Definability and Arithmetic

Abstract

The purpose of this article is to give an overview of some very active interactions between first order definability and arithmetic. However, even to get started we need to clarify what we mean by “first order definability”. The reader who is already familiar with these notions can safely skip directly to the second section. For the experts, we would like to clarify that this introductory note is not intended to be a comprehensive survey, and many interesting topics are left out by space limitations: uniformity and effectivity in unlikely intersections, arithmetic of complex function fields, Diophantine sets of infinite extensions ofQ, etc. Furthermore, at several points we prefer to discuss less-general theorems for the sake of