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Uncountable sets such as or may be asserted to be subcountable. [ 11 ] [ 12 ] This is a notion of size that is redundant in the classical context, but otherwise need not imply countability. The existence of injections from the uncountable 2 N {\displaystyle 2^{\mathbb {N} }} or N N {\displaystyle {\mathbb {N} }^{\mathbb {N} }} into N ...
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 ...
A set is countable if it can be enumerated, that is, if there exists an enumeration of it. Otherwise, it is uncountable. For example, the set of the real numbers is uncountable. A set is finite if it can be enumerated by means of a proper initial segment {1, ..., n} of the natural numbers, in which case, its cardinality is n.
Linguistic prescriptivists usually say that fewer and not less should be used with countable nouns, [2] and that less should be used only with uncountable nouns. This distinction was first tentatively suggested by the grammarian Robert Baker in 1770, [ 3 ] [ 1 ] and it was eventually presented as a rule by many grammarians since then.
The collection of subsets of which are countable or whose complements are countable is a σ-algebra (which is distinct from the power set of if and only if is uncountable). This is the σ-algebra generated by the singletons of X . {\displaystyle X.} Note: "countable" includes finite or empty.
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]
In order theory and model theory, branches of mathematics, Cantor's isomorphism theorem states that every two countable dense unbounded linear orders are order-isomorphic.For instance, Minkowski's question-mark function produces an isomorphism (a one-to-one order-preserving correspondence) between the numerical ordering of the rational numbers and the numerical ordering of the dyadic rationals.
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.