Cantor's Diagonal Argument Why Cantor's Diagonal Argument to prove that real number set is not countable, cannot be applied to natural numbers? Indeed, if we cancel the "0." in the proof, the list contains all natural numbers, and the argument can be applied to this set.

The list contains all natural numbers, but also quite a few more.
Natural numbers are terminating strings of digits, that is they are of finite length. Cantors diagonal argument enumerates all strings of digits, especially non-terminating ones.  And yes, the set of those is uncountable, whereas the set of terminating strings is in indeed countable.
