This part:

> Here's how: consider the set of numbers that cannot be described using any finite collection of symbols. Such numbers must exist *because there are only a countably infinite number of numbers that can be described using a finite collection of symbols*, but there are an uncountably infinite number of real numbers.

That’s clearly wrong. The real numbers between 0 and 1 (an uncountable collection) are describable with a finite collection of symbols - the digits between 0 and 9. Each real number is an equivalence class of sequences with the same limit. (It’s been a while since I took real analysis but I believe this is the construction due to Weierstrass?)