TITLE: Is $\mathbb{N}$ impossible to pin down?
QUESTION [36 upvotes]: I don't know if this is appropriate for math.stackexchange, or whether philosophy.stackexchange would have been a better bet, but I'll post it here because the content is somewhat technical.
In ZFC, we can prove that the (second-order) Peano axioms have a model, call it $\mathbb{N} = (N,0,S)$. Furthermore, $\mathbb{N}$ is unique up to isomorphism. Thus, it would seem that we've pinned down $\mathbb{N}$.
However, if ZFC is consistent, then it has some very peculiar models. In particular, it has a model $(V_0,\in_0)$ whose "native" $\mathbb{N}$, let call it $\mathbb{N}_0$, is a model for the sentence "ZFC is consistent," and another model $(V_1,\in_1)$ whose "native" $\mathbb{N}$, lets call it $\mathbb{N}_1$, is a model for the sentence "ZFC is inconsistent."
But in reality, in other words, for the "real" $\mathbb{N}$, only one of these can be sentences can be true. Furthermore, the objects $\mathbb{N}_0$ and $\mathbb{N}_1$ aren't isomorphic, since the set of sentences they satisfy are different.
So it seems to me that $\mathbb{N}$ cannot truly be pinned down. Furthermore, its categoricity is, in some sense, illusory. Is this correct, or am I missing something?

REPLY [2 votes]: There's actually two distinct ways that the structure $\mathbf{N}=(\mathbb{N},S,0,+,\times)$ cannot be pinned down.

The first-order theory of $\mathbf{N}$ cannot be enumerated by a Turing machine. This can be deduced as a corollary of Godel's first incompleteness theorem.
The first-order theory of $\mathbf{N}$ isn't categorical, meaning that it doesn't pin down the structure $\mathbf{N}$ up to isomorphism. This can be viewed as a corollary of the upward Lowenheim-Skolem theorem, but it's best viewed as a corollary of the much stronger statement that the first-order theory of $\mathbf{N}$ fails even to be $\omega$-categorical.

As a general rule, whenever a non-expert mentions Godel's incompleteness theorems, I always begin by explaining this distinction; it's an empirical fact that doing so vastly increases the probability that the ensuing discussion will actually be enjoyable and thought-provoking :)