Infinite, finite and arbitrarily large models.

Question

How is it possible to have a sentence of First Order Logic with identity such that it has both finite and infinite models, but not arbitrarily large models?

Edited: (arbitrarily large -finite- models)

Answer 1

Let $\varphi$ be any sentence that is only satisfied by infinite models and let $\psi$ be the sentence $\forall x\forall y (x=y)$. Then every model of $\varphi\vee\psi$ is either infinite or has at most one element.

Answer 2

Your statement is false by upward Lowenheim-Skolem. It states that in FOL if a $\tau$-theory $T$ has an infinite model, then it has a model of any cardinality $\kappa \geq \max\{|\tau|,\aleph_0\}$.