# [FOM] infinitary logic and `core mathematics'

John Baldwin jbaldwin at uic.edu
Sun Apr 9 19:05:26 EDT 2006

```I have just posted the article:
The complex numbers and \\ complex exponentiation\\ Why Infinitary Logic
is necessary!

at http://www2.math.uic.edu/~jbaldwin/model.html

Here is the first paragraph which I hope justifies my calling the
attention of fom members to the article.

In this article we discuss some of the uses of model theory to
investigate the structure of the field of complex numbers with
exponentiation and associated algebraic groups. After a sketch of
some background material on the use of first order model theory in
algebra, we describe the inadequacy of the first order framework
for studying complex exponentiation.  Then, we discuss the
Zilber's program for understanding complex exponentiation using
infinitary logic and the essential role of understanding models in
cardinality greater than \$\aleph_1\$. This analysis has inspired a
number of algebraic results; we summarize some of them. We close
by discussing some consequences on `semiabelian varieties' of
the work on the model theory of uncountable models in infinitary
logic.  We place in context seminal works of Zilber
\cite{Zilbercatex,Zilbercovers,Zilberpseudoexp, Zilbersa} and
Shelah \cite{Sh48,Sh87a,Sh87b}.  The earlier works in Zilber's
program use model theory to formulate problems concerning complex
exponentiation; this motivates work in complex analysis, algebraic
geometry and number theory.  But in \cite{Zilbersa} the
interaction between core mathematics and model theory goes both
ways; the deep work of Shelah is exploited to obtain an
equivalence between categoricity conditions and non-trivial
arithmetic properties (in the sense of a number theorist) of
certain algebraic groups.

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
```