What we're going to look at in this Review here: Everything and More: A Compact History of(infinity), by David Foster Wallace, publ. by W.W. Norton & Company, 320 p., ISBN: 0393003388.
DFW, as you maybe already know, is also the author of a novel, The Broom of the System; two collections of stories/weird ephemera, Girl with Curious Hair and Brief Interviews with Hideous Men; a collection of mostly New Journalism-type essayistic pieces, A Supposedly Fun Thing I'll Never Do Again; and Infinite Jest, which many people, Your Reviewer Here included, regard as One of the Best Books Ever Written. Everything and More is part of Norton's only-slightly-self-aggrandizingly named "Great Discoveries" series, which pairs high-profile/big-name authors with historical-scientific subjects, some synergistic resonance with their previous writings being the underlying logic behind said pairings.
Thus: the author of Infinite Jest writing about the mathematical concept of infinity. Get it?
So to categorize a little bit further, Everything and More is primarily a history of the life/ideas of one Georg Cantor (1845-1918), a towering figure in 19c mathematics, and one of the first people to coherently integrate infinity into modern math's schema. Aside from just generally revolutionizing math and inventing set theory, which if you don't know what that is, don't worry about it, Cantor was also a first-class flake. He spent a lot of his non-math time delving into mysticism, and was particularly obsessed with proving that F. Bacon authored the plays of W. Shakespeare (which fact curiously gets just a footnote in DFW's text).
Then there's the horribly sad factoid that Cantor spent his final years in a sanitarium writing letters to his wife begging for his release. All of which is to say, Cantor was a figure of surpassing pathos, a prime example of the John Nash-A Beautiful Mind paradigm that Hollywood seems to make so much hay of. That same brilliant/nuts high-wire act makes Cantor a great subject for math pop-histories, which DFW's book emphatically is not.
DFW is, instead, concerned with Cantor's ideas, which mostly had to do with number theory and infinite sets, which again, don't worry about it. Let's allow off the bat that a lot of the stuff in Everything and More is going to seem esoteric or just plain weird to readers without a semi-rigorous grounding in math. (And, in the interest of disclosure, let's stipulate that Your Reviewer Here--henceforth "YRH"--hit a wall math-wise in Calc II and is in no way representing himself as having such a background). The reader is further asked to bear with DFW's stylistic tics, e.g., copious footnoting, tortured-to-the-brink-of-incoherent syntax, and semi-legible shorthand, of which latter YRH is attempting a non-malevolent and mostly affectionate parody in this Review.
To see where Cantor is coming from, let's maybe start off with an example DFW uses, Zeno's Paradox, about Achilles and the Tortoise, since that's one a lot of people are familiar with or will at least recollect vaguely from 5thG word problems. Recall: Achilles and Tortoise are, for some reason, engaged in a foot race. Achilles, sporting chap that he is, gives Tortoise a head start. Now the paradox: No matter how fast Achilles runs to catch up, Tortoise will, at the same time, be putting new distance between them. Ergo, Achilles can't win. Thus, also, the trouble with infinity: No matter how close two numbers are to each other, there'll always be an infinity of other numbers between them. So that nice Number Line you probably recall from kindergarten is actually shot through with a bunch of infinitely deep holes.
As DFW points out, this turned out to be a major problem. The Greeks, who pretend that infinity isn't a quantifiable thing and is thus nothing to worry about, get around the aforementioned minefield by fudging; but, by Cantor's time, mathematicians realize that not having a definition of infinity--or even, really, of numbers themselves--isn't going to fly any longer. What good is math, after all, if it can't confirm the most easily observable physical facts, i.e. that Achilles is going to whoop Tortoise's ass?
Which is where G. Cantor comes in. "You might recall infinitesimals from college math," DFW writes on page 32. "You may well however not recall--probably because you were not told--that infinitesimals made the foundations of calculus extremely shaky and controversial for 200 years, and for much the same reason that Cantor's transfinite math was met with such howling skepticism in the late 1800s: nothing has caused math more problems--historically, methodologically, metaphysically--than infinite quantities." What was genius and generally math-world-rocking about Cantor's theory of transfinite numbers was that it allowed mathematicians to deal with these troublesome infinites as though they were more or less regular old numbers, and not some airy abstraction to be fudged and/or ignored.
Of course, all this is a radically condensed version of DFW's explication of Cantor's work. But YRH is guessing you don't have the patience for much more detail. Like a lot of higher math, Cantor's theories can seem pretty intuitive and straightforward on the face of things, but then get hairy and brutal and, frankly, unparsable when you jump into the technical nitty-gritty. And this is where the patience/fortitude thing really comes into play w/r/t Everything and More. Meaning which, your average math-averse reader's response to all this is understandably going to be, WTF?