Rating: 
Summary: Where's the beef ?
Review: A very wordy, but interesting book. The author intertwines the subject matter around a very well developed background, in fact a little too well developed. The main story is about the human mind's inability to contemplate infinity . The author points out that there are warnings in the Kabbalah, the Jewish book of mysticism, about peering into this aspect of mathematics. The famous mathematician Georg Cantor is credited with discovering and pioneering this area of mathematics. Mysteriously every time Cantor attempted to seriously delve into infinity theory he experienced a mental breakdown. Kurt Goebel another famous mathematician was also mentally affected by working in this area. The very basics of what they were considering was performing mathematical operations on magnitudes of infinities. Infinities of different sizes. It is a strange concept but something akin to asking, "what set has more members, a set of all the integers or the set of odd integers "? Once infinities are ordered then they may be ranked and considered eligible for mathematical operations. This is the second book I have read by Mr. Aczel. I believe the title of this book should have lead to the "meat" of the story, but the background history and the biography of Georg Cantor is given the same weight as the "mystery of the Aleph". The same was true of another book by Mr.Aczel, "God's Equation" the background material overshadowed the story. I would prefer a little more in-depth text about the heart of the matter.
Rating: 
Summary: Great book, but really no mystery.
Review: Aczel has a way of making the most mundane really interesting. He wrote one of the best non-fiction books I have ever read in Fermat's Last Theorem. With this book, he continues his combination of a math book with the movement, clarity, and pace of a novel. The unfortunate part about this book is that no matter how good Aczel tries, the subject of infinity is not as interesting.This is a chronicle with some mathematicians' obsession with, and the history of, the concept of infinity. First, he goes into the ancient origins of infinity, from Pythagoras to the discovery of irrational numbers, to the Jewish sect that dealt with infinity. Then he explains Galileo, Bolzano, and others' perspectives and expansion of infinity, including set theory. The most important part deals with George Cantor, a man who expanded upon, and ultimately went insane trying to figure out infinity. He details his steps towards madness, and his views on set theory, it's axioms, and the concept of order of infinity. Unfortunately, though told brilliantly told, doesn't capture the interest that Fermat's Last Theorem elicited. The concept of infinity, and it's order, is not that interesting a subject, as it lacks a practical purpose, and is not a problem that many have dealt with, educational or otherwise. Overall though, it's a very good book, and I highly recommend it.
Rating: 
Summary: Aczel's books are disappointing
Review: Aczel has gotten a lot of press, so I read two of his books to evaluate whether to recommend them to my students. I do not recommend his work to anyone. His topics are interesting and important, but his treatment is very disappointing. Light and fluffy writing can help ease laypeople into reading science, if it is organized and accurate. Aczel's is not. He wanders among ideas without developing them in enough depth to appreciate. He poses interesting questions as if he intends to answer them, but does not. Read Simon Singh or Timothy Ferris instead. Just as easy to read, and far more coherent and satisfying.
Rating: 
Summary: Superb!
Review: Aczel writes in a way that is utterly profound, expounding on the mysteries of the maker of all infinity, the Blessed Ein Sof (the true essence of which can never be known to humans), at the same time delving into the mysteries of transfinite theory, and its creator, Georg Cantor, who went mad trying to concieve of that which can never be fully understood. Of all of the books that I have read, this is by far one of the most profound. You don't have to a Kabbalist or lover of mathematics to enjoy this book, Aczel writes in a way that makes the macrocosm of infinity easily understandable by the avearge reader. However, love of Kabbalah and math is a plus in understanding the profundity of the message the book conveys.
Rating:
Summary: Interesting book, but not necessary
Review: Aczel wrote this book in layman terms and he did a good job explaining the concept of transfinite numbers.
However, the association of the concept of infinity with another concept, God, is totally nonsense. (Same with the application of the Incompleteness Theorem to god.)
Rating:
Summary: Mystery revealed
Review: Aczel's wonderful book is an exploration of the development of the idea of infinity through the ages. He takes us from its origins in the ancient world, right up to the frontiers of modern set theory, stopping along the way to explore non-mathematical ideas of infinity as well, such as the Jewish notion of Ein Sof (without limit) in the Kabbalah. But what really makes the book shine is the human side of the drama. Just as in his "Fermat's Last Theorem", Aczel's understanding of the lives and motivation of the cast of characters is what makes the book such a page turner. We are drawn into the intrigues of the lives of these men - their rivalries, aspirations, triumphs, and disappointments. The central character in the drama is Georg Cantor, the father of modern set theory, who died in an asylum, frustrated by his unsuccessful attempts to prove his continuum hypothesis. For the lay reader, this is a great introduction into the ideas of the transfinite. For the more mathematically inclined reader, this book may kindle a desire to explore these subjects in more depth. As for me, I've taken my old set theory tome off the shelf for a reread. Thanks, Amir!
