Real analysis through modern infinitesimals


    • Vakil, Nader


Real analysis through modern infinitesimals

Nader Vakil

(Encyclopedia of mathematics and its applications / edited by G.-C. Rota, [140])

Cambridge University Press, 2011

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Includes bibliographical references (p. [554]-556) and index



Real Analysis Through Modern Infinitesimals provides a course on mathematical analysis based on Internal Set Theory (IST) introduced by Edward Nelson in 1977. After motivating IST through an ultrapower construction, the book provides a careful development of this theory representing each external class as a proper class. This foundational discussion, which is presented in the first two chapters, includes an account of the basic internal and external properties of the real number system as an entity within IST. In its remaining fourteen chapters, the book explores the consequences of the perspective offered by IST as a wide range of real analysis topics are surveyed. The topics thus developed begin with those usually discussed in an advanced undergraduate analysis course and gradually move to topics that are suitable for more advanced readers. This book may be used for reference, self-study, and as a source for advanced undergraduate or graduate courses.


  • Preface
  • Introduction
  • Part I. Elements of Real Analysis: 1. Internal set theory
  • 2. The real number system
  • 3. Sequences and series
  • 4. The topology of R
  • 5. Limits and continuity
  • 6. Differentiation
  • 7. Integration
  • 8. Sequences and series of functions
  • 9. Infinite series
  • Part II. Elements of Abstract Analysis: 10. Point set topology
  • 11. Metric spaces
  • 12. Complete metric spaces
  • 13. Some applications of completeness
  • 14. Linear operators
  • 15. Differential calculus on Rn
  • 16. Function space topologies
  • Appendix A. Vector spaces
  • Appendix B. The b-adic representation of numbers
  • Appendix C. Finite, denumerable, and uncountable sets
  • Appendix D. The syntax of mathematical languages
  • References
  • Index.

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