Russell's Paradox
<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 Fränkel 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)
Nearby terms:
Russell « Russell, Bertrand « Russell's Attic «
Russell's Paradox » rusty iron » rusty memory »
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