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AuthorGoldblatt, Robert. author
TitleLectures on the Hyperreals [electronic resource] : An Introduction to Nonstandard Analysis / by Robert Goldblatt
ImprintNew York, NY : Springer New York : Imprint: Springer, 1998
Connect tohttp://dx.doi.org/10.1007/978-1-4612-0615-6
Descript XIV, 293 p. online resource

SUMMARY

There are good reasons to believe that nonstandard analysis, in some verยญ 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 exยญ press our intuitions, and so the subject poses some unique and challenging pedagogical issues. The most fundamental ofthese is how to turn the transยญ 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


CONTENT

I 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


Mathematics Functions of real variables Mathematics Real Functions



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