Given a set $A$ with cardinality $c$, there is a subset of $A$ having any cardinality less than $c$.
There is an injection from the subset $B$ to the set $A$, namely, the identity in which each element of $B$ maps to itself in $A$ (no axiom of choice needed; it is like choosing from pairs of shoes).
For $C$ a subset of $B$, and $B$ a subset of $A$, if $C$ and $B$ have different cardinality the identity injection shows this and also which one is smaller. The same can be applied to a set with size $2^c$ (the power set of $A$), and so on.
In this way all the cardinalities that are reached by Cantor's theorem are comparable. In a set of these cardinalities, that one which has an injection into all the others is least, so they are well ordered also. Cantor's Theorem reaches to aleph-omega, and large cardinal axioms are needed to go beyond that, so where is room left for incomparable cardinalities?
If one existed it would have to be the cardinality of a subset of one of the above cardinalities, so, it would be comparable to the others. Where am I going wrong?