>>12672702model theory and metamathematics, as anon here
>>12672729 suggests. It is not category theory tho, as anon here claims
>>12673263>>12673502This
>>12679296The adequate way would be to define the natural numbers as any Peano system, where the latter is an infinite set that satisfies the Peano axioms. It is easy to prove that any two Peano systems are "equivalent". Then the naturals are just a way to build a Peano system, which actually does make sense if you look up what the ordinals are.
This is the thing, we really don't care we build the numbers, no one uses the set form of the numbers, it is just like making a model, no one cares which material did you use, just that it serves its purpose. Moreover no one "checks" if a set is a number since quickly we replace all set notions for arithmetic ones.
Finally if you have more doubts about the whole "iterative set" thing, I recommend you to read about the foundation and constructibility axioms
