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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 56,779 ratings · 5,487 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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After spending years working in the corporate world, Robert Goldblatt finally decided to pursue a lifelong passion for writing. They currently reside in a quiet suburb where they spend most of their time outlining new stories, reading extensively, and occasionally attempting to bake.

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David Baker
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Honestly one of the most premium reading experiences lately. The author orchestrated a breathtaking finale. Truly a work of art. An effortless 5 stars.
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Masterpiece from start to finish. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. Worth every moment spent reading.
Rebecca Carter
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Excellent, though slightly flawed. It held my rapt attention, even when certain twists were telegraphed. A wonderful companion for a quiet weekend.
Jessica Reyes
Jessica Reyes November 09, 2024
A decent, if standard, experience. It verged on repetitive in the second act. A more concise edit would have served it well. Suitable if your reading queue is empty.
Ashley  Parker
Ashley Parker August 28, 2025
A very strong entry. The middle had a deliberate pace, but the climax was phenomenal. I would highly recommend this to discerning readers.
Nancy Turner
Nancy Turner December 15, 2024
A very strong entry. A brilliant concept executed with flair. Just a few pacing choices I would change. A very distinguished novel.
Donna Johnson
Donna Johnson June 26, 2025
Exceeded all my expectations. The prose is exquisite. It’s accessible yet undeniably sophisticated. I am now a lifelong fan of this author.
Emily Flores
Emily Flores February 03, 2025
Honestly one of the most premium reading experiences lately. The prose is exquisite. It’s accessible yet undeniably sophisticated. A true literary gem.
Gary Rivera
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A highly sophisticated read. The middle had a deliberate pace, but the climax was phenomenal. I would highly recommend this to discerning readers.
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Emily Taylor July 24, 2024
Masterpiece from start to finish. I adored how the story unfolded. It kept me captivated without sacrificing elegance. A true literary gem.
Eric Clark
Eric Clark December 12, 2024
A very strong entry. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. I would highly recommend this to discerning readers.
Rebecca Lee
Rebecca Lee January 13, 2025
Such a rich narrative. The prose is exquisite. It’s accessible yet undeniably sophisticated. I will undoubtedly revisit this masterpiece.
Donna Nguyen
Donna Nguyen September 13, 2024
A profoundly satisfying read. The author orchestrated a breathtaking finale. Truly a work of art. A true literary gem.
Jacob Baker
Jacob Baker March 30, 2024
A very strong entry. The middle had a deliberate pace, but the climax was phenomenal. A solid 4 stars from me.
Sharon Torres
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Exceeded all my expectations. I adored how the story unfolded. It kept me captivated without sacrificing elegance. An effortless 5 stars.
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Matthew Williams July 13, 2025
Honestly one of the most premium reading experiences lately. I became deeply invested in the protagonist's journey. A masterclass in storytelling. An absolute must-read for connoisseurs of the genre.
Jennifer Hill
Jennifer Hill February 24, 2024
Absolutely brilliant. I became deeply invested in the protagonist's journey. A masterclass in storytelling. An effortless 5 stars.
Linda Miller
Linda Miller March 26, 2024
A very strong entry. The middle had a deliberate pace, but the climax was phenomenal. Quite impressive overall.
Brian  Parker
Brian Parker December 27, 2024
Took a moment to build, but the payoff was spectacular. A brilliant concept executed with flair. Just a few pacing choices I would change. Quite impressive overall.
Stephanie Rivera
Stephanie Rivera October 31, 2025
A highly sophisticated read. The protagonist was compelling, though some subplots felt slightly overshadowed. A very distinguished novel.
Margaret Rodriguez
Margaret Rodriguez March 17, 2024
It was adequate. It verged on repetitive in the second act. A more concise edit would have served it well. Unlikely to reread, but a reasonable expenditure of time.
Emily Moore
Emily Moore January 20, 2024
I was completely blown away. I adored how the story unfolded. It kept me captivated without sacrificing elegance. An absolute must-read for connoisseurs of the genre.
Patricia Turner
Patricia Turner November 21, 2024
Exceeded all my expectations. The author orchestrated a breathtaking finale. Truly a work of art. Worth every moment spent reading.
Ryan Lewis
Ryan Lewis October 25, 2024
Excellent, though slightly flawed. The middle had a deliberate pace, but the climax was phenomenal. I would highly recommend this to discerning readers.
Deborah Baker
Deborah Baker January 25, 2024
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.
John Rodriguez
John Rodriguez September 29, 2024
Absolutely brilliant. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An absolute must-read for connoisseurs of the genre.
David Wright
David Wright December 14, 2024
Masterpiece from start to finish. I became deeply invested in the protagonist's journey. A masterclass in storytelling. I will undoubtedly revisit this masterpiece.
Cynthia Campbell
Cynthia Campbell October 28, 2025
I was completely blown away. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I am now a lifelong fan of this author.
Emily Flores
Emily Flores August 28, 2025
I find myself somewhat conflicted. It possessed moments of brilliance, but the narrative felt slightly disjointed ultimately. Unlikely to reread, but a reasonable expenditure of time.
Andrew  Davis
Andrew Davis April 26, 2024
I thoroughly enjoyed the elegance of it. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. I would highly recommend this to discerning readers.
Rebecca Jackson
Rebecca Jackson January 17, 2025
I thoroughly enjoyed the elegance of it. The middle had a deliberate pace, but the climax was phenomenal. A solid 4 stars from me.
Jacob Roberts
Jacob Roberts January 12, 2025
Not entirely to my refined tastes. It verged on repetitive in the second act. A more concise edit would have served it well. A fair 3 stars.
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A profoundly satisfying read. I adored how the story unfolded. It kept me captivated without sacrificing elegance. I am now a lifelong fan of this author.
Donald Collins
Donald Collins July 16, 2025
A highly sophisticated read. The protagonist was compelling, though some subplots felt slightly overshadowed. A very distinguished novel.
David Evans
David Evans June 25, 2025
Such a rich narrative. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I will undoubtedly revisit this masterpiece.
Daniel Gomez
Daniel Gomez September 23, 2025
A profoundly satisfying read. I became deeply invested in the protagonist's journey. A masterclass in storytelling. I am now a lifelong fan of this author.
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