# Definition:Countable Set/Countably Infinite/Definition 2

## Definition

Let $S$ be a set.

$S$ 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 a countable set to be what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a countably infinite set.

That is, they use **countable** to describe a set 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 set **can be counted**, which, plainly, a finite set can be, it is suggested that this interpretation may be counter-intuitive.

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

## Also known as

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

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

## Also see

## Sources

- 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): Appendix: Elementary set and number theory - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Definition $2.1.7$