Are Undefinable Numbers Useful in Mathematics?

[Stephen Tashi]

doesn't it restrict all possible formalisms, and thus all numbers that can be represented with said formalisms, to a countable set?

Suppose we think of "the real numbers" as a reality existing outside of mathematics - think of them as points on a line (the intuitive idea of a line, not a formally defined one.) When we represent some of these points as numbers in the usual manner, we can imagine picking any point on the line we wish and calling it "zero". To talk about "the set of numbers defined by all possible systems of representing them by finite strings of symbols", raises the question of whether two copies of the same system of representaion necessarily represent the same set of numbers. (For example, do two copies of the usual representation of numbers refer to the same "zero"?)

Suppose we ditch the Platonic idea of a number line existing outside of mathematics. We still need some way to determine if string a1 in system S1 means the same number as string a2 in system S2. So not only do we need to talk about all possible systems of reprsenting numbers as strings; we also need to talk about all possible ways of relating strings in one system to strings in another.

 

[Warp]

Some mathematicians disliked real numbers in the past (and who knows, there may still be some today) because there are "too many" of them. I'm starting to understand them.