Different models of complete theory modeled by ($\mathbb{N}$, $\leq$) This is a homework excercise.
I have a FO language $\mathcal{L}$ without identity ($"="$) with a binary predicate symbol $P$, an $\mathcal{L}$-structure $\mathcal{N}$ with ground set $
\mathbb{N}$ and $P_\mathcal{N} = \{(a,b) \in \mathbb{N}^2 : a \leq b\}$ and a theory $T = \{\phi \mid \mathcal{N} \models \phi\}$. Are there any (countably infinite) models of $T$ besides $\mathcal{N}$ which are not isomorphic to $\mathcal{N}$?
I believe there aren't -- could you please give me a hint on how to prove that? I am also curious what happens when $\mathcal{L}$ is with identity ($"="$).
 A: As a matter of fact, there are models of $T$ which are not isomorphic to $\mathcal{N}$. Proving their existence is not hard, depending on what you know already:


*

*If you're familiar with the Compactness Theorem, consider the language $L'=(P, c)$ - $L$ augmented with a new constant symbol - and the theory $T^+$ consisting of $T$ together with axioms asserting "there are at least $n$ distinct elements $a$ satisfying $P(a, c)$." Note that this applies to absolutely any first-order theory: if a first-order theory has at least one infinite model, then it has more than one - in fact, proper class many.

*If you're not familiar with the compactness theorem, this gets a bit messier, but it's not too bad. In fact, $T$ has lots of models not isomorphic to $\mathcal{N}$ which are easy to describe (this is not true of other theories about natural numbers; see e.g. Tennenbaum's Theorem). As an example, consider a linear order which looks line "$\mathbb{N}+\mathbb{Z}$" - for instance, the underlying set is $(\mathbb{N}\times\{0\})\cup(\mathbb{Z}\times\{1\})$, and the ordering $P$ is given by $(a, b)P(c, d)$ iff $(i)$ $b=0$ and $d=1$, or $(ii)$ $b=d$ and $a<c$ in the usual sense. Showing that such a structure satisfies $T$ can be proved by a variety of methods - for instance, quantifier elimination, or Ehrenfeucht-Fraisse games.
