Are there more transcendental numbers or irrational numbers that are not transcendental? This is not a question of counting (obviously), but more of a question of bigger vs. smaller infinities. I really don't know where to even start with this one whatsoever. Any help? Or is it unsolvable?

The set of algebraic numbers $\mathbb A$ is countable, so $\mathbb A\cap (\mathbb R\setminus \mathbb Q)$ is also countable. On the other hand, the set of transcendental numbers $\mathbb R\setminus\mathbb A$ must be uncountable, so $$|\mathbb R\setminus\mathbb A|>|\mathbb A\cap(\mathbb R\setminus\mathbb Q)|.$$
