FOM: Complex vs. Real, and stability

[John Baldwin]

On Wed, 26 Jan 2000, Yoav wrote:

>  
> In answer to Mark Steiner's question concerning the model therotic difference 
> between C (the complex numbers) and R (the reals):
>  
and continued with some important points.  

Let me summarise the model
theoretic distinction a bit more broadly.  The complex field
(algebraically closed field) is a aleph one categorical theory and the
prototype of structure theories in model theory.  The real field
(real closed field) is unstable. However, the invention of o-minimality
provided significant mathematical tools for analyzing real fields.  Model
theoretic methods have established mathematical results in both areas
while clarifying, what seem to me to be fom-questions, about the role of
order in a mathematical structure.

Slightly more technically one could distinguish ( a few years ago) the
model theoretic from the algebraic geometric approach to these subjects
by the fact that model theorists could not distinguish a set definable
by an equation (i.e. a variety) from one which used negations, while this
distinction is fundamental to geometry.  The invention of Zariski
geometries by Hrushovski-Zilber remedied this difficulty and led to
significant mathematical advances. I mention this to fom because
of the `foundational significance' of finding a non-syntactic meaning
of `not' and a possible (though speculative) connection with the
questions Harvey was raising about diagrams in Euclidean geometry as
a distinguished class of formulas. (This last is only analogy).