We know $\Bbb R$ is bigger than $\Bbb Q$ because its cardinality is bigger. We know that $\Bbb R^2$ is bigger than $\Bbb R$, which is bigger than $[0, 1]$ because the latter can be thought of as a subset of the former. However, the question is, how much bigger?
There is a notion of additive index, multiplicative index, and cardinality index. For example:
- [0, 1] has additive index $2$ in [0, 2]. This is because [0, 2] is a union of 2 [0, 1]'s. Thanks to Captain Lama's comment, when we assign a measure on $\Bbb R$ and both sets are measurable, the additive index is simply the quotient of the measures of the two sets.
- [0, 1] has additive index $\aleph_0$ in $\Bbb R$.
- $\Bbb R$ has multiplicative index $\Bbb Z$/2 in $\Bbb R^2$, since $\Bbb R^2$ is a 2-dimensional vector space over $\Bbb R$. However, if we were looking at additive index, then $\Bbb R$ would be identified with $\Bbb R$ x {$0$}, and thus has additive index $\aleph_1$ in $\Bbb R^2$.
- Similarly, $\Bbb R$ has multiplicative index $\Bbb Z$ in $\Bbb R^\Bbb Z$; however, the cardinality of both is the same.
- Cardinality, then, is something to measure only the really really big differences between sets.
The most interesting index I've run into in math is, the multiplicative index. For example:
if $L$ is a finite-dimensional vector space over $K$ without additional structure, then we can define multiplicative index as simply $\Bbb Z/[L:K]$.
More generally, if $L$ is a Galois extension of $K$, then the multiplicative index can be defined to be the Galois group of $L/K$. In particular, this would allow differentiating things of countable multiplicative index, say those with Galois group $\Bbb Z$ x $\Bbb Z/2$ from those with Galois group $\Bbb Z$, or $\hat{\Bbb Z}$.
My questions is: how to define this multiplicative index when: 1) the extension is not Galois, and 2) cardinality is the same.
For example, $~~~~\bar{\!\!\!\!\!\!\!\!\!\rm if}$ denotes the algebraic closure, then we know $\bar{\Bbb Q}\cap\Bbb R/\Bbb Q$ is a countable extension. However, since the automorphisms of $\Bbb R$ are all trivial, this extension is not Galois. Is there a way to associate an index to this extension? What about to various extensions of $\bar{\Bbb Q}\cap\Bbb R$?