# FOM: Why *should* infinitesimals be definable?

Moshe' Machover moshe.machover at kcl.ac.uk
Thu Nov 27 18:44:13 EST 1997

```971127mm.txt

I can't quite understand why some people are so keen to have a *definable*
infinitesimal. Why *should* there be one?

In many years of teaching Nonstandard Analysis to maths and fom graduates,
I've never been asked  by a student to provide an example of an
infinitesimal. But if I were to be asked, I would say: I *can't* give you
an actual example. These things do not really exist; they are just
fictions--but very, very useful fictions.

`...  je leur te'moignai, que je ne croyais point qu'il y
eu^t
des grandeurs ve'ritablement infinies et ve'ritablement
infinite'simales, que ce n'e'taient que des fictions mais des
fictions utiles pour abre'ger et pour parler
universellement...'
(Leibniz)

To `abre'ger' I would add also `simplifier'.  Of course Harvey Friedman is
right: infinitesimals are as undefinabe as non-principal ultrafilter. In
fact, infinitesimals *are* ultrafilters in disguise (and the non-szero
ones are non-principal). But the beauty of this disguise is that if we
count points as objects of type 0 then ultrafilters are of type 2 (sets of
set of points); while infinitesimals are *points, ie *objects of type 0.

This is why, as I mentioned in a previous posting, the definition of
compactness which is an AE^2 statement can be reduced in NSA to a *AE^0
statement:

` ...  de sorte qu'on ne diffe`re du style d'Archime`de [ie
delta-epsilon] que dans les expressions, qui sont plus directes dans
notre me'thode et plus conformes a` l'art d'inventer.'
(id.)

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