Definition:Countable Set

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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 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

  • Results about countable sets can be found here.