# numerable set

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

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

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, it is not hard to construct an element 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 $X$ is enumerable. There exists numerable sets that are not enumerable.

 Title numerable set Canonical name NumerableSet Date of creation 2013-03-22 16:01:32 Last modified on 2013-03-22 16:01:32 Owner juanman (12619) Last modified by juanman (12619) Numerical id 11 Author juanman (12619) Entry type Definition Classification msc 97A80 Related topic Calculus Related topic TopicsOnCalculus Related topic Denumerable Related topic Countable Defines enumeration Defines enumerable