Over signature $S= \{f \}$, $n_f=1$, find a formula $\phi$ so that for every structure, $M \vDash \phi \implies|M|$ is infinite

Question

Over signature $S= \{f \}$, $n_f=1$, find a formula $\phi$ so that for every structure, $M \vDash \phi \implies |M|$ is infinite

I'm trying to solve this for a really long time. I tried to perhaps think of infinite orbits, however I can't see how to define this in a finite formula. Any assistance will be great!

Answer 1

HINT: If $X$ is a finite set, then every function $f\colon X\to X$ is injective if and only if it is surjective. What sort of condition would imply that $X$ is infinite? Now use the fact that you have a function in your language to talk about.