[FOM] Infinitesimal calculus

[Harvey Friedman]

> On May 25, 2009, at 10:23 PM, David Ross wrote:
>
> Harvey Friedman wrote:
>
>> The usual systems - the system of reals, with the field operations,
>> and a predicate for the integers - are second order categorical. This
>> is the right kind of categoricity to use for such a discussion.
>
> Why is second-order categorical the "right kind"?  In other words,  
> what are
> the criteria used to judge that it is "right"?
>
> 2nd order logic gives the ability to express Dedekind completeness,  
> but why
> should that one property of R have so much power in determining what  
> is the
> right model for the reals?

Second order categoricity is fundamental to mathematician's acceptance  
of the fundamental structures of mathematics. E.g., all of the major  
number systems, and also liberal portions of the set theoretic  
universe such as V(omega + omega) are nicely second order categorical,  
and that fact is generally taught in various guises. It is the best  
explanation as to why mathematicians are not going to overhaul the  
foundations of real analysis.

>> 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.
>
> See however Kanovei and Shelah, "A definable nonstandard model of the
> reals", JSL 2004.

I was thinking that Kanovei's participation in the old FOM discussion  
led to this paper. However, the paper does say

"The problem of the existence of a definable proper elementary  
extension of IR was communicated to one of the authors (Kanovei) by V.  
A. Uspensky."

Here is the abstract from the web:

A definable nonstandard model of the reals
Vladimir Kanovei and Saharon Shelah

Source: J. Symbolic Logic Volume 69, Issue 1 (2004), 159-164.

Abstract
We prove, in ZFC, the existence of a definable, countably saturated  
elementary extension of the reals.

*******

As interesting as this construction is, it doesn't change the picture  
of mathematicians greatly preferring to stick with the usual  
foundations of analysis.

Another property of the reals (with just the field structure, or with  
constants for all multivariate relations) is that it is rigid - no  
automorphisms. We can ask if their model is provably rigid, and if  
not, whether we can also have that?

The model they construct provably, in ZFC, has cardinality > c. If we  
demand that ZFC also prove that the cardinality is c, then can this be  
done?

There is a reason to be interested in cardinality c. It is a weak form  
of saying that every "real number" is determined by countably much  
information. A stronger condition would be that ZFC proves that some  
definition defines a bijection between the power set of omega and the  
points in the model.

An interesting question is: how does second order categoricity fit  
into this? It seems plausible that the construction is, provably in  
ZFC, also second order categorical using a single second order sentence.

Harvey Friedman