# Dictionary Definition

countable adj : that can be counted; "countable
sins"; "numerable assets" [syn: denumerable, enumerable, numerable]

# User Contributed Dictionary

## English

### Adjective

- Capable of being counted; having a quantity or a numerical attribute.
- math of a set Having a bijection with a subset of the natural numbers.
- math of a set Countably infinite; having a bijection with the natural numbers.
- linguistics of a noun Freely usable with the indefinite article and with numbers, and therefore having a plural form.

#### Antonyms

#### Derived terms

#### Translations

capable of being counted

- Dutch: telbaar, telbare
- French: dénombrable
- Italian: enumerabile
- Slovene: števen , števna , števno

having a bijection with a subset of the natural
numbers

- Czech: spočetný
- Finnish: numeroituva
- French: dénombrable
- German: abzählbar
- Hungarian: megszámolható
- Italian: finito o numerabile, contabile (uncommon anglicism)
- Slovene: števen , števna , števno
- Swedish: uppräknelig

freely used with numbers and the definite
article

- Czech: počitatelný
- Dutch: telbaar, telbare
- Estonian: loenduv
- French: dénombrable
- Italian: numerabile
- Japanese: 可算 (かさん, kasan)
- Russian: исчисляемый
- Slovene: števen , števna , števno
- Swedish: räknebar

### See also

# Extensive Definition

In mathematics, a countable set
is a set with the same
cardinality (i.e.,
number of
elements) as some subset
of the set of natural
numbers. The term was originated by Georg
Cantor; it stems from the fact that the natural numbers are
often called counting numbers. A set that is not countable is
called uncountable.

Some authors use countable set to mean a set with
exactly as many elements as the set of natural numbers. The
difference between the two definitions is that under the former,
finite
sets are also considered to be countable, while under the
latter definition, they are not considered to be countable. To
resolve this ambiguity, the term at most countable is sometimes
used for the former notion, and countably infinite for the
latter.

If f is also surjective,
thus making f bijective, then S is called
countably infinite or
denumerable.

As noted above, this terminology is not
universal: some authors define denumerable to mean what is here
called "countable"; some define countable to mean what is here
called "countably infinite".

The next result offers an alternative definition
of a countable set S in terms of a surjective function:

THEOREM: Let S be a set. The following statements
are equivalent:

- S is countable, i.e. there exists an injective function f\colon S \to \mathbb.
- Either S is empty or there exists a surjective function g\colon \mathbb \to S.

## Gentle introduction

A set 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 \. 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 ellipsis ('…') are used, if the writer believes
that the reader can easily guess what is missing; for example, \
presumably denotes the set of integers from 1 to 100. Even in
this case, however, it is still possible to list all the elements,
because the set is finite; it has a specific number of
elements.

Some sets are infinite; these sets have more than
n elements for any integer n. For example, the set of natural
numbers, denotable by \, has infinitely many elements, and we can't
use any normal number to give its size. Nonetheless, it turns out
that infinite sets do have a well-defined notion of size (or more
properly, of cardinality, which is the technical term for the
number of elements in a set), and not all infinite sets have the
same cardinality.

To understand what this means, we must first
examine what it doesn't mean. For example, there are infinitely
many odd integers, infinitely many even integers, and (hence)
infinitely many integers overall. However, it turns out that the
number of odd integers, which is the same as the number of even
integers, is also the same as the number of integers overall. This
is because we arrange things such that for every integer, there is
a distinct odd integer: … −2 → -3,
−1 → −1,
0 → 1, 1 → 3,
2 → 5, …; or, more generally,
n → 2n + 1. What we have done
here is arranged the integers and the odd integers into a
one-to-one correspondence (or bijection), 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.

However, not all infinite sets have the same
cardinality. For example, Georg Cantor (who introduced this branch
of mathematics) demonstrated that the real numbers
cannot be put into one-to-one correspondence with the natural
numbers (non-negative integers), and therefore that the set of real
numbers has a greater cardinality than the set of natural
numbers.

