[FOM] Infinitesimal calculus
Vaughan Pratt
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?
Vaughan Pratt
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