[FOM] Nonstandard analysis

Harvey Friedman friedman at math.ohio-state.edu
Wed Jul 30 18:29:46 EDT 2003

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

>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

> 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

More information about the FOM mailing list