Does "$\dots$" imply countability?

Question

I am given an arbitrary set $S$.

If I say the following:

"Suppose that the elements of $S$ are labeled $x_1,x_2,x_3,\dots,$"

am I notationally implying that the number of elements in $S$ is countable?

This issue came up when I was grading papers for an introductory proofs class.

If the situation I gave above does imply that $S$ is a countable set, are there any situations in which we use "$\dots$" to mean uncountably many things (since indexing with $1$, $2$, and $3$ perhaps affects my example)?

In general I can only think of using "$\dots$" in situations where I am performing an operation countably many times - say $$1+\frac{1}{4}+\frac{1}{9}+\dots$$

Answer 1

Yes, I'd say so. If you want arbitrary indexing you should say something like $x_i, i \in I$ and then maybe say something about how big $I$ is, whether it's equipped with a total order, etc.

Answer 2

I'd say you are justified in taking the notation $x_1, x_2, x_3 \ldots$ to mean a countable set, because the usual notation for a possibly non-countable set is something like $\{x_\alpha\}$ where the index is a member of some arbitrary set $J$.