[FOM] Infinitesimal calculus

[Vaughan Pratt]

On 5/24/2009 10:09 PM, Harvey Friedman wrote:
> Specifically, I raised the point that there is no definition in the
> language of set theory which, in ZFC, can be proved to form a system
> having the required properties.

What was the crux of the obstacle?

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.)

Are there other requirements that run into problems?

> I then considered whether there is a definition in the language of set
> theory which, in ZFC, can be proved to form a set (or even class) of
> systems having the required properties, all of which are isomorphic. I
> think there were similar negative results.

If they were all isomorphic wouldn't they all encounter the above 
problem, which one would assume to be preserved by isomorphism?

Vaughan Pratt