Everything And More David Foster Wallace

How to Read Wallace's Infinity Book Without Wasting Your Time

The book is formally titled Everything And More: A Compact History of Infinity, published in 2002. It covers the mathematical development of infinity from ancient Greece through Cantor, Riemann, and into modern set theory. David Foster Wallace was working on a larger project about infinity when he died in 2008, and this book grew out of that research. It is not a pop-science overview in the popular sense. It is a serious attempt to make the mathematics itself legible to someone with only high school algebra and trigonometry. I bought the hardcover on release day, read it straight through in three days, and then put it on the shelf. Two years later I went back and actually read it. The difference in what I got out of it was enormous. That is the honest pattern with this book: the first pass is an exercise in frustration, and the second pass is where the math starts to click. Most people stop after the first pass and conclude the book is either too hard or not worth it. Neither conclusion is quite right. Wallace's approach to Cantor's diagonal argument is different from how you will see it in most textbooks. He does not present the proof as a neat three-step logical sequence. He builds it the way the people who actually discovered it had to: by making the mistakes, hitting the objections, and circling back. You watch him do this over about forty pages. It is exhausting in the moment. When you come back to it, you understand the diagonal proof better than you would have from any cleaner exposition because you have lived through the failure modes that the proof has to answer.

The practical difficulty is pace. A single page of Wallace can contain one paragraph of actual mathematical argument and two pages of digression that are funny, autobiographical, or about philosophy. The digressions are not padding. They are his method for forcing you to confront the conceptual obstacles that standard textbook treatments glide over. But they are also genuinely optional in terms of the logical chain. If you are trying to follow the mathematics alone, you can often skip the tangents and return to them later. I did that on my third read-through and cut the time spent on the cardinality sections roughly in half. Here is a specific edge case that almost made me abandon the book entirely. In the chapter on countable and uncountable infinities, Wallace spends a long section on why the set of all finite strings of digits is countable while the set of all infinite decimal expansions is not. I kept getting tripped up because I was conflating two different questions: how many elements are in each set, and whether you can list them in a sequence. Those are the same question in this context, but Wallace separates them verbally in a way that confused me for pages. My workaround was to write out the definitions on paper with my own words before going back to the text. Once I had my own version of "countable means there exists a bijection with the natural numbers" scrawled next to my forehead, the rest of that section cleared up in about twenty minutes. It was the first time in the entire book that something finally landed. One thing that catches people off guard is the treatment of actual vs. potential infinity. You will get a lot of hand-waving about this in other books. Wallace does not. He traces the debate from Aristotle through the nineteenth century with actual primary source references. The arc of the argument is: potential infinity was acceptable to most mathematicians for years, then Cantor insisted on actual infinity as a completed object, and the resistance to that was not just mathematical conservatism but a legitimate philosophical problem that had real consequences for how analysis was formalized. This is not a side note. It is central to understanding why the math looks the way it does. If you skip it, you will think Cantor's diagonal argument is a trick. It is not. It is a consequence of treating infinity as something you can quantify over directly.

Another counter-intuitive point that beginners miss: Wallace's discussion of the Banach-Tarski paradox is not really about the paradox itself. It is about the Axiom of Choice, and the fact that the paradox forces you to admit you have made a choice you cannot fully justify. The book uses the paradox as pressure to make you feel the cost of accepting the Axiom of Choice, which most people accept without thinking. By the time he is done, you understand that the Axiom of Choice is not a theorem. It is an assumption you carry into your work, and it has real consequences you do not always want. That is the deeper point, and it is easy to miss if you are only reading for the theorem statements. The book has serious limitations. The humor does not land for everyone. If you do not find Wallace's self-deprecating asides amusing, the reading experience becomes slower than it needs to be. The mathematics is accurate but occasionally sloppy in the way that happens when a writer is covering a huge historical sweep in a single volume. There are places where he simplifies a proof beyond what is strictly correct, and there are a handful of errors in the typeset equations that readers have caught and reported. None of them are fatal to the argument, but if you are using this as a serious mathematical reference, you should cross-check the harder proofs against a textbook like Halmos' Naive Set Theory or Kunen's Set Theory. The density is the main bottleneck. A chapter that should take an hour will take three if you are stopping to work through every claim. I estimate that a careful first reading runs about twenty-five to thirty hours total across the whole book. A surface reading without engaging with the math is closer to six hours, but you will get far less out of it. There is no way to speed through this and retain the understanding.

For people who want the mathematical content without the biographical material, the 2011 revised edition does not change the substance but removes some of the earlier cut content. There is no abridged version that is actually faithful to the mathematics. If you want a shorter companion, Greg Chaitin's work on algorithmic information theory gets at similar territory from a different angle, though it is not a replacement. E.T. Bell's Men of Mathematics is older and less rigorous but useful for the historical framing. Wallace's book remains the best single-volume treatment of the subject aimed at a non-specialist audience, which is why people keep asking about it. I would recommend the following order of engagement. Read the introduction. Then go straight to the chapter on Cantor and do not look at anything else until you can explain the diagonal argument in your own words without looking at the book. After that, return to the historical chapters in sequence. Skip the biographical sections on your first pass if they slow you down. Come back to them once the math is working. The book rewards the second pass much more than the first. There is a PDF circulating on several file-sharing sites. I have not checked its accuracy against the published text, and I would not vouch for it. The publisher is Grand Central Publishing. The ISBN for the hardcover is 978-0-446-53207-8. If you are going to read this, reading the real book is worth the cover price because you will need to slow down and reread sections, and having the physical copy makes that easier than navigating a screen.