enow.com Web Search

Search results

  1. Results from the WOW.Com Content Network
  2. Cantor's theorem - Wikipedia

    en.wikipedia.org/wiki/Cantor's_theorem

    Cantor's theorem and its proof are closely related to two paradoxes of set theory. Cantor's paradox is the name given to a contradiction following from Cantor's theorem together with the assumption that there is a set containing all sets, the universal set. In order to distinguish this paradox from the next one discussed below, it is important ...

  3. Cantor's diagonal argument - Wikipedia

    en.wikipedia.org/wiki/Cantor's_diagonal_argument

    An illustration of Cantor's diagonal argument (in base 2) for the existence of uncountable sets. The sequence at the bottom cannot occur anywhere in the enumeration of sequences above. An infinite set may have the same cardinality as a proper subset of itself, as the depicted bijection f(x)=2x from the natural to the even numbers demonstrates ...

  4. Controversy over Cantor's theory - Wikipedia

    en.wikipedia.org/wiki/Controversy_over_Cantor's...

    So Cantor's argument implies that the set of all subsets of N has greater cardinality than N. The set of all subsets of N is denoted by P ( N ), the power set of N . Cantor generalized his argument to an arbitrary set A and the set consisting of all functions from A to {0, 1}. [ 4 ]

  5. Cantor's paradox - Wikipedia

    en.wikipedia.org/wiki/Cantor's_paradox

    This fact is a direct consequence of Cantor's theorem on the cardinality of the power set of a set. Proof: Assume the contrary, and let C be the largest cardinal number. Then (in the von Neumann formulation of cardinality) C is a set and therefore has a power set 2 C which, by Cantor's theorem, has cardinality strictly larger than C .

  6. Cardinality of the continuum - Wikipedia

    en.wikipedia.org/wiki/Cardinality_of_the_continuum

    Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if, and only if, there exists a bijective function between them. Between any two real numbers a < b , no matter how close they are to each other, there are always infinitely many other real numbers, and Cantor showed that they are as many as those ...

  7. Paradoxes of set theory - Wikipedia

    en.wikipedia.org/wiki/Paradoxes_of_set_theory

    However, Cantor's theorem proves that there are uncountable sets. The root of this seeming paradox is that the countability or noncountability of a set is not always absolute, but can depend on the model in which the cardinality is measured. It is possible for a set to be uncountable in one model of set theory but countable in a larger model ...

  8. Uncountable set - Wikipedia

    en.wikipedia.org/wiki/Uncountable_set

    The best known example of an uncountable set is the set ⁠ ⁠ of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers ⁠ ⁠ (see: (sequence A102288 in the OEIS)), and the set of all subsets of the set ...

  9. Cantor's first set theory article - Wikipedia

    en.wikipedia.org/wiki/Cantor's_first_set_theory...

    [proof 1] This case handles case 1 and case 3 of Cantor's 1874 proof. In the second case, which handles case 2 of Cantor's 1874 proof, P is dense in [a, b]. The denseness of sequence P is used to recursively define a sequence of nested intervals that excludes all the numbers in P and whose intersection contains a single real number in [a, b].