FOM: From philosophy to `applied' model theory

[John Baldwin]

This is a further response to the comments of Steve Simpson and Dave
Marker on the use of the term
applied model theory.  It arises because these  remarks coincided with
my writing a review of a paper called `Finite Covers' by Evans, McPherson
and Ivanov.  This topic shows one example of how `philosophical' questions
in model theory have evolved into technical mathematics.

In his thesis, Morley posed the question.  Can an aleph-1 categorical
theory be finitely axiomatizable?

This question has some philosophical content.  Bill Tait put it, do we
know all the  ways a single sentence can demand infinity:
  discrete linear order, dense linear; a pairing function.  (I think these
basically remain the only examples.)  
(Note that even a requirement of completeness complicates
the issue:  Asserting that a unary function is 1-1 and only one element
does not have a predecessor forces an infinite model without dealing
with how many finite cycles are permitted.  Morley asks more strongly
that the sentence be aleph_1 categorical.)

This question split into two parts depending on whether the theory was
required to be aleph_0 categorical. If not, Peretyatkin gave a
counterexample and in fact has constructed a machine (deriving from Hanf)
 for converting non-finitely axiomatized aleph_1 categorical theories to
finitely axiomatized ones.  The question of whether there is
a finitely axiomatizable strongly minimal set remains open; conceivably
this would require a `new' axiom of infinitity.

But, the proof (Zilber, Cherlin-Harrington-Lachlan) that there is
 no finitely axiomatizable theory categorical in all infinite
cardinalities, gave birth to `geometric stability theory'.
The proof required that one analyze strictly minimal sets (every definable
subset is finite or cofinite and there is no definable 
non-trivial equivalence relation)
and how the models of a totally categorical theory were constructed from
strictly minimal sets.

The analysis of strictly minimal sets was done in two ways. Cherlin and
Mills (independently) saw how to the get the required result from a part
of the classification of finite simple groups. 

 (So here is a use of nontrivial `core' (well, maybe not, according to
Bourbaki) mathematics in model
theory. This is a better example than the one I mentioned last week.  In
particular, Pillay pointed out I quoted the wrong application.)
In the other direction, Zil'ber's method  for this  analysis eventually
 worked and gave a different proof of the classification of finite two transitive
groups. (Don't hold me too tightly to the exact class of groups.)

The study of how the models of a totally categorical theory were built up
led to Zil'ber's notion of binding groups and to the Ahlbrandt-Ziegler
(and many more) analysis of finite covers.   A finite-one surjection 
$\pi: C \rightarrow W$ is a {\em  finite cover} if there is a
$0$-definable equivalence relation on $C$, whose classes are the inverse
images of points, and any relation on $W^n$ ($C^$) which is $0$-definable in the
two-sorted structure $(C,W,\pi)$ is already $0$-definable in $W$ ($C$).


The study of finite covers can be viewed as a branch of permutation group
theory; it uses techniques like Pontriagin duality and cohomology of
discrete groups.  On the other hand, Shelah notions like strong types and
the finite equivalence relation theorem play a central role (and have
motivated analogous constructions on the permutation group side.) The survey I
reviewed referred to over 20 papers analyzing these covers.  This work is
certainly a blend of model theory and group theory.  But the problems grew
up within the area so the description `applied model theory' is strained.


In closing let me point out one further development.  Suppose we weaken
finitely axiomatizable to `axiomatizable by a family of 
sentences in k-variables'.  For totally categorical theories, this is
just a reformulation of the finite axiomatizability problem.  For
general aleph_1 categorical theories, Shawn Hedman has done some
interesting though, so far, technical work.  He conjectures no non-locally
modular aleph_1 categorical theory can be finitely variable
axiomatizable.  This generalizes Abraham Robinson's question of whether
the complex field can be so axiomatized.  

One could imagine the answers to these questions
being characterized as either pure or applied model theory (depending in
some cases on which way the questions were answered).