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.

Cantor's diagonal argument is one of contradiction.  You start with the assumption that your set is countable and then show that the assumption isn't consistent with the conclusion you draw from it, where the conclusion is that you produce a number from your set but isn't on your countable list.  Then you show that for any 
If you wanted to do Cantor's diagonal argument, you'd start by assuming the Natural numbers are countable, then try to create a contradiction by producing a number that isn't on the countable list.
Suppose the natural numbers are countable, and we list them then perform Cantor's diagonal technique, producing a string of digits $x$.  $x$ either has an infinite number of non-zero digits or $x$ has a finite number of non-zero digits.
If $x$ has a finite number of non-zero digits, then $x$ is on the list (and it's a Natural number), so there no contradiction.  If $x$ has an infinite number of non-zero digits, then $x$ isn't a Natural number (and it isn't on the list), so there is no contradiction.
Either way, we can't find a contradiction by using Cantor's diagonal argument on the assumption that the Natural numbers are countable, so we can't conclude from that argument that the Natural numbers are uncountable.
The most important part here is to notice that any natural number must have a finite number of non-zero digits.
