[FOM] Infinitesimal calculus
pratt at cs.stanford.edu
Mon May 25 16:26:50 EDT 2009
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?
More information about the FOM