Comment by triclops200

It's an axiom (the axiom of choice, actually). A valid way of viewing an axiom is not dissimilar to a "modeling requirement" or an "if statement". By that I mean, for example with the axiom of choice: it is just a formal statement version of "assume that you can take an element from a (possibly infinite) collection of sets such that you can create a new set (the new set does not have to be unique)." It makes intuitive sense for most finite sets we deal with physically, and, for infinite sets, it can actually make sense in a way that actually successfully predicts results that do hold in the real world and provides a really convenient way to define a lot of consistent properties of the continuum itself.

However, if you're dealing with a problem where you can't always usefully distinguish between elements across arbitrary set-like objects; then it's not a useful axiom and ZFC is not the formalism you want to use. Most problems we analyze in the real world, that's actually something that we can usefully assume, hence why it's such a successful and common theory, even if it leads to physical paradoxes like Banac-Tarsky, as mentioned.

Mathematicians, in practice, fully understand what you mean with your complaint about "completion," but, the beauty of these formal infinities is the guarantee it gives you that it'll never break down as a predictive theory no matter the length of time or amount of elements you consider or the needed level of precision; the fact that it can't truly complete is precisely the point. Also, within the formal system used, we absolutely can consistently define what the completion would be at "infinity," as long as you treat it correctly and don't break the rules. Again, this is useful because it allows you to bridge multiple real problems that seemingly were unrelated and it pushes "representative errors" to those paradoxes and undefined statements of the theory (thanks, Gödel).

If it helps, the transfinite cardinalities (what you call infinity) you are worried about are more related to rates than counts, even if they have some orderable or count-like properties. In the strictest sense, you can actually drop into archimedian math, which you might find very enjoyable to read about or use, and it, in a very loose sense, kinda pushes the idea of infinity from rates of counts to rates of reaching arbitrary levels of precision.