I wish to learn nonstandard analysis. Are there any good book recommendations? I'm familiar with the ZFC system, and learnt analysis the classical way. I've found some undergraduate texts, but they are too verbose.
If there are applications to complex or functional analysis, that would be great.
Thanks in advance.
This one sounds like what you want:
Arkeryd, Cutland and Henson: Nonstandard Analysis, Theory and Applications.
I took a course as undergraduate which followed (parts of) this book - I first accompanied it with the more friendly written Goldblatt to get some feeling for the subject, then switched to this one, when I also started finding Goldblatt too "verbose". It found it very well readable.
Cutland has produced other enjoyable writings and there also is Loeb, Wolff: Nonstandard analysis for the working mathematician, which I haven't read but which follows a very similar agenda at a very similar pace, according to the table of contents.
Enjoy!
I learned the material first from Robinson's own book, simply titled Non-Standard Analysis, which I quite liked. A few years later, I read Goldblatt's Lectures on the Hyperreals (link to table of contents of the book), which I would heartily recommend. Having read that, I would very much recommend Non-Archimedean fields and asymptotic expansions by Robinson and Lightstone, which seems to be seriously under-appreciated [only a few model theorists seem to have recently dug it up]; not the best introduction to non-standard analysis, but to me the best introduction to its connections with the rest of analysis.
Hey, I just peeked at your MathOverflow page and saw that you are interested in "spatial and visual arguments". So I tell you something else (which you didn't ask for):
There also is another version of analysis with nilpotent infinitesimals, i.e. elements which are not zero, but some power of which is zero. In classical logic this contradicts the field axioms, but in intuitionistic logic it can be done. J.L. Bell's Primer of Infinitesimal Anlysis develops basic analysis on these grounds, by assuming (axiomatically) that you have something like the real numbers with nilpotents. Proofs become much easier even than in Nonstandard Analysis. Only in an appendix he addresses the existence of models for his axioms - they live in toposes.
As is very nicely laid out in the preface of Moerdijk/Reyes' "Models for Smooth Infinitesimal Analysis", it is these infinitesimals which were (implicitly) used by classical geometers like Cartan, and are (implicitly) used by physicists until today. They illustrate their point with a visual proof of Stokes' theorem using nilpotents.
In the settings of Moerdijk/Reyes (which are certain toposes) there also exist real numbers which combine the two kinds of infinitesimals, nilpotents and invertibles.
Robert's book Nonstandard Analysis (Dover Publications) is where I learned nsa - it presents (slightly informally) Nelson's IST set theory, covers a selection of basic real analysis in a n-s way, then looks at some applications. You have to watch out for a few typos in the second half of the book, but it is short and easy to read. It won't teach you any model theory though.
I loved Goldblatt's book, "Lectures on the Hyperreals".
For a more sophisticated treatment, don't overlook "Nonstandard Analysis: Theory and Applications", edited by Henson (first chapter available there) and others. It seems at first blush like a collection of articles, and it is, but they are introductions to various uses of NSA and came across quite well to me. That is, the article on topology assumes that you know some standard topology, and the article on probability assumes that you know some measure theory and probability, and so on.
I learned the basics from the introductory chapters of the book
The rest of the book is toward mathematical physics but the introduction is mathematically clean (using ultrafilters language), precise and useful. When I go back I always return to that book though I read parts of many others on the topic. There is a Russian translation as well (a translation is often a sign of an importance of the book).
I should add to those Martin Davis's "Applied Nonstandard Analysis", available from Dover Publications.It's exposition is clear and logical and begins with ultrafilers by using Zarmello-Frankel set theory.
I have learned some internal set theory (IST) from Lutz and Goze's Nonstandard analysis: a practical guide with applications. It is jam-packed with lots of interesting material, and has a nifty proof of the inverse function theorem. However, since it is a bunch of lecture notes, it is not as coherent as some other books, such as Robert's Nonstandard analysis or Nelson's own papers, his own unfinished book at http://www.math.princeton.edu/~nelson/books/1.pdf, or the probability book mentioned above.
So if you want to get excited about IST and get fun ideas for using it, read Lutz and Goze. To understand it, read Nelson or Robert.
In the study of non-standard analysis, I found the published works of Prof. Robert A. Herrmann very instructive.
If you can read French, the book Analyse Non Standard by Diener and Reeb is very beautifully written, and has some material that I don't believe has appeared anywhere else. It's out of print but one can find a copy on Amazon or a pdf online easily enough.