FOM: Godel on the foundations of nonstandard analysis

[Jon Barwise]

Moshe Machover writes:
>
>I'd be interested to hear FOM contributors' views on the foundational
>status of nonstandard analysis.
>

This is a very interesting question, about which I had time to write.  Let
me instead point to an intriguing passage from Godel 1961, in Vol III of
the collected works, page 377.

"Indeed, mathematics has evolved into ever higher abstractions, away from
matter and to ever greater clarity in its foundations (e.g. by [giving] an
exact foundation of the infinitesimal calculus [and] the complex
numbers)--thus, away from skepticism."

Comment: One interesting feature of Robinson's foundation of the
infinitesimal calculus is the lack of cateogoricity, which some find
troubling. It might be taken as a sign that there is not a unique concept
of infinitesimal, but rather competing conceptions, each shown to be
consistent by the competing models. What makes this somewhat
unsatisfactory, as compared with the natural number, the reals, or the
complexes, is in the latter case we have some assurance that we are all
talking about the same things.

How is that for brief?  Jon