Skip to main content

Part of the book series: Mathematics and Its Applications ((MAIA,volume 510))

  • 551 Accesses

Abstract

In this chapter, we develop the general framework of nonstandard analysis and the necessary logic for the transfer principle. The reader who has read the first chapter of this book will appreciate that Skolem functions will no longer be needed to replace the existential quantifier. The results obtained in the last chapter using our simple transfer principle will still be valid, since the transfer principle used here extends that simple one. The outline of this chapter is similar to that of Chapter 2 of the author’s book with Albert E. Hurd, [4].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. M. Anderson, A non-standard representation for Brownian motion and Itô integration, Israel J. Math. 25(1976), 15–46.

    Article  MathSciNet  MATH  Google Scholar 

  2. N. G. de Bruijn and P. Erdős, A color problem for infinite graphs and a problem in the theory of relations, Proc. Kon. Nederl. Akad. v. Wetensch. Ser. A 54(1951), 371–373.

    Google Scholar 

  3. C. C. Chang and H. J. Keisler, Model theory, North-Holland, Amsterdam, 1973.

    MATH  Google Scholar 

  4. A. E. Hurd and P. A. Loeb, An Introduction to Nonstandard Real Analysis, Academic Press, Orlando, 1985.

    MATH  Google Scholar 

  5. P. A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211(1975), 113–122.

    Article  MathSciNet  MATH  Google Scholar 

  6. P. A. Loeb and H. Osswald, Nonstandard integration theory in topological vector lattices, Monatshefte fur Math. 124(1997), 53–82.

    Article  MathSciNet  MATH  Google Scholar 

  7. W. A. J. Luxemburg, A general theory of monads, in: Applications of Model Theory to Algebra, Analysis, and Probability, W. A. J. Luxemburg (ed.), Holt, Rinehart, and Winston, New York, 1969.

    Google Scholar 

  8. A. Robinson, On generalized limits and linear functional, Pacific J. Math. 14(1964), 269–283.

    MathSciNet  MATH  Google Scholar 

  9. —, Non-standard Analysis, North-Holland, Amsterdam, 1966.

    MATH  Google Scholar 

  10. A. Robinson and E. Zakon, A set-theoretical characterization of enlargements, in: Applications of Model Theory to Algebra, Analysis, and Probability, W. A. J. Luxemburg (ed.), Holt, Rinehart, and Winston, New York, 1969.

    Google Scholar 

  11. K. Stroyan and W. A. J. Luxemburg, Introduction to the Theory of Infinitesimals, Academic Press, New York, 1976.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Loeb, P.A. (2000). An Introduction to General Nonstandard Analysis. In: Loeb, P.A., Wolff, M. (eds) Nonstandard Analysis for the Working Mathematician. Mathematics and Its Applications, vol 510. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4168-0_2

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-4168-0_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-0-7923-6341-5

  • Online ISBN: 978-94-011-4168-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics