Finite and Infinite in Logic

Abstract

Classical logic generally focuses on infinite structures, such as the integers. However, in many applications, including computer science, finite structures are the primary focus. In this article we consider the following question: what happens to logic when structures are restricted to be finite? We survey several central results of finite model theory --- the study of logic on finite structures. We start with the observation that many results that provide crucial tools in the general case, such as the Compactness theorem, fail in the finite case. We then present several key results of finite model theory that provide specific tools to replace some of the failed classical theorems. We focus on Ehrenfeucht-Fraissé games and 0--1 laws.