# FOM: why formal power series do not work

Kanovei kanovei at wminf2.math.uni-wuppertal.de
Wed Nov 12 13:58:36 EST 1997

>it occurred to me that polynomials
>with real coefficients, more precisely formal power series (since
>1/(1+x) must be infinite: 1 - x + x^2 - x^3 + ...) which are allowed to
>begin at a negative power of x (since 1/x must be x^{-1}), ought to be
>able to serve as the canonical model Jon is looking for.

>Vaughan

Try to define 2^z for the z being x^{-1}.
Get a series with inf. many negative powers.
You may admit such -- now you
have the next problem, how to linearly order

The point is that one needs not only to define
infinitesimals but then to work with them.
(Isn't here a distinction foundations and
the *other* math ?)
Euler etc. worked with this stuff quite a bit
and what they sometimes did is puzzling even
from the modern NSA standpoint (see my paper
on the Euler sin factorization in RMS, 1988, no 4).

Therefore one needs infinitesimals obeying
some rules. NSA provides the complete solution
(Transfer). There is no such a simple way
(as the *polynomial* model) to get an enlargement
fully satisfying Transfer.