A set is countable if: (1) it is finite, or (2)
it has the same cardinality (size) as the set of natural numbers.
Equivalently, a set is countable if it has the same cardinality as
some subset of the set of
natural numbers. Otherwise, it is uncountable.

## More formal introduction

It might then 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 is not tenable, however, under the natural definition of size.To elaborate this we need the concept of a
bijection. Do the sets
and have the same size?

- "Obviously, yes."
- "How do you know?"
- "Well, it's obvious. Look, they've both got 3 elements."
- "What's a 3?"

This may seem a strange situation but, although a
"bijection" seems a more advanced concept than a "number", the
usual development of mathematics in terms of set theory defines
functions before numbers, as they are based on much simpler sets.
This is where the concept of a bijection comes in: define the
correspondence

- a ↔ 1, b ↔ 2, c ↔ 3

Since every element of is paired with precisely
one element of (and vice versa) this defines a bijection.

We now generalize this situation and define two
sets to be of the same size if (and only if) there is a bijection
between them. For all finite sets this gives us the usual
definition of "the same size". What does it tell us about the size
of infinite sets?

Consider the sets A = , the set of positive
integers and B = , the
set of even positive integers. We claim that, under our definition,
these sets have the same size, and that therefore B is countably
infinite. Recall that to prove this we need to exhibit a bijection
between them. But this is easy, using n ↔ 2n, so that 1 ↔ 2, 2 ↔ 4,
3 ↔ 6, 4 ↔ 8, ...

As in the earlier example, every element of A has
been paired off with precisely one element of B, and vice versa.
Hence they have the same size. This gives an example of a set which
is of the same size as one of its proper subsets, a situation which
is impossible for finite sets.

Likewise, the set of all ordered
pairs of natural numbers is countably infinite, as can be seen
by following a path like the one in the picture: The resulting
mapping is like this: 0 ↔ (0,0), 1 ↔ (1,0), 2 ↔ (0,1), 3 ↔ (2,0), 4
↔ (1,1), 5 ↔ (0,2), 6 ↔ (3,0) … It is evident that this mapping
will cover all such ordered pairs.

Interestingly: if you treat each pair as being
the numerator and
denominator of a
vulgar
fraction, then for every positive fraction, we can come up with
a distinct number corresponding to it. This representation includes
also the natural numbers, since every natural number 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
true also for all rational numbers, as can be seen below (a more
complex presentation is needed to deal with negative
numbers).

THEOREM: The Cartesian
product of finitely many countable sets is countable.

This form of triangular mapping recursively generalizes to
vectors of
finitely many natural numbers by repeatedly mapping the first two
elements to a natural number. For example, (0,2,3) maps to (5,3)
which maps to 41.

Sometimes more than one mapping is useful. This
is where you map the set which you want to show countably infinite,
onto another set; and then map this other set to the natural
numbers. For example, the positive rational
numbers can easily be mapped to (a subset of) the pairs of
natural numbers because p/q maps to (p, q).

What about infinite subsets of countably infinite
sets? Do these have fewer elements than N?

THEOREM: Every subset of a countable set is
countable. In particular, every infinite subset of a countably
infinite set is countably infinite.

For example, the set of prime
numbers is countable, by mapping the n-th prime number to n:

- 2 maps to 1
- 3 maps to 2
- 5 maps to 3
- 7 maps to 4
- 11 maps to 5
- 13 maps to 6
- 17 maps to 7
- 19 maps to 8
- 23 maps to 9
- etc.

What about sets being "larger than" N? An obvious
place to look would be Q, the set of all rational
numbers, which is "clearly" much bigger than N. But looks can
be deceiving, for we assert

THEOREM: Q (the set of all rational numbers) is
countable.

Q can be defined as the set of all fractions a/b
where a and b are integers and b > 0. This can be mapped onto
the subset of ordered triples of natural numbers (a, b, c) such
that a ≥ 0, b > 0, a and b are coprime, and c ∈ such that c = 0
if a/b ≥ 0 and c = 1 otherwise.

