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  2. 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 ...

  3. Cantor's diagonal argument - Wikipedia

    en.wikipedia.org/wiki/Cantor's_diagonal_argument

    In which case, if P 1 (S) is the set of one-element subsets of S and f is a proposed bijection from P 1 (S) to P(S), one is able to use proof by contradiction to prove that |P 1 (S)| < |P(S)|. The proof follows by the fact that if f were indeed a map onto P ( S ), then we could find r in S , such that f ({ r }) coincides with the modified ...

  4. Aleph number - Wikipedia

    en.wikipedia.org/wiki/Aleph_number

    Notably, 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: For any natural number , we can consistently assume that =, and moreover it is possible to assume that is as least as large as any cardinal number we like.

  5. Cantor's first set theory article - Wikipedia

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

    Cantor's article is short, less than four and a half pages. [A] It begins with a discussion of the real algebraic numbers and a statement of his first theorem: The set of real algebraic numbers can be put into one-to-one correspondence with the set of positive integers. [3]

  6. Infinite set - Wikipedia

    en.wikipedia.org/wiki/Infinite_set

    Both Burton and Rogers use finite sets to start to explain infinite sets using proof concepts such as mapping, proof by induction, or proof by contradiction. [5] [7] Mathematical trees can also be used to understand infinite sets. [8] Burton also discusses proofs of infinite sets including ideas such as unions and subsets. [5]

  7. Paradoxes of set theory - Wikipedia

    en.wikipedia.org/wiki/Paradoxes_of_set_theory

    B. Russell: The principles of mathematics I, Cambridge 1903. B. Russell: On some difficulties in the theory of transfinite numbers and order types, Proc. London Math. Soc. (2) 4 (1907) 29-53. P. J. Cohen: Set Theory and the Continuum Hypothesis, Benjamin, New York 1966. S. Wagon: The Banach–Tarski Paradox, Cambridge University Press ...

  8. Unit interval - Wikipedia

    en.wikipedia.org/wiki/Unit_interval

    In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1. Its standard orientation goes from 0 to 1. The unit interval is a totally ordered set and a complete lattice (every subset of the unit interval has a supremum and an infimum).

  9. Natural number - Wikipedia

    en.wikipedia.org/wiki/Natural_number

    Starting at 0 or 1 has long been a matter of definition. In 1727, Bernard Le Bovier de Fontenelle wrote that his notions of distance and element led to defining the natural numbers as including or excluding 0. [23] In 1889, Giuseppe Peano used N for the positive integers and started at 1, [24] but he later changed to using N 0 and N 1. [25]