It is well-known that if the natural (partial) order on the class of cardinal numbers is a linear order, then it is in fact a well-order and the axiom of choice holds. I was, however, interested in how much choice we can recover given some linear ordering, or better — a well-ordering — of the class of cardinals.
I couldn't figure out by myself any results, but I would imagine it at least implies that there are no amorphous sets.
Are there any results known about this? To be more specific, let me ask the following question:
Is the axiom of countable choice implied by the existence of a linear ordering on the class of all cardinals? How about the existence of a well-order on this class?
Thanks in advance for all the feedback.