A mathematical set is countable if either it is finite or it can be put in one to one correspondence with the set of natural numbers. 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 (the number of elements of the set) is not greater than that of the natural numbers. A countable set that is not finite is said to be countably infinite; for example the set of all natural numbers or all rational numbers.

The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers.

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.

The terms enumerable and denumerable may also be used, referring to countable and countably infinite, respectively. Definitions vary and care is needed respecting the difference with recursively enumerable.

Definition

is countable if:

• Its cardinality (aleph-null), the cardinality of the set of natural numbers.

• There exists an injective function.

• is empty or there exists a surjective function.

• There exists a bijective.

• is either finite) or countably infinite.

All of these definitions are equivalent.

is countably infinite if:

• .

• There is an injective and surjective (and therefore bijective.

• has a one-to-one correspondence.

• is listed.

A set is uncountable.. In models where the axiom of choice, the so-called Dedekind finite infinite sets.

History

In 1874, in his first set theory article, Cantor proved that the set of real numbers 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 is a collection of elements, called roster form. presumably denotes the set of integers from 1 to 100. Even in this case, however, it is still possible".

Some sets are infinite, (see picture). What we have done here is arrange the integers and the even 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. 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.

Georg Cantor 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. Cantor's diagonal argument provides a formal proof by using a reductio ad absurdum strategy. It begins with the assumption that the set of real numbers within the interval (0,1) is countable. If this were true, every real number in that range could be arranged into a sequential list (s1,s2,s3, ...) where each number is represented as an infinite decimal expansion. This structured list is the basis for constructing a specific real number that is logically guaranteed to be missing from the sequence, thereby invalidating the initial premise of countability.

is a real number between 0 and 1 that does not appear in the assumed to complete list, it follows that bijection cannot exist between the natural numbers and the real numbers.

Formal overview

is countable if there exists a bijection and a subset of the natural numbers is paired with precisely one, and is countable. Similarly we can show all finite sets to be countable.

is countably infinite if there is a bijection, the set of positive integers 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 pairs of natural numbers (the Cartesian product) is countably infinite, as can be seen by following a path like the one in the picture: The resulting mapping This mapping covers all such ordered pairs.

This form of triangular mapping recursively-tuples-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 integers and the set of all rational numbers. But looks can be deceiving. If a pair is treated as the numerator and denominator of a vulgar fraction. 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 numbers is countable.

is one-to-one mapped to the set of natural numbers. For example, the set of positive rational numbers. 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.

We need the axiom of countable choice to index all simultaneously.

This set is the union of the length-1 sequences, the length-2 sequences, the length-3 sequences, and so on, each of which is a countable set (finite Cartesian product). Thus the set is 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 is its power set. 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.

The set of real numbers is uncountable, and so is the set of all infinite sequences 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 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 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, for example:

Well-orders (see also ordinal number):

• 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

• 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).