Does "$\dots$" imply countability? 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 $$1+\frac{1}{4}+\frac{1}{9}+\dots$$

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. 
