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Why the domain of sequences has been chosen to be the set of natural numbers, and not for example the set of real numbers ?

Are there advantages from the fact that $\mathbb{N}$ is a countable set ?

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    $\begingroup$ I would say: it is more the other way around. If $f:I\to X$ is a function and $I$ is infinite and countable and equipped with a well-order then it can be looked at as a sequence. $\endgroup$ – drhab Sep 10 '18 at 8:33
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    $\begingroup$ There is also the notion of a "net" (basically a sequence with some arbitrary ordered set as domain). $\endgroup$ – Janik Sep 10 '18 at 8:36
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The short answer is: because that's the way sequences are defined. But, in fact, the general concept of sequence is: a sequence is a function whose domain is an ordered set $(S,\preccurlyeq)$ which, as an ordered set, is isomorphic to $(\mathbb{N},\leqslant)$. So, the basic idea is that we can talk about the first element of the sequence, the second element of the sequence, and so on.

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The name says it, in a sequence a term has a next term. In an uncountable universe, there are no next.

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