Consider the elementary theory of the category of sets (ETCS). Inside this framework, we have that $(\mathbb N, 0, s)$ is a natural number objet and that :
- it is a model of the Peano axioms,
- it is unique up to isomorphism.
In other words, it seems to me that the ETCS provides a categorical characterization of the standard natural numbers (in the sense that is shows that the Peano axioms have an essentially unique model). But the ETCS is a first-order theory and seems to "contain" the usual Peano axioms, so does the uniqueness of the natural number object contradict the Gödel's incompleteness theorem ?
Even though ETCS proves that $\mathbb N$ is unique, that only means that each model of ETCS has a unique object in it that it thinks is a model of the Peano axioms. Different models of ETCS may have different $\mathbb N$s that are not isomorphic when we compare them at the metalevel (which is, of course, the only way we can compare them).
This is not different from the situation in ZF(C) wherein we can prove that the (second-order) Peano axioms have a unique model up to isomorphism.