Prove that $(\mathbb{N},\le)$ and $(\mathbb{Z}, \le)$ are not order isomorphic I want to prove that $(\mathbb{N},\le)$ and $(\mathbb{Z}, \le)$ are not order isomorphic. So what I want to show is that the following is not true:
$$x \le_\mathbb{N} y \iff f(x) \le_\mathbb{Z} f(y)\qquad\forall x,y \in \mathbb{N}$$
But I think it is true, isn't it? Or have I done a mistake in the definition of the orde isomorphism?
I could also want to show that $\mathbb{N}$ and $\mathbb{Z}$ are not bijections. Is this what I have to do rather than show that it contradicts the definition of orde isomorphisms? Do I have to show that $\mathbb{N}$ does not have a surjection $\phi$ to $\mathbb{Z}$?

$\mathbb{N}$ has a minimal element; $\mathbb{Z}$ doesn't. An order isomorphism preserves minimality of an individual element. Can you show this?
