Infinitary Logic Has No Expressive Efficiency Over Finitary Logic

We can measure the complexity of a logical formula by counting the number of alternations between existential and universal quantifiers. Suppose that an elementary first-order formula $\varphi$ (in $\mathcal{L}_{\omega,\omega}$) is equivalent to a formula of the infinitary language $\mathcal{L}_{\infty,\omega}$ with $n$ alternations of quantifiers. We prove that $\varphi$ is equivalent to a finitary formula with $n$ alternations of quantifiers. Thus using infinitary logic does not allow us to express a finitary formula in a simpler way.Comment: 19 page