The theory of the class of finite sets with at least n elements

Question

Let $n$ be a positive integer. Let $C$ be the class of finite sets with at least $n$ elements. Now, by Lowenheim-Skolem, that class is not a first-order axiomatizable class in the language of pure equality. However, it does have a theory $Th(C)$ associated with it. Is that theory axiomatized by the sentence that says there are at least $n$ elements?

Answer 1

Yes. Let $\varphi$ be the sentence "there are at least $n$ elements". Then $\varphi\in\mathrm{Th}(C)$. Conversely, let $\psi\in\mathrm{Th}(C)$. We would like to show that $\varphi\models \psi$.

Since $\psi$ is true of all finite sets of size at least $n$, by compactness or by taking an ultraproduct, $\psi$ is true of some infinite set. But all infinite sets are elementarily equivalent, so $\psi$ is true of all infinite sets. But then $\psi$ is true of all sets of size at least $n$, i.e. of all models of $\phi$, which is what we wanted to show.