Is the class of cardinals totally ordered?
Intuitively, it seems like for any sets $A,B$ either $\lvert A\rvert\leq \lvert B\rvert$ or $\lvert B \rvert \leq \lvert A\rvert$. How can I prove this?
Using the definition of cardinality, the problem reduces to proving that for all sets $A,B$, there is either an injection from $A$ to $B$ or from $B$ to $A$. However, I don't see how to proceed from there. Is AC necessary?