Proving that $\langle \omega+1, \leq \rangle$ and $\langle \omega + \omega^*, \leq \rangle$ are not elementarily equivalent

Question

This is an exercise from Kees Doet's Basic Model Theory. The basic idea is to show that an ordering of type $\omega + 1$ is not isomorphic to an ordering of type of $\omega + \omega^*$, where $\omega^*$ is $\omega$ with the order reversed. He suggests that we do this by finding a counterexample showing that they're not elementarily equivalent (or by finding a formula that defines a set of elements in one structure which has no analogue in the other).

The first thing that came to my mind was showing that one was well-founded but the other was not, but being well founded is not a first-order property, so it wouldn't quite fit the bill. Another thing that came to my mind was that every element in $\omega + \omega^*$ either has finitely many predecessors or finitely many successors, but that this doesn't seem to hold for $\omega + 1$. I'm not sure if this first-order definable, though. Any hints?

Answer 1

The second structure has a second greatest element. The first does not. You can write that in first order.

Answer 2

I would say that $\omega +1$ has a limit point. That is, an element which is not the minimum but has no predecessor.