Again, excellent point! These numbers clearly show why* my reasoning is missing a rather large portion of this set.
[edit] - fixed typo
[edit] - fixed typo
One you fix a finite set of symbols, let's say ascii [1], the amount of numbers you can describe with it with strings of finite length is different from the amount you can describe if you allow also infinite length.
The difference between finite length and infinite length in this problem is very important.
[1] I'm not sure about unicode /s