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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 $\kappa\leq^*|X|$?

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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 $\kappa$".

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  • $\begingroup$ Typo! ${{}}{{}}$ $\endgroup$ Jul 20, 2023 at 15:53
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    $\begingroup$ Where? I was typing on my phone, I have an excuse! $\endgroup$
    – Asaf Karagila
    Jul 20, 2023 at 16:13
  • $\begingroup$ Ah, found it. Ducking auto portrait. $\endgroup$
    – Asaf Karagila
    Jul 20, 2023 at 17:21

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