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Let $P$ denotes the property that if there exists a surjection from set $A$ to set $B$, then there exists an injection from $B$ to $A$. It's apparent that $P$ can be proved in ZFC. My question is that if $P$ and AC are equivalent in the ZF theory?

Asaf Karagila
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Censi LI
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    Short answer, we don't know. See also this and this. Probably a few other related links that I've missed. – Asaf Karagila May 15 '15 at 12:18
  • @AsafKaragila It surprises me that this question is still open. Anyway thank you very much. – Censi LI May 15 '15 at 12:40
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    I have a plan to try and solve it, but it might take another 10-20 years or so. But as for the surprise, if there's one thing that I've learned in the past five years of being a grad student and trying to figure out mathematical research, is that it can be very easy to ask questions that can be very hard to answer; it can even be easy to ask natural question that arise organically from research (as opposed to just shifting variables of a statement randomly). The tricky part is to find a question that is both natural, non-trivial and that you can still answer it. There lies great mathematics. – Asaf Karagila May 15 '15 at 12:44

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