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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 ...
Such sets are now called uncountable sets, and the size of infinite sets is treated by the theory of cardinal numbers, which Cantor began. Georg Cantor published this proof in 1891, [1] [2]: 20– [3] but it was not his first proof of the uncountability of the real numbers, which appeared in 1874.
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. [a] Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time ...
For each countable ordinal β choose a real number x β that is not in any of the sets S α for α < β, which is possible as the union of these sets has measure 0 so is not the whole of R. Then the uncountable set X of all these real numbers x β has only a countable number of elements in each set S α, so is a Sierpiński set.
One of the earliest results in set theory, published by Cantor in 1874, was the existence of different sizes, or cardinalities, of infinite sets. [2] An infinite set is called countable if there is a function that gives a one-to-one correspondence between and the natural numbers, and is uncountable if there is no such correspondence function.
The concept of countability led to countable operations and objects that are used in various areas of mathematics. For example, in 1878, Cantor introduced countable unions of sets. [67] In the 1890s, Émile Borel used countable unions in his theory of measure, and René Baire used countable ordinals to define his classes of functions. [68]
For each Borel set B, there is some countable ordinal α B such that B can be obtained by iterating the operation over α B. However, as B varies over all Borel sets, α B will vary over all the countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is ω 1, the first uncountable ordinal.
In 1891, with the publication of Cantor's diagonal argument, he demonstrated that there are sets of numbers that cannot be placed in one-to-one correspondence with the set of natural numbers, i.e. uncountable sets that contain more elements than there are in the infinite set of natural numbers. [9]