- 0 maps to (0,1,0)
- 1 maps to (1,1,0)
- −1 maps to (1,1,1)
- 1/2 maps to (1,2,0)
- −1/2 maps to (1,2,1)
- 2 maps to (2,1,0)
- −2 maps to (2,1,1)
- 1/3 maps to (1,3,0)
- −1/3 maps to (1,3,1)
- 3 maps to (3,1,0)
- −3 maps to (3,1,1)

- 1/4 maps to (1,4,0)
- −1/4 maps to (1,4,1)
- 2/3 maps to (2,3,0)
- −2/3 maps to (2,3,1)
- 3/2 maps to (3,2,0)
- −3/2 maps to (3,2,1)
- 4 maps to (4,1,0)
- −4 maps to (4,1,1)
- ...

By a similar development, the set of algebraic
numbers is countable, and so is the set of definable
numbers.

THEOREM: (Assuming the
axiom of countable choice) The union
of countably many countable sets is countable.

For example, given countable sets a, b, c
...

Using a variant of the triangular enumeration we
saw above:

- a0 maps to 0

- a1 maps to 1
- b0 maps to 2

- a2 maps to 3
- b1 maps to 4
- c0 maps to 5

- a3 maps to 6
- b2 maps to 7
- c1 maps to 8
- d0 maps to 9

- a4 maps to 10
- ...

Note that this only works if the sets a, b, c,...
are disjoint. If not,
then the union is even smaller and is therefore also countable by a
previous theorem.

THEOREM: The set of all finite-length sequences of natural numbers is
countable.

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.

THEOREM: The set of all finite subsets of the natural numbers is
countable.

If you have a finite subset, you can order the
elements into a finite sequence. There are only countably many
finite sequences, so also there are only countably many finite
subsets.

## Further theorems about uncountable sets

- The set of real numbers is uncountable (see Cantor's first uncountability proof), and so is the set of all sequences of natural numbers and the set of all subsets of N (see Cantor's diagonal argument).

## Minimal model of set theory is countable

If there is a set which 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 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 which are- subsets of M, hence countable,
- but uncountable from the point of view of M,

The minimal standard model includes all the
algebraic
numbers and all effectively computable transcendental
numbers, as well as many other kinds of numbers.

## Total orders

Countable sets can be totally ordered in various ways, e.g.:- well
orders (see also ordinal
number):
- the usual order of natural numbers
- the integers in the order 0, 1, 2, 3, .., -1, -2, -3, ..

- other:
- the usual order of integers
- the usual order of rational numbers

countable in Arabic: مجموعة عدودة

countable in Bengali: গণনাযোগ্য সেট

countable in Bulgarian: Изброимо множество

countable in Czech: Spočetná množina

countable in Danish: Tællelig mængde

countable in German: Abzählbarkeit

countable in Spanish: Conjunto numerable

countable in Persian: مجموعه شمارا

countable in French: Ensemble dénombrable

countable in Galician: Conxunto contábel

countable in Georgian: თვლადი სიმრავლე

countable in Korean: 가산 집합

countable in Icelandic: Teljanlegt mengi

countable in Italian: Insieme numerabile

countable in Hebrew: קבוצה בת מנייה

countable in Lithuanian: Skaiti aibė

countable in Lombard: Cungjuunt cüntàbil

countable in Dutch: Aftelbare verzameling

countable in Japanese: 可算集合

countable in Norwegian: Tellbar

countable in Polish: Zbiór przeliczalny

countable in Portuguese: Conjunto contável

countable in Russian: Счётное множество

countable in Simple English: Countable set

countable in Slovak: Spočítateľná množina

countable in Serbian: Пребројиво бесконачан
скуп

countable in Finnish: Numeroituva joukko

countable in Swedish: Uppräknelig

countable in Tamil: எண்ணுறுமையும்
எண்ணுறாமையும்

countable in Chinese:
可數集