FOM: why formal power series do not work

[Kanovei]

>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 
your infinitesimals. 

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.

Vladimir