Many conjectures about the natural numbers have the property that for any particular natural number, it is decidable (under some set of axioms like ZFC) whether or not the conjecture holds for that number. Nonexistence of odd perfect numbers and the Goldbach conjecture have this property, for instance.
When is it possible for the conjecture itself to be undecidable in ZFC? Intuitively I would expect never, since if it were undecidable I could find a counterexample in the universe where the conjecture is false, "port" it to the universe where it is true, and get a contradiction.
The top answer in this question, however, writes that this intuitive argument is flawed, since the natural numbers in one universe might contain elements that aren't natural numbers in the other. I don't understand how this is possible: can't I define $\mathbb{N}$ in a way that's provably unique (up to isomorphism) under ZFC? If so, how can adding an independent axiom to ZFC change the makeup of $\mathbb{N}$? Is there an easy example of how this can occur?