PlanetMath: numerable set

(more info)

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numerable set

(Definition)

Let be a set. An enumeration on is a surjection from the set of natural numbers $\mathbb{N}$ to .

A set is called numerable if there is a bijective enumeration on .

It is easy to show that $\mathbb{Z}$ and $\mathbb{Q}$ are numerable.

It is a standard fact that $\mathbb{R}$ is not numerable. For, if we suppose that the numbers [0,1] were countable, we can arrange them in a list (given by the supposed bijection).

Representing them in a binary form, anyone would be able to construct an object in [0,1], which is not in the list.

This contradiction implies that [0,1] $\subset\mathbb{R}$ is not numerable.

Remark. If the enumeration $\mathbb{N}\to X$ is furthermore a computable function, then we say that is enumerable. There exists numerable sets that are not enumerable.