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Lectures on the Hyperreals: An Introduction to Nonstandard Analysis (Graduate Texts in Mathematics, 188)

Lectures on the Hyperreals: An Introduction to Nonstandard Analysis (Graduate Texts in Mathematics)

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Lectures on the Hyperreals: An Introduction to Nonstandard Analysis (Graduate Texts in Mathematics, 188)

Lectures on the Hyperreals: An Introduction to Nonstandard Analysis (Graduate Texts in Mathematics)

4.85 94,612 ratings · 7,917 reviews
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Overview: 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 set-theoretic approach to enlargements more axiomatic than the usual one based on superstructures.

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Published October 1, 1998
ISBN 2245546
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Robert Goldblatt is a writer and avid reader based in the Pacific Northwest. Before pursuing writing full-time, they spent over a decade working in education. When not working on their next book, Robert Goldblatt can usually be found hiking with their dog or trying out new coffee shops in their neighborhood.

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Such a rich narrative. I adored how the story unfolded. It kept me captivated without sacrificing elegance. I will undoubtedly revisit this masterpiece.
Steven Carter
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Masterpiece from start to finish. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. An absolute must-read for connoisseurs of the genre.
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Took a moment to build, but the payoff was spectacular. The middle had a deliberate pace, but the climax was phenomenal. A wonderful companion for a quiet weekend.
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Honestly one of the most premium reading experiences lately. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. Worth every moment spent reading.
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Excellent, though slightly flawed. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. A wonderful companion for a quiet weekend.
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Excellent, though slightly flawed. A brilliant concept executed with flair. Just a few pacing choices I would change. A wonderful companion for a quiet weekend.
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A decent, if standard, experience. It possessed moments of brilliance, but the narrative felt slightly disjointed ultimately. Perhaps more aligned with a different audience's tastes.
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I thoroughly enjoyed the elegance of it. A brilliant concept executed with flair. Just a few pacing choices I would change. Quite impressive overall.
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A highly sophisticated read. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. A very distinguished novel.
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Took a moment to build, but the payoff was spectacular. It held my rapt attention, even when certain twists were telegraphed. A very distinguished novel.
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Honestly one of the most premium reading experiences lately. The prose is exquisite. It’s accessible yet undeniably sophisticated. I am now a lifelong fan of this author.
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Masterpiece from start to finish. I adored how the story unfolded. It kept me captivated without sacrificing elegance. A true literary gem.
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Took a moment to build, but the payoff was spectacular. It held my rapt attention, even when certain twists were telegraphed. Quite impressive overall.
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Absolutely brilliant. The prose is exquisite. It’s accessible yet undeniably sophisticated. An absolute must-read for connoisseurs of the genre.
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An acceptable diversion. The tempo was somewhat languid, and I found my mind wandering. Not poor, simply average. Perfectly average.
Christopher Wilson
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A profoundly satisfying read. I adored how the story unfolded. It kept me captivated without sacrificing elegance. An absolute must-read for connoisseurs of the genre.
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A profoundly satisfying read. I became deeply invested in the protagonist's journey. A masterclass in storytelling. I will undoubtedly revisit this masterpiece.
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Absolutely brilliant. The prose is exquisite. It’s accessible yet undeniably sophisticated. A true literary gem.
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Such a rich narrative. The prose is exquisite. It’s accessible yet undeniably sophisticated. I am now a lifelong fan of this author.
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A profoundly satisfying read. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. Worth every moment spent reading.
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Masterpiece from start to finish. The author orchestrated a breathtaking finale. Truly a work of art. Worth every moment spent reading.
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I was completely blown away. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An absolute must-read for connoisseurs of the genre.
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Such a rich narrative. I became deeply invested in the protagonist's journey. A masterclass in storytelling. A true literary gem.
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A decent, if standard, experience. Certain acts were fascinating, while others felt like a slight endeavor to complete. A fair 3 stars.
Jessica Evans
Jessica Evans January 25, 2024
Not entirely to my refined tastes. Certain acts were fascinating, while others felt like a slight endeavor to complete. Perfectly average.
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