# Teaching Infinitesmals and Non-Standard Analysis

This question is asked from a self-teacher standpoint(I am currently trying to learn more about non-standard analysis on my own), but I'd think it could be applicable to educators also.

What are good textbooks for non-standard analysis that goes in depth on how to construct the hyperreals and explores the properties of them? I have been working with Keisler's great calculus book based on infinitesimals with associated monograph. This doesn't seem to go over some of what I think are the more interesting topics like microcontinuity and other properties that the hyperreals have that the reals do not or are extensions with some changes.

Has anyone taught a course in this and can recommend best ways to learn it and good books to go with it?

• Are you teaching the course from the perspective of a logician, or an analyst?
– user507
Commented Mar 29, 2014 at 22:10
• @Ben Crowell from the perspective of a constructive analyst Commented Mar 29, 2014 at 22:22
• I recommend John Bell's Primer of Smooth Infinitesimal Analysis. It deals with a different number system than the hyperreals, but also with infinitesimals. From your perspective of constructive analysis, you might like both the logic and the smoothness of the functions.
– user173
Commented Jun 2, 2014 at 2:54

Alain Robert's book on non-standard analysis (following Edward Nelson's IST approach) was what finally convinced me that non-standard analysis could be packaged in an effective form. It is a small book, and the narrative is very unpretentious and informal, yet touches several further topics.

"Nonstandard analysis in practice", edited by Diener and Diener is very good.

"Nonstandard methods in stochastic analysis and mathematical physics", edited by Albevario, Fenstad, et al.

A. Robinson's original book is in fact quite good, too.

• At this link you can find a (work in progress) report about nonstandard analysis in statistics. Might be interesting! : stat.umn.edu/geyer/nsa Commented Mar 30, 2014 at 20:34
• @KjetilBHalvorsen, I know Charlie Geyer a little, as the stat dept is just next door to the math dept here in MN... but I was unaware that he had written this interesting manuscript! :) Commented Mar 30, 2014 at 21:22

"Infinitesimal Calculus" by James Henle A short but complete introduction to Calculus which starts by defining a hyper-real number system. Very enjoyable.

I had the pleasure of taking a course using this as a textbook, taught by the author's brother, Michael Henle, at Oberlin College. It was not my first calculus course though. I found the hyper-real number system stuff fascinating. The use of it as a basis for Calculus was boring to me at that point.

Yes, I have taught a course on this and am currently teaching a course on this, including microcontinuity you mentioned; see here. An important distintion here is between infinitesimal calculus and infinitesimal analysis. In the former, it is inappropriate to dwell on constructions of number systems, whether real or hyperreal. In the latter of course this is eminently appropriate. The ultrapower approach is more accessible to the broader mathematical public than Robinson's original approach via the compactness theorem of first order logic, in my experience.