I think I basically agree that this is how one should consider this.
But I think there is a reasonable defence of the “ZF-universe as somewhat transcendent entities” and that is that we do virtually all of our actual maths in ZF-universes, by saying that ultimately we will be able to appeal to ZF-axioms. This makes ZF-objects pretty different from groups. E.g. I think there’s a pretty tight analogy between forcing in ZF and galois extensions (just forcing is much more complicated), but the consequences of forcing for how we do the rest of maths can be somewhat deep (e.g. CH is doomed in ZFC, consequences about Turing computability, etc). So the mystical reputation is somewhat deserved. Woodin would defend some much more complicated and involved version of this as I discussed in my post about the constructible universe.
But I agree ultimately, with our current understanding of ZF-universes that they are just another mathematical object, they just happen to be an object that we use to do other maths with, and we can step outside those objects these days with large cardinal axioms, if we’d like, and analyze other consequences of them. It’s pretty similar to how we started viewing logic and logics after Gödel’s results (i.e. clearly first-order logic is very useful because of compactness/completeness, but that doesn’t mean “Second order logic is wrong.”)
I think I basically agree that this is how one should consider this.
But I think there is a reasonable defence of the “ZF-universe as somewhat transcendent entities” and that is that we do virtually all of our actual maths in ZF-universes, by saying that ultimately we will be able to appeal to ZF-axioms. This makes ZF-objects pretty different from groups. E.g. I think there’s a pretty tight analogy between forcing in ZF and galois extensions (just forcing is much more complicated), but the consequences of forcing for how we do the rest of maths can be somewhat deep (e.g. CH is doomed in ZFC, consequences about Turing computability, etc). So the mystical reputation is somewhat deserved. Woodin would defend some much more complicated and involved version of this as I discussed in my post about the constructible universe.
But I agree ultimately, with our current understanding of ZF-universes that they are just another mathematical object, they just happen to be an object that we use to do other maths with, and we can step outside those objects these days with large cardinal axioms, if we’d like, and analyze other consequences of them. It’s pretty similar to how we started viewing logic and logics after Gödel’s results (i.e. clearly first-order logic is very useful because of compactness/completeness, but that doesn’t mean “Second order logic is wrong.”)