[FOM] Nonstandard Methods

Harvey Friedman friedman at math.ohio-state.edu
Wed Jul 30 09:43:15 EDT 2003

```Reply to Baldwin Nonstandard Analysis, and Lemberg Infinitesimals.

Before replying, let me first say that we should draw a distinction between

1. Nonstandard analysis.
2. Nonstandard methods (models).

Obviously 2 is much broader than 1. 1 can be viewed as 2 adapted with
applications to analysis in mind - with various notions of "applications".

If we are talking about 2, then I do little else but live and swim and
struggle in this every day for several decades, including yesterday and
today. In particular, I make a living out of using 2 in order to obtain
independence results of various kinds. But I also have used it for
miscellaneous purposes. Notably, my Theorem that "interpretability and
relative consistency are equivalent", where I build interpretations - in the
classical sense of Tarski - out of the structure of the relevant nonstandard
models. Also, I use them to prove statements in the integers in Boolean
relation theory from Mahlo cardinals of every finite order. Of course, the
proof of their independence from ZFC, or even from Mahlo cardinals of just
some finite order, is drenched with wall to wall manipulations and
constructions of nonstandard models.

A. Robinson is considered the pioneer in 1, but not so clearly in 2. For 2,
one should cite

T. Skolem, Fund. Math., vol. 23, 1934, 150-161.

where he constructs a nonstandard model of Peano arithmetic - presumably the
true sentences of arithmetic? - via an ultrapower construction.

I assume there has been an English translation? Presumably in:

T. Skolem, Peano's axioms and models of arithmetic. In: Mathematical
Interpretations  of Formal Systems (Studies in Logic and the Foundations of
Mathematics 10), North Holland, Amsterdam, 1955, p. 1-14.

Another early result predating A. Robinson's nonstandard analysis is that of
Spekker and McDowell, 1959, where they showed that every model of PA has an
elementary end extension. Did they do it for arbitrary models, or did they
use countability, which is not needed?

Also, who is credited for analyzing the unique order type of any countable
nonstandard model of PA?

My uses of nonstandard models do not build on the directions that A.
Robinson took them when he focused on nonstandard analysis.

On 7/29/03 10:05 PM, "John T. Baldwin" <jbaldwin at uic.edu> wrote:

> As an example of type 2 nonstandard analysis in Friedmans classification:
> (Nonstandard analysis as a development in standard mathematics which
> purports to simplify proofs in standard mathematics, replacing them with
> simpler proofs in standard mathematics.)
>
> I want to point to the exhaustive analysis of finite structure: Annals of Math
> Studies 152
> Finite Structures with few types by Cherlin and Hrushovski.

Didn't know it was available. They were preparing this for a long time.

> I won't at this time attempt to summarise the book but point to page 11:
>
> 'We will make use of nonstardard terminology as a convenient way of dealing
> with `large' integers.
> (See Fried and Jarden, Field Arithmetic)...  The method is based on the idea
> of replacing the standard
> model of set theory in which one usually works by a proper elementary
> extension, the "enlargement" in which
> there are "new" (hence, infinite) integers.' Cherlin and Hrushovski

Is this anything more than convenient expositional terminology, that can be
routinely removed without any work? I haven't looked to see, and Cherlin is
out of the country right now. This is presumably nonstandard models rather
than nonstandard analysis.
>
> I write only to point out two examples of this usage; Friedman points out the
> iversion of Gromov's theorem
> on groups of polynomial growth proved by Wilkie and Van den Dries.

Again, I would like to see an analysis of just what the relationship is
between nonstandard arguments and standard arguments here. Again, I assume
that this is nonstandard models rather than nonstandard analysis.

>Benedikt
> uses the techniques described above routinely
> in the study of database models. I am sure there are other examples.

Same question. Again presumably nonstandard models rather than nonstandard
analysis.

One can be sensitive to the distinction between a serious use of nonstandard
analysis, and merely expositional terminology.

HOWEVER: The various conservative extension theorems are not routine, at
least in the sense that they use technical constructions from math logic,
and show that there could be various interesting examples where the removal
of nonstandard methods is not at all routine. Perhaps even impossible
without mathematically unnatural technical machinations.

It would be interesting is "routine" and "not routine" could be subject to
foundational investigation, in this context.

On 7/30/03 1:33 AM, "Alexander M Lemberg" <sandylemberg at juno.com> wrote:

> It is true that NSA has provided a framework for efficient and intuitive
> proofs of certain results in analysis. However, I don't regard the system
> itself, burdened as it is with its galaxies and so on as compact or
> efficient.

Is "galaxies" a technical term now in nonstandard analysis?
>
> More significantly, does it reflect the original heuristic motivations
> for infinitesimals and origins of calculus? I believe that  the theory of
> "smooth analysis" does so to a far greater extent.

I would like to see you say something for the FOM about what the theory of
smooth analysis is. E.g., I am not sure if you just mean Bishop style
constructive analysis, or something else.

>... But I believe that the rival  the theory of smooth analysis
> goes more deeply into the heart of the matter.

I am sure that the FOM would like to hear what the heart of the matter
really is.

Harvey Friedman

```