Let be a set. An enumeration on is a surjection from the set of natural numbers to .
A set is called numerable if there is a bijective enumeration on .
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 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] is not numerable.
Remark. If the enumeration is furthermore a computable function, then we say that is enumerable. There exists numerable sets that are not enumerable.
|Date of creation||2013-03-22 16:01:32|
|Last modified on||2013-03-22 16:01:32|
|Last modified by||juanman (12619)|