# Definition:Countable Set

## Definition

Let $S$ be a set.

### Definition 1

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

- $f: S \to \N$

### Definition 2

$S$ is **countable** if and only if it is finite or countably infinite.

### Definition 3

$S$ is **countable** if and only if there exists a bijection between $S$ and a subset of $\N$.

## 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

- Equivalence of Definitions of Countable Set
- Surjection from Natural Numbers iff Countable
- Countable Set equals Range of Sequence

- Results about
**countable sets**can be found**here**.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): Notation and Terminology - 1989: George S. Boolos and Richard C. Jeffrey:
*Computability and Logic*(3rd ed.) ... (next): $1$ Enumerability