Can the number of numbers in two intervals over $\mathbb{R^+}$ be compared?

Question

Every number in the interval $[2.1,4]$ can be mapped to its square in the interval $[4.41, 16]$. Conversely, every number in the interval $[4.41,16]$ can be mapped to its respective square root in the interval $[2.1,4]$.

If every number in the interval $[4.41,16]$ has a unique corresponding square root in the interval $[2.1,4]$, this would imply that the number of numbers in both intervals are the same magnitude of infinity.

This implies comparing the length of two intervals does not indicate which interval has more numbers in it, as $[4.41, 16]$ is obviously longer than $[2.1, 4]$, yet has the same number of numbers.

Answer 1

Cardinality strips the set of its structure. This means that we don't care that $[0,1]$ is an interval of length $1$ and $\Bbb R$ is an infinite interval. This is the idea behind cardinality in the first place, stripping down any pre-existing structure.

On the other hand, we do have a way of comparing sizes of sets of real numbers which assigns intervals their length, the Lebesgue measure (or the Borel measure). This is a very different method of measuring sizes of sets.

Answer 2

This is correct. Much more simply, you can use a linear transformation to make the correspondence between any two intervals in $\Bbb R$ as long as the end points are of the same type-closed or open. If they are not, it is only two points, which is clearly negiligible. There are easy patch-ups available.