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 $ℝ$ 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]$\subset ℝ$ 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