Who proved the rationals are countable?i

Who proved the rationals are countable? (No fair looking this up.) If you're not sure, guess.

I'm wondering if people know the answer to this one; via some informal surveying it seems less well known that I'd expect. Then again, I didn't know it.

On a related note, the usual proof that N × N is countable (where N = {1, 2, 3, 4, ...} is the natural numbers) goes by giving an explicit enumeration of the pairs of natural numbers, namely,

(1,1), (2,1), (1,2), (3,1), (2,2), (1,3), (4,1), (3,2), (2,3), (1,4), ...

where we list all pairs of integers summing to 2, then to 3, then to 4, then to 5, ... Thus we list pairs of integers in order of their "size". Alternatively, we're enumerating pairs of sets in order of their "size", where sets are only distinguished up to cardinality.

Combinatorialists are often concerned with classes of objects where there are a finite number of objects of size n for any integer n... of course this means that any such class is in fact countable. I don't think I'd heard this stated explicitly, probably because it's not particularly important for doing combinatorics.

For the answer to the question: see the comments.

infinity times two

Infinity, from the classic math television show "Square One".



I got this from meep, who points out: "A little ditty about infinity - and is good about showing infinity as a process, not a number." This is definitely true; although I don't want to transcribe the full lyrics, the Big Idea here is that infinity is not just a very large number. As the Hitchhiker's Guide to the Galaxy says (I don't know exactly where), "Infinity is just so big that, by comparison, bigness itself looks really titchy." In Chapter 24 of the first book, there's the following:

The car shot forward straight into the circle of light, and suddenly Arthur had a fairly clear idea of what infinity looked like.
It wasn't infinity in fact. Infinity itself looks flat and uninteresting. Looking up into the night sky is looking into infinity — distance is incomprehensible and therefore meaningless. The chamber into which the aircar emerged was anything but infinite, it was just very very big, so that it gave the impression of infinity far better than infinity itself.

This makes quite a bit of sense; psychologically I don't think that we're built to understand the concept of infinity (or infinitesimals, for that matter), because we evolved to deal with medium-sized things. Mathematics, in a sense, is profoundly un-evolutionary, in that we've advanced it so far that it goes far beyond any intuition a "normal" human being might have. This probably explains why it's so hard for a lot of people to learn.

(One day I might write the Mathematical Hitchhiker's Guide to the Hitchhiker's Guide to the Galaxy, or something like that. There are a lot of math jokes in that series.)

Note: could you please let me know if you can see the embedded YouTube video in this post? I'm not sure if I did it correctly.