FOM: Complex vs. Real, and stability
John Baldwin
jbaldwin at math.uic.edu
Wed Jan 26 12:42:21 EST 2000
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).
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