TITLE: How to divide aleph numbers
QUESTION [6 upvotes]: Recently, I was wondering how division of aleph numbers would work. First, I thought about how finite cardinality division would work. What I came up with was that the result of $A/B$ where $A$ and $B$ are both cardinalities, is the number of times that each element of $B$ had to be mapped to an element of $A$ in order to ensure that all elements of $A$ were mapped to.
Extending this to aleph numbers, specifically, I thought that $\aleph_1/\aleph_0$ would be $\aleph_1$. The reasoning behind this is that there are an infinite ($\aleph_1$) number of real numbers between any two natural numbers. As such, that mapping would need to be applied.
Can anyone validate this idea? Is this how division of aleph numbers actually works, or am I totally off base?
Thanks.

REPLY [3 votes]: If you're looking for something compatible with cardinal multiplication, you'll have to deal with the fundamental problem that $\kappa\cdot\lambda=\max\{\kappa,\lambda\}$ whenever $\kappa,\lambda$ are well-orderable cardinals and at least one of them is an aleph. That sort of absorption means--for example--that $\aleph_0\cdot\aleph_2=\aleph_1\cdot\aleph_2=\aleph_2\cdot\aleph_2=\aleph_2$, so even trying to define $\aleph_2/\aleph_2$ in some way compatible with cardinal multiplication is problematic. Now, one could choose a convention for $\lambda/\kappa$ in instances that $\kappa\leq\lambda$--say, for example, that it's always just $\lambda$--but trying to compatibly define it when $\kappa>\lambda$ is fruitless.