[FOM] infinitary logic and core mathematics

[John Baldwin]

The study of the first order theory of the complex numbers with 
exponentiation is model theoreically intractable.  By taking the zero's of 
the exponential function, one codes in arithmetic and the model theoretic 
notions 
of dimension are destroyed.

Zilber conjectures that this difficulty can be resolved by requiring the
kernel of the exponential to be isomorphic to (Z,+).  This is  a simple 
sentence in L omega-1 omega.

In the 70's Shelah discovered both stability theory and how to apply it in 
the infinitary case. These two discoveries were too big a bite. After a 
generation first order stability has been digested and with Zilber's 
insight there are now possible applications of infinitary model theory
to central mathematical structures.

For further information see the surveys and talks of Zilber:

http://www.maths.ox.ac.uk/~zilber/surveys.html

and particularly my first talk

  "The Complex Numbers and Exponentiation: Why Infinitary Logic is 
necessary"  Columbian Mathematical Association, August 2005

http://www2.math.uic.edu/~jbaldwin/talks.html


In fact one must go a bit beyond L omega_1 omega and make at least of use
of the `there exist uncountably many' quantifier.



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