Set theory. What are the best interpretations of a countable set. tldr: What are the set theory interpretations?
In science we have formalism (Schrodinger equation) and interpretations: 


*

*Copenhagen

*Many-worlds interpretation

*Transactional interpretation of quantum mechanics


My colleague and I might believe different interpretations (and argue about it), but our results of the calculation are the same as long as we accept the same formalism. 
Likewise there is a set theory formalism (e.i. ZFC axioms). I understand the formalism and can answer exam questions (just derive your answer from ZFC). But the only interpretation I know is the "infinite number of mathematicians in an hotel with infinite number of booked rooms".  And this interpretation doesn't feel right. 
Are there better alternative interpretations for a set theory? 

Update: I see there is uncertainty on what I am asking about. Well the question is this. When you are teaching set theory you can go for formal definitions of ZFC with formal derivation of results. Or, you can employ the "infinite mathematicians in infinitely sized hotel". Are there other models of teaching set theory?
 A: Actually I just wanted to make it a comment, but a become a bit to long and maybe it even worth an answer:
I guess you are talking about Hilbert's hotel here, but that really is just to show how "unintuitive" infinity can become, especially when talking about countable sets because one maybe feels like to understand why the natural numbers are infinite but why they really should be the "same amount" as for example the rational numbers is quite baffling. 
I think on thing to keep in mind here, is that this is mathematics and similar to your argument with quantum mechanics (where for daily life it doesn't matter what you think you are "actually" calculating) even more so this true for mathematics. And while as a physicist it is not unlikely that someone comes up to you and ask you about how finally to think about this wave function or space time or whatever, it is more than unlikely than anyone asks, what really is $\pi$? Or $0$. Sure we will say $0 := \emptyset$ and then define all natural numbers and then define the integers and then define rational numbers and finally $\pi$ is a map of a equivalence class of some Cauchy sequence of equivalence classes of rational numbers which are equivalence classes $\dots$. But of course on can wonder, that is super technical and even though it is consistent with modern set theory, it really doesn't answer at all "What is $\pi$?" And does it really exists? Maybe in a different "meta"physical world $\dots$.
From my (not so super vast) experience most mathematician don't care to much for these question or at least do not discuss them. And again very much different to physics it really wouldn't even make that big of a difference. Where as an Interpretation of QM really is of fundamental importance to not so few people.
Not sure if this addresses you question. But in conclusion I am pretty sure the approach is the former one with (depending on the teacher) some few heuristic arguments. After all setting up things rigorous and derive results is what mathematics is about, isn't?
