For any wf $\mathcal{B}$, apparently there is only a finite number of interpretations of a wf $\mathcal{B}$ on a domain with cardinality $k \in \mathbb{N}$. But why?
I received the following hint: " Let's fix a wf $\mathcal{B}$, a number $k \in \mathbb{N} $ and an interpretation with domain $D$ containing $k$ elements. How many different interpretations of the symbol in the language of $K$ that occur in the wf $\mathcal{B}$ can you have?"
How do I know how many of them? I know that any theory that has a model has a denumerable model (by Skolem-Löwenheim Theorem). But denumerable is not necessarily finite. Should I show by contradiction that $\mathcal{B}$ cannot have infinite interpretations when $D$ is finite?