Overview
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 more axiomatic set- theoretic approach to enlargements than the usual one based on superstructures.
Full Product Details
Author: Robert Goldblatt
Publisher: Springer-Verlag New York Inc.
Imprint: Springer-Verlag New York Inc.
Edition: 1998 ed.
Volume: 188
Dimensions:
Width: 15.50cm
, Height: 1.90cm
, Length: 23.50cm
Weight: 1.360kg
ISBN: 9780387984643
ISBN 10: 038798464
Pages: 293
Publication Date: 01 October 1998
Audience:
College/higher education
,
Professional and scholarly
,
Postgraduate, Research & Scholarly
,
Professional & Vocational
Format: Hardback
Publisher's Status: Active
Availability: In Print
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Reviews
R. Goldblatt Lectures on the Hyperreals An Introduction to Nonstandard Analysis Suitable for a graduate course ... could be covered in an advanced undergraduate course ... The author's ideas on how to achieve both intelligibility and rigor ... will be useful reading for anyone intending to teach nonstandard analysis. -AMERICAN MATHEMATICAL SOCIETY
R. Goldblatt Lectures on the Hyperreals An Introduction to Nonstandard Analysis Suitable for a graduate course ... could be covered in an advanced undergraduate course ... The author's ideas on how to achieve both intelligibility and rigor ... will be useful reading for anyone intending to teach nonstandard analysis. --AMERICAN MATHEMATICAL SOCIETY
R. Goldblatt <p>Lectures on the Hyperreals <p>An Introduction to Nonstandard Analysis <p> Suitable for a graduate course . . . could be covered in an advanced undergraduate course . . . The authora (TM)s ideas on how to achieve both intelligibility and rigor . . . will be useful reading for anyone intending to teach nonstandard analysis. a AMERICAN MATHEMATICAL SOCIETY
