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Just recently I became slightly interested in non-standard analysis. After a preliminary check at the subject there seem to be at least two relatively common ways of establishing the framework: Hyperreals constructed via a non-principal ultrafilter and internal set theory.

My questions are the following:

(1) What are the pros/cons of the various approaches (especially in practice, I don't really have a background/great interest in logic)?

(2) Is one of the approaches more expressive than the others? (I.e. are there situations where one of the approaches can carry out the argument in the non-standard setting, whereas the other approach needs to resort to the classical way of proving?)

(3) Reading suggestions are welcome, although I already saw a list on MathOverflow.

J. J.
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  • I answered this at http://math.stackexchange.com/questions/405492/a-few-questions-on-nonstandard-analysis/405506#405506 – Mikhail Katz Aug 17 '14 at 08:38
  • @user726947: Thanks! One thing that bothers me about the differences between the ultrafilter construction and IST is that you can iterate the ultrafilter construction to get positive numbers smaller than any hyperreal. Can you do something similar with IST? – J. J. Aug 17 '14 at 16:39
  • Certainly. The big authority on this is Hrbacek. The reference that's more or less standard is the book by Kanovei and Reeken: Kanovei, Vladimir; Reeken, Michael Nonstandard analysis, axiomatically. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2004. xvi+408 pp. ISBN: 3-540-22243-X. Also, Terry Tao has blogged extensively about just this aspect of the hyperreals. – Mikhail Katz Aug 18 '14 at 07:30
  • @user72694: Thank you again. I did not find any blog articles by Tao discussing the axiomatic approach, but I had a look at that book. I think that the ultrafilter track seems clearer to me, so I'm going to proceed with that for the time being. – J. J. Aug 19 '14 at 06:59
  • I was referring to the idea of a hierarchical structure on infinitesimals, where one level is so far from the other that it "can't be reached" by ordinary means. The reference to Tao's book can be found at http://en.wikipedia.org/wiki/Criticism_of_non-standard_analysis – Mikhail Katz Aug 19 '14 at 10:11

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For reading, Ed Nelson's book "Radically Elementary Probability Theory" is great. He uses "internal set theory". And he applies it to give, as he says, "radically elementary" proofs of very powerful, modern versions of the central limit theorem.

I'm not enough of an expert to address your other questions, other than to share my general impression that the differences in the various approaches are more about establishing the foundations of non-standard analysis, and matter less in its applications.

Lee Mosher
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  • Thanks for your answer, I actually knew about Ed Nelson's book already and I agree that it seems very nice. I was looking at a bit more in depth answer, though, so I'm not accepting this one yet. (Upvoted, though.) – J. J. Aug 17 '14 at 16:38