[FOM] Infinitesimal calculus

[A J Franco de Oliveira]

At 21:26 25-05-2009, you wrote:


>One can't have a Dedekind-complete ordered field that contains
>infinitesimals since the infinitesimals (defined as those numbers
>sandwiched between the positive and negative standard rationals) don't
>have a sup, and the positive rationals don't have an inf.
>(Cauchy-completeness doesn't seem to run into this problem.)

Yes you can. There is no problem at all if the infinitesimals do not 
form a set (as do, for instance, the reals between 0 and 1). Recall 
that this is what happens in E. Nelson's Internel Set Theory IST (a 
conservative extension of ZFC in an extension of the language of ZFC 
with a new unary predicate standard(x)). An approach to 
infinitesimals in a simplified version of IST has been tried and 
tested and the subject of several
books and papers by the French school of NSA created by Reeb, Lutz, 
the Dieners, etc.
ajfo