FOM: model theory

[Dave Marker]

Steve:

Let me come to Anand's defence.

You clearly have no idea what is going on in model theory.
Much (but certainly not all) of what I have to say below has
little to do with the foundations of mathematics, but I am replying
because I find your ignorance quite shocking.

Fifteen years ago the applied model theory/pure model theory dichotemy
was quite clear. Applied model theory was mostly applications of
quantifier elimination to algebra. Pure model theory was working on
a general theory of structures. Starting in the early 80's the
distinction began to blur.
Here are some major trends.

1) Algebraic concerns began to arise in "pure model theory"
  -the classification theorem for finite simple groups  entered into
the Cherlin-Harrington-Lachlan proof of the non finite
axiomatizability of totally categorical theories. (Zilber's proof
used some analytic number theory). This has led to an extensive
interaction between model theory and the study of permutaiton groups.
  -representation theory of modules comes into the proof of
Vaught's conjecture for finite rank superstable theories
  -Geometric Stability Theory: Zilber began studying the combinatorial
geometry of strongly minimal sets. This was continued, generalized and
extended by Hrushovski. A key insight is that one can often recognize
algebraic structure from the combinatorial geometry of model theoretic
dependence (forking). Often one finds that intersting model theoretic
behaviour is caused by the presence of definable or interpretable
groups. These are profound insights.  They have
completely changed the way one attacks problems in pure model theory.
They have also had rich applications (which I will get to in 3)
below). I also completely agree with Anand that these results are
telling us something  about how mathematics works. While this does
not fit your narrowly conceived view of foundations, I agree with
Anand that they are foundational.


2) The study of algebraic structure under model theoretic assumptions.
This begins with Macintyre's theorem that omega-stable fields
are algebraically closed. One tries to deduce good structure theory
from model assumptions.  Some big programs:
  -The study of groups of finite Morley rank. The Cherlin conjecture
is that the simple ones are algebraic groups. There are many
parallels to the study of algebraic and finite groups.
  -The study of groups interpretable in o-minimal structures.
Pillay proved they are topological groups (Lie groups if over the
reals).  Much more work has been done.
  -The study of stable groups and stable fields.

3) Algebraic problems where pure model theory plays a role.
  -the structure of  omega-stable groups gives rise to alot of
results in differential algebra (an in particular new approaches to
differential Galois theory). (Our understanding of strongly minimal
sets also has many useful consequences).
   -the Hrushovski-Zilber analysis of Zariski geometries (a result I
still view as "foundational")
  -Hrushovski's striking application of stability theory to the
Mordell-Lang conjecture for function fields. To prove the Manin-Mumford
conjecture for  number fields he (in part with Chatzidakis) had to
extend many   stability theoretic ideas to an unstable context.
These turned out to be the "simple" theories previously introduced by
Shelah. The interplay between the algebraic examples and the general
theory have led to important developments on both sides.

Certainly there is still work purely in the Robinson tradition of
applied model theory. The study of o-minimal expansions of the reals
is the most striking example (though even there Wilkie's first proof
of the model completeness of the reals with exponentiation used
stability inspired  dimension theoretic ideas). [A side
comment: O-minimality is an interesting hybrid as it was inspired by
both "applied" (van den Dries) and "pure" (Pillay-Steinhorn) concerns.

Another example is Hrushovski's (and Macintyre's) study of
ultraproducts of algerbaically closed fields of char p with the
Frobenius automorphism.

There is also work in pure model theory: general classificaiton theory
with a set theoretic flavor (the Helskinki school is a doing alot of
this) and  Fraise style combinatorial constructions (though these are
inspired by questions in geometric model theory).  Finite model
theory is also an exciting subject (though this is more closely
related to combinatorics, probablity and computer science than
classical logic even here there are connections to pure model
theory--generic structures and 0-1 laws).

Conservatively I would guess   that at least 80% of current research
in model theory has  a significant algebraic or geometric component.

-Dave Marker