numerable set

Let X be a set. An enumeration on X is a surjection from the set of natural numbers to X.

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

It is easy to show that and are numerable.

It is a standard fact that is not numerable. For, if we suppose that the numbers [0,1] were countableMathworldPlanetmath, 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 contradictionMathworldPlanetmathPlanetmath implies that [0,1] is not numerable.

Remark. If the enumeration X is furthermore a computable functionMathworldPlanetmath, 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