enow.com Web Search

Search results

  1. Results from the WOW.Com Content Network
  2. Cardinality - Wikipedia

    en.wikipedia.org/wiki/Cardinality

    The continuum hypothesis says that =, i.e. is the smallest cardinal number bigger than , i.e. there is no set whose cardinality is strictly between that of the integers and that of the real numbers. The continuum hypothesis is independent of ZFC , a standard axiomatization of set theory; that is, it is impossible to prove the continuum ...

  3. Cardinal number - Wikipedia

    en.wikipedia.org/wiki/Cardinal_number

    A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4. Aleph-null, the smallest infinite cardinal. In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set.

  4. Aleph number - Wikipedia

    en.wikipedia.org/wiki/Aleph_number

    Notably, ℵ ω is the first uncountable cardinal number that can be demonstrated within Zermelo–Fraenkel set theory not to be equal to the cardinality of the set of all real numbers 2 ℵ 0: For any natural number n ≥ 1, we can consistently assume that 2 ℵ 0 = ℵ n, and moreover it is possible to assume that 2 ℵ 0 is as least as large ...

  5. Cardinality of the continuum - Wikipedia

    en.wikipedia.org/wiki/Cardinality_of_the_continuum

    The set of real algebraic numbers is countably infinite (assign to each formula its Gödel number.) So the cardinality of the real algebraic numbers is . Furthermore, the real algebraic numbers and the real transcendental numbers are disjoint sets whose union is .

  6. Continuum hypothesis - Wikipedia

    en.wikipedia.org/wiki/Continuum_hypothesis

    Hence, the set {banana, apple, pear} has the same cardinality as {yellow, red, green}. With infinite sets such as the set of integers or rational numbers, the existence of a bijection between two sets becomes more difficult to demonstrate. The rational numbers seemingly form a counterexample to the continuum hypothesis: the integers form a ...

  7. Transfinite number - Wikipedia

    en.wikipedia.org/wiki/Transfinite_number

    Any finite natural number can be used in at least two ways: as an ordinal and as a cardinal. Cardinal numbers specify the size of sets (e.g., a bag of five marbles), whereas ordinal numbers specify the order of a member within an ordered set [9] (e.g., "the third man from the left" or "the twenty-seventh day of January").

  8. Cantor's theorem - Wikipedia

    en.wikipedia.org/wiki/Cantor's_theorem

    As a consequence, the cardinality of the real numbers, which is the same as that of the power set of the integers, is strictly larger than the cardinality of the integers; see Cardinality of the continuum for details.

  9. Beth number - Wikipedia

    en.wikipedia.org/wiki/Beth_number

    Because of Cantor's theorem, each set in the preceding sequence has cardinality strictly greater than the one preceding it. For infinite limit ordinals, the corresponding beth number is defined to be the supremum of the beth numbers for all ordinals strictly smaller than :