Yeah for sure. If a sentence S is undecidable from your axioms for the natural numbers then there are two models A and B satisfying those axioms where A satisf

Yeah for sure. If a sentence S is undecidable from your axioms for the natural numbers then there are two models A and B satisfying those axioms where A satisfies S and B satisfies not S. So which one is the standard natural numbers, is it A or B? Either A or B will be an example of something that satisfies your definition of natural numbers and yet is not the natural numbers.