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Nonstandard Analysis, Axiomatically

  • Vladimir Kanovei
  • Michael Reeken

Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages I-XVI
  2. Vladimir Kanovei, Michael Reeken
    Pages 1-10
  3. Vladimir Kanovei, Michael Reeken
    Pages 11-52
  4. Vladimir Kanovei, Michael Reeken
    Pages 53-82
  5. Vladimir Kanovei, Michael Reeken
    Pages 83-129
  6. Vladimir Kanovei, Michael Reeken
    Pages 131-178
  7. Vladimir Kanovei, Michael Reeken
    Pages 179-217
  8. Vladimir Kanovei, Michael Reeken
    Pages 219-255
  9. Vladimir Kanovei, Michael Reeken
    Pages 257-287
  10. Vladimir Kanovei, Michael Reeken
    Pages 289-315
  11. Vladimir Kanovei, Michael Reeken
    Pages 317-388
  12. Back Matter
    Pages 389-410

About this book

Introduction

In the aftermath of the discoveries in foundations of mathematiC's there was surprisingly little effect on mathematics as a whole. If one looks at stan­ dard textbooks in different mathematical disciplines, especially those closer to what is referred to as applied mathematics, there is little trace of those developments outside of mathematical logic and model theory. But it seems fair to say that there is a widespread conviction that the principles embodied in the Zermelo - Fraenkel theory with Choice (ZFC) are a correct description of the set theoretic underpinnings of mathematics. In most textbooks of the kind referred to above, there is, of course, no discussion of these matters, and set theory is assumed informally, although more advanced principles like Choice or sometimes Replacement are often mentioned explicitly. This implicitly fixes a point of view of the mathemat­ ical universe which is at odds with the results in foundations. For example most mathematicians still take it for granted that the real number system is uniquely determined up to isomorphism, which is a correct point of view as long as one does not accept to look at "unnatural" interpretations of the membership relation.

Keywords

Nonstandard ZFC calculus differential equation forcing foundations model theory problems set theory

Authors and affiliations

  • Vladimir Kanovei
    • 1
  • Michael Reeken
    • 2
  1. 1.IITP, Institute for Information TransmissionMoscowRussian Federation
  2. 2.Bergische Universität WuppertalFB C MathematikWuppertalGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-08998-9
  • Copyright Information Springer-Verlag Berlin Heidelberg 2004
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-06077-9
  • Online ISBN 978-3-662-08998-9
  • Series Print ISSN 1439-7382
  • Buy this book on publisher's site
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