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Consider