Does there exist a formula of first-order logic that is satisfiable only on structures with infinite domains?

Question

I am curious to know if there exists a formula of first-order logic without identity =, that is satisfiable only on structures with infinite domain.

Does there exist such a formula, and if so, what is it?

Answer 1

Given any sentence with equality which is satisfied only in infinite structures, you can get one without equality by replacing $=$ with an equivalence relation everywhere. That is, replace every instance of $=$ by a new binary relation symbol $R$ and add a subsentence saying $R$ is an equivalence relation to your sentence.

Answer 2

If the semantics does not allow for the empty structure, you can see :

• Stephen Cole Kleene, Mathematical logic (1967 - Dover ed 2002), page 293 :

$\forall x \lnot R(x,x) \land \forall x \forall y \forall z (R(x,y) \land R(y,z) \to R(x,z)) \land \forall x \exists y R(x,y)$.