Let me consider the reals between [0;1] and work in binary numeration system. I order the list like this: first number is 1, second is 0.0, third is 0.10. For the next numbers, the rule is that all the diagonal decimal digits are 0's. Cantor's diagonal number will then be 0.111111....=0.(1)=1. So, he failed to produce a number which is not on my list.