[FOM] Infinitesimal calculus

[Harvey Friedman]

> On May 25, 2009, at 4:26 PM, Vaughan Pratt wrote:
>
> 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?

Without reconstructing any proofs and seeing if I an do better, I  
think I remember that:

there is no definition which, provably in ZFC, defines a proper  
elementary extension of (R,Z,+,dot).

Abraham Robinson's foundations for nonstandard analysis use an  
elementary extension not only of (R,Z,+,dot), but also, if I recall,  
an elementary extension of R with constants for every multivariable  
predicate on R.

>> 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?

No. E.g., might be a model of ZFC with a set of structures which is  
definable in the model, yet none of the structures are definable in  
the model.

Harvey Friedman