> All real numbers, and all numbers of any other variety, can be written with a finite number of symbols. That's what it means to give something a name. This i

> All real numbers, and all numbers of any other variety, can be written with a finite number of symbols. That's what it means to give something a name. This is untrue and not particularly hard to prove by contradiction. Suppose every elements in R can be named by a finite number of symbols. You can build a bijection between R and the names of its elements (by definition a name points to a unique element and if elements have multiple names, you can easily well order a finite number symbols by building a lexicographic order and only consider the smallest name). Or, you can also easily build a bijection between a finite number of symbols and N. That's just an encoding like ASCII or UTF-8. Therefore, the set of names is countably infinite. Yet R is uncountable (see Cantor's diagonal argument). Therefore, by contradiction, there has to exist elements in R which can't be named by a finite number of symbols.