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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
is countable if either it is finite or it can be made in
one to one correspondence In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
with the set of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...
s. 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, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
(its number of elements) is not greater than that of the natural numbers. A countable set that is not finite is said countably infinite. The concept is attributed to
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s.


A note on terminology

Although the terms "countable" and "countably infinite" as defined here are quite common, the terminology is not universal. An alternative style uses ''countable'' to mean what is here called countably infinite, and ''at most countable'' to mean what is here called countable. To avoid ambiguity, one may limit oneself to the terms "at most countable" and "countably infinite", although with respect to concision this is the worst of both worlds. The reader is advised to check the definition in use when encountering the term "countable" in the literature. The terms ''enumerable'' and denumerable may also be used, e.g. referring to countable and countably infinite respectively, but as definitions vary the reader is once again advised to check the definition in use.


Definition

The most concise definition is in terms of
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
. A set S is ''countable'' if its cardinality , S, is less than or equal to \aleph_0 ( aleph-null), the cardinality of the set of natural numbers \N = \. A set S is ''countably infinite'' if , S, = \aleph_0. A set is ''
uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...
'' if it is not countable, i.e. its cardinality is greater than \aleph_0; the reader is referred to Uncountable set for further discussion. For every set S, the following propositions are equivalent: * S is countable. * There exists an injective function from to \N. * S is empty or there exists a surjective function from \N to S. * There exists a
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
mapping between S and a subset of \N. * S is either finite or countably infinite. Similarly, the following propositions are equivalent: * S is countably infinite. * There is an injective and surjective (and therefore
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
) mapping between S and \N. * S has a one-to-one correspondence with \N. * The elements of S can be arranged in an infinite sequence a_0, a_1, a_2, \ldots, where a_i is distinct from a_j for i\neq j and every element of S is listed.


History

In 1874, in his first set theory article, Cantor proved that the set of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s is uncountable, thus showing that not all infinite sets are countable. In 1878, he used one-to-one correspondences to define and compare cardinalities. In 1883, he extended the natural numbers with his infinite ordinals, and used sets of ordinals to produce an infinity of sets having different infinite cardinalities.


Introduction

A ''
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
'' is a collection of ''elements'', and may be described in many ways. One way is simply to list all of its elements; for example, the set consisting of the integers 3, 4, and 5 may be denoted , called roster form. This is only effective for small sets, however; for larger sets, this would be time-consuming and error-prone. Instead of listing every single element, sometimes an ellipsis ("...") is used to represent many elements between the starting element and the end element in a set, if the writer believes that the reader can easily guess what ... represents; for example, presumably denotes the set of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
s from 1 to 100. Even in this case, however, it is still ''possible'' to list all the elements, because the number of elements in the set is finite. If we number the elements of the set 1,2, and so on, up to n, this gives us the usual definition of "sets of size n". Some sets are ''infinite''; these sets have more than n elements where n is any integer that can be specified. (No matter how large the specified integer n is, such as n=10^, infinite sets have more than n elements.) For example, the set of natural numbers, denotable by , has infinitely many elements, and we cannot use any natural number to give its size. It might seem natural to divide the sets into different classes: put all the sets containing one element together; all the sets containing two elements together; ...; finally, put together all infinite sets and consider them as having the same size. This view works well for countably infinite sets and was the prevailing assumption before Georg Cantor's work. For example, there are infinitely many odd integers, infinitely many even integers, and also infinitely many integers overall. We can consider all these sets to have the same "size" because we can arrange things such that, for every integer, there is a distinct even integer: \ldots \, -\! 2\! \rightarrow \! - \! 4, \, -\! 1\! \rightarrow \! - \! 2, \, 0\! \rightarrow \! 0, \, 1\! \rightarrow \! 2, \, 2\! \rightarrow \! 4 \, \cdots or, more generally, n \rightarrow 2n (see picture). What we have done here is arrange the integers and the even integers into a ''one-to-one correspondence'' (or ''
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
''), which is a function that maps between two sets such that each element of each set corresponds to a single element in the other set. This mathematical notion of "size", cardinality, is that two sets are of the same size if and only if there is a bijection between them. We call all sets that are in one-to-one correspondence with the integers ''countably infinite'' and say they have cardinality \aleph_0.
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
showed that not all infinite sets are countably infinite. For example, the real numbers cannot be put into one-to-one correspondence with the natural numbers (non-negative integers). The set of real numbers has a greater cardinality than the set of natural numbers and is said to be uncountable.