Rating:
Summary: Finite, all too finite
Review: I really expected to like this book. God, infinity, Kabbalah -- how could you _miss_? It was okay, I guess. So what happened? Well, frankly, although the biographical information on Georg Cantor and Kurt Goedel is pretty good and the mathematical history is reliable, there's no real meat in the discussions of either infinity or the Kabbalah. Every time I thought Aczel was really going to get rolling and make a profound connection, he sort of petered out and changed the subject. It's too bad, because Aczel really does have an important point lurking in here: the mathematics of infinity really does provide a window into the Ein Sof, and there probably is a connection (both historical and deeper) between the Kabbalistic and the Cantorian uses of the Hebrew letter alef. I'd have enjoyed some more thorough exposition, even at an elementary level, of both sides of this equation. But for that, the reader will have to look (for infinity) to Rudy Rucker's _Infinity and the Mind_ or (slightly more elementary) Eli Maor's _To Infinity and Beyond_, or (for the rest) to any of numerous sources on Kabbalah. This book is only about a quarter-inch deep. On the plus side, though, I will say that this isn't a bad book for somebody who has never encountered the subject(s) before. Just don't expect a lot of specificity; Aczel usually doesn't offer much more than vague allusions.
Rating:
Summary: Complete Nonsense
Review: I would never have imagined that a quest this abstract could entail a story so human. Yet, that is exactly what Amir Aczel provides in this smooth tale of the many humbling encounters with the realm of infinity. There are two lessons from this compact survey on the effort expended and the toll imposed on those bold enough to go where no person can go. First, the urge to comprehend infinity is an ancient quest and inextricably tied to the effort to ascertain the nature of God. Second, getting to know infinity can be massively bad for one's mental health. Mr. Aczel manages an almost impossible task (infinity tends to do that) in this text. He is (a) attempting to survey an enormous amount of the history mathematics and, to some extent, religion, and (b) providing a glimpse into the lives of those mathematicians that have ventured into this field. At the heart of this book is Georg Cantor, founder of modern set theory. Cantor sought to transcent an intuitive understanding of infinity. He sought an ordered system; specifically he sought to prove what became known as the continuim hypothesis: basically, that the lowest order of infinity (some cardinal numbere) was followed by the cardinal number, c (thus permitting Cantor to give ordere to his transfinite numbers). Against this hypothesis stood the possibility, urged by any number of Cantor's opponents, of infininty somewhere before one reached c. The search to prove what Godel later demonstrated to be an undecidable hypothesis may well have led Cantor (and Godel for that matter) to madness. At minimum it may have activated any underlying predisposition to mental illness in both men. They were not, as Aczels's discussion of the Kabbalists shows, inifinity's first victims. Aczel has provided a balanced and very human exploration into a topic that draws its victims as a moth to the flames.
Rating:
Summary: On the inifinity that comes next...
Review: I would never have imagined that a quest this abstract could entail a story so human. Yet, that is exactly what Amir Aczel provides in this smooth tale of the many humbling encounters with the realm of infinity. There are two lessons from this compact survey on the effort expended and the toll imposed on those bold enough to go where no person can go. First, the urge to comprehend infinity is an ancient quest and inextricably tied to the effort to ascertain the nature of God. Second, getting to know infinity can be massively bad for one's mental health. Mr. Aczel manages an almost impossible task (infinity tends to do that) in this text. He is (a) attempting to survey an enormous amount of the history mathematics and, to some extent, religion, and (b) providing a glimpse into the lives of those mathematicians that have ventured into this field. At the heart of this book is Georg Cantor, founder of modern set theory. Cantor sought to transcent an intuitive understanding of infinity. He sought an ordered system; specifically he sought to prove what became known as the continuim hypothesis: basically, that the lowest order of infinity (some cardinal numbere) was followed by the cardinal number, c (thus permitting Cantor to give ordere to his transfinite numbers). Against this hypothesis stood the possibility, urged by any number of Cantor's opponents, of infininty somewhere before one reached c. The search to prove what Godel later demonstrated to be an undecidable hypothesis may well have led Cantor (and Godel for that matter) to madness. At minimum it may have activated any underlying predisposition to mental illness in both men. They were not, as Aczels's discussion of the Kabbalists shows, inifinity's first victims. Aczel has provided a balanced and very human exploration into a topic that draws its victims as a moth to the flames.
Rating:
Summary: Trying to hard to ride the Kabbalah trend
Review: It is highly ironic that at the start of this book the author chides those celebrates who like to play Kabbalah just because of its new found trendyness because this is exactly what seems to be going on in this book.
While Aczek does an excellent job of looking at the history of the concept of infinity as well as providing a clear survey of the mathematical principles involved but then basically shoots himself in the foot by trying to shove the Kabbalah in where it doesn't belong. Aczek himself may have recognized this because despite putting it into the title he never really spends much time developing the connection in the book except through somewhat veiled references and hints (no doubt because he knew if he tried to show a connection more clearly it could be torn apart by anyone who read the book.)
I think that Janna Levin offered a much better explication for the seeming connection between theoretical mathematicians and madness in her book How the Universe Got Its Spots when she wrote "The lore is that their theories drove them mad, though I suspect they were just
lonely, isolated by what they knew."
This would have been a much better book without the unrelated Kabbalaistic elements tacked on - but then again it might not have sold as many copies (and one gets the impression that's why it was included in the first place.)
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