Countable set

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“Countable” redirects here. For the linguistic concept, see Count noun.

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.

Note that countable set is sometimes given an alternate definition: a set with 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.

1 Definition
2 Gentle introduction
3 A more formal introduction
4 Further theorems about uncountable sets
5 The minimal model of set theory is countable
6 See also

[edit] Definition

A set S is called countable if there exists an injective function

$f\colon S \to \mathbb{N}$

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 we have called "countable"; some define countable to mean what we have 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 $f\colon S \to \mathbb{N}$ .
Either S is empty or there exists a surjective function $g\colon \mathbb{N} \to S$ .

[edit] Gentle introduction

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The elements of a finite set can be listed, say { a₁, a₂, ..., a_n }. However, insofar as a set is a logical description of the properties of its members, it need not be finite. To understand this, imagine that I ask you: how many words can you make out of Scrabble tiles if you are allowed to ask me for more pieces no matter how many you used up? The answer? As many as you like; you can go forever. But that doesn't mean they won't each of them be a word made out of Scrabble tiles, rather than apple pies or racing cars. Thus an infinite set is still a set, insofar as it is a tool for separating out things with different properties.

Now what is a countably infinite set?

Technically, a countably infinite set is any set which, in spite of its boundlessness, can be shown equinumerous to the natural numbers — nothing more, nothing less. This makes it possible to set apart elements of a countably infinite set using natural numbers as indices, and in turn puts the logic associated with them in very close proximity to the logic associated with the natural numbers themselves; and this makes such sets easily logically tractable.

[edit] A 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 { 1, 2, 3 } and { a, b, c } 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 { a, b, c } is paired with precisely one element of { 1, 2, 3 } (and vice versa) this defines a bijection.

We now generalise this situation and define two sets to be of the same size precisely when 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 = { 1, 2, 3, ... }, the set of positive integers and B = { 2, 4, 6, ... }, 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: 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 this one:

$\begin{matrix} (0,0) & \rightarrow & (0,1) & & (0,2) & \rightarrow & (0,3) & \\ & \swarrow & & \nearrow & & \swarrow & & \\ (1,0) & & (1,1) & & (1,2) & & \ddots & \\ \downarrow & \nearrow & & \swarrow & & & & \\ (2,0) & & (2,1) & & \ddots & & & \\ & \swarrow & & & & & & \\ (3,0) & & \ddots & & & & & \\ \downarrow & & & & & & & \\ \vdots & & & & & & & \end{matrix}$

The resulting mapping is like this: 0 ↔ (0,0), 1 ↔ (0,1), 2 ↔ (1,0), 3 ↔ (2,0), 4 ↔ (1,1), 5 ↔ (0,2), … 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 possible fraction, we can come up with a distinct number corresponding to it. Since every natural number is also a fraction N/1, we can conclude that there are exactly as many rational numbers as there are positive integers.

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 b > 0, a and b are coprime, and c ∈ {0, 1} 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 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.

[edit] Further theorems about uncountable sets

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

Remember our example of the Scrabble words. Although we can keep asking for more tiles from the bag, each word we form is finitely long. The number of possible words is the same as the number of natural numbers. If we permit infinitely long words, then the number of possible "words" is greater than this.

In fact, with infinitely long words, the number of words is the same as the number of real numbers.

We noted earlier that there are no more fractions than there are natural numbers. The decimal expansion of a fraction is always a finitely long decimal number followed by a repeating decimal.

0.33333333333 ...
12.648986986986986986 ...
1.75

Let's say we use our decimal point to also indicate the start of the repeater:

..3
12.648.986
1.75.

Then we can express any fraction using a finitely long decimal expansion with repeating bit. It's clear that this is the same situation as with our finitely long scrabble words, and so once again the number of possible fractions is not greater than the number of natural numbers.

[edit] The 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. It includes all the algebraic numbers and all effectively computable transcendental numbers, as well as many other kinds of numbers.