I think that this is correct and a consequence of Gödel's incompleteness theorems, although my knowledge about this is quite sketchy.

What I was referring to more (and should have clarified better) was that there are models of arithmetic that can characterise the natural numbers uniquely (up to isomorphism) - translated to type theory, it is possible in languages like Coq and Lean to fully define what a natural number is without including any extra objects - even if you won't be able to prove all theorems about them. Notably, though, first order theories are not sufficient to define the natural numbers, for every set of first-order axioms about the natural numbers, there are "non-standard models" that can look quite weird. You need second-order logic or something equivalent in power to prevent that (the difficult axiom here is the axiom of induction which needs to be able to quantify over predicates).