Lectures on the Hyperreals: An Introduction to Nonstandard Analysis

Graduate Texts in Mathematics #188

by Robert Goldblatt

This is an introduction to nonstandard analysis based on a course of lectures given several times by the author. It is suitable for use as a text at the beginning graduate or upper undergraduate level, or for self-study by anyone familiar with elementary real analysis. It presents nonstandard analysis not just as a theory about infinitely small and large numbers but as a radically different way of viewing many standard mathematical concepts and constructions; a source of new ideas, objects and proofs; and a wellspring of powerful new principles of reasoning (transfer, overflow, saturation, enlargement, hyperfinite approximation, etc.).

The book begins with the ultrapower construction of hyperreal number systems and proceeds to develop one-variable calculus, analysis and topology from the nonstandard perspective, emphasizing the role of the transfer principle as a working tool of mathematical practice. It then sets out the theory of enlargements of fragments of the mathematical universe, providing a foundation for the full-scale development of the nonstandard methodology. The final chapters apply this to a number of topics, including Loeb measure theory and its relation to Lebesgue measure on the real line, Ramsey's theorem, nonstandard constructions of p-adic numbers and power series, and nonstandard proofs of the Stone representation theorem for Boolean algebras and the Hahn-Banach theorem.

Features of the text include an early introduction of the ideas of internal, external, and hyperfinite sets, and a set-theoretic approach to enlargements more axiomatic than the usual one based on superstructures.

Genres: Mathematics, Logic

307 pages, Hardcover

First published October 1, 1998

Reviews

[Hyung Mook Kang · Rating 5 out of 5]

The best introduction to nonstandard analysis. Very elementary and self-contained, self-studying book.