Lectures on the Hyperreals: An Introduction to Nonstandard Analysis / Edition 1by Robert Goldblatt, P. R. Halmos, F. W. Gehring, S. Axler
Pub. Date: 10/01/1998
Publisher: Springer New York
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,
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 the 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
What are the Hyperreals?- Large Sets.- Ultrapower Construction of the Hyperreals.- The Transfer Principle.- Hyperreals Great and Small.- Convergence of Sequences and Series.- Continuous Functions.- Differentiation.- The Riemann Integral.- Topology of the Reals.- Internal and External Sets.- Internal Functions and Hyperfinite Sets.- Universes and Frameworks.- The Existence of Nonstandard Entities.- Permanence, Comprehensiveness, Saturation.- Loeb Measure.- Ramsey Theory.- Completion by Enlargement.- Hyperfinite Approximation.- Books on Nonstandard Analysis.
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