This is, perhaps, more a question of philosophy of math rather than math itself. While it may be trivial to most people fluent in math, it was a bit surprising for me to learn recently that the set of algebraic numbers is countable. However, I quickly realized why that is: Each algebraic number can be expressed with a finite polynomial, and the set of all possible finite polynomials (and, more generally, the set of all possible finite strings) is countable. Therefore the set of all algebraic numbers is countable. Then I realized that this can be further extended. The set of all numbers that can be defined with a finite closed-form expression is likewise also countable. The same is true for all numbers that can be defined with a finite analytical expression. In fact, we can generalize this the most by saying: Let's define a number as "expressible" if it can be expressed (unambiguously) with a finite amount of information. (It doesn't matter if you need a special formalism to formulate the expression, as long as you can define said formalism with a finite amount of information. Basically, the formalism becomes part of the definition of the number.) The set of all "expressible" numbers is countable. Now, I understand perfectly that this set is most probably ill-defined. However, let's assume for the moment that it is well-defined. This means that there are real numbers that can not be expressed in any way with a finite amount of information. (Because the set of all "expressible" numbers is countable, that means that there are real numbers that do not belong to that set.) The (mostly philosophical) question becomes: Are these "non-expressible" numbers useful in any way? They cannot be the answer to any problem (because a number that's the answer to a problem belongs to the "expressible" set: It can be defined by the problem in question.) These numbers cannot be used in any way, at least not individually, because there's no way you can define them. At most you can define them in (uncountably) infinite sets, but never individually. If they can't be the answer to any problem, are they useful?