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Robert Goldblatt: "Lectures on the Hyperreals - An Introduction to Nonstandard Analysis"

Springer Graduate Texts in Mathematics 188.

This 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.).

The book begins with the ultrapower construction of hyperreal number systems, and proceeds to develop one-variable calculus, analysis and topology from the nonstandard perspective, emphasizing the role of the transfer principle as a working tool of mathematical practice. It then sets out the theory of enlargements of fragments of th! e mathematical universe, providing a foundation for the full-scale development of the nonstandard methodology. The final chapters apply this to a number of topics, including Loeb measure theory and its relation to Lebesgue measure on the real line, Ramsey's Theorem, nonstandard constructions of p-adic numbers and power series, and nonstandard proofs of the Stone representation theorem for Boolean algebras and the Hahn-Banach theorem. Features of the text include an early introduction of the ideas of internal, external and hyperfinite sets, and a more axiomatic set- theoretic approach to enlargements than the usual one based on superstructures.

Table of Contents:

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. Non-Standard 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.

Index

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