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1. The Free On-line Dictionary of Computing (30 December 2018)
Russell's Paradox

    A paradox (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.


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