$\mathbb{N}$ is not definable in a non-standard model 
Let $\mathcal{M}$ be a non-standard model of arithmetic and consider the set of natural numbers $\mathbb{N}$. Show that $\mathbb{N}$ is NOT a definable subset of $\mathcal{M}$.

My attempt so far is to note that since $\mathcal{M}$ is a model in $Th(\mathbb{N})$ then every definable subset must have a least element. Moreover, since definable sets are closed under complement if $\mathbb{N}$ is definable then $M\setminus\mathbb{N}$ should be too, but $M\setminus \mathbb{N}$ is a set of infinite numbers and so, it doesn't have a least element.
What I want to understand is why every definable subset in $Th(\mathbb{N})$ must have a least element?
 A: Let $X \subseteq \mathcal{M}$ be defined by some formula $\phi_X$. Then, since every subset of $\mathbb{N}$ has a least element (it's well ordered) we have
$$\mathbb{N} \models \exists a . \phi_X(a) \land \forall x . (\phi_X(x) \to a \leq x)$$
This is a first order sentence saying that $a$ is a minimal element of $X^{(\mathbb{N})}$.
But now, we find $\exists a . \phi_X(a) \land \forall x . (\phi_X(x) \to a \leq x)$ is in $\text{Th}(\mathbb{N})$, and so any model of $\text{Th}(\mathbb{N})$ must also satisfy it. In particuler, $\mathcal{M}$ models it, and so $X$ has a least element in $\mathcal{M}$.
This is an application of transfer in first order logic. If we know something about some model of $\text{Th}(\mathbb{N})$, and everything in sight is first order, then we can transfer that result to other models of $\text{Th}(\mathbb{N})$ as well. So by proving the theorem for some special choice of model (in this case we chose $\mathbb{N}$, because it's well ordered), we actually get to claim the result for all models.
Indeed, we had a definable set $X$. Then we transferred our question about it $\mathbb{N}$, where the problem was easy to solve, then we transfered the result back to $\mathcal{M}$.
More generally, you can run this argument with any complete theory. You can transfer first order properties from one model to another. This is one of the big ways that model theory is used in "mainstream" mathematics.

I hope this helps ^_^
