If by any chance you're not familiar with science-writer Charles Seife's work, I highly recommend it (one of my favorite science expositors). More recently he put up a simple 'tweet' that caught my eye, noting that he was now in possession of "David Foster Wallace's annotations" to his (Seife's) book "Zero: the biography of a dangerous idea." He actually linked to an image (pdf) of one of the pages, which included the following passage, with Wallace's simple annotation, "Heavy." :-)
"How big are the rational numbers? They take up no space at all. It's a tough concept to swallow, but it's true.Fascinating stuff… and, 'heavy' indeed!; all the moreso, no doubt, for someone who was essentially an English major/novelist/writer, like Wallace, with a brilliant/eclectic mind (he did deeply study logic and mathematics in college though).
Even though there are rational numbers everywhere on the number line, they take up no space at all. If we were to throw a dart at the number line, it would never hit a rational number. Never. And though the rationals are tiny, the irrationals aren't, since we can't make a seating chart and cover them one by one; there will always be uncovered irrationals left over. Kronecker hated the irrationals, but they take up all the space in the number line.
The infinity of the rationals is nothing more than a zero."
About 3 years after Seife's "Zero" came out, Wallace produced "Everything and More: A Compact History of Infinity" -- a remarkable take (300 pages that he calls "a booklet"), from a non-mathematician, on the whole history and thought surrounding mathematical infinity. I happen to like this quirky volume a lot for its sheer uniqueness (I still can't imagine the mind that could write such a book, and if you are familiar with Wallace's innovative writing style, you know it is unique!), but I've seen professional mathematicians give the volume both negative and positive reviews. In any event, we may now have at least some inkling as to what moved Wallace to even tackle such subject matter. As Paul Harvey would say, perhaps from Wallace's Seife-annotations we will get a glimpse of "the rest of the story."
[-- Tragically, Wallace took his own life at age 46 in 2008, after lifelong bouts with depression.]