Russell's Paradox

Russell's paradox

[′rəs·əlz ′par·ə‚däks] (mathematics) The paradox concerning the concept of all sets which are not members of themselves which forces distinctions in set theory between sets and classes.

Russell's Paradox

(mathematics)A logical contradiction in set theorydiscovered by Bertrand Russell. If R is the set of all setswhich don't contain themselves, does R contain itself? If itdoes then it doesn't and vice versa.

The paradox stems from the acceptance of the followingaxiom: If P(x) is a property then

x : P

is a set. This is the Axiom of Comprehension (actually anaxiom schema). By applying it in the case where P is theproperty "x is not an element of x", we generate the paradox,i.e. something clearly false. Thus any theory built on thisaxiom must be inconsistent.

In lambda-calculus Russell's Paradox can be formulated byrepresenting each set by its characteristic function - theproperty which is true for members and false for non-members.The set R becomes a function r which is the negation of itsargument 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 aman who shaves all men in Seville who don't shave themselves,and only those men, who shaves the barber?" This can be takensimply as a proof that no such barber can exist whereasseemingly obvious axioms of set theory suggest the existenceof the paradoxical set R.

Zermelo Fr?nkel set theory is one "solution" to thisparadox. Another, type theory, restricts sets to containonly elements of a single type, (e.g. integers or sets ofintegers) and no type is allowed to refer to itself so no setcan contain itself.

A message from Russell induced Frege to put a note in hislife's work, just before it went to press, to the effect thathe now knew it was inconsistent but he hoped it would beuseful anyway.

单词	russell's paradox
释义	Russell's paradox Russell's paradox n (Logic) logic the paradox discovered by Bertrand Russell in the work of Gottlob Frege, that the class of all classes that are not members of themselves is a member of itself only if it is not, and is not only if it is. This undermines the notion of an all-inclusive universal class Russell's Paradox Russell's paradox [′rəs·əlz ′par·ə‚däks] (mathematics) The paradox concerning the concept of all sets which are not members of themselves which forces distinctions in set theory between sets and classes. Russell's Paradox (mathematics)A logical contradiction in set theorydiscovered by Bertrand Russell. If R is the set of all setswhich don't contain themselves, does R contain itself? If itdoes then it doesn't and vice versa. The paradox stems from the acceptance of the followingaxiom: If P(x) is a property then x : P is a set. This is the Axiom of Comprehension (actually anaxiom schema). By applying it in the case where P is theproperty "x is not an element of x", we generate the paradox,i.e. something clearly false. Thus any theory built on thisaxiom must be inconsistent. In lambda-calculus Russell's Paradox can be formulated byrepresenting each set by its characteristic function - theproperty which is true for members and false for non-members.The set R becomes a function r which is the negation of itsargument 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 aman who shaves all men in Seville who don't shave themselves,and only those men, who shaves the barber?" This can be takensimply as a proof that no such barber can exist whereasseemingly obvious axioms of set theory suggest the existenceof the paradoxical set R. Zermelo Fr?nkel set theory is one "solution" to thisparadox. Another, type theory, restricts sets to containonly elements of a single type, (e.g. integers or sets ofintegers) and no type is allowed to refer to itself so no setcan contain itself. A message from Russell induced Frege to put a note in hislife's work, just before it went to press, to the effect thathe now knew it was inconsistent but he hoped it would beuseful anyway.
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