Classical Model Theory of Fields

Summary

We begin with some thoughts on how model theory relates to other parts of mathematics, and on the indirect role of Gödel's incompleteness theorem in this connection. With this in mind we consider in Section 2 the fields of real and p-adic numbers and show how these algebraic objects are understood model-theoretically: theorems of Tarski, Kochen, and Macintyre. This leads naturally to a discussion of the famous work by Ax, Kochen and Ershov in the mid sixties on henselian fields and its numbertheoretic implications.

In Section 3 we add analytic structure to the real and p-adic fields, and indicate how results such as the Weierstrass preparation theorem can be used to extend much of Section 2 to this setting. Here we make contact with the theory of subanalytic sets developed by analytic geometers in the real case.

In Section 4 we focus on o-minimal expansions of the real field that are not subanalytic, such as the real exponential field (Wilkie's theorem). We indicate in a diagram the main known o-minimal expansions of the real field. We also provide a translation into the coordinate-free language of manifolds via “analytic-geometric categories”. (This has been found useful by geometers.)

1. Introduction

In model theory we associate to a structure M invariants of a logical nature like Th(M), the set of first-order sentences which are true in M. Other invariants of this kind are the category of definable sets and maps over M or over M^eq and the category of definable groups and definable homomorphisms over M or over M^eq. If we are lucky we can find a well-behaved notion of dimension for the objects in these categories, which make these objects behave more or less like algebraic varieties and algebraic groups.