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  2. Cardinality - Wikipedia

    en.wikipedia.org/wiki/Cardinality

    If | X | ≤ | Y | and | Y | ≤ | X |, then | X | = | Y |. This holds even for infinite cardinals, and is known as Cantor–Bernstein–Schroeder theorem . Sets with cardinality of the continuum include the set of all real numbers, the set of all irrational numbers and the interval [ 0 , 1 ] {\displaystyle [0,1]} .

  3. Cardinal number - Wikipedia

    en.wikipedia.org/wiki/Cardinal_number

    A bijective function, f: XY, 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. 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. The theorem is named for Georg Cantor, who first stated and proved it at the end of the 19th century.

  5. Power set - Wikipedia

    en.wikipedia.org/wiki/Power_set

    Generally speaking, X Y is the set of all functions from Y to X and | X Y | = | X | | Y |. Cantor's diagonal argument shows that the power set of a set (whether infinite or not) always has strictly higher cardinality than the set itself (or informally, the power set must be larger than the original set).

  6. 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 ...

  7. Cardinality of the continuum - Wikipedia

    en.wikipedia.org/wiki/Cardinality_of_the_continuum

    In fact, the cardinality of ℘ (), by definition , is equal to . This can be shown by providing one-to-one mappings in both directions between subsets of a countably infinite set and real numbers, and applying the Cantor–Bernstein–Schroeder theorem according to which two sets with one-to-one mappings in both directions have the same ...

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  9. Controversy over Cantor's theory - Wikipedia

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

    This larger set consists of the elements (x 1, x 2, x 3, ...), where each x n is either m or w. [3] Each of these elements corresponds to a subset of N—namely, the element (x 1, x 2, x 3, ...) corresponds to {n ∈ N: x n = w}. So Cantor's argument implies that the set of all subsets of N has greater cardinality than N.