What about using Lebesgue outer measure? The interval $[0,1]$ has Lebesgue outer measure 1 (the proof of this uses compactness of $[0,1]$, which can be proved just from the completeness of $\mathbb{R}$).

On the other hand, it is a direct consequence of the definition that any countable set has Lebesgue outer measure 0. 

Hence, $[0,1]$ is uncountable and so is $\mathbb{R}$.