You can’t talk about what the natural numbers are and are not without some form of set theory.
But you can talk about some of the properties they have, and quite often that is all we care about.
Also, the stronger your system is, the more likely it is that your formulation is inconsistent (and if the system is inconsistent, you’re definitely not describing anything meaningful). I’m much more confident that first-order Peano arithmetic is consistent than I am that first-order ZFC set theory is consistent.
But you can talk about some of the properties they have, and quite often that is all we care about.
Also, the stronger your system is, the more likely it is that your formulation is inconsistent (and if the system is inconsistent, you’re definitely not describing anything meaningful). I’m much more confident that first-order Peano arithmetic is consistent than I am that first-order ZFC set theory is consistent.