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Encyclopedia > Cantor's paradox

Cantor's paradox, also known as the paradox of the greatest cardinal, demonstrates that there is no cardinal greater than all other cardinals—that the class of cardinal numbers is infinite. It is named for Georg Cantor, who is often credited with first identifying the paradox in 1899 (or between 1895 and 1897). In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. ... Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845 – January 6, 1918) was a mathematician who was born in Russia and lived in Germany for most of his life. ...

Contents


Cantor's Paradox

  1. Assume that there exists a set of all sets (call it set B).
  2. B must therefore be its own power set, i.e. the set of subsets of itself.
  3. By Cantor's theorem, the cardinality (or, informally, size) of any power set is always greater than the cardinality of the set associated with it.
  4. Thus, the cardinality of the power set of B must be greater than the cardinality of B.
  5. But, by #2, B is its own power set. Therefore, B must have greater cardinality than itself. This is a contradiction.
  6. Therefore, we conclude that the initial assumption must be false, and no such set can exist.

In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ... Note: in order to fully understand this article you may want to refer to the set theory portion of the table of mathematical symbols. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... Broadly speaking, a contradiction is an incompatibility between two or more statements, ideas, or actions. ...

Historical Note

While Cantor is usually credited with first identifying this property of cardinal sets, some mathematicians award this distinction to Bertrand Russell, who defined a similar theorem in 1899 or 1901. The Right Honourable Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970), was an influential British logician, philosopher, and mathematician, working mostly in the 20th century. ...


Sources

  • [[|Anellis, I.H.]] () ( 1991). "" Drucker, Thomas [ "The first Russell paradox," Perspectives on the History of Mathematical Logic], 33-46Cambridge, Mass.: Birkäuser Boston. Inc.. ..
  • [[|Moore, G.H. and Garciadiego, A.]] () ({{{Month}}} ). [ Burali-Forti's paradox: a reappraisal of its origins]. Historia Math 8 (): 319-350. .

External links

  • An Historical Account of Set-Theoretic Antinomies Caused by the Axiom of Abstraction Justin T Miller
  • http://planetmath.org/encyclopedia/CantorsParadox.html

  Results from FactBites:
 
Encyclopedia: Paradox (837 words)
Paradoxes which are not based on a hidden error generally happen at the fringes of context or language, and require extending the context (or language) to lose their paradox quality.
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paradox paradox is an apparently true statement or group of statements that seems to lead to a contradiction or to a situation that defies intuition.
Talk:Cantor's diagonal argument - Wikipedia, the free encyclopedia (6085 words)
There are other diagonalization proofs which share essential properties with the Cantor diagonal proof: they include the halting problem argument, standard proofs for Godel's incompleteness theorem and Tarski's theorem on the indefinability of truth, Curry's paradox (and Russell's paradox for that matter).
Cantor's diagonal argument does not also work for fractional rational numbers because the "anti-diagonal real number" is indeed a fractional irrational number --- hence, the presence of the prefix fractional expansion point is not a consequence nor a valid justification for the argument that Cantor's diagonal argument does not work on integers.
Cantor's uncountable reals R on [0,1) are an abstract concept, shown to not be (Peano-) 'countable', via his diagonal argument.
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