TITLE: Earliest diagonal proof of the uncountability of the reals.
QUESTION [12 upvotes]: I cited the diagonal proof of the uncountability of the reals as an example of a `common false belief' in mathematics, not because there is anything wrong with the proof but because it is commonly believed to be Cantor's second proof. The stated purpose of the paper where Cantor published the diagonal argument  is to prove the existence of uncountable infinities, avoiding the theory of irrational numbers. I have no problem believing that Cantor himself realized that a diagonal proof of the uncountability of R was possible but I have not even found an allusion to this in his collected works. The earliest appearance in print that I know is on page 43 of The theory of sets of points by W. H. Young and Grace Chisholm Young (1906). I would be very grateful for any reference to some scrap of paper where Cantor himself mentions the possibility of using the diagonal method to prove the set of reals uncountable.

REPLY [2 votes]: The evidence that you look for can already be found in the second paragraph of Cantor's original paper. There he states "Es läßt sich aber von jenem Satze ein viel einfacherer Beweis liefern, der unabhängig von der Betrachtung der Irrationalzahlen ist.", meaning that his diagonal argument supplies a much simpler proof of the theorem proved in his first paper on the uncountability of the real numbers. This does not only mention but declare that his method can be applied to real numbers.
Probably in order to emphasize its independence of numbers, Cantor did not use numerals 0 and 1, but m and w which (and this answers your last remark) in German are abbreviations of male and female. So he had a good substitution for 0 and 1 or up and down or yes and no - and he deliberately or unconsciously circumvented the problem that this proof without reservations, as stated in his original text, would fail on binary sequences.