[FOM] Uncountable structures and `core mathematics'

[John Baldwin]

Here are some interactions between the study of the `uncountable' and core 
mathematics.

1) Zilber's attempt to find an axiomatization for the complex 
numbers with exponentiation (necessarily in infinitary 
logic) has the interesting twist that the canonicity (his word 
-categoricity in power is the precise interpretation) in aleph_1 is 
rather
straightforward algebra. To extend to structures of cardinality the 
continuum requires the notion of excellence - a notion of how to 
amalgamate n-dimensional systems of models. And then the results extend to 
arbitrary cardinality.  This theory was first established  by Shelah in a 
more general situation but the connections with `core math' are due to 
Zilber.

2)  In particular, in `Model Theory, Geometry, and Arithmetic of the 
universal cover of a semi-abelian variety' Zilber establishes an 
equivalence between `categoricity in uncountable powers of a certain 
infinitary sentences' and `arithmetic properties of semi-abelian 
varieties.  The notion of `arithmetic' here is that of `arithmetic 
algebraic geometry' not various logical hierarchies.

3)  See http://www.maths.ox.ac.uk/~zilber/
and http://www2.math.uic.edu/~jbaldwin/org/res.html
for more detailed accounts.

I have a couple of follow-ups on this that I should post in a few days.

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