Countable set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and—although the counting may never finish—every element of the set is associated with a unique natural number.

Some authors use countable set to mean countably infinite alone.[1] To avoid this ambiguity, the term at most countable may be used when finite sets are included and countably infinite, enumerable,[2] or denumerable[3] otherwise.

Georg Cantor introduced the term countable set, contrasting sets that are countable with those that are uncountable (i.e., nonenumerable or nondenumerable[4]). Today, countable sets form the foundation of a branch of mathematics called discrete mathematics.

If such an f can be found that is also surjective (and therefore bijective), then S is called countably infinite.

This terminology is not universal. Some authors use countable to mean what is here called countably infinite, and do not include finite sets.

Alternative (equivalent) formulations of the definition in terms of a bijective function or a surjective function can also be given.[7] See also § Formal overview without details below.

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.[8] In 1878, he used one-to-one correspondences to define and compare cardinalities.[9] 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.[10]

To understand what this means, we first examine what it does not 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 even integers, which is the same as the number of odd integers, is also the same as the number of integers overall. This is because we can arrange things such that, for every integer, there is a distinct even integer:

However, not all infinite sets have the same cardinality. For example, Georg Cantor (who introduced this concept) 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 (i.e., denumerable).[12] Equivalently, a set is countable if it has the same cardinality as some subset of the set of natural numbers. Otherwise, it is uncountable.

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 is not tenable, however, under the natural definition of size.

To elaborate this, we need the concept of a bijection. 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

We now generalize this situation; we define that two sets are 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".

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 is an example of a set of the same size as one of its proper subsets, which is impossible for finite sets.

Likewise, the set of all ordered pairs of natural numbers (the Cartesian product of two sets of natural numbers, N × N) is countably infinite, as can be seen by following a path like the one in the picture:

The Cantor pairing function assigns one natural number to each pair of natural numbers
0 ↔ (0, 0), 1 ↔ (1, 0), 2 ↔ (0, 1), 3 ↔ (2, 0), 4 ↔ (1, 1), 5 ↔ (0, 2), 6 ↔ (3, 0), ....

If a pair is treated as the numerator and denominator of a vulgar fraction (a fraction in the form of a/b where a and b ≠ 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 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.

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 numbers 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.

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

This form of triangular mapping recursively generalizes to n-tuples of natural numbers, i.e., (a1, a2, a3, ..., an) where ai and n are natural numbers, by repeatedly mapping the first two elements of a 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 following theorem concerns infinite subsets of countably infinite sets.

Theorem: Every subset of a countable set is countable. In particular, every infinite subset of a countably infinite set is countably infinite.[13]

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

There are also sets which are "naturally larger than" N. For instance, Z the set of all integers or Q, the set of all rational numbers, which intuitively may seem much bigger than N. But looks can be deceiving, for we assert:

Theorem: Z (the set of all integers) and Q (the set of all rational numbers) are countable.

These facts follow easily from a result that many individuals find non-intuitive.

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.

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

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.

We need the axiom of countable choice to index all the sets a, b, c, ... simultaneously.

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.

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.

The following theorem gives equivalent formulations in terms of a bijective function or a surjective function. A proof of this result can be found in Lang's text.[3]

(Basic) Theorem: Let S be a set. The following statements are equivalent:

Cantor's theorem asserts that if A is a set and P(A) is its power set, i.e. the set of all subsets of A, then there is no surjective function from A to P(A). A proof is given in the article Cantor's theorem. As an immediate consequence of this and the Basic Theorem above we have:

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

The proofs of the statements in the above section rely upon the existence of functions with certain properties. This section presents functions commonly used in this role, but not the verifications that these functions have the required properties. The Basic Theorem and its Corollary are often used to simplify proofs. Observe that N in that theorem can be replaced with any countably infinite set.

Proof: The restriction of an injective function to a subset of its domain is still injective.

Proposition: If A and B are countable sets then AB is countable.[17]

Proof: Let f: AN and g: BN be injections. Define a new injection h: ABN by h(x) = 2f(x) if x is in A and h(x) = 2g(x) + 1 if x is in B but not in A.

Proposition: The Cartesian product of two countable sets A and B is countable.[18]

Proof: Observe that N × N is countable as a consequence of the definition because the function f : N × NN given by f(m, n) = 2m3n is injective.[19] It then follows from the Basic Theorem and the Corollary that the Cartesian product of any two countable sets is countable. This follows because if A and B are countable there are surjections f : NA and g : NB. So

is a surjection from the countable set N × N to the set A × B and the Corollary implies A × B is countable. This result generalizes to the Cartesian product of any finite collection of countable sets and the proof follows by induction on the number of sets in the collection.

Proposition: The integers Z are countable and the rational numbers Q are countable.

Proof: The integers Z are countable because the function f : ZN given by f(n) = 2n if n is non-negative and f(n) = 3n if n is negative, is an injective function. The rational numbers Q are countable because the function g : Z × NQ given by g(m, n) = m/(n + 1) is a surjection from the countable set Z × N to the rationals Q.

Proposition: If An is a countable set for each n in N then the union of all An is also countable.[20]

Proof: This is a consequence of the fact that for each n there is a surjective function gn : NAn and hence the function

given by G(n, m) = gn(m) is a surjection. Since N × N is countable, the Corollary implies that the union is countable. We use the axiom of countable choice in this proof to pick for each n in N a surjection gn from the non-empty collection of surjections from N to An.

A topological proof for the uncountability of the real numbers is described at finite intersection property.

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:

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.

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.