# Definition:Countably Infinite Set/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 **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

## Sources

- 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): Appendix: Elementary set and number theory - 1979: John E. Hopcroft and Jeffrey D. Ullman:
*Introduction to Automata Theory, Languages, and Computation*... (previous) ... (next): Chapter $1$: Preliminaries: $1.4$ Set Notation: Infinite sets - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Definition $2.1.7$