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.
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.
1) I can find that number also in the set between 1 and 4.
2) But I can't find that number in the set between 2 and 4.
To myself numbers are always relative to one another, don't exist outside the mind, and the set between 2 and 4 is twice the set between 1 and 2.
But like I said, it's a personal spin on it.
f(x) = (x-1)/9 + 1, maps bijectively from [1,4] to [1,4/3]
g(x) = (x-1)/9 + 4/3, maps bijectively from [1,4] to [4/3,5/3]
h(x) = (x-1)/9 + 5/3 maps bijectively from [1,4] to [5/3,2]
Therefore, [1,2] must contain three times as many numbers as [1,4], right?
It doesn't work like that.