[FOM] foundations and model theory II

John T. Baldwin jbaldwin at uic.edu
Wed Jul 9 22:08:24 EDT 2003

One of the important discoveries of the middle 20th century is the
futility of a trying to find a general foundations of mathematics.

I have tried to figure out just how to read this sentence so that it is 
even partly true. This has proved difficult.

Baldwin: As I replied to Shipman, the intent of this rather extravagant 
remark is that the foundational enterprise has
separated itself from much of what mathematicians do.  

Baldwin in another place

> The general explanation by model theorists of the Borel Tits Theorem 
> is a more
> direct contribution to the foundations of algebraic geometry and group 
> representations than results about measurable cardinals.
> (See Poizat: Stable groups  Chapter 4.)

To examine just what you mean by "foundations" here, could you explain 
further? Is there information of a "foundational" character about 
algebraic geometry and group representations that comes from this work 
in model theory that the algebraic geometers and group representation 
theorists understand and did not know?

Baldwin replies.  This will be the extent of my reply to the general 
question.  Marker's note explained the importance of  Chow's theorem in 
terms of
the difference beween algebraic and transcendental methods.  That was 
pretty convincing as a foundational issue to the model theorists then 
and now.

Let me continue briefly on the other example.  Algebraic groups (of an 
algbebraically closed field) are defined in a fairly elaborate way as 
groups that admit certain kinds of represnetations.
The Hrushovski-Weil theorem gives a simple definition: groups definable 
in an algebraically closed field.  (For this this to be satisfying you 
have to be comfortable
with definablitity; few algebraists are now. But we are surprised from 
time to time to walk down the hall and hear the algebraic geometers 
tossing around words
like `inducition on quantifiers'.  So the sociological answer may 
change.)  This allows a conceptual proof of the Borel Tits theorem.  In 
particular, one which does
not involve extensive case analsyis as all algebraic proofs do.

I strongly recommend Poizat's book -- now available in English.  The 
introductions to the various topics are not technical and will give you 
a feel for the issues.

The foundational character is in the analysis of the FUNDAMENTAL 
CONCEPTS OF THE SUBJECT UNDER STUDY -which are groups, group morphisms
etc.  not sets and orderings.

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