first order logic models Let $\Gamma$ be the set of sentences containing the formula
$\exists x(x\neq f^n(x)) $
for each $n>0$. 
I'm trying to get a feel for the models of $\Gamma$; in particular, is there a single sentence of FOL that true in exactly the models of $\Gamma$. Thanks.
 A: If there were a single sentence $\phi$ that is equivalent to the infinite collection of sentences $\exists x\,(f^n(x)\neq x)$, which I'll abbreviate as $\beta_n$, then, by the completeness theorem for first-order logic, there would be a formal deduction of $\phi$ from the $\beta_n$'s.  That deduction, being a finite list of sentences, would use only finitely many of the $\beta_n$'s.  Pick a number $m$ bigger than all the finitely many $n$'s such that $\beta_n$ is used in your deduction.  Consider a model consisting of $m$ elements and a function $f$ that permutes them cyclically --- a single cycle of length $m$.  This model satisfies all the $\beta_n$'s used in your deduction (in fact, it satisfies $\beta_n$ for every $n$ that isn't divisible by $m$), so, thanks to that deduction, it satisfies $\phi$.  But it doesn't satisfy $\beta_m$.  So  $\phi$ is not equivalent to the collection of all your $\beta$'s.
A: $$
\forall n \exists x (n > 0 \implies x \neq f^n(x))
$$

For all $n$ bigger than $0$ there exists $x$ which is not equal to $f^n(x)$.

