[FOM] Explicit constructions

John T. Baldwin jbaldwin at uic.edu
Thu Jul 3 16:01:09 EDT 2003

Friedman suggests I hold the following view:

Baldwin views model theory as a focused mathematical subject that
studies the definable sets and relations in arbitrary relational
structures. Any issues about concreteness or explicitness are not
relevant. What functions and sets are to be considered is
explicitly not within the province of model theory.

Sometimes model theory is useful in core mathematics and sometimes
it is not. But model theory is just as intrinsically important as
most other focused mathematical subjects, and needs no
justification in terms of "general intellectual interest" and the
like. Nor does it need any justification in terms of what it can
do for foundations of mathematics. As such, it is no different
than other focused mathematical subjects.

end quote.

I don't have a big quarrel with this characterization.  I think it
does miss the psychological leanings of most model theorists
towards foundational issues (as opposed to FOM -- check the
archives long ago for a too exhaustive discussion of fom vrs FOM).
To stoke the controversy very briefly, I'll outline in more detail
a `foundational' view of model theory.

One of the important discoveries of the middle 20th century is the
futility of a trying to find a general foundations of mathematics.
One of the essential contributions of the model theoretic view
point is to investigate the foundations of different mathematical
subjects via different specific formalisms. In a trite way one
studies algebraic geometry as the first order theory of
algebraically closed fields.  But this viewpoint has led to ways
to understand related topics in diophantine geometry.  Model
theory studies both the comparisons between various systems (thus
the stability hierarchy, o-minimality etc) and the analysis of
particular systems - e.g. the intensive work on expansions of the
real numbers or groups of finite Morley rank.

The key step in this analysis is to determine the subject. Thus,
in studying algebraically closed fields we are studying sets with
two binary operations and some constants; we are NOT studying the
representation of the complex numbers on what ever cardinal
happens to be the cardinality of the continuum.

Another important insight is that to study the structure of
definable subsets of a single object (e.g. the complex field C) it
is useful to study the family of all structures elementarily
equivalent to C.

A third is that certain similarities appear when studying
different systems - thus the study of all aleph_1 categorical
structures led to detailed investigation of the geometry of
strongly minimal sets.  These investigations then had specific
applications in Diophantine geometry.  These developments are
symbiotic not teleological.   Different model theorists (at
different times) focus on the specific theories (what Friedman
calls applied model theory) and the development of the general
theory.  These different foci enrich one another.  The 90's were a
high point in the study of specific theories.  There seems to be
more of a movement to the general situation in the last few years
for two reasons.  The work of e.g. Buechler, Lessmann, Pillay, Ben
Yaacov ... has tied together new insight into Shelah's pioneering
work of the 70's on nonelementary classes with developments in
simple theories, and studies of the expansions of models.  Zilber
has discovered that infinitary logic and Shelah's theory of
excellent classes provides a fruitful framework for investigation
of the complex exponential field.

 A example of the kind of foundational issues which arise is, to recall 
an old
Fom chestnut - Chow's Theorem.

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