[Math] Meaning of quote: “model theory = algebraic geometry – fields”

Question

On the wikipedia article for model theory, it says that a modern definition of model theory is "model theory = algebraic geometry – fields" and cites Hodges, Wilfrid (1997). A shorter model theory. Cambridge: Cambridge University Press.

I don't have access to the book and it doesn't really elaborate. What exactly is Hodges talking about? What does modern model theory look like and study? How accurate is this claim (perhaps too argumentative for here?)? What do other people have to say?

Answer 1

The exact quote is:

In 1973 C.C. Chang and Jerry Keisler characterised model theory as

universal algebra plus logic

They meant the universal algebra to stand for structures and the logic to stand for logical formulas. This is neat, but it might suggest that model theorists and universal algebraists have closely related interest, which is debatable. Also it leaves out the fact that model theorists study the sets definable in a single structure by a logical formula. In this respect model theorists are much closer to algebraic geometers, who study sets of points definable by equations over a field. A more up-to-date slogan might be that model theory is

algebraic geometry minus fields

In fact some of the most striking successes of model theory have been theorems about the existence of solutions of equations over fields. Examples are the work of Ax, Kochen and Ershov on Artin's conjecture in 1965, and the proof of the Mordell-Lang conjecture for function fields by Hrushovski in 1993. These examples lie beyond the scope of this book.

As for my interpretation, I think it's rather clear: model theorists study objects which, are (very much like, or just generalizations of those) in the scope of algebraic geometry when we consider theories of fields or rings (like formulas and equations, definable sets and algebraic varieties, definable groups and algebraic groups, types and ideals), and frequently use methods inspired by or directly generalised from those used in algebraic geometry.

So in a way, model theory extends algebraic geometry beyond the case of fields.

How accurate it is depends on what kind of model theory we are talking about. From my perspective, modern model theory has strong ties to algebraic geometry, but also to other branches of algebra, as well as to analysis (real, complex, functional, Lie groups), descriptive set theory, computer science (though that's closer to finite model theory, which is a rather different animal) and likely quite a few others I've missed.