Rating: 
Summary: Fascinating and approachable
Review: The topic matter at hand here -- infinity -- is very subtle. The book is appropriately detailed, at times hilarious, and very colorful. Occasionally requires re-reading of certain parts (at least for me!), but the overall experience is most enjoyable. Mr. David Foster Wallace's writing style -- sometimes hard to decipher, sometimes too detailed, sometimes not detailed enough -- required a little bit of adjustment from me, as a reader. I'm certain, however, that the style is appropriate to the subject, if not a byproduct of infinity's complexity as a concept and as a historical development. Some of these copmlexities -- such as the wonderful footnotes and quasi-sidebar discussions -- add even more color and detail to the story. This book is very approachable, even though it requires a little patience at times. It's accessible for even those who (like me) don't have a very strong mathematical background. This is a fascinating topic, and a very well written book. Buy it and enjoy it.
Rating: 
Summary: Less than the sum of its parts
Review: There are two books here, that should complement each other, but, in reality, turn out to be mutually exclusive. Which is unfortunate, since they're both pretty good books in their own right.
The first book is your standard David Foster Wallace long essay/argument/explication. If you've read and enjoyed 'A Supposedly Fun A Thing . . .etc., etc. etc.' you'll recognize this easily. It's in DFW's distinctive voice, awash in digressions and asides and continually undercutting itself. And using little tricks like breaking up sentences into fragments which you'd think is just a clever trick but, to me, is just really enjoyable. (And, if you haven't noticed already, DFW's style is, to me, kind of infectious ' a lot of my writing is kind of influenced by it.) Anyway, this first book is a long chatty essay about infinity in general and Cantor's struggles with it in particular.
This is where the second book pops up. It's basically a math textbook. A focused one, yes, but a textbook nonetheless. Meaning that there are a lot of equations and Greek letters and the like. DFW continually claims that most of this math should be accessible to anyone with some high-school math and maybe a semester or two of college calculus. I fit that description to a T, and I should admit that I struggled at times.
But I think the reason I struggled was that the two books ' chatty essay and dry textbook ' undermine each other. The equations are stumbling blocks in the flow of DFW's prose. And the second guessing and asides in DFW's prose (which work just fine in an essay about pomo lit and television or state fairs or such) really get on your nerves. Because when you hit the hundredth iteration of DFW saying, 'well, what I'm about to explain isn't entirely accurate but . . .' you just want to scream 'where in hell can I find a book that explains this all accurately'' (Which, given the complexity of all this, might be a tall order.)
Frankly, I think this could have been a hell of a book if DFW's editors had let him blow it up to Infinite Jest size. Then he could have had room for more historical context, actual biographical details. As well as the space to go through the assorted proofs and equations in a more complete way. As well as the space for an index, table of contents and glossary (because the emergency glossaries, while well intentioned, don't quite cut it.)
But as it is, you basically have to read this thing twice. Once for pleasure and once for the math. If that sounds like your cup of tea (and you're a DFW fan), give this a read.
Rating: 
Summary: Everything and More: A Compact History of Infinity
Review: Wallace's writing about math isn't new-his novel Infinite Jest (1996) and some of his essays include a more than superficial treatment of the subject. Here, however, he digs as deeply into it as is possible for a nonprofessional math geek faced with a page limit, and the result is classic DFW: engaging, self-conscious, playful, and often breathless. This second installment in the "Great Discoveries" series traces the history of infinity from the Greeks to the calculus, culminating in a discussion of Georg Cantor's (1845-1918) groundbreaking work with transfinite numbers. Unfortunately, context requires Wallace to bulldoze heroically through a couple thousand years of logic, geometry, and number theory, which, even with "emergency glossaries" and frequent cross-referencing tips, can make for some teeth-grindingly dense passages. In one of the 400-plus footnotes, he writes, "It's true that it would be nice if you've had some college math, but please rest assured that considerable pains have been taken and infelicities permitted to make sure it's not required." For devout Wallace fans, it won't matter either way. Readers looking to soak up some pure abstraction, however, may just need to read the book twice. Luckily, they couldn't have been blessed with a more talented or stimulating guide. Enthusiastically recommended for all libraries.
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