The axiom $W\_\kappa$, for $\kappa$ a cardinal, is the statement that for all sets $X$, either $|X|\leq\kappa$ (that is, there is an injection $X\to\kappa$) or $\kappa\leq|X|$. Is there literature on the dual notion $W^\*\_\kappa$ that for all $X$, $|X|\leq^\*\kappa$ (that is, there is a surjection $\kappa\to X$, or $X$ is empty) or $\kappa\leq^\*|X|$?



References for the axiom of surjective comparability
Not too much that I'm aware of. It comes up in the case of $\omega$, since that would imply that even if Dedekind finite sets exist, they can at least be mapped onto $\omega$ (equivalently, "the power set of any infinite set is Dedekind infinite").


In the general case, this isn't as helpful, and there's not too much we can say (except, perhaps, there are no $\kappa$-amorphous sets).


Do note, of course, that if $\kappa$ is an $\aleph$, then $|X|\leq^\*\kappa$ if and only if $|X|\leq\kappa$, so the real difference is in the other direction, which we can restate as "any set that cannot be well ordered can be mapped onto $\kappa$".


