I don't think the author even knows who his intended audience is. My suggestion is no one. But if you must, it would probably help if you've had some calculus and a little bit of set theory and logic. Although the other doesn't seem to think this is necessary.
The book, which the author calls a booklet, is a 305 page history of the mathematical concept of infinity, complete with citations, bibliography, and copious footnotes. The book is divided into seven sections each with several subsections (though no table of contents is given). The book begins roughly with Zeno's paradoxes and ends with the Continuum Hypothesis and Goedel's and Cohen's proofs.
The book is full of historical and biographical information (not all well-known) about key mathematicians in the development of the concept of infinity. There are numerous references to original mathematical sources as well as current pop literature on math and science.
This book is actually more of a meta-book than a book, as the author seems to talk almost as much about the book, his audience, the writing, the editing, etc. as he does about the subject matter at hand. There are numerous footnotes, abbreviations, emergency glossaries, asides, and references.
While I did learn a few things from this book (that is, if I trust that the author is accurate) and while the material is right up my alley (as I always enjoy thinking about set theory, analysis, infinity, etc.), this must be one of the worst books I have ever read. Let's begin with the fact that the author is not a mathematician (but rather a proclaimed amateur mathematician), and probably this is why he continually gets the math terribly wrong. I found more serious mathematical mistakes in this book than I care to list. I'm trying to forget most of the errors, and I really didn't catalog them, but just to give you a taste, I'll try to recall a couple. For example, he gets the definition of "uniformly convergent" wrong (he thinks it just means "convergent"). He has a completely nonsensical presentation of an example involving finding the area under a curve, in which he proves exactly what he postulated, and during the course of the proof he gets several things terribly wrong. He even repeatedly states the Continuum Hypothesis incorrectly (which you would think would be important to get right considering this is a book about infinity). He says that the Continuum Hypothesis is the statement that the cardinality of the set of reals equals the cardinality of the power set of the set of natural numbers, i.e. 2 raised to the aleph-nought. He also says that 2^{aleph-nought} = aleph-one, the next cardinal number after aleph-nought. This is awful. But there are many, many more mathematical errors he makes throughout the book.
Another serious flaw of this book is the author's perception of what mathematics is. He has the attitude that subjects like calculus and analysis are extremely abstract and difficult and boring, and he repeatedly asks his readers for forgiveness when he tries (and fails) to explain something slightly complicated. The author also seems to be an extreme Platonist to the point where he doesn't even understand the axiomatic approach and what mathematicians do in their work.
Mathematical issues aside, the book is a mess. It reads like a very rough first draft written by someone who didn't bother to make an outline first. He even repeatedly says things like, "oh, I guess I should have told you this fact in the last section," or "maybe footnote 37 on page 63 should have mentioned this." There are what he calls "Emergency Glossaries" inserted here and there throughout the text. The author uses a vast number of abbreviations, some of which he warns the reader about at the beginning of the text and some of which he doesn't. He just seems to ramble on and on, and it is all very annoying. And by the way, enough with the footnotes already. Some sentences have two or three footnotes, and some of the footnotes spill over to another page. If you really need that many footnotes and asides, then your book is disorganized.
The author does pretend to have a sense of humor, and maybe other readers beside myself could appreciate it. But I just found him to be pretentious and annoying. His extreme penchant for abbreviations is matched only by his extreme penchant for annoying literary/writing terms and obscure technical vocabulary. (I'm sure he made up a few words.) Of course, maybe my frustration with him getting the math wrong is spilling over to other areas and I'm just lashing out.
The pedagogy is also unsound. The author has a knack for making things complicated when they aren't, for leaving out important details that are easily explained, and for inserting meaningless details that should be left out. I never really understood who he was writing for. A person who didn't major in math will probably be lost, and a math major will be bored by the math.
Somehow, despite the complete failure of this book, I am intrigued by David Foster Wallace. He's an enigma. His book shows that he has done a ton of reading on the subject. He's even consulted the orginal mathematical papers. He seems to be familiar with some advanced mathematical topics. And yet, despite all of this, he is terrible at math. A friend has recommended some other Wallace non-fiction books on non-mathematical topics. I'll probably pick them up. I expect them to either be very good or another train wreck. Either way I'll be entertained.