The rest of this comment might end up being more appropriate for a later post, but I’m leaving it here anyway. I was really upset when I learned that the rationals are countable. It offended my sensibilities. I was taking a Moore method class basically on naive set theory and how to write proofs, and one of the things we were supposed to work on was whether the set of ordered pairs in NxN was countable. I tried everything I could think of to show that they weren’t. I was OK with 2N, 1/2N, etc. being countable (even though 2N was smaller in some sense), but I felt like an “infinite multiple” of an infinite set could not possibly be countable. Multiplying something by an infinite something just HAD to ratchet up the “degree of infiniteness”. It was probably one of the first times my mathematical intuition was dead wrong.

I still have trouble with uncountableness when I really think about it. Basically, any of the numbers that we do think about are countable. So we’ve got the rationals. Then we can add the algebraic numbers, which are most of the first irrationals we think about. Then we can add pi and all the rational multiples of it and numbers algebraic over Q[pi], and then the same for e. Those are all countable sets. What’s left? Measure theoretically, everything! But you can’t say what it is. (I guess you could go with the infinite sequences of decimal/binary/whatever representations.) It’s almost like the act of naming or recognizing these numbers moves them to the “countable” pile in the big pile of numbers. Recently I’ve gotten a little more comfortable with just thinking about the reals as the completion of the rationals, which are the really meaningful numbers. The reals are the mortar holding the bricks together.