(mathematics) | **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 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. | |