In fact, the Peano axioms are enough to represent the natural numbers, and they are much less general than ZFC. The construction with finite ordinals in ZF(C) c

In fact, the Peano axioms are enough to represent the natural numbers, and they are much less general than ZFC. The construction with finite ordinals in ZF(C) constructs a model for it within ZF(C). If you prefer something else than set theory, you should still be able to construct natural numbers satisfying the Peano axioms. In other words, any reasonable axiomatization of mathematics will give you the naturals.