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Amir Aczel

The mystery of the aleph

Infinity has been a source of much confusion over the ages. In The mystery of the aleph Amir Aczel traces our gradual acceptance of the concept. He starts with the ideas of Zeno and Pythagoras and goes on to consider the Jewish mystical system known as the Kabbalah. He shows how Galileo began to take infinity seriously, accepting actual as well as potential infinities. Aczel then moves on to the nineteenth century, when infinitesimals were put on a firmer footing, but most importantly of all he describes the life of Georg Cantor, and his work on transfinite set theory. The last few chapters look at later developments of this theory, and in particular the work of Kurt Gödel.

I would say that the reader needs some mathematical background to be able to follow this book. It's harder than God's Equation where biographical details help the reader along, but then biographies of those working with the infinite tend to be a bit depressing. But I think that someone who has pursued mathematics to a reasonably advanced level at school should have no problems with this book, and that it would be a useful introduction to the concept of infinity for those wanting to go on to study mathematics or physical science at university level.

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Paperback 272 pages
ISBN: 0743422996
Salesrank: 274192
Weight:0.53 lbs

Published: 2001 Washington Square Press

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Amazon.co.uk info

Paperback 272 pages
ISBN: 0743422996
Salesrank: 142196
Weight:0.53 lbs

Published: 2002 Pocket Books

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Paperback 272 pages
ISBN: 0743422996
Salesrank: 275549
Weight:0.53 lbs

Published: 2001 Washington Square Press

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Product Description
From the end of the 19th century until his death, one of history's most brilliant mathematicians languished in an asylum. The Mystery of the Aleph tells the story of Georg Cantor (1845-1918), a Russian-born German who created set theory, the concept of infinite numbers, and the "continuum hypothesis," which challenged the very foundations of mathematics. His ideas brought expected denunciation from established corners - he was called a "corruptor of youth" not only for his work in mathematics, but for his larger attempts to meld spirituality and science.

Product Description

From the end of the 19th century until his death, one of history's most brilliant mathematicians languished in an asylum. The Mystery of the Aleph tells the story of Georg Cantor (1845-1918), a Russian-born German who created set theory, the concept of infinite numbers, and the "continuum hypothesis," which challenged the very foundations of mathematics. His ideas brought expected denunciation from established corners - he was called a "corruptor of youth" not only for his work in mathematics, but for his larger attempts to meld spirituality and science.

Very Good book about the Infinity and the Continuum Hypothesis ****
I enjoyed reading the book very much. In general, any book about numbers, and its development in human history amazes me. The story of Mr. Aczel is not an exception. The struggle of Georg Cantor is clearly elaborated in details in this book. How he goes mad to prove the cardinality of the real numbers. How many are there really? The book also makes a good job in demonstrating the early studies of Jewish mystics to explore the infinity and God. Yet one thing that looked weak to me is the connection between infinite numbers, God and Kabbalah. I wish the author used more examples to give the message that he is trying to make. I am personally very interested in Sufism (a sister trend or movement) so for me reading the book was definitely informative and inspiring to a certain degree. Yet I expect similar books from Mr. Aczel and especially a follow-up sequel to this book to make things clear.

Very Good book about the Infinity and the Continuum Hypothesis ****

I enjoyed reading the book very much. In general, any book about numbers, and its development in human history amazes me. The story of Mr. Aczel is not an exception. The struggle of Georg Cantor is clearly elaborated in details in this book. How he goes mad to prove the cardinality of the real numbers. How many are there really? The book also makes a good job in demonstrating the early studies of Jewish mystics to explore the infinity and God. Yet one thing that looked weak to me is the connection between infinite numbers, God and Kabbalah. I wish the author used more examples to give the message that he is trying to make. I am personally very interested in Sufism (a sister trend or movement) so for me reading the book was definitely informative and inspiring to a certain degree. Yet I expect similar books from Mr. Aczel and especially a follow-up sequel to this book to make things clear.

