Then it said without proof that μ is finitely additive, but not σ-additive.

As I did not get why I tried to prove it by myself and I tried to show that $S$ is a semi-ring, I guess that is important before I start with the other proof.

We have $\mathrm{\varnothing}\in \mathbb{Q}$ and furthermore $(\mathbb{Q}\cap {I}_{1})\cap (\mathbb{Q}\cap {I}_{2})=\mathbb{Q}\cap {I}_{1}\cap {I}_{2}$ and the union of two closed intervals is either an interval or the disjoint union of two intervals. Then $(\mathbb{Q}\cap {I}_{1})\setminus (\mathbb{Q}\cap {I}_{2})$ is also an interval or the disjoint union of two intervals.

Now the proof. I do not quite understand how it cannot be $\sigma $-additive. Does it have something in common with Cantor sets? I don´t know how to start the proof here. Any help or explanation (maybe an idea for the beginning of a proof) is appreciated. If there is a proof...