# Definition:Countable Set/Definition 1

## Definition

Let $S$ be a set.

$S$ is **countable** if and only if there exists an injection:

- $f: S \to \N$

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

## Also see

## Sources

- 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 4$: Number systems $\text{I}$: A set-theoretic approach - 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $2.4$ - 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): Appendix $\text{A}$: Set Theory: Countable Sets