early maths theory *****
Very interesting history of the development of mathematical ideas,especially the existence of irrational numbers,and the idea infinity can be approached and used but never reached.....

early maths theory *****

Very interesting history of the development of mathematical ideas,especially the existence of irrational numbers,and the idea infinity can be approached and used but never reached.....

mathematics, cantor and mysticism ****
I started reading this book on the plane that took me to my new home in New Jersey. I finished it about a month later. I am a slow reader and I also was very busy getting settled into my new job. As I prepared to write my review for Amazon I looked at the many other reviews that had already been written and I found that they were quite mixed. Some raved about it and some hated it. There were many good points on both sides. I hope my review adds something new for potential readers to think about. I am a mathematician by training. I have a bachelor's degree in mathematics and also a masters degree. In my university education I learned about algebra and analysis and did have some acquaintance with the results of Cantor on transfinite numbers. I also knew some things about the axiom of choice, the continuum hypothesis and the Hahn-Banach theorem. I got this education in the late 1960s and early 1970s. In the mid 1970s I went on to Stanford where I studied Operations Research and Statistics eventually leading me to a career as a statistician. I had not given much thought to these mathematical ideas in a long time. While at Stanford, I did hear about Paul Cohen who was then considered to be a star in the Mathematics Department because of his great discoveries in set theory and logic at an early age. This book provided me with an interesting reminder of my past education and cleared up a few ideas in logic that had been puzzling to me. At first I thought I was going to hear about the life story of Georg Cantor, the father of transfinite numbers. I was pleasantly surprised to find out that the book develops ideas about infinity and infinite numbers going back to the time of the Greeks and the discovery of irrational numbers by the Pythagorean school. Aczel also discusses the lives of Galileo and Bolzano and their contributions to mathematics. I was aware of the one-to-one correspondence between the integers and the square of the integers. The fact that the discovery goes back to Galileo was news to me. While I knew of Galileo for his invention of useful telescopes and his contributions to astronomy, I had no idea that he had made such a fundamental contribution to mathematics. As with some of the other reviewers, I find the discussion of the Kabbalah somewhat weak and perhaps misplaced. I also think there is a mathematical error in this chapter. Aczel states that there are 10 permutations of the arrangement of the Hebrew name for God, YHVH, and he places importance on the number 10. He enumerates the permutations to be YHVH, YVHH, VYHH, VHYH, HVYH, HYVH, HVHY, HYHV, HHYV AND HHVY. This puzzled me. As I thought about my combinatorial mathematics I thought the correct answer should be 12. I tried a complete enumeration myself and found 12. It seems that Aczel missed YHHV and VHHY. Aside from this, the discussion of mathematics is generally good. It is not detailed and is written in a popular style to be readible to a general audience. The heart of the book is the life of Georg Cantor. Cantor aided by the work of Galileo and Bolzano and his teacher Karl Weierstrass made the breakthroughs that led to the development of transfinite numbers and modern set theory. He worked mostly in isolation at Halle University and was frustrated by never being granted an appointment at University in Berlin where most of the famous mathematicians of the time resided. His conflict with Kronecker is discussed and the support he got from Mittag-Leffler is also covered. Aczel provides background to varying degrees on all the mathematicians that he discusses and we feel that we understand their personalities and the underlying reasons for the positions that they took. Cantor's bouts with insanity are also described. Although it could be simply that he was suffering from manic depression (a disorder that was not understood at the time), Aczel attributes Cantor's insanity to the frustration of his efforts to cope with infinity. Certainly there must have been frustration over his inability to prove the continuum hypothesis (later determined to be unprovable) and the lack of universal acceptance of his ideas in the mathematical community. However, I agree with some of the other reviewers who think that Aczel's thesis, that doing mathematical research on infinity might induce insanity, is a bit farfetched. In covering the life of Kurt Godel, a important successor to Cantor, Aczel points to Godel's bouts with insanity to try to reinforce this thesis. Godel did not have the same issues in his life history that Cantor had. Still, other mathematicians that worked in this area including Russell and Cohen never had similar bouts. Coverage of the work of Godel and Cohen brings the reader up to the current state of knowledge about transfinite numbers and set theory. For the mathematically inclined there is an appendix at the end that provides statements of Zermelo's axioms that are the basis of modern set theory. It is within this system that the axiom of choice and the continuum hypothesis are both consistent and independent and therefore can neither be proven to be true or false. If you like reading about the history of mathematics and the personalities of important mathematicians you will enjoy this book inspite of a few flaws.

