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'numerable set'
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| Title of object: |
numerable set |
| Canonical Name: |
NumerableSet |
| Type: |
Definition |
| Created on: |
2006-06-20 15:05:21 |
| Modified on: |
2007-10-16 17:11:19 |
| Classification: |
msc:97A80 |
| Keywords: |
Analysis |
| Defines: |
enumeration, enumerable |
| Synonyms: |
numerable set=countable |
Preamble:
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Content:
Let $X$ be a set. An \emph{enumeration} on $X$ is a surjection from the set of natural numbers $\mathbb{N}$ to $X$.
A set $X$ is called \emph{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, 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.
\textbf{Remark}. If the enumeration $\mathbb{N}\to X$ is furthermore a computable function, then we say that $X$ is \emph{enumerable}. There exists numerable sets that are not enumerable. |
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