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18 of 18 people found the following review helpful
5.0 out of 5 stars Infinity analyzed, August 24, 2008
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This review is from: Infinity: Beyond the Beyond the Beyond (Paperback)
Nobody explains mathematical ideas for the layman as does Lillian R. Lieber. And the fanciful illustrations that always accompany her work, done by Hugh Gray Lieber, are amusing and informative.

Infinity: Beyond the Beyond the Beyond presents an account of how mathematics has learned to deal with the infinite, primarily through the work of Georg Cantor. Controversial at first, Cantor's set theory and transfinite arithmetic are now part of the foundations of modern mathematics. Perhaps the most startling idea to be had from this book is that infinite sets are not all the same size.

I have before me a copy of the 1953 original, as well as the 2007 abridgement. Aside from the fact that the older book is a hardcover, the abridgement is the better book. The editor, Barry Mazur, a mathematician at Harvard, has removed the dated, nonmathematical introductory material and the chapters on calculus. This book is now a superb layman's guide to the mathematics of transfinities.

If you would like more biography and less mathematics, you might try The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity, by Amir D. Aczel. And two magazine articles are worth seeking out: "Georg Cantor and the Origins of Transfinite Set Theory," by Joseph W. Dauben, Scientific American, June 1983; and "Non-Cantorian Set Theory," by Paul J. Cohen and Reuben Hersh, Scientific American, December 1967.

Note: In 1900, David Hilbert put forth a list of the 23 most important unsolved problems in mathematics. At the head of the list was Cantor's continuum hypothesis. The problem was still open when the Liebers wrote their book. In 1963, Stanford University mathematician Paul Cohen, extending work of Kurt Godel, proved that the continuum hypothesis is actually independent of the generally accepted axioms of set theory, and earned the Fields medal for it.
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10 of 10 people found the following review helpful
5.0 out of 5 stars Infinity, November 26, 2003
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Brian Marley (seattle, wa, usa) - See all my reviews
This is a great book. I first found it in my high school library. For the uninitiated, who would have thought there were different levels of infinity? This book explains infinity in a readable and entertaining way. It is too bad this book is out of print as I suspect it would still be in high demand. It would make a great title for a book club. Somebody needs to republish it!
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16 of 18 people found the following review helpful
5.0 out of 5 stars A Wonderful Book!, May 30, 2001
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"maguszero" (Greenville, SC USA) - See all my reviews
Lillian Lieber and her husband Hugh created some of the most wonderful books in the fields of mathematics, logic, and relativity. Although some of my fondest childhood memories are the hours I spent trying to fully grasp the meaning in her books, I find these same books to be no less enjoyable today as an adult. I cannot recommend her books highly enough.
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9 of 11 people found the following review helpful
5.0 out of 5 stars I can still remember, February 21, 2002
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A guy named Joe (PennSylvania, USA) - See all my reviews
As an Army brat, I found this book in the school library on the Naval base in Tianan, Tiawan in 1958.
As a 10th grader with a fondness for math, it was great. I think I'd seen a little bit about transfinite numbers in George Gamow's "1 2 3 Infinity", but this was an amazing tour of transfinite numbers, written so it could be understood by T C Mits. I learned a lot from it -- a real mind stretcher. I later recognized other books by the same author by the illustrations -- If you know her other books, nothing more need be said.
I've not seen the book in over 40 years, but decided I needed to find a copy -- it's one of the favorite books I read before college. I was looking at my copy of "The Education of T.C.Mits" and decided to see what I could find.
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10 of 13 people found the following review helpful
5.0 out of 5 stars Beware! Beware!, July 6, 2008
This review is from: Infinity: Beyond the Beyond the Beyond (Paperback)
Beware! This is not Lillian Lieber's original work. It has been abridged. Approximately one third of the original text and presumably the drawings have disappeared. In the forward, Barry Mazur, states plainly that he zapped Lillian's preface, chapter 1, one half of chapter 17, and all of chapters 18 through to 24. Gone is Lillian's introduction to SAM, Lillian's spirit creature of Science, Art, and Mathematics. Why did Mazur do this? He thought the Liebers digressed too much. He wanted them to stay on track with the main subject, transfinite mathematics. He thought that some of their worldly concerns speak less to a modern audience than they did to their readers in 1953. However we have to take Dr. Mazur's word for it, as the sections are deleted and you can no longer judge for yourself. Despite my misgivings I give a 5 star rating as what is left is still beautiful. However you may wish to try the used book market to get the original version.
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3 of 3 people found the following review helpful
4.0 out of 5 stars Looking at what infinity means, February 26, 2011
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This review is from: Infinity: Beyond the Beyond the Beyond (Paperback)
Explains a difficult and strange subject clearly and is amplified by associated and delightful drawings that illustrate and translate the mathematical points into artistic impression. The authors proceed slowly, step by step, and the work does not require advanced mathematical knowledge, just the ability to open the mind to different and new ways of understanding concepts.
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4 of 5 people found the following review helpful
5.0 out of 5 stars Great introduction to a topic not well understood., February 21, 2009
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This review is from: Infinity: Beyond the Beyond the Beyond (Paperback)
This is a wonderful book for an introduction to a topic that many don't realize how rich it is. Many think of infinity as just a really big number and it is so much more. The explanations of how big infinity is and that there is more than one infinity are clear and accessible to anyone. I particularly enjoy the formatting of the text. The text is formatted as prose so each page holds a small bite size portion of the concept she is explaining. Instead of feeling like you are presented with a mound of information to comprehend in just one page you are looking at it broken down over several pages. It makes you feel like you are ready poetry. The entire series that Lieber has written is just wonderful. A definite buy.
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1 of 1 people found the following review helpful
5.0 out of 5 stars A delightful book explaining Cantors mathematical looking at the Infinity, May 25, 2013
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This review is from: Infinity: Beyond the Beyond the Beyond (Paperback)
I first read this book 5 years ago, but now I have just have been reading it once again, and I again find the book to be fine by its easy understandable text, and the many drawings, by which trying to show geometrical and speculatively how to look, and think, about the Infinitive. And thereby being helpful especially to persons who during self-study are trying to learn about the mathematical behind the Infinity which Cantor proved, and how Cantor probably was speculating when he came to these ideas and conclusions.

But if it come to having more reading about the Infinity with even les mathematic (But personally I like heavy mathematic and formulas), then I will recommend these books: Amir D. Aczel: "The Mystery of the Aleph, Mathematics, the Kabbalah, and the Search for Infinity", and Eli Maor: "To Infinity and Beyond, A Cultural History of the Infinity". But if interested in much mathematic and geometry (much parallel to how we are watching it used in fractals) then we for example have a 152 sides, A5, book, written by Leo Zippin: "Uses of Infinity", first printed in 1962, and mine, from Dover, in 2000.

But now a day we in many books are reading about the Infinity, especially since we around 90 years ago started reading, thinking, and speculating about the Infinity of parallel Universes.
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5.0 out of 5 stars Five Stars, July 6, 2014
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This review is from: Infinity: Beyond the Beyond the Beyond (Paperback)
Thanks!
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Infinity: Beyond the Beyond the Beyond
Infinity: Beyond the Beyond the Beyond by Lillian Rosanoff Lieber (Paperback - November 1, 2007)
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