mathematics, cantor and mysticism ****

I started reading this book on the plane that took me to my new home in New Jersey. I finished it about a month later. I am a slow reader and I also was very busy getting settled into my new job. As I prepared to write my review for Amazon I looked at the many other reviews that had already been written and I found that they were quite mixed. Some raved about it and some hated it. There were many good points on both sides.
I hope my review adds something new for potential readers to think about.

I am a mathematician by training. I have a bachelor's degree in mathematics and also a masters degree. In my university education I learned about algebra and analysis and did have some acquaintance with the results of Cantor on transfinite numbers. I also knew some things about the axiom of choice, the continuum hypothesis and the Hahn-Banach theorem. I got this education in the late 1960s and early 1970s. In the mid 1970s I went on to Stanford where I studied Operations Research and Statistics eventually leading me to a career as a statistician. I had not given much thought to these mathematical ideas in a long time.

While at Stanford, I did hear about Paul Cohen who was then considered to be a star in the Mathematics Department because of his great discoveries in set theory and logic at an early age.

This book provided me with an interesting reminder of my past education and cleared up a few ideas in logic that had been puzzling to me.

At first I thought I was going to hear about the life story of Georg Cantor, the father of transfinite numbers. I was pleasantly surprised to find out that the book develops ideas about infinity and infinite numbers going back to the time of the Greeks and the discovery of irrational numbers by the Pythagorean school.

Aczel also discusses the lives of Galileo and Bolzano and their contributions to mathematics. I was aware of the one-to-one correspondence between the integers and the square of the integers. The fact that the discovery goes back to Galileo was news to me. While I knew of Galileo for his invention of useful telescopes and his contributions to astronomy, I had no idea that he had made such a fundamental contribution to mathematics.

As with some of the other reviewers, I find the discussion of the Kabbalah somewhat weak and perhaps misplaced. I also think there is a mathematical error in this chapter. Aczel states that there are 10 permutations of the arrangement of the Hebrew name for God, YHVH, and he places importance on the number 10. He enumerates the permutations to be YHVH, YVHH, VYHH, VHYH, HVYH, HYVH, HVHY, HYHV, HHYV AND HHVY. This puzzled me. As I thought about my combinatorial mathematics I thought the correct answer should be 12. I tried a complete enumeration myself and found 12. It seems that Aczel missed YHHV and VHHY.

Aside from this, the discussion of mathematics is generally good. It is not detailed and is written in a popular style to be readible to a general audience. The heart of the book is the life of Georg Cantor. Cantor aided by the work of Galileo and Bolzano and his teacher Karl Weierstrass made the breakthroughs that led to the development of transfinite numbers and modern set theory. He worked mostly in isolation at Halle University and was frustrated by never being granted an appointment at University in Berlin where most of the famous mathematicians of the time resided. His conflict with Kronecker is discussed and the support he got from Mittag-Leffler is also covered.

Aczel provides background to varying degrees on all the mathematicians that he discusses and we feel that we understand their personalities and the underlying reasons for the positions that they took. Cantor's bouts with insanity are also described. Although it could be simply that he was suffering from manic depression (a disorder that was not understood at the time), Aczel attributes Cantor's insanity to the frustration of his efforts to cope with infinity. Certainly there must have been frustration over his inability to prove the continuum hypothesis (later determined to be unprovable) and the lack of universal acceptance of his ideas in the mathematical community.

