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Related to Russell's Paradox: Russell's viper

[′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.

(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 Fr?nkel 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.
References in periodicals archive ?
One mayor issue around Frege's Theorem is why Frege did not choose (HP) as an axiom, even after knowing of Russell's Paradox.
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But to our analysis it is important that young Wittgenstein declares that he has solved Russell's paradox--'Herewith Russell's paradox vanishes' (1922, prop.
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For example, while Russell's paradox was inspired by one of Cantor's, his paradoxical set is not "universally" large in the style of Cantor's greatest number in [sections] 2.
Russell's paradox is a well-known example: is the set of all and only those sets that are not members of themselves a member of itself or not?
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But McGuinness himself is prone to speak in terms of race; he also has a notable enthusiasm for military history, explaining the Austrian eastern campaigns with more care than the logic of Russell's Paradox.
Referencing various Borges texts (most notably "The Book of Sand," "The Library of Babel," and "Pascal's Sphere"), Martinez nevertheless addresses three recurrent themes, all three of which come together in the short story "The Aleph": different models of infinity, the sphere whose center is everywhere and circumference nowhere, and Russell's paradox.
He illustrates the former with Chomsky's 'colorless green ideas sleep furiously' and the latter by arguing that formal systems would not recognize the nonsense of the 'truthful liar' conundrum or Russell's paradox.
Annex 2: The Proof of Russell's Paradox in First Order Logic

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