From: ara@zurich.ai.mit.edu (Allan Adler)
Subject: Re: nonstandard analysis
Organization: M.I.T. Artificial Intelligence Lab.
	<1992Dec9.151343.23419@ulrik.uio.no>
	<1g5pnaINN2mu@hilbert.math.ksu.edu>
	<COLUMBUS.92Dec10092145@strident.think.com>
Date: Wed, 16 Dec 1992 06:51:26 GMT

In article <COLUMBUS.92Dec10092145@strident.think.com> columbus@strident.think.com (Michael Weiss) writes:

   Nonstandard analysis led to the first proof of the theorem on polynomially
   compact operators, since reproved by standard techniques.  I'm curious---
   any other comparable success stories in recent years?


Caterina Kiefe and I proved some time ago that with respect to a certain
language, Ax's theory of pseudofinite fields (ie. the  infinite models of
the theory of finite fields) is the model completion of the theory of
procyclic fields. One of the key elements of the proof was carried out
by adapting Abraham Robinson's method of defining the Krull topology
on the Galois group of an infinite normal extension using nonstandard
analysis.

I think the term "nonstandard analysis" is overworked. The
entire theory belongs in the context of saturated models, including
ultraproducts and it is not really natural to separate it from that
context. Don't just study nonstandard analysis, study logic and model 
theory. And instead of asking what we can do with nonstandard analysis,
ask what are the applications of logic and model theory to other
parts of mathematics and vice versa.

Another application was a theorem of Lenore Blum and myself, in which
we proved that a differentially closed differential field has no
strongly normal extensions. In this case, we had to use ultraproducts,
since the argument was not apparent using only saturated models.

Something that has the flavor of nonstandard analysis in some respects
is the book Synthetic Differential Geometry, by Anders Kock. He
begins the book by talking about R, which is supposed to be the real
numbers, at least in spirit, but which he describes as a ring. And
he introduces an axiom about R, which I hope I am remembering correctly:

Axiom: Let D denote the set of elements x of R such that x^2=0.
       Let f:D-->R be any function. Then there are unique elements a,b
       of R such that f(d)=ad+b for all d in D.

D is like the monad of zero and the above axiom gives a way of looking
at any function from R to itself as looking like a linear function in the
monad of any point. 

It is wonderful to be able to differntiate all functions in this way.

The only trouble is that one can easily prove that there is no such
ring, by taking the function f to be  f(d) = 0 if d=0 and =1 if d is
not equal to 0. The condition that a,b are unique implies that D has
elements other 0, then a.0+b=0 implies b=0 and then a.d=1 for d a nonzero
element of D yields a contradiction.

Kock notes, however, that the only reason the proof works is that we
used the law of the excluded middle. If we do not accept the law
of the excluded middle, we cannot prove there is no such ring,
he claims.

To support this claim, he constructs a model of intuitionistic set
theory (alias an elementary topos) in which there is a ring object R
in which the axiom is true. He also embeds the category of manifolds
in such a topos and shows that one can prove theorems in differntial
geometry by working in this topos.

Nice book.

Allan Adler
ara@altdorf.ai.mit.edu
Newsgroups: sci.math