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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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  • $\begingroup$ Also math.stackexchange.com/q/393196/622 and other questions on the "Linked" menu to the right in that link. $\endgroup$
    – Asaf Karagila ♦
    Commented Feb 5, 2016 at 10:23
  • $\begingroup$ There's probably a question about this on this site as well. $\endgroup$
    – Asaf Karagila ♦
    Commented Feb 5, 2016 at 10:23

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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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    $\begingroup$ That is correct. $\endgroup$
    – Asaf Karagila ♦
    Commented Feb 5, 2016 at 10:22
  • $\begingroup$ So the statement is strictly "in between" ${\sf ZF}$ and ${\sf ZFC}$ it appears... $\endgroup$
    – Dominic van der Zypen
    Commented Feb 5, 2016 at 10:47

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