Graduate Texts in Mathematics Ser.: Lectures on the Hyperreals : An Introduction to Nonstandard Analysis by Robert Goldblatt (1998, Hardcover)

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About this product

Product Identifiers

PublisherSpringer New York

ISBN-10038798464X

ISBN-139780387984643

eBay Product ID (ePID)11038300140

Product Key Features

Number of PagesXiv, 293 Pages

LanguageEnglish

Publication NameLectures on the Hyperreals : an Introduction to Nonstandard Analysis

SubjectAlgebra / General, Mathematical Analysis

Publication Year1998

TypeTextbook

AuthorRobert Goldblatt

Subject AreaMathematics

SeriesGraduate Texts in Mathematics Ser.

FormatHardcover

Dimensions

Item Height0.3 in

Item Weight48 Oz

Item Length9.3 in

Item Width6.1 in

Additional Product Features

Intended AudienceScholarly & Professional

LCCN98-018388

ReviewsR. GoldblattLectures on the HyperrealsAn 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 authora'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, 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

Series Volume Number188

Number of Volumes1 vol.

IllustratedYes

Table Of ContentI Foundations.- 1 What Are the Hyperreals?.- 2 Large Sets.- 3 Ultrapower Construction of the Hyperreals.- 4 The Transfer Principle.- 5 Hyperreals Great and Small.- II Basic Analysis.- 6 Convergence of Sequences and Series.- 7 Continuous Functions.- 8 Differentiation.- 9 The Riemann Integral.- 10 Topology of the Reals.- III Internal and External Entities.- 11 Internal and External Sets.- 12 Internal Functions and Hyperfinite Sets.- IV Nonstandard Frameworks.- 13 Universes and Frameworks.- 14 The Existence of Nonstandard Entities.- 15 Permanence, Comprehensiveness, Saturation.- V Applications.- 16 Loeb Measure.- 17 Ramsey Theory.- 18 Completion by Enlargement.- 19 Hyperfinite Approximation.- 20 Books on Nonstandard Analysis.

SynopsisThis 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.)., There are good reasons to believe that nonstandard analysis, in some ver- sion or other, will be the analysis of the future. KURT GODEL This book is a compilation and development of lecture notes written for a course on nonstandard analysis that I have now taught several times. Students taking the course have typically received previous introductions to standard real analysis and abstract algebra, but few have studied formal logic. Most of the notes have been used several times in class and revised in the light of that experience. The earlier chapters could be used as the basis of a course at the upper undergraduate level, but the work as a whole, including the later applications, may be more suited to a beginning graduate course. This prefacedescribes my motivationsand objectives in writingthe book. For the most part, these remarks are addressed to the potential instructor. Mathematical understanding develops by a mysterious interplay between intuitive insight and symbolic manipulation. Nonstandard analysis requires an enhanced sensitivity to the particular symbolic form that is used to ex- press our intuitions, and so the subject poses some unique and challenging pedagogical issues. The most fundamental ofthese is how to turn the trans- fer principle into a working tool of mathematical practice. I have found it vi Preface unproductive to try to give a proof of this principle by introducing the formal Tarskian semantics for first-order languages and working through the proofofLos's theorem., Based on the author's lectures, this book introduces nonstandard analysis as a radically different way of viewing mathematical concepts and constructions, a source of new ideas, objects and proofs and a wellspring of powerful new principles of reasoning., There are good reasons to believe that nonstandard analysis, in some ver sion or other, will be the analysis of the future. KURT GODEL This book is a compilation and development of lecture notes written for a course on nonstandard analysis that I have now taught several times. Students taking the course have typically received previous introductions to standard real analysis and abstract algebra, but few have studied formal logic. Most of the notes have been used several times in class and revised in the light of that experience. The earlier chapters could be used as the basis of a course at the upper undergraduate level, but the work as a whole, including the later applications, may be more suited to a beginning graduate course. This prefacedescribes my motivationsand objectives in writingthe book. For the most part, these remarks are addressed to the potential instructor. Mathematical understanding develops by a mysterious interplay between intuitive insight and symbolic manipulation. Nonstandard analysis requires an enhanced sensitivity to the particular symbolic form that is used to ex press our intuitions, and so the subject poses some unique and challenging pedagogical issues. The most fundamental ofthese is how to turn the trans fer principle into a working tool of mathematical practice. I have found it vi Preface unproductive to try to give a proof of this principle by introducing the formal Tarskian semantics for first-order languages and working through the proofofLos's theorem.

LC Classification NumberQA331.5