[FOM] Infinitesimal calculus

[David Ross]

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?

> 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.

David Ross