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I would like to design a one-semester elective course on non-standard analysis for mathematics majors at my department. The target audience will be junior and senior students who, by the time they can take this course, are supposed already have taken the mandatory courses on calculus, advanced calculus; and real analysis if they are seniors. (However, their knowledge on mathematical logic is almost non-existent unless they have taken a course on the topic.)

I have not determined a course syllabus yet. At the end of the course, I want the students to learn

  • the foundations such as the construction of hyperreal field, the transfer principle etc.
  • some of elementary calculus and some results regarding real analysis covered using NSA,
  • some more "sophisticated" uses of NSA outside calculus (if time left)

In short, I don't aim to re-teach the whole elementary calculus to students but rather aim to show how it could have been done differently and why these techniques are useful in general.

Admittedly, I am no expert on the topic. Thus, I would like to ask those who have been more involved with non-standard analysis which textbooks are appropriate for purposes of such a course. (Any suggestions to the list above regarding the aims of the course are also welcome.)

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    $\begingroup$ Nonstandard Analysis by Alain Robert is a Dover Book that covers the Internal Set Theory approach to nonstandard analysis instead of the hyperreal approach. It doesn't quite fit what you want, but the last half of the book is dedicated to applications outside typical calculus so it might be worth flipping through just for ideas. $\endgroup$
    – GPhys
    Feb 7, 2018 at 18:29
  • $\begingroup$ Yes, A. Robert's book is wonderful, making Ed Nelson's IST approach seem completely reasonable and so on. :) That is, it makes the idea appear helpful (for doing other things!) rather than just an extra piece of baggage. Many NSA sources don't seem to have that attitude. Also, J. Keisler has a serious calculus book based on NSA, with a theoretical "instructors' manual" doing some background. $\endgroup$
    – paul garrett
    Jul 30, 2022 at 20:50

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One fine text is the book by Goldblatt:

Goldblatt, Robert. Lectures on the hyperreals. An introduction to nonstandard analysis. Graduate Texts in Mathematics, 188. Springer-Verlag, New York, 1998.

The book keeps the logic requirements to a minimum and gradually develops whatever material is necessary to get there. Goldblatt gives several advanced applications, including a proof of Ramsey's theorem that is particularly transparent.

I have a set of notes from which I have been teaching a graduate course for several years already. These can be found here.

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