$\tau = \{f\}$, where $f$ is a unary function symbol. $\operatorname{Im}(f)$ is finite. Is the class of $\tau$-structures based on these properties axiomatizable, finitely axiomatizable, or not axiomatizable?
At first I thought that since the Image is finite, the theory would be finite, so that I could prove it would not be axiomatizable with the Löwenheim–Skolem theorem, but I was told the theory wouldn't actually be finite. Any help or hints would be greatly appreciated.
Hint : use the compactness theorem, noting that "finite" gives no restriction on (finite) size