FM-Representability and Beyond

Abstract

This work concerns representability of arithmetical notions in finite models. It follows the paper by Marcin Mostowski [1], where the notion of FM–representability has been defined. We discuss how far this notion captures the methodological idea of representing infinite sets in finite but potentially infinite domains.

We consider mainly some weakenings of the notion of FM–representability. We prove that relations weakly FM–representable are exactly those being $\Sigma_{\rm 2}^{\rm 0}$–definable. Another weakening of the notion, namely statistical representability, turns out to be equivalent to the original one. Additionally, we consider the complexity of sets of formulae naturally defined in finite models. We state that the set of sentences true in almost all finite arithmetical models is $\Sigma_{\rm 2}^{\rm 0}$–complete and that the set of formulae FM–representing some relations is $\Pi^{0}_{3}$–complete.