Model Theory Guidance in Number Theory?

This short note mentions several areas of number theory and related parts of mathematics where model theory can potentially offer important new insights. Many of the listed above situations are very well known to number theorists. In some of them one can feel important similarities between two mathematical theories, which are still not formalized and well understood. A model theoretical analysis may provide a valuable help. The main reason to hope for such developments involving model theory, for example as a bridge between two currently separated areas in mathematics, is that in the situations discussed below it is natural to anticipate existence of certain common structures remaining invisible at the current level of knowledge. Model theoretical analysis could help to reveal some of those structures. In some of the situations one seeks a more algebraic construction lying behind analytical objects. And it is well known that model theory does provide a sort of algebraization of analytical constructions. It is also appropriate to recall that Poizat compares the inclusion model theory – mathematical logic with the inclusion arithmetic – mathematics. This note is an extended version of a talk given at a conference in spring 2005 inside the INI programme Model Theory and Applications to Algebra and Analysis.