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  2. Cardinality - Wikipedia

    en.wikipedia.org/wiki/Cardinality

    The cardinality of the natural numbers is denoted aleph-null (), while the cardinality of the real numbers is denoted by "" (a lowercase fraktur script "c"), and is also referred to as the cardinality of the continuum.

  3. 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 ...

  4. 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.

  5. 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 ...

  6. Equinumerosity - Wikipedia

    en.wikipedia.org/wiki/Equinumerosity

    In his controversial 1878 paper, Cantor explicitly defined the notion of "power" of sets and used it to prove that the set of all natural numbers ⁠ ⁠ and the set of all rational numbers ⁠ ⁠ are equinumerous (an example where a proper subset of an infinite set is equinumerous to the original set), and that the Cartesian product of even a ...

  7. 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 .

  8. Continuum (set theory) - Wikipedia

    en.wikipedia.org/wiki/Continuum_(set_theory)

    The cardinality of the continuum is the size of the set of real numbers. The continuum hypothesis is sometimes stated by saying that no cardinality lies between that of the continuum and that of the natural numbers , ℵ 0 {\displaystyle \aleph _{0}} , or alternatively, that c = ℵ 1 {\displaystyle {\mathfrak {c}}=\aleph _{1}} .

  9. Glossary of number theory - Wikipedia

    en.wikipedia.org/wiki/Glossary_of_number_theory

    class number 1. The class number of a number field is the cardinality of the ideal class group of the field. 2. In group theory, the class number is the number of conjugacy classes of a group. 3. Class number is the number of equivalence classes of binary quadratic forms of a given discriminant. 4. The class number problem. conductor

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