I think you (and some other commenters) are responding to something different - potential vs actual infinity - not being able to write down all natural numbers, but we can potentially write any number(with some assumptions like an infinite universe).
Whereas the point being made is different and needs some math background which is the work of Cantor. Countable can include all natural numbers that you count until infinity, and the reals are uncountable in that they exceed even this.
The proof of this fact(Reals are uncountable), has the same idea involved in Turing machines halting problem and also Godel's theorem. The general version is the power set of a set(set of subsets of a set) cant be put in one to one correspondence with original set. If you assume such a correspondence, then () leads to a contradiction.
Reals are sequences of naturals which include sequences of 1's and 0's which can be interpreted as subsets of naturals(a sequence represents the subset of indexes which get assigned the value 1). So reals are bigger than the power set of naturals.
Also, the infinite countable union of countable sets is countable. The number of grid points(integer coordinates) on a plane is the same as the number of grid points on a line. You can set up a zig-zag correspondence. A salesman can visit all grid points on a plane in infinte time. This is used to prove rationals are countable. So even countably infinitely different symbolic systems doesnt help.