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 Frankel 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)