Call for Registration: 31st Novembertagung on the History and Philosophy of Mathematics

[Mikhail Katz]

Monroe Eskew wrote:

"Of course, nonstandard analysis is perfectly compatible with ZF, and
it was through set-theoretic methods that Robinson gave nonstandard
analysis a firm foundation and ignited a renaissance of infinitesimal
reasoning.  Are you suggesting that it is preferable to leave
infinitesimals as unformalized or as logically primitive?"

Thanks for your question.  I have a few comments.  The set-theoretic
constructions that Robinson used to give a firm foundation to
nonstandard analysis (enlargements and/or superstructures) cannot be
carried out in ZF or even ZF + DC; they require stronger forms of AC:
at the least the existence of nonprincipal ultrafilters, and for its
full strength, the Prime Ideal Theorem.

Simpson introduced a useful distinction between ordinary mathematics
and set-theoretic mathematics.  While the use of AC in set-theoretic
mathematics is generally accepted, ordinary mathematics (calculus,
real and complex analysis, differential equations...) traditionally
requires nothing more than DC.

Some (many?) mathematicians have reservations about nonstandard
analysis because (in Robinson's framework) even its most elementary
parts, such as the very existence of infinitesimals, depend on strong
forms of AC.  The goal of our paper is to show that such criticism is
in fact incorrect.

The axiomatic approach to NSA treats the infinitesimal (more
precisely, the standardness predicate) as primitive.  In contrast with
the original Robinsonian framework, it does not a priori require AC,
and thus makes it possible to analyze the role of AC in nonstandard
mathematics.

The theories SPOT (conservative over ZF) and SCOT (conservative over
ZF + DC) show that nonstandard methods in ordinary mathematics are
just as effective as the standard ones.  On the other hand, the gains
are well-known at the level of both pedagogy and economy in the area
that Terry Tao referred to as "epsilon management".