All rational numbers of the unit interval [0, 1] can be covered by countably many intervals, such that the $n$-th rational is covered by an interval of measure $1/10^n$. There remain countably many complementary intervals of measure $8/9$ in total.
Does each of the complementary intervals contain only one irrational number? Then there would be only countably many which could be covered by another set of countably many intervals of measure $1/9$.
Is there at least one of the complementary intervals countaining more than one irrational number? Then there are at least two irrational numbers without a rational between them. That is mathematically impossible.
My question: Can this contradiction be formalized in ZFC?