Correct me if I'm wrong, but I believe the cardinality of the two sets you describe are equal, as there exists a bijective mapping between them, meaning there are an equal "number" of real numbers in both.
Not only does there exist a bijective mapping between [1,2] and [1,4], there exists infinite different bijective mappings between a subset of [1,2] and [1,4].
i.e.: One could map bijectively from [1,1.5] to [1,4] and map bijectively from [1.5,2] to [1,4] (1)
To talk about there being "twice as much" in one uncountable infinity than in another uncountable infinity is nonsense, since you can't apply words like "twice", since the infinities can't be counted.
