FOM: OTT; Goedel's hype

[Moshe' Machover]

>Bill Tait:
>
>P.S. Moshe': What does OTT mean?

Over the top, ie hype.  The quotations (Baldwin and mine) from Goedel are
from his Forward to the second edition of Robinson's book.

The reasons why I think Goedel's dictum is OTT are explained in ch.  11 of
the Course in ML by John Bell and me, as well as in a paper on NSA in the
British J Phil Sc., vol 44, 1993, pp.205-21.  Briefly, they have to do with
non-canonicity.

If you define, say, continuity in NS terms then in order to show that it
is an invariant property of the objects in question (the map and the point
at which it is continuous), independent of the choice of enlargement, then
the only reasonable way to do so is to show the NS def is equivalent to
the standard one. For this reason I think NSA can never *replace* the
standard thing.

But the conservativeness results quoted by HF do not militate against the
enormous potential usefulness of NSA or its (possibly) foundational role.
(And, BTW, Robinson was not dismayed by these results. He fully realized
from the start that NSA was conservative.)

The point is that the NS formulas expressing a given property or relation
are often drastically simpler than the standard ones.  This aids
visualization, understanding and invention.  A clear example is the
definition of compactness (in general topology) which standardly is Pi^2_2
(modulo the notion of finiteness of a set of points).  The NSA equivalent
is Pi^1_2 (mod the notion of standard point).  This is a formidable
reduction in complexity, and it does help clarify the notion of compactness
and make it much easier to use.  So proofs are often considerably simplified
and clarified.  Compare Hirschfeld's NS version of the solution to
Hilbert's 5th problem with the original M-Z solution!

In addition, NSA allows (via the Lob measure and nonstandard hull) a
fantastically easy route to constructing standard artefacts, such as
measures with `made to measure' desirable properties.

I'm not sure what the foundational import of all this is. But I feel there
is some. A technique that offers such far-reaching clarification and
simplification is no `mere' technique.

As an example of a brilliant use of NSA by a good analyst, let me mention
Edward Nelson's Radiaclly Elementary Probability Theory, Annals of Math
Studies 117, 1987. Henson's work on linear spaces may also be mentioned.


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