0%

russell's paradox

Russell's paradox
R r

Transcription

    • US Pronunciation
    • US IPA
    • US Pronunciation
    • US IPA

Definitions of russell's paradox words

  • noun russell's paradox a paradox of set theory in which an object is defined in terms of a class of objects that contains the object being defined, resulting in a logical contradiction. 1
  • noun Definition of russell's paradox in Technology (mathematics)   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. 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. 1
  • noun russell's paradox 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 0
See all 3 definitions of russell's paradox

Information block about the term

Origin of russell's paradox

First appearance:

before 1920
One of the 12% newest English words
First recorded in 1920-25; first proposed by Bertrand Russell

Historical Comparancy

Parts of speech for Russell's paradox

noun
adjective
verb
adverb
pronoun
preposition
conjunction
determiner
exclamation
Full grammatical hierarchy of Russell's paradox

russell's paradox popularity

This term is known only to a narrow circle of people with rare knowledge. Only 2% of English native speakers know the meaning of this word.
According to our data most of word are more popular. This word is almost not used. It has a much more popular synonym.

See also

Matching words

Was this page helpful?
Yes No
Thank you for your feedback! Tell your friends about this page
Tell us why?