However, I agree with some of the other reviewers who think that Aczel's thesis, that doing mathematical research on infinity might induce insanity, is a bit farfetched. In covering the life of Kurt Godel, a important successor to Cantor, Aczel points to Godel's bouts with insanity to try to reinforce this thesis. Godel did not have the same issues in his life history that Cantor had. Still, other mathematicians that worked in this area including Russell and Cohen never had similar bouts.

Coverage of the work of Godel and Cohen brings the reader up to the current state of knowledge about transfinite numbers and set theory. For the mathematically inclined there is an appendix at the end that provides statements of Zermelo's axioms that are the basis of modern set theory. It is within this system that the axiom of choice and the continuum hypothesis are both consistent and independent and therefore can neither be proven to be true or false.

If you like reading about the history of mathematics and the personalities of important mathematicians you will enjoy this book inspite of a few flaws.

Well-written and interesting, but somewhat superficial ***
First, the good news. Aczel's book -- part biography, part history of infinity, part primer of some of the more challenging concepts in mathematics -- is engaging and well written. Much better written, in fact, than many similar books on the history of or on topics in mathematics that I've read. He has a lively style that keeps you turning the pages, and he is generally very good at simplifying complex axioms and proofs for the layperson. The short précis of the concepts of infinity among the ancient Greeks and Jews is pretty captivating subject matter, too. And the short biographies of the key mathematicians chasing the infinite are all sound and worthwhile. Now, the bad news. Considering that the subtitle of the book invokes the Kabbalah, Aczel gives it rather short shrift. He endeavors to summarize the subject, particularly in relation to things infinite, but does so too carelessly. I wanted more elaboration on that. Then he attempts to bring the Kabbalah back from time to time, as with Cantor's debatably Jewish heritage and with the diaspora of the Jews during World War II, but these connections are only hinted at. They feel superficial and without the persuasive weight to justify their inclusion. Also, I feel Aczel is a bit too baldly assertive in blaming Cantor and Gödel's mental problems on their struggles with the Continuum Hypothesis. Might it not have been the other way around, latent mental instability leading these two men to that particular compulsive struggle? I understand that pointing a finger at Infinity and shouting "j'accuse!" makes for more dramatic nonfiction, but it comes at a cost in accracy, doesn't it? Still, despite these complaints, I can marginally recommend the book as an interesting read on the history of the notion of infinity. Or at least parts of that history.

Well-written and interesting, but somewhat superficial ***

First, the good news. Aczel's book -- part biography, part history of infinity, part primer of some of the more challenging concepts in mathematics -- is engaging and well written. Much better written, in fact, than many similar books on the history of or on topics in mathematics that I've read. He has a lively style that keeps you turning the pages, and he is generally very good at simplifying complex axioms and proofs for the layperson. The short précis of the concepts of infinity among the ancient Greeks and Jews is pretty captivating subject matter, too. And the short biographies of the key mathematicians chasing the infinite are all sound and worthwhile.

Now, the bad news. Considering that the subtitle of the book invokes the Kabbalah, Aczel gives it rather short shrift. He endeavors to summarize the subject, particularly in relation to things infinite, but does so too carelessly. I wanted more elaboration on that. Then he attempts to bring the Kabbalah back from time to time, as with Cantor's debatably Jewish heritage and with the diaspora of the Jews during World War II, but these connections are only hinted at. They feel superficial and without the persuasive weight to justify their inclusion. Also, I feel Aczel is a bit too baldly assertive in blaming Cantor and Gödel's mental problems on their struggles with the Continuum Hypothesis. Might it not have been the other way around, latent mental instability leading these two men to that particular compulsive struggle? I understand that pointing a finger at Infinity and shouting "j'accuse!" makes for more dramatic nonfiction, but it comes at a cost in accracy, doesn't it?

