There are only finitely many atoms in the line segment, so finitely many ways to write a line segment, no?

Why are you implementing the idea that we must be able to write the line segment with only atomic precision (as if even that is possible)?

I could have just said I'm using a language where each element of $\mathcal{P}(\mathbb{N})$ is a sentence of the language representing $\frac{1}{3}$.

The point, for the OP's sake, is that we can represent numbers using whatever notation we want. The numbers are what they are, just like Maschke is who he is (no matter what names I choose to call him). :ninja: