Cantor’s argument about the infinities of the Real Numbers shows that there is something wrong about the entire way the proof is set up. We don’t know if it is because there are infinities within infinities or if there is something wrong about the argument that we can treat an infinite set of numbers, each of which are infinite in length, the same way as if it were a conveniently finite set of numbers each of which were conveniently "finite in length".

A similar argument holds (with a slight modification) for an exploration of a continuous span. There are infinite numbers in between two points on a number line does not mean that the irrationals are more uncountable than the rationals. You might be able to use this as a basis for an argument to show that there must be more irrationals than rationals in a continuous span, but you can not complete the proof based on a feasible counting of the infinite. Even if there are more of them that does not mean that they are more uncountable. You might argue that there are infinite irrationals between any two rational numbers. OK, so what? There are also infinite rationals between any two rational number. The act of “counting” the infinite, while acceptable as an idealization of a practical action, is going to be fraught with possible complications and you should be smart enough to think about it.

In my opinion, this is a good case when a cautionary sense learned from the constructivist's argument should kick in and make you at least pause before you wander down an infinite path of infinities and accept it on face value.