What tells us that the structure of the cardinals is "discrete"? I'm not using the words "discrete" and "dense" with their formal meanings.
Maybe I have this confusion because I'm using concepts that I've not formalized (in my mind) but in other words why can we talk about successor of a infinite cardinal?
Who tells us that there isn't a cardinal $\kappa$ such that $\aleph_\alpha\lt\kappa\lt\aleph_\alpha^+$?
For exaple when we talk about $\mathbb Q$ we can't find the successor of a number $q^+\in\mathbb Q$ such that $\nexists r (q\lt r \lt q^+)$.
If I must define what I mean with successor I wold say this:
Let be $(A,\lt) $ a total strict order and $a\in A$ I define $succ_\lt(a)=a^+$ only if $\nexists b\in A(a\lt b\lt succ_\lt(a))$
How matematicians know that if cardinalities are linearly ordered exists a successor (like for finite sets) with the properties that I've defined? What if infinite cardinals have the structure of the rational numbers?... then we can find betwen two infinite sets always an infinite number of infinite sets with intermediate sizes? Why is not possible?