Finite theory with no finite model

Question

Suppose that $\mathcal{L}=\{f\}$, where $f$ is a unary function symbol. My question is: show that there is a finite $\mathcal{L}$-theory with no finite model?

I think that I can interpret $f$ as successor function and the theory be the $Th(\mathbb Z ,s)$. Is my answer true?

Answer 1

The emphasis is finite theory, though; $Th(\mathbb{Z}, s)$ consists of infinitely many sentences! So you need to pick out finitely many sentences which already force the structure to be infinite.

HINT: think about surjections and injections - what's a property infinite sets have, about these types of functions, which finite sets don't?

Answer 2

If we consider the following statements about $f$, then any model of the conjunction of these sentences must be infinite.

• $f$ is injective.
• $f$ is not surjective.