With AC, we have that cardinals are well-ordered. Without AC, we can have two sets which are incomparable, meaning that, at best, cardinals are only partially ordered.
But do we still have, without AC, that cardinals are at least partial-well-ordered, meaning there is no infinite descending chain of cardinals? Or can we have such infinite descending chains?
I had the thought that the existence of an infinite, Dedekind-finite set would suffice to prove that cardinals are not in general partial-well-ordered, since with those we have the property that if we remove any one element we get a strictly smaller cardinal. If we can show that this strictly smaller cardinal is always also Dedekind-finite, then we could remove another element to get an even smaller one, and so on. However, I am never quite sure how much this kind of reasoning really works in such bizarre models of ZF, so could use some clarity.