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.

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.