“Nessuno riuscirà a cacciarci dal Paradiso che Cantor ha creato per noi.”

I feel the same way about advanced mathematics and advanced physics.
The concepts and ramifications fascinate me, but I don't want to spend the
time and energy it would take to fully comprehend the details.
The author did an excellent job focusing on the background of the
discussions about infinity, Cantor's life, and a few mathmaticians that followed
in his footsteps. He explained just enough of infinity math so that the layman
could understand the concepts without getting bogged down in the details.

If you like this one than I would recommend trying your hand at
'The Dancing Wu Li Masters' which is a layman's discussion of
quantum physics. And if you really want a brain bender try
'Godel Escher Bach' which talks about the connections between
mathematics, art and music.

Interesting, or a caution of staring into infinity.

It is Aczel what else needs to be said. This is a good book that is a fun read.

An interesting history of people and ideas that I really wish were better written. I found the writing stripped down and formulaic, which is too bad since the topic itself seems filled with interesting possibilities.

I confess I didn't hold out much hope for this book when I began to read it,several years after it came into my possession,but I ended up enjoying it despite my reservations. Much of the math,even "dumbed down" to layman levels by Mr. Aczel was still over my head. But the history of the search for an understanding of infinity,and the connections to such mystic disciplines as Kabbalah really held my interest. I'm a sucker for those seeking ultimate truth. Even if in the end they fall a bit short. The search goes on. To infinity(or in the spirit of the book infinities) and beyond.

A brief but entertaining narrative of the history of thought concerning certain ideas of infinity and set theory. Being aimed at the general reader, coverage of the mathematics is substantially constrained and the remainder is spent mostly on biographical sketches. In addition, the connection to the kabbalah is tenuous and fragmentary.

Fine, but hard to get through. Not an ideal audiobook

Enlightening and easy to grasp, even for someone who isn’t mathematically inclined
like me :)

A somewhat flawed, magical, fascinating read

Aczel's fascinating book is a short narrative history of the concept of infinity (the aleph) with a concentration on its mathematical development, especially through Galileo, Cantor, Gödel, Paul Cohen and others, meshed with some very interesting material from the ancient Greeks and the Kabbalists who associated infinity with their ideas of God. He includes some material on how strikingly difficult it was for Cantor and others to go against established ideas. I think it was also Aczel's intent to force the reader to think about infinity as something spiritual. At least his book had that effect on me.

God is infinity, the ancient Kabbalists proclaimed, and indeed an all-powerful, all-knowing, immovable yet irresistible God may be something akin to infinity. God is perhaps a higher order of infinity, above the infinity of the transcendental numbers: infinity to the infinite power, one might say, and having said that, one might dismiss it all from the mind as being hopelessly beyond all comprehension. Yet, here, Amir Aczel brings us back. Cantor showed that we can think about infinity, at least to the extent that we can prove differences among infinities. We can, as it were, and from a distance, make distinctions about something we cannot really grasp. In a sense it is similar to contemplating what is beyond the big bang, or imagining the world below the Planck limit. Our minds were not constructed to come to grips with such things, yet maybe we can know something indirectly.

Maybe. In science what we know is forever subject to revision; but in mathematics we are said to have eternal knowledge. When it is proven (barring error) it is settled. Still, might mathematics exist beyond even the furthest reach of the human mind with a higher order affecting our proofs? Beyond the infinities might there exist something more "irrational" more completely "transcendent" than we can imagine even in our wildest fantasies?

At any rate, reading Aczel's magical book, I am persuaded to think so. And I can understand how New Agers and Kabbalists can become so enamored of numbers that they slip quite imperceptibly into numerology. (Numerology being to mathematics what astrology is to astronomy.)

Where I think Aczel is off the mark is in suggesting that it was concentration on the continuum that led to the ill mental health of Georg Cantor and Kurt Gödel. The old saw about thinking so long and hard on a subject leading to madness is something however that won't go away. In chess we have the preeminent examples of Paul Morphy and Bobby Fischer, both towering genius like Cantor and Gödel, who slipped into delusion and paranoia some believe after plummeting the depths of Caissa. With the great strides being made in neuroscience today, we might one day understand what these men had in common besides great intelligence and the ability to concentrate to an extraordinary degree.

There is a lot of interesting material throughout the book. I was especially intrigued with an implication of the fact that an infinite number of steps (e.g., 1/2 + 1/4 + 1/8...etc.--convergence) could lead to a finite sum. (p. 12) This really implies to my mind that we can relate in some sense to the idea of infinity. I contrasted this with Aczel's assertion on page 90 that if one could choose at random a number on the real line, that number would be "transcendental with a probability of one" (missing by force any of an infinity of rational numbers). However, as Aczel points out elsewhere, one cannot actually choose a number randomly out of an infinite collection!

I also liked the report about the exasperated Paris Academy in the nineteenth century passing "a law stating that purported solutions to the ancient problem" of squaring the circle "would no longer be read by members of the academy." (p. 89) This reminded me of the action by the U.S. Patent Office some many years ago of refusing to accept patent applications for perpetual motion machines.

Aczel gives Cantor's proof of a higher order of infinity for transcendental numbers on page 115. It is a very beautiful proof that can be understood with very little knowledge of math. On page 112 he gives Cantor's equally beautiful proof that rational numbers are as infinite as whole numbers. However his gloss at the top of the next page I think contains some typographical error that makes it not helpful. But perhaps I am wrong. (Maybe somebody knows and would tell me.) There is also some confusion about when Gödel married Adele on pages 198 and 200, and there are perhaps too many typos in the book, e.g., on the first sentence of page 162 the word "of" is missing, and on page 164 the word "way" (or something similar) should follow the word "humiliating." Also note Michael R. Chernick's correction in his review below showing the two missing permutations for the Hebrew word for God that Aczel left out on page 32.

Despite these flaws, this is overall an extremely engaging book and a delight to read.

--Dennis Littrell, author of “The World Is Not as We Think It Is”

A fantastic historical perspective of numbers, infinity, and the giants of mathematics and philosophy that were responsible for them.

About the mathematical development of infinity. A few irrelevant bits about the Kabbalah jammed in, sillyness. Otherwise mildly interesting- Then stupidly stopped mid-stream after the end of the work of Kanter.

Overall a stupid book written on behalf of a publisher