Lectures on the Hyperreals : An Introduction to Nonstandard Analysis Export

@book{citeulike:566503, abstract = {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.}, author = {Goldblatt, Robert}, citeulike-article-id = {566503}, citeulike-linkout-0 = {http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09-20&path=ASIN/038798464X}, citeulike-linkout-1 = {http://www.amazon.de/exec/obidos/redirect?tag=citeulike01-21&path=ASIN/038798464X}, citeulike-linkout-2 = {http://www.amazon.fr/exec/obidos/redirect?tag=citeulike06-21&path=ASIN/038798464X}, citeulike-linkout-3 = {http://www.amazon.co.uk/exec/obidos/ASIN/038798464X/citeulike00-21}, citeulike-linkout-4 = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/038798464X}, citeulike-linkout-5 = {http://www.worldcat.org/isbn/038798464X}, citeulike-linkout-6 = {http://books.google.com/books?vid=ISBN038798464X}, citeulike-linkout-7 = {http://www.amazon.com/gp/search?keywords=038798464X&index=books&linkCode=qs}, citeulike-linkout-8 = {http://www.librarything.com/isbn/038798464X}, day = {01}, howpublished = {Hardcover}, isbn = {038798464X}, keywords = {book, nonstandard\_analysis}, month = {October}, posted-at = {2006-03-28 05:03:03}, priority = {0}, publisher = {Springer}, series = {Graduate Texts in Mathematics}, title = {Lectures on the Hyperreals : An Introduction to Nonstandard Analysis}, url = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/038798464X}, year = {1998} }

TY - BOOK ID - citeulike:566503 L3 - citeulike-article-id:566503 N2 - 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. SE - Graduate Texts in Mathematics SN - 038798464X TI - Lectures on the Hyperreals : An Introduction to Nonstandard Analysis PB - Springer KW - book KW - nonstandard_analysis AU - Goldblatt, Robert PY - 1998/10/01/ UR - http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09-20&path=ASIN/038798464X ER -