[FOM] RE: FOM Nonstandard Methods

[Matt Insall]

[Friedman]
Perhaps Godel thought that if one
thinks nonstandardly, taking seriously infinitesmials, then one might be
able to come up with new axioms that allow one to prove new concrete
statements.

[Insall]
In my dissertation, I did come up with a new definition.  (I would argue
that
this corresponds to a new axiom, though I doubt that what I came up with
fits
what Harvey is here referring to.)  The new definition provided a context in
which one may prove a concrete result in universal algebra.  The definition
to which I refer is the following:

A variety V of algebras is a _strong variety_ provided that for any
subvariety
W of V, and for any collection C of algebras in W, if S is the sum in the
category
V of the algebras in C, then S is a member of W.

Using nonstandard methods, and taking seriously hyperfinite sets (which are
related
to infinitesimals), I proved the following theorem:

Let V be a strong variety, and let C be a collection of locally finite
algebras in V.
Let S be the sum in the category V of the algebras in C.  Then S is locally
finite.

The intuition behind the proof was the following:

Extend the collection C to an hyperfinite collection D of hyperfinite
algebras.
Form the sum of the algebras in D.  Show that this sum is an hyperfinite
algebra,
and that it is an extension of the sum of C.

The devil is in the details.

Reference:
M. Insall, ``Nonstandard Methods and Finiteness Conditions in Algebra'', in
Zeitschr. fur Mathe. Logik und Grundlagen der Mathematik 37 (1991), pages
525 to 532.