Formal overview

By definition, a set S is ''countable'' if there exists a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
between S and a subset of the natural numbers \N=\. For example, define the correspondence Since every element of S=\ is paired with ''precisely one'' element of \, ''and'' vice versa, this defines a bijection, and shows that S is countable. Similarly we can show all finite sets are countable. As for the case of infinite sets, a set S is countably infinite if there is a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
between S and all of \N. As examples, consider the sets A=\, the set of positive
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
s, and B=\, the set of even integers. We can show these sets are countably infinite by exhibiting a bijection to the natural numbers. This can be achieved using the assignments ''n'' ↔ ''n+1'' and ''n'' ↔ 2''n'', so that Every countably infinite set is countable, and every infinite countable set is countably infinite. Furthermore, any subset of the natural numbers is countable, and more generally: The set of all
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
s of natural numbers (the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\tim ...
of two sets of natural numbers, \N\times\N is countably infinite, as can be seen by following a path like the one in the picture: The resulting mapping proceeds as follows: This mapping covers all such ordered pairs. This form of triangular mapping recursively generalizes to n-
tuple In mathematics, a tuple is a finite ordered list ( sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is de ...
s of natural numbers, i.e., (a_1,a_2,a_3,\dots,a_n) where a_i and n are natural numbers, by repeatedly mapping the first two elements of an n-tuple to a natural number. For example, (0, 2, 3) can be written as ((0, 2), 3). Then (0, 2) maps to 5 so ((0, 2), 3) maps to (5, 3), then (5, 3) maps to 39. Since a different 2-tuple, that is a pair such as (''a'', ''b''), maps to a different natural number, a difference between two n-tuples by a single element is enough to ensure the n-tuples being mapped to different natural numbers. So, an injection from the set of n-tuples to the set of natural numbers \N is proved. For the set of n-tuples made by the Cartesian product of finitely many different sets, each element in each tuple has the correspondence to a natural number, so every tuple can be written in natural numbers then the same logic is applied to prove the theorem. The set of all
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
s \Z and the set of all
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...
s \Q may intuitively seem much bigger than \N. But looks can be deceiving. If a pair is treated as the numerator and
denominator A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
of a vulgar fraction (a fraction in the form of a/b where a and b\neq 0 are integers), then for every positive fraction, we can come up with a distinct natural number corresponding to it. This representation also includes the natural numbers, since every natural number n is also a fraction n/1. So we can conclude that there are exactly as many positive rational numbers as there are positive integers. This is also true for all rational numbers, as can be seen below. In a similar manner, the set of
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the p ...
s is countable. Sometimes more than one mapping is useful: a set A to be shown as countable is one-to-one mapped (injection) to another set B, then A is proved as countable if B is one-to-one mapped to the set of natural numbers. For example, the set of positive
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...
s can easily be one-to-one mapped to the set of natural number pairs (2-tuples) because p/q maps to (p,q). Since the set of natural number pairs is one-to-one mapped (actually one-to-one correspondence or bijection) to the set of natural numbers as shown above, the positive rational number set is proved as countable. With the foresight of knowing that there are uncountable sets, we can wonder whether or not this last result can be pushed any further. The answer is "yes" and "no", we can extend it, but we need to assume a new axiom to do so. For example, given countable sets \textbf,\textbf,\textbf,\dots Using a variant of the triangular enumeration we saw above: *''a''0 maps to 0 *''a''1 maps to 1 *''b''0 maps to 2 *''a''2 maps to 3 *''b''1 maps to 4 *''c''0 maps to 5 *''a''3 maps to 6 *''b''2 maps to 7 *''c''1 maps to 8 *''d''0 maps to 9 *''a''4 maps to 10 *... This only works if the sets \textbf,\textbf,\textbf,\dots are disjoint. If not, then the union is even smaller and is therefore also countable by a previous theorem. We need the axiom of countable choice to index ''all'' the sets \textbf,\textbf,\textbf,\dots simultaneously. This set is the union of the length-1 sequences, the length-2 sequences, the length-3 sequences, each of which is a countable set (finite Cartesian product). So we are talking about a countable union of countable sets, which is countable by the previous theorem. The elements of any finite subset can be ordered into a finite sequence. There are only countably many finite sequences, so also there are only countably many finite subsets. These follow from the definitions of countable set as injective / surjective functions. Cantor's theorem asserts that if A is a set and \mathcal(A) is its power set, i.e. the set of all subsets of A, then there is no surjective function from A to \mathcal(A). A proof is given in the article Cantor's theorem. As an immediate consequence of this and the Basic Theorem above we have: For an elaboration of this result see
Cantor's diagonal argument In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, was published in 1891 by Georg Cantor as a m ...
. The set of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s is uncountable, and so is the set of all infinite
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
s of natural numbers.


Minimal model of set theory is countable

If there is a set that is a standard model (see inner model) of ZFC set theory, then there is a minimal standard model (''see'' Constructible universe). The
Löwenheim–Skolem theorem In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem. The precise formulation is given below. It implies that if a countable first-orde ...
can be used to show that this minimal model is countable. The fact that the notion of "uncountability" makes sense even in this model, and in particular that this model ''M'' contains elements that are: * subsets of ''M'', hence countable, * but uncountable from the point of view of ''M'', was seen as paradoxical in the early days of set theory, see Skolem's paradox for more. The minimal standard model includes all the
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the p ...
s and all effectively computable
transcendental number In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classes ...
s, as well as many other kinds of numbers.


Total orders

Countable sets can be
totally ordered In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...
in various ways, for example: * Well-orders (see also
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least n ...
): **The usual order of natural numbers (0, 1, 2, 3, 4, 5, ...) **The integers in the order (0, 1, 2, 3, ...; −1, −2, −3, ...) *Other (''not'' well orders): **The usual order of integers (..., −3, −2, −1, 0, 1, 2, 3, ...) **The usual order of rational numbers (Cannot be explicitly written as an ordered list!) In both examples of well orders here, any subset has a ''least element''; and in both examples of non-well orders, ''some'' subsets do not have a ''least element''. This is the key definition that determines whether a total order is also a well order.


See also

*
Aleph number In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named ...
* Counting * Hilbert's paradox of the Grand Hotel * Uncountable set


Notes


Citations


References

* * * * * * Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. (Paperback edition). * * * * {{Set theory Basic concepts in infinite set theory Cardinal numbers Infinity