Still, despite these complaints, I can marginally recommend the book as an interesting read on the history of the notion of infinity. Or at least parts of that history.

Interesting Book on Mathematics and a Mathematician ****
Aczel offers an interesting book on Georg Cantor, the "father" of set theory (a branch of mathematics). The book covers a number of interesting topics, including a unique overview of infinity, a description of the inception of a new branch of mathematics and a mini-biography of Georg Cantor. I recommend this book to readers interested in mathematics and mathematicians.

Interesting Book on Mathematics and a Mathematician ****

Aczel offers an interesting book on Georg Cantor, the "father" of set theory (a branch of mathematics). The book covers a number of interesting topics, including a unique overview of infinity, a description of the inception of a new branch of mathematics and a mini-biography of Georg Cantor.

I recommend this book to readers interested in mathematics and mathematicians.

Contemplating the Infinite ****
One of only a handful of absorbing books that I have read in one sitting. It started extremely well so I was a little shaken when I read on page 24 how Archimedes compared the volumes of a sphere and a cone (it was a sphere and a cylinder I believe). However, rest assured, this was the only glitch I found in an excellent exposition of the Infinite from a set-theoretical approach. As this book is not particularly mathematical, and is extremely well-written, it should appeal to those readers with a strong interest in mathematics, without the mathematical hardware behind them. Aczel's book makes a great accompaniment to Rudy Rucker's "Infinity and the Mind", which is much more mathematical, and covers many more aspects of Infinity. I give Aczel's book 4 stars, as Rucker's is out there on its own with 5 for this particular topic area. To complete the trilogy I recommend Maor's "To Infinity and Beyond", which is also a very easy read without being too mathematical.

Contemplating the Infinite ****

One of only a handful of absorbing books that I have read in one sitting. It started extremely well so I was a little shaken when I read on page 24 how Archimedes compared the volumes of a sphere and a cone (it was a sphere and a cylinder I believe). However, rest assured, this was the only glitch I found in an excellent exposition of the Infinite from a set-theoretical approach.
As this book is not particularly mathematical, and is extremely well-written, it should appeal to those readers with a strong interest in mathematics, without the mathematical hardware behind them. Aczel's book makes a great accompaniment to Rudy Rucker's "Infinity and the Mind", which is much more mathematical, and covers many more aspects of Infinity. I give Aczel's book 4 stars, as Rucker's is out there on its own with 5 for this particular topic area.
To complete the trilogy I recommend Maor's "To Infinity and Beyond", which is also a very easy read without being too mathematical.

Defies Intuition but it's well worth a try. ****
Once again Amir Aczel has provided us with an enthusiastic and intriguing look at a fascinating subject.Trying to come to terms with the Infinite is a difficult task that, as explained in the book, has claimed many victims. Aczel does a wonderful job turning the reader into a victim, although thankfully not as critical as the likes of Cantor and Godel. Much like his previous books Aczel blends the science( in this case mind boggling mathematics)into a fascinating background, paying great attention to the lives and characters of those concerned. The biographies of Gallileo and Kurt Godel are particularly interesting, especially as one would think that there was little left to know about them. However the centre of attention is the life and work of Georg Cantor, the mathematician synonomous with discoveries concerning the infinite. It would be difficult to find a more interesting and bizarre story than that of Georg Cantor but the real source of intrigue are his ideas and those of others concerning the subject of infinity. Added to that is a touch of mystisism in the form of "Kabalah" making Infinity an even more awesome concept. While the book is written in an entertaining and absorbing style with ideas explained simply and concisley, much contemplation is required by the reader. I personally would reccomend that one take the neccesary time in order to try come to terms with a concept that "defies intuition". Four stars then, and the only reason I didn't give it five is because I can still sleep at night...barely.

