Well-foundedness of cardinals and the axiom of choice Without axiom of choice, it is not generally true that the class of all cardinals (in this question we consider Scott cardinal rather than cardinals as ordinals) is not well-founded under the ordinary cardinality comparison. However, we also know that it is fine to assume the well-foundedness of the class of all cardinals.
Under ZF with this assumption, we can prove that every infinite set is Dedekind-infinite as follows: for infinite set $X$, consider the collection of cardinals
$$\mathcal{A} = \{|A| : A\subseteq X \text{ and $A$ is infinite}\}.$$
From assumption, $\mathcal{A}$ is well-founded. If $|B|$ is a minimal element, then $B$ should be Dedekind-infinite, since $|B|-1 = |B|$. (where $|B|-1$ is a cardinality of the set $B$ except one element in $B$.)
I wonder we can prove a stronger result; for example, the axiom of choice follows from that the class of cardinals is well-founded? I would appreciate your answer.
 A: This is an open problem.
It was shown that for every $\kappa$, $\sf DC_\kappa$ cannot prove that the cardinals are well-founded. While not enough to conclude the principle is equivalent to the axiom of choice ($\sf BPI$ does not follow from $\sf DC_\kappa$ either), it is worth remarking that we really don't know much about this principle.
A very recent paper gave a nice survey of this problem and related results:

Paul Howard, Eleftherios Tachtsis, "No decreasing sequence of cardinals", Archive for Mathematical Logic, First online: 28 December 2015.

Let me finish by stating that generally speaking the structure of the cardinals is a bit of a wild beast when it comes to the axiom of choice. We don't have good techniques to control it very well in order to produce separating models for much awaited-results (e.g. the Partition Principle is a statement about the structure of the cardinals). So we mainly know how to violate things wildly (e.g. embed partial orders into the cardinals of a model), but not how to fine tune this in order to produce nice results.
