Lectures on the Hyperreals : an Introduction to Nonstandard Analysis /
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 analys...
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| Format: | eBook |
| Language: | English |
| Published: |
New York, NY :
Springer New York,
1998.
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| Series: | Graduate texts in mathematics ;
188. |
| Subjects: | |
| Online Access: | Connect to the full text of this electronic book |
| Summary: | 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. |
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| Item Description: | Electronic resource. |
| Physical Description: | 1 online resource (xiv, 293 pages) |
| ISBN: | 9781461206156 (electronic bk.) 1461206154 (electronic bk.) |
| ISSN: | 0072-5285 ; |