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.)