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  2. Cardinality - Wikipedia

    en.wikipedia.org/wiki/Cardinality

    There are two ways to define the "cardinality of a set": The cardinality of a set A is defined as its equivalence class under equinumerosity. A representative set is designated for each equivalence class. The most common choice is the initial ordinal in that class. This is usually taken as the definition of cardinal number in axiomatic set theory.

  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. Cardinal assignment - Wikipedia

    en.wikipedia.org/wiki/Cardinal_assignment

    The oldest definition of the cardinality of a set X (implicit in Cantor and explicit in Frege and Principia Mathematica) is as the set of all sets that are equinumerous with X: this does not work in ZFC or other related systems of axiomatic set theory because this collection is too large to be a set, but it does work in type theory and in New ...

  5. Von Neumann cardinal assignment - Wikipedia

    en.wikipedia.org/wiki/Von_Neumann_cardinal...

    The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers.For a well-orderable set U, we define its cardinal number to be the smallest ordinal number equinumerous to U, using the von Neumann definition of an ordinal number.

  6. Cardinal function - Wikipedia

    en.wikipedia.org/wiki/Cardinal_function

    Cardinal functions are widely used in topology as a tool for describing various topological properties. [2] [3] Below are some examples.(Note: some authors, arguing that "there are no finite cardinal numbers in general topology", [4] prefer to define the cardinal functions listed below so that they never taken on finite cardinal numbers as values; this requires modifying some of the ...

  7. Equinumerosity - Wikipedia

    en.wikipedia.org/wiki/Equinumerosity

    An attempt to define the cardinality of a set as the equivalence class of all sets equinumerous to it is problematic in Zermelo–Fraenkel set theory, the standard form of axiomatic set theory, because the equivalence class of any non-empty set would be too large to be a set: it would be a proper class.

  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. Controversy over Cantor's theory - Wikipedia

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

    This argument can be improved by using a definition he gave later. The resulting argument uses only five axioms of set theory. Cantor's set theory was controversial at the start, but later became largely accepted. Most modern mathematics textbooks implicitly use Cantor's views on mathematical infinity.