Here is what I have come up with. It seems pretty sound to me, but I have little experience in pure math and want to be sure. (I know the easiest way is to show that R is not countable, Q is, so the irrationals, R-Q, is not, but I was wondering about this proof.) Consider the following numbers: 0.1223334444... 0.112222333333... 0.111222222... ... (When numbers of two or more digits are encountered, they repeat in the same way, e.g. in the first number we would have ...99999999910101010101010101010.... ) These numbers are irrational, as they are non-repeating infinite decimals. They also form an countably infinite set, as they are infinite but can be listed. However, for obvious reasons, ∏ is not included. Nor is e, √2, or many other irrationals. Since the set of the irrationals contains more elements than a countable infinite set, it is uncountable. QED Thanks for looking.