[FOM] finite axiomatizability and categoricity

[John T. Baldwin]

Sentences versus Theories

I summarize below how the investigation of a foundational question
-- the expressive power of a single sentence --led model theorists
to extremely technical work connected with group theory and
geometry.

For most model theoretic purposes, the distinction between a
single sentence and an infinite conjunction of sentences is
pointless.  The basic model theoretic constructions: (generalized
completeness), compactness, omitting types, Ehrenfeucht-Mostowski
models, quantifier elimination cannot tell the difference.

But, one important distinction has motivated fundamental work.

What are the structural implications of a single sentence
axiomatizing a complete theory with only infinite models.

The underlying intuition is that such a sentence must in some
sense encode an order and this is incompatible with the strongest
kind of structures -- namely categoricity in power.

CATEGORICITY:

Morley asked in 1962 whether such a sentence can be
aleph_1-categorical.

Answers:

 Peretyatikin showed there is a sentence which is aleph_1
categorical but not aleph_0 categorical.  It seems the fundamental
idea is the existence a bijection from 3 space to 2-space.  See
Hodges Model Theory for a summary.

Thus, an underlying order is not essential for an axiom of
infinity but perhaps it is if aleph_0 categoricity is also
required.

Zilber (Cherlin-Harrington-Lachlan) showed there is no sentence
which is categorical in every infinite power. This is a very deep
argument including the analysis of the types of geometries on a
strongly minimal set and analysis of how an aleph_1 categorical
model is constructed from its basic strongly minimal set. There
were at least two major spin-offs from this analysis.

Spinoff I: The Cherlin-Hrushovski-Lachlan analysis of structures
which can be closely approximated by finite substructures and the
refinement to study the class of finite approximations.  This
study has close connection with finite group theory.  From a model
theoretic view the context widened from aleph_1-categorical (and
hence omega-stable) to stable and then to simple.  The recent
monograph by Cherlin and Hrushovski is the latest word.  Finite
Structures with Few Types; Annals of Math Studies Princeton 2003.

Spinoff II: Geometric Stability theory:  Buechler,Hrushovski,
Pillay, .... In particular the applications to Diophantine
geometry eventually arise from the analysis of how strongly
minimal sets can fail to be orthogonal.  See Bouscaren Model
Theory and Algebraic Geometry Springer LNM 1696 or Pillay's
Geometric Stability Theory


MORE GENERAL CONSIDERATIONS

 The underlying philosophical question continues to
stimulate work.  Macpherson has conjectured: every finitely
axiomatized aleph_0 categorical theory with infinite models has
the strict order property.

The strict order property is a particular strong way to be
unstable. It is open whether every finitely axiomatized aleph_0
categorical theory with infinite models is unstable.