[FOM] uncountable structures and core mathematics II

[John Baldwin]

This is not a direct response to Harvey's questions since none of his 
categories seem to me to express what I am about to describe.

The distinction between `pure' and `applied' model theory dates from the 
very beginnings of the field (say the compactness theorem versus 
quantifier elimination for the reals).  This distinction was summarised
in the 70's as `East Coast' and `West Coast' Model Theory.

The talk called `Tameness' at 
http://www2.math.uic.edu/~jbaldwin/talks.html
is very light reading (at least the beginning) and

gives what I hope is an entertaining survey of this distinction.  I then 
proceed to discuss how the study of structures with cardinality greater 
than the continuum can influence `core mathematics'.

In particular, I point out the (almost universally accepted) fact that
there have been crucial methodological contributions. That is, techniques 
invented by Shelah to study uncountable models were essential to 
Hrushovski's solution of the geometric Mordell-Lang conjecture.  And they
continue to play a crucial role in the work of Pillay, Scanlon, etc.
in differential field theory and compact manifolds.

But I also advance the more tenuous proposition that there are direct 
connections between model theoretic properties of larger models and
currently popular mathematics.

The results discussed are mostly in ZFC -but some crucial points require 
that cardinal exponential is a strictly increasing function (at least up 
to $\aleph_omega$.


The underlying argument is that the understanding of uncountable 
structures only advanced from the combinatorial to the algebraic with the 
development of stability theory.  This subject is well-known for first 
order logic.  I will discuss the role of infinitary logic  in a 
further note.  A good starting place is the survey by Grossberg:

http://www.math.cmu.edu/~rami/Rami-NBilgi.pdf

The actual mathematical exploration of 
Cantor's paradise is the task of the next century.

John T. Baldwin
Director, Office of Mathematics Education
Department of Mathematics, Statistics, 
and Computer Science  M/C 249
jbaldwin at uic.edu
312-413-2149
Room 327 Science and Engineering Offices (SEO)
851 S. Morgan
Chicago, IL 60607

Assistant to the director
Jan Nekola: 312-413-3750