Defies Intuition but it's well worth a try. ****

Once again Amir Aczel has provided us with an enthusiastic and intriguing look at a fascinating subject.Trying to come to terms with the Infinite is a difficult task that, as explained in the book, has claimed many victims. Aczel does a wonderful job turning the reader into a victim, although thankfully not as critical as the likes of Cantor and Godel. Much like his previous books Aczel blends the science( in this case mind boggling mathematics)into a fascinating background, paying great attention to the lives and characters of those concerned. The biographies of Gallileo and Kurt Godel are particularly interesting, especially as one would think that there was little left to know about them. However the centre of attention is the life and work of Georg Cantor, the mathematician synonomous with discoveries concerning the infinite. It would be difficult to find a more interesting and bizarre story than that of Georg Cantor but the real source of intrigue are his ideas and those of others concerning the subject of infinity. Added to that is a touch of mystisism in the form of "Kabalah" making Infinity an even more awesome concept.

While the book is written in an entertaining and absorbing style with ideas explained simply and concisley, much contemplation is required by the reader. I personally would reccomend that one take the neccesary time in order to try come to terms with a concept that "defies intuition". Four stars then, and the only reason I didn't give it five is because I can still sleep at night...barely.

A worthwhile read.... ****
Though it's been some time since I read this book, I recall that I was (and still am) fascinated with its subject matter. The mathematics of the continuum is fascinating enough on its own, even if it may be somewhat incomprehensible to many. But when the philosophy of the continuum is brought into the picture, all kinds of interesting things happen. Another reviewer said that connections between the mathematics of the continuum and Jewish mysticism are without basis. I disagree. I feel that Mr Aczel has provided a very sound, logical basis for his statements. Additionally, if you're looking for a thorough treatment of the mathematics of the continuum, you need only look at the title of this book to realize that it will not be found within its pages. Titles with the words "Mystery" and "Kabbalah" must be given the benefit of an open mind. Any mathematician should understand that.

A worthwhile read.... ****

Though it's been some time since I read this book, I recall that I was (and still am) fascinated with its subject matter. The mathematics of the continuum is fascinating enough on its own, even if it may be somewhat incomprehensible to many. But when the philosophy of the continuum is brought into the picture, all kinds of interesting things happen. Another reviewer said that connections between the mathematics of the continuum and Jewish mysticism are without basis. I disagree. I feel that Mr Aczel has provided a very sound, logical basis for his statements.

Additionally, if you're looking for a thorough treatment of the mathematics of the continuum, you need only look at the title of this book to realize that it will not be found within its pages. Titles with the words "Mystery" and "Kabbalah" must be given the benefit of an open mind. Any mathematician should understand that.

Interesting book, but not necessary ***
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.)

Interesting book, but not necessary ***

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.)

Complete Nonsense *
Although the book's content about mathematics is informative, its association of abstract mathematics with Judaism is completely ridiculous. The author tries hard to establish a link between the ponderings of some ancient Rabbis and Cantor's astonishing accomplishments and between some religious concepts and mathematical infinity. Please do not read this book if you are in any way serious about mathematics or science. Over and over again, the author promotes god and the Jewish religion whereas I was just interested in the exciting mathematics of the period. I consider this book propaganda for Judaism and mysticism. It is one of those instances where a religious zealot tries to cast his absurd views into science.

Complete Nonsense *

Although the book's content about mathematics is informative, its association of abstract mathematics with Judaism is completely ridiculous. The author tries hard to establish a link between the ponderings of some ancient Rabbis and Cantor's astonishing accomplishments and between some religious concepts and mathematical infinity. Please do not read this book if you are in any way serious about mathematics or science. Over and over again, the author promotes god and the Jewish religion whereas I was just interested in the exciting mathematics of the period. I consider this book propaganda for Judaism and mysticism. It is one of those instances where a religious zealot tries to cast his absurd views into science.

On the inifinity that comes next... ****
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.

On the inifinity that comes next... ****

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.

Where's the beef ? ***
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.

Where's the beef ? ***

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.