> All real numbers, and all numbers of any other variety, can be written with a finite number of symbols.

This is false. In some sense, there exist numbers that can't be referred to. We can refer to the set of real numbers as a whole, but not some of the elements. https://en.wikipedia.org/wiki/Definable_real_number

But then there are issues with defining "definable numbers", which complicates things by a lot. https://mathoverflow.net/questions/44102/is-the-analysis-as-...