mathematics> 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 Frnkel 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.
(2000-11-01)