The reason different "languages" are referred to is to try to prevent paradoxes like "this sentence is false," or "all sentences that do not refer to themselves" (does this sentence refer to itself? If not, then it does!). The mathematical analogue would be a type theory.

And yet, this attempt to prevent paradoxes didn't really work - arithmetic is only allowed to talk about numbers, not about itself, but Gödel's theorem is all about using numbers to talk about the proof system that's trying to prove things about numbers. If the Gödel sentence is true but unprovable (or false but nonstandard, or whatever), why not just let things talk about themselves, and call "this sentence is false" false but nonstandard (or whatever)? We've kind of lost our motivation for having this hierarchy of languages in the first place.