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I am a high school rising senior with an interest in mathematics, and I will be taking AP calculus AB next year. I have been doing research online, and recently came across hyperreal numbers, which I believe (correct me if I'm wrong) to be an idea featuring in non-standard analysis. My question is this: What are the best introductory books for learning non-standard analysis (furthermore, does the Dover books on Mathematics series have any such books). I do have a basic understanding of proofs, but know no more computational math than simple limits, if that helps.

Thanks

Suganth Kannan
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Conan G.
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    I'd recommend Goldblatt's Lectures on the Hyperreals. You can get it free if you know how to mooch off a university library, which isn't too hard. But understand that the books you find will be of a very different flavor than you're used to from your calculus. This is not to discourage you by any means, but to inform you that any books you find will be "higher" math books, and there'll likely be some difficulty transitioning, or at least tedium. Best of luck in your foray! – AJY Jul 07 '16 at 19:41
  • It depends a bit on what you mean by "learning nonstandard analysis". First of all I think you really mean "learning hyperreal analysis" in this case. There is another theory involving infinitesimals but not infinities which is called "smooth infinitesimal analysis"; I would claim that this is farther removed from standard analysis but there is still interest there. I would probably recommend avoiding it: its motivation, results, and methodology are significantly different from standard analysis. – Ian Jul 07 '16 at 19:44
  • (Cont.) But within hyperreal analysis there are basically two significantly different constructions: there are Robinson-type constructions and there is Nelson's Internal Set Theory. Ordinarily the construction wouldn't make any real difference, but actually in this case the choice of construction affects the "interface" that you use to work with the hyperreal structure that you constructed. – Ian Jul 07 '16 at 19:44
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    To see how nonstandard analysis is applied to calculus, get Jerome Keisler's book Elementary Calculus, which uses Robinson's nonstandard approach throughout. I'm not sure what's best at a more foundational level. Here's one of the simplest examples of the "transfer principle" of nonstandard analysis, which gives some idea of the flavor of the reasoning involved: If $n$ is a positive integer, which may be infinite, then every "internal" one-to-one function from ${1,2,3,\ldots,n}$ to ${1,2,3,\ldots,n,n+1,n+2}$ omits exactly two members of the latter set from its image$,\ldots\qquad$ – Michael Hardy Jul 07 '16 at 19:45
  • (Cont.) Anyway, the one elementary reference on hyperreal analysis is Keisler's calculus book, which is a Dover book: https://www.amazon.com/Elementary-Calculus-Infinitesimal-Approach-Mathematics/dp/0486484521/ref=sr_1_1?ie=UTF8&qid=1467920700&sr=8-1&keywords=keisler+calculus I should say that this is a calculus book, it is not really an analysis book. If you want something more theoretical, you should clarify your question accordingly. – Ian Jul 07 '16 at 19:45
  • $\ldots,$even though if $n$ is infinite, you can have a one-to-one function from the former set to the latter that omits infinitely many of the latter from its image.$\qquad$ – Michael Hardy Jul 07 '16 at 19:45
  • @MichaelHardy Sorry that we split up each other's continued comments. – Ian Jul 07 '16 at 19:46
  • Maybe this is helpful: http://math.stackexchange.com/questions/714646/construction-of-the-hyperreal-numbers – Thomas Jul 07 '16 at 19:54
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    My advice: Learn calculus first. Then try some of the books suggested in these comments. – GEdgar Jul 07 '16 at 21:06
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    @GEdgar : Keisler's book, which I mentioned, is not a bad place to learn calculus. – Michael Hardy Jul 07 '16 at 22:55
  • Jerry Keisler’s Elementary Calculus: An Infinitesimal Approach is freely available here. As @MichaelHardy said, it’s not a bad source for learning calculus. It covers all of the usual topics; it just does so from a non-standard point of view. If you want a bit more of the theoretical background, the companion Foundations of Infinitesimal Calculus is freely available here. – Brian M. Scott Jul 08 '16 at 22:19

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To answer one of your specific questions, there is indeed a Dover Books on Mathematics on the topic of Nonstandard Analysis: Nonstandard Analysis by Alain M. Robert.

However, this book is written for somebody who has already taken a basic intro to real analysis course and I imagine it would be more difficult than necessary for somebody who has not otherwise been introduced to calculus. Further, there are other books, like Keisler's, that are written on a more introductory level. Additionally, I would note the book by Robert takes the less common Internal Set Theory approach to nonstandard analysis which differs from the approach taken using hyperreals. You may wish to research which approach you are interested in before starting.

GPhys
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  • Regarding your second to last sentence: IST is a formal framework for the hyperreals. To put it another way, IST is to Robinson as the Cauchy completion of $\mathbb{Q}$ is to Dedekind cuts. They describe the same theory through different constructions. The analogy breaks down a bit because you do still use the "interface" given by the construction itself in each construction (for example the notion of "internal" persists throughout a IST-based hyperreal theory). Still, the two are ultimately equivalent even though there is some slightly nontrivial translation between them. – Ian Jul 07 '16 at 22:39
  • @Ian In IST you don't use the terminology hyperreals or construct them, and that's what I mean to say by referring to the Robinson approach as the one using hyperreals (as opposed to suggesting the IST approach is somehow deficient). – GPhys Jul 07 '16 at 22:44