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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.
(2000-11-01)
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Common Misspellings > |
Russell's Paradox uRussell's Paradox uRssell's Paradox RuRssell's Paradox ussell's Paradox RRussell's Paradox Rsusell's Paradox Rususell's Paradox Rssell's Paradox Ruussell's Paradox Russsell's Paradox Rusell's Paradox Russsell's Paradox Rusesll's Paradox Russesll's Paradox Rusell's Paradox Russsell's Paradox Russlel's Paradox Russelel's Paradox Russll's Paradox Russeell's Paradox Russelll's Paradox Russel's Paradox Russelll's Paradox Russel'ls Paradox Russell'ls Paradox Russel's Paradox Russelll's Paradox Russells' Paradox Russell's' Paradox Russells Paradox Russell''s Paradox Russell' sParadox Russell's sParadox Russell' Paradox Russell'ss Paradox Russell'sP aradox Russell's P aradox Russell'sParadox Russell's Paradox Russell's aPradox Russell's PaPradox Russell's aradox Russell's PParadox Russell's Praadox Russell's Paraadox Russell's Pradox Russell's Paaradox Russell's Paardox Russell's Parardox Russell's Paadox Russell's Parradox Russell's Pardaox Russell's Paradaox Russell's 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Russell'z Paradox Russell'sz Paradox Russell'zs Paradox Russell'x Paradox Russell'sx Paradox Russell'xs Paradox Russell'c Paradox Russell'sc Paradox Russell'cs Paradox Russell' Paradox Russell's Paradox Russell' s Paradox Russell's 0aradox Russell's p0aradox Russell's 0paradox Russell's )aradox Russell's p)aradox Russell's )paradox Russell's oaradox Russell's poaradox Russell's oparadox Russell's laradox Russell's plaradox Russell's lparadox Russell's Pqradox Russell's Paqradox Russell's Pqaradox Russell's Pwradox Russell's Pawradox Russell's Pwaradox Russell's Psradox Russell's Pasradox Russell's Psaradox Russell's Pxradox Russell's Paxradox Russell's Pxaradox Russell's Pzradox Russell's Pazradox Russell's Pzaradox Russell's P radox Russell's Pa radox Russell's P aradox Russell's Peradox Russell's Piradox Russell's Poradox Russell's Puradox Russell's Pa4adox Russell's Par4adox Russell's Pa4radox Russell's Pa$adox Russell's Par$adox Russell's Pa$radox Russell's Pa5adox Russell's 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