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| Let $X$ be a set. An \emph{enumeration} on $X$ is a surjection from the set of natural numbers $\mathbb{N}$ to $X$. |
Let $X$ be a set. An \emph{enumeration} on $X$ is a surjection from the set of natural numbers $\mathbb{N}$ to $X$. |
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| A set $X$ is called \emph{numerable} if there is a bijective enumeration on $X$. |
A set $X$ is called \emph{numerable} if there is a bijective enumeration on $X$. |
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| It is easy to show that $\mathbb{Z}$ and $\mathbb{Q}$ are numerable. |
It is easy to show that $\mathbb{Z}$ and $\mathbb{Q}$ are numerable. |
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| 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). |
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). |
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Representing them in a binary form, it is not hard to construct an element in [0,1], which is not in the list.
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Representing them in a binary form, anyone would be able to construct an object in [0,1], which is not in the list.
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| This contradiction implies that [0,1]$\subset\mathbb{R}$ is not numerable. |
This contradiction implies that [0,1]$\subset\mathbb{R}$ is not numerable. |
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| \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. |
\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. |
