We know that the Cantor's cardinality of a countable set is $\aleph_0$ and the cardinality of continuum is $2^{\aleph_0}=\aleph_0^{\aleph_0}$. Unfortunately, this measure is based on the idea of bijection, so sets of different volume can have the same cardinalities.
On the other hand, there is an idea of "numerocities" (which requires ordered or metric space), linked to the divergent integrals and series. For a subset of non-negative integers $S$ the numerocity is represented by a generally divergent series:
$$\sum_{k=0}^\infty p_S(k)$$
where $p_S(k)$ is the membership function, equal to $1$ if $k\in S$ and $0$ otherwise. It allows to compare (using properties of divergent series) such sets as even and odd numbers, prime numbers, etc, giving more precise notion of the set's size than cardinality.
The question is, can we somehow introduce a concept similar to numerocity to the uncountable sets, in such a way that it would reflect the volume of those sets? In other words, the set $[0,2)$ should have twice he numerocity of $[0,1)$. This is desirable...
First, let us agree on some symbols. Let use $\omega_-=\sum_{k=1}^\infty 1$ for numerocity of natural numbers, and $\omega_+=\sum_{k=0}^\infty 1=\omega_-+1$ for numerocity of non-negative integers.
The following is pure handwaving, please don't beat me hard.
First, let us consider the binary representation of the reals from $[0,1]$: $0.1101001...$. Here each digit can be either $0$ or $1$. Since we start from position $1$ and have positions corresponding to all natural numbers, we can say we have $\omega_-$ positions. So, the whole numerocity is $2^{\omega_-}$. Of course, if we take another base rather than $2$, we will have different expression for that numerocity. But this should not be confusing because numerocity is often dependent on ordering and filtering. So, if we make the same infinite set more dense, its numerocity changes (grows).
Can we somehow establish the expression for numerocity of reals on an interval, that would not depend on the chosen base for representation?
As I see it, the only way to avoid dependence on the base is to allow the base to go to infinity. So, we transcend to a representation in an infinite base. The digits start from $0$ but can be any natural number. As such (because we start from $0$), we have $\omega_+$ digits. And the whole numerocity is $\omega_+^{\omega_-}$. Please don't ask me how a particular number would be represented in such system. One can think of it as of some kind of limit. Also, since we do not have the biggest digit, the number $1$ does not belong here, so this numerocity represents the range $[0,1)$ rather than $[0,1]$. I would point out that the expression of the form $\omega_+^{\omega_-}$, $\omega_-^{\omega_-}$ etc, often appear in the theory of divergent series, so this is not something "unseen". It also fits well with the similar expression for the cardinality of continuum, which is a similar but less refined measure. The numerocity of the whole real line then would be expressed as $(\omega_++\omega_-)\omega_+^{\omega_-}$.
I understand, these are very loose deas, particularly the idea of "infinite base" for a digital representation is unjustified. But I wonder whether some grains of it can be put on solid ground.
