Main Question or Discussion Point
I am sure you are all familiar with this. The number generated by picking different integers along the diagonal is different from all other numbers previously on the list. But you could just put this number as next element on the list. Of course that just creates a new number which is missed, but if you successively kept putting the number missed as indicated by the diagonal wouldn't you eventually hit all real numbers on the interval (0,1)? I mean isn't it the same as saying that we haven't hit the rational number 5/32 but that it is coming later in the sequence we use to pair the rationals with natural numbers.