Today I did the “ is uncountable” proof in class. I got a few “understanding nods”, and only one person seemed to be falling asleep (visibly). One person stormed out the moment I finished the proof, but I think it was because I went about 30 seconds over, rather than being angry at Cantor. All in all, I think that was success. Last time I did this a few years ago to a similar audience I got many angry stares. I think they were also not small Kroneckers, but more angry as in “I have to learn this nonsense?” I always find that this theorem has a quality about it that makes people either like it a lot or hate it a lot. I think much more so than anything else they will see in this class. I already had a question in office hours whose answer depended on the axiom of choice. I guess that is more subtle than uncountability of , but also I don’t talk about AC in class.
I did something like Cantor’s original proof of 1874. I looked at Wikipedia (which has a simplified version of Cantor’s proof) and I looked at baby Rudin, which has essentially the same proof in completely different language and in greater generality. Though I did it without using contradiction, since proofs by contradiction are evil.
Proof: Given a countable set , you construct a sequence of closed intervals such that , and such that . The intersection of all the intervals is nonempty (compactness) and can’t contain any element of the countable set. QED
That’s more analysis-like than the diagonal proof, which is more computer-science-like.