# Axiomatizability in monadic second-order logic

For my thesis in finite model theory I'm considering some basic classes of structures, and I want to show in which logical systems they can or cannot be axiomatized. I now consider the class $$\text{EVEN}=\{\mathcal{A}=(A)\mid|A|\text{ is even}\}$$ of structures with no constants/relations and an even domain. It is easily seen, using an Ehrenfeucht-Fraïssé-game, that $\text{EVEN}$ is not first-order-axiomatizable.

Considering a second-order language, one can give the axiom $$\exists R[\forall a\exists b[aRb\wedge bRa\wedge\forall c[aRc\rightarrow c=b]]]$$ which indicates I can make pairs in my structure.

What I am now interested in, is proving or disproving the axiomatizability of $\text{EVEN}$ in monadic-second-order logic, i.e. second order logic with only 1-ary relation symbols. Is there an easy/intuitive way to see this?

The answer is no. This follows from a result of Skolem, which implies that any sentence of monadic second order logic is equivalent to a (finite) propositional combination of sentences $L_n$, $n = 1, 2, \ldots$, where $L_n$ means the universe has at least $n$ elements. See my answer to Is there any formula of monadic second-order logic that is only satisfied by an infinite set? for a reference and a bit more detail.