In ZFC set theory, what is called a class is a notion of the metalanguage, not the object language, right? (the collection defined by some formula)
And if so, if we talk about axiomatic set theory, and formalize exactly this reasoning, then do we have to introduce sets on this metalanguage level and its formalization (using ZFC again) what where classes before, are now sets.
So by this process of formalising the metalanguage and making it the object language of the next "layer" we get an infinite of models of (ZFC-) set theory, where even the proper classes become sets in "the new viewpoint".
But does this make any sense? By this process every model is enlarged, so there is no "largest" one.
Has someone thought about this? Or does it make no sense what I am talking about (making classes sets again...)?