Many will surely interpret this as a trivial question but I've found myself stuck on it for a while now. The structures in question are linear orderings and the signature consists of only the symbol $<$, interpreted in the usual way. The problem is to prove that all (non-parametrically) definable subsets of $\mathbf{\omega}$ are either finite or cofinite. $\mathbf{\zeta}$ is defined to be the order type of the integers and the hint given is to look at $\mathbf{\omega}+\mathbf{\zeta}$ (as we know $\mathbf{\omega}\equiv\mathbf{\omega}+\mathbf{\zeta}\,$).
The closest I could get to finding an answer was in this question as part of example 2 in the chosen solution - which I quote:
It is easy to show that all definable subsets over the empty set are either finite or cofinite.
Any hints would be much appreciated.
Hint: There is an automorphism of $\omega+\zeta$ that is the identity on $\omega$ and shifts each element of $\zeta$ forward by $1$. What does this tell you about the definable subsets of $\omega+\zeta$? Can you write a first-order property that all such subsets have, which when interpreted in $\omega$ means they are either finite or cofinite?