# Definition:Countably Infinite/Set

## Definition

Let $S$ be a set.

### Definition 1

$S$ is **countably infinite** if and only if there exists a bijection:

- $f: S \to \N$

where $\N$ is the set of natural numbers.

### Definition 2

$S$ is **countably infinite** if and only if there exists a bijection:

- $f: S \to \Z$

where $\Z$ is the set of integers.

An infinite set is **countably infinite** if it is countable, and is uncountable otherwise.

## Cardinality

The cardinality of a countably infinite set is denoted by the symbol $\aleph_0$ (**aleph-null**).

## Also defined as

Some sources define **countable** to be what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as **countably infinite**.

That is, they use **countable** to describe a collection which has *exactly the same* cardinality as $\N$.

Thus under this criterion $X$ is said to be countable if and only if there exists a bijection from $X$ to $\N$, that is, if and only if $X$ is equivalent to $\N$.

However, as the very concept of the term **countable** implies that a collection **can be counted**, which, plainly, a finite can be, it is suggested that this interpretation may be counter-intuitive.

Hence, on $\mathsf{Pr} \infty \mathsf{fWiki}$, the term **countable** will be taken in the sense as to **include** the concept of finite, and countably infinite will mean a **countable** collection which is specifically *not* finite.

## Also known as

When the terms **denumerable** and **enumerable** are encountered, they generally mean the same as **countably infinite**.

Sometimes the term **enumerably infinite** can be seen.

Some modern pedagogues (for example Vi Hart and James Grime) use the term **listable**, but this has yet to catch on.

## Also see

- Infinite Set has Countably Infinite Subset: the Axiom of Countable Choice implies that $\aleph_0$ is the
**smallest**possible cardinality of an infinite set.

- Results about
**countably infinite**can be found**here**.

## Sources

- 1989: George S. Boolos and Richard C. Jeffrey:
*Computability and Logic*(3rd ed.) ... (previous) ... (next): $1$ Enumerability - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**countable (denumerable; enumerable)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**countable (denumerable, enumerable)** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**countably infinite** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**enumerable**