russell's paradox

1 definition found

russell's paradox - Free On-line Dictionary of Computing (26 May 2007) :

  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)