I have always been both fascinated and frustrated by nonstandard analysis. The fascination stems from the amazing vision of a consistent theory of infinitesimals that would realize Leibniz's dreams. One can't help but hope, when one hears that this subject exists, that it will provide easier entrance into analysis than post-Weierstrassian epsilontics.
The frustration, on the other hand, comes when I actually try to learn it. A metamathematical law of conservation of difficulty seems to apply here: the subtle play of quantifiers of standard analysis gets replaced with subtle logical constraints having to do with "transitively bounded sentences," languages, and interpretations. It seems the nonstandard approach is not quite the Leibnizian paradise one has dreamed of.
Given that, the question that seems to pose itself is "well, then what is it good for?" After all, most of us have no choice but to learn the epsilontic formalism. If learning and using nonstandard analysis means having to acquire proficiency in a whole new formalism, and if the theorems to be proved are, in the end, the same, what do we gain?
One of the great virtues of Martin Väth's Nonstandard Analysis is that the author is aware of this question and proposes an answer. He argues that constructing nonstandard worlds requires an appeal to the axiom of choice, so that it is built into the very foundation of the method. As a result, it becomes possible to give nonstandard "constructive" definitions of such things as non-measurable sets. These definitions are not, of course, constructive in the normal sense, relying on infinite integers and similar objects, but they can give some insight as to what is actually going on.
An example of this which I found particularly striking is Väth's Theorem 7.17:
A function f: D → R is uniformly continuous if and only if the relations x, y ∈ *D and x ≅ y imply *f(x) ≅ *f(y).
(Here * is the embedding of standard real analysis into nonstandard, so *D contains nonstandard points and *f is the nonstandard extension of f. The relation a ≅ b means that a – b is infinitesimal, i.e., that a and b are infinitely close.) This looks a lot like a local description of a global property, which is worth thinking about.
The book opens with a fairly compact (and therefore dense) introduction to the logical underpinnings of nonstandard analysis. This is followed by a chapter describing nonstandard real analysis. Many of the theorems in this chapter are like the one quoted above, that is, they prove the equivalence of a standard and a nonstandard characterization. That means that this book is not an account of what analysis would look like if we "went non-standard." Instead, it emphasizes the comparison between the two versions of the theory.
Then we're back to logic for a discussion of enlargements and saturated models, which allow Väth to do some interesting things with topology and functional analysis in the last chapters.
For me, the logical chapters were fairly heavy going. The book is well-written in the traditional terse way of advanced mathematics texts, and there are occasional moments of insight that I found very helpful. The English has its rough spots — it is clearly not the author's native language — but only very rarely did I feel confused about what was meant.
Overall, this is a very nice introduction to the subject, especially for readers who have the required mathematical maturity.
Fernando Q. Gouvêa heard about nonstandard analysis as an undergraduate, rushed to the library, and checked out Robinson's book. The love-hate relationship so generated still continues.