Suppose we have a denumerable set $S$ with a total order $\preceq$. Is there necessarily an isomorphism $\phi:S\to\mathbb{N}$ such that $x\preceq y \Leftrightarrow \phi(x)\leq\phi(y)$?
Specifically in the case of $S=\mathbb{Q}$ I can't find one, so I wonder about the general principle.
If we allow the image to be $\mathbb{Q}$ then it seems possible, but I'm not sure it will work if we restrict it to the naturals (or even the integers).