$\omega +1 $ is not isomorphic to $\omega$ (in the well-ordering by $\varepsilon$)

Question

• $\omega +1 $ is not isomorphic to $\omega$ (in the well-ordering by $\varepsilon$).

I see that $\omega +1$ does have maximal element but $\omega$ is not so there is no ismorphism between $\omega +1 $ and $\omega $ but how can I write as proof this?

Answer 1

The following is how you might translate your reasoning into a slightly more formal proof.

Suppose towards contradiction that $\varphi\colon \omega+1\to \omega$ is an isomorphism. Notice that $\omega\in \omega+1$ is a maximal element.

Since order isomorphisms map maximal elements to maximal elements, it follows that $\varphi(\omega)$ is maximal in $\omega$. But $\omega$ has no maximal element, a contradiction, which finishes the proof.

As suggested by Henno Brandsma in the comments, this proof presupposes that we already know that being a maximal element is preserved by order isomorphism. If that is not completely clear, the proof of this fact should be included.