Cardinality definitions in variants of ZF theory, yet again As has been discussed here before, it is possible to define cardinality (a function $|\cdot|$ from the universal class to a class of "cardinals" such that $x\approx y\iff|x|=|y|$) using either AC or regularity. I saw somewhere else the statement that the following, strictly weaker than AC or regularity, is sufficient:
Every set is equinumerous to some well-founded set.
I don't know if that statement is provable in ZF without regularity. Is it known whether cardinality in the sense I have described must exist in ZF without regularity?
 A: First of all, much like Andres Caicedo writes in the comments, it is consistent with $\sf ZF$ that there is no choice function on the class of Scott cardinals (i.e. cardinals defined using Scott's trick). In fact it is consistent that there is no such function on a set of cardinals as well.
This is due to Jech and Sochor, although they claimed that the proof is trivially achieved from their embedding theorem. Pincus later showed they are wrong, and gave an improved embedding theorems which allowed this to be transfered from $\sf ZFA$. He later wrote a paper about cardinal representatives where he showed some conditions on the existence of a choice function on Scott cardinals. I am unaware of anyone else working on this topic since then.
As for the statement "Every set is equinumerous to a well-founded set", I don't think it is sufficient, and here is an example which I haven't sat down to write in details, but seems to work:
Consider the case where we have the axiom of choice and one Quine atom $x=\{x\}$ which generates the universe using iterated power set operations. Force (with symmetries of course) over the well-founded part in a way which violates the existence of cardinal representatives.
Note that the Quine atom $x$ still generates the universe. It is not hard to show that any set is equinumerous to a well-founded set (note that $V_1$ is a singleton, and therefore $\mathcal P^\alpha(x)\approx V_{1+\alpha}$).
