[FOM] Nonstandard analysis

[Harvey Friedman]

Reply to Baldwin Nonstandard Analysis.

On 7/30/03 4:50 PM, "John T. Baldwin" <jbaldwin at uic.edu> wrote:

> I think a key point is that nonstandard analysis builds in `2nd order
> properties' by taking not just
> a model but a least a fragment of the hierarchy of sets above that model --
> then forming a nonstanddard model
> of the entire edifice.  This leads to a notion of `internal set' - which does
> not arise in more ordinary
> model theoretic investigations.

Absolutely. Although I was mainly thinking that specific developments are
made that are clearly only motivated by the developments in analysis. Of
course, that's why rich "internal sets" are needed.

> 
> Cherlin-Hrushovski page 10 again:
> 
> "Since the extension is elemetary, all notions of set theory continue to have
> meaning, and (more or
> less) their usual properties.  In particular, for any set S occuring in the
> enlargement there is an
> associated collection of `all' subsets of S in the sense of the enlargement;
> this will not actually contain all
> subsets of S in general, and those which are in fact present in the
> enlargement are called `internal'."
> 
> 
> Having said that I do not deny that Friedman's suggestion that this may be
> only a manner of speaking.

I will talk to Cherlin about this.

> I do know that the
> Benedikt example is simply a way of
> expressing compactness and we presented it in a more standard way in our
> paper:
> Baldwin,J.T. and Benedikt,M." ,"Stability
> theory, permutations of indiscernibles and Embedded finite models,
> Transactions of the American
> Mathematical Society, 2000}

That's what I was guessing about Cherlin/Hrushovski, but I do not know.
> 
> Returning to a still earlier Friedman classification: He distinguieshed
> 
> 1. Nonstandard analysis as a way of providing a model for the informal
> reasoning that was prominent among mathematicians after the calculus but
> before the advent of epsilon-delta (or arguably predating the calculus).
> 
> 2. Nonstandard analysis as a development in standard mathematics which
> purports to simplify proofs in standard mathematics, replacing them with
> simpler proofs in standard mathematics.
> 
> 3. Nonstandard analysis as a certain approach(es) to certain specialized
> branches of mathematics, replacing certain standard theorems in certain
> settings in standard mathematics with alternative standard theorems in
> alternative settings in standard mathematics.
> 
> 
> I still oontend that Cherlin-Hrushovski, Gromov-Van Den Dries-Wilkie, Benedikt
> fall into category 2 --although
> of course except in the case of Van Den Dries-Wilkie the standard theorem is
> new.  

I wasn't so much interested in denying the power of nonstandard analysis for
2, as to be interested in examining some basic example here on the FOM, to
the extent possible.

I know it first hand with a range of uses of nonstandard models - not
nonstandard analysis - in the case of independence results and some other
uses.

>And certainly
> a key claim of `nonstandard analyis' as a manner of speaking is that the ease
> of expression allows
> the discovery of theorems.  Robinson's celebrated work on the Bernstein
> problem certainly fits into that
> vein.
> 
Of course, at some level, this already happened with the discovery of the
calculus. 

> 
> Harvey,  could you look at statement 3) again.  I thought I understood what
> you were saying on the first
> two readings but I don't see it now.

3 refers to new formulations of existing theorems, where the formulations
involve infinitesimals. I gather that Keisler has done this kind of thing a
lot, especially in mathematical economics and I gather in stochastic
differential equations. I remember vividly that in mathematical economics,
an agent is modeled as an infinitesimal, so there are infinitely many
agents, and the whole market is nonstandardly finite.
> 
> 
Harvey Friedman