HERE IS QUITE A CONUNDRUM:
Can we meaningfully speak of the "infinity-th" and "infinity+1-th" term of the sequence of natural numbers?
If not, then what do we in fact mean by "all" (as distinct from "any" or "each") when applied to an "infinite" set?
Given that a real number constructed via the diagonal construction on a F I N I T E set, of n reals, can always be added to the list at position n+1 to give a list of n+1 reals, why couldn't a real number constructed via the diagonal construction simply be included in the "infinite" list of reals at "position" "infinity+1" ??? (Which is to say that, in the "infinite" case, no real could be constructed outside the infinite list of reals at all!)
Also, in the case of the natural numbers, if a number m, is defined as the sum from 1 to n of the first n natural numbers, then m is a natural number that is not in the list of the first n natural numbers. If you make this construction on the "entire" set of "all" natural numbers, then by construction, there is always a natural number m that is not counted in the set, but this surely does not imply that the set of natural numbers is uncountable. (Or does it?)
Why then is this argument considered valid in the case of reals? is it not equally fallacious in both cases?