Can we claim that there are infinite number of objects when the set of the objects does not exist?
For example, there is no set of all sets, but can we still say that there are infinitely many sets (of any kind)? And what would that mean? How would we say that formally?
One way is just to say: look, there are infinitely many subsets of integers, so, of course, there are infinitely many sets of any kind overall. But saying that, we must rely on something like "a superset of an infinite set is infinite" or something similar. And what would that mean to say that there are infinitely many sets?
Another example is from here: provide-different-proofs-for-the-following-equality. The set of all proofs of a given theorem is not defined, but we can clearly describe an infinite set of such proofs (which is NOT a subset of the set of all proofs because such a set does not exist). Can we still claim that there are infinitely many proofs of the theorem, and what would that claim mean exactly?
P.S. If you are aware of any literature discussing this or similar questions, you may just guide me through the literature by providing some references. I would definitely appreciate it.