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**Lectures on the hyperreals. An introduction to nonstandard analysis.**
*(English)*
Zbl 0911.03032

Graduate Texts in Mathematics. 188. New York, NY: Springer. xiv, 289 p. (1998).

This is a textbook on nonstandard analysis for advanced undergraduate and beginning graduate students. The author develops all the necessary tools from logic and model theory to make his book fully selfcontained. Historical references and philosophical digressions are dispersed throughout the text. As usual, the ultrapower construction is used to provide the student right from the beginning with a non-archimedean extension \(^{*}\mathbb{R}\) of the reals \(\mathbb{R}\). A restricted version of Łoś theorem then allows for a nonstandard treatment of the basic calculus. The author is here quite complete and discusses all topics that are usually covered in freshmen calculus. Of course, nonstandard proofs are not always “simpler” or more intuitive than the conventional ones. For example, the nonstandard proof of Bolzano-Weierstrass, i.e., that a monotone increasing and bounded sequence converges towards its least upper bound, is actually quite cumbersome. But it illustrates the nonstandard definition of convergency. On the other hand, many of the nonstandard proofs that involve uniform continuity or compactness are transparent and simple.

The core of the book are two chapters on “Internal and External Entities” and “Nonstandard Frameworks”. The material treated in these chapters is quite technical and shows that there is a price to pay for entering the world of nonstandard mathematics. For some mathematicians the price might be too high, however. But the author explains the logic apparatus and set-theoretic formalism in full detail in order to justify the existence of enlargements, as well as of higher-order structures with nice saturation properties. The pace is gentle and the author provides a lot of guidance and motivation for the reader. In a final chapter on “Applications”, the author gives a fairly complete exposition of the Loeb measure and derives from it Lebesgue measure. This is a substantial application of nonstandard methods and illustrates the usefulness of nonstandard concepts, like saturation, comprehension and overflow. Other applications are more algebraic and include Ramsey theory, some Boolean algebras, vector spaces, and the Hahn-Banach Theorem. The list of references contains about forty items. They are all texbooks or conference proceedings. However, historical references are given at the end of some chapters in form of endnotes.

The reviewer likes the book. It provides the interested reader with a firm foundation of the set-theoretic methods for the construction of nonstandard models. With the appropriate background of the classical theory, a reader will be well prepared to study and investigate on his own deeper and more advanced topics of conventional mathematics.

The core of the book are two chapters on “Internal and External Entities” and “Nonstandard Frameworks”. The material treated in these chapters is quite technical and shows that there is a price to pay for entering the world of nonstandard mathematics. For some mathematicians the price might be too high, however. But the author explains the logic apparatus and set-theoretic formalism in full detail in order to justify the existence of enlargements, as well as of higher-order structures with nice saturation properties. The pace is gentle and the author provides a lot of guidance and motivation for the reader. In a final chapter on “Applications”, the author gives a fairly complete exposition of the Loeb measure and derives from it Lebesgue measure. This is a substantial application of nonstandard methods and illustrates the usefulness of nonstandard concepts, like saturation, comprehension and overflow. Other applications are more algebraic and include Ramsey theory, some Boolean algebras, vector spaces, and the Hahn-Banach Theorem. The list of references contains about forty items. They are all texbooks or conference proceedings. However, historical references are given at the end of some chapters in form of endnotes.

The reviewer likes the book. It provides the interested reader with a firm foundation of the set-theoretic methods for the construction of nonstandard models. With the appropriate background of the classical theory, a reader will be well prepared to study and investigate on his own deeper and more advanced topics of conventional mathematics.

Reviewer: K.Kaiser (Houston)