Infinitary Logic Has No Expressive Efficiency Over Finitary Logic

TL;DR

This paper proves that infinitary logic does not provide any expressive efficiency over finitary logic regarding the complexity of formulas measured by quantifier alternations.

Contribution

It establishes that any elementary first-order formula equivalent to an infinitary formula with n alternations can also be expressed with n alternations in finitary logic.

Findings

Infinitary logic does not simplify the expression of formulas compared to finitary logic.

Equivalent formulas in infinitary logic can be translated into finitary formulas with the same number of quantifier alternations.

The complexity measure based on quantifier alternations remains consistent between infinitary and finitary logic.

Abstract

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.