Consider the statement
For any infinite set $X$ there is an injection $\varphi$ from $(X\times\{0\}) \cup (X\times\{1\})$ into $X$.
Does this imply the ${\sf AC}$?
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asked Feb 5, 2016 at 9:12
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$\begingroup$ Also math.stackexchange.com/q/393196/622 and other questions on the "Linked" menu to the right in that link. $\endgroup$– Asaf Karagila ♦Feb 5, 2016 at 10:23
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$\begingroup$ There's probably a question about this on this site as well. $\endgroup$– Asaf Karagila ♦Feb 5, 2016 at 10:23
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1 Answer
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It's way over my head, but it seems that this question was answered in the negative by Gershon Sageev, An independence result concerning the axiom of choice, Ann. Math. Logic 8 (1975), 1-184.
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$\begingroup$ So the statement is strictly "in between" ${\sf ZF}$ and ${\sf ZFC}$ it appears... $\endgroup$ Feb 5, 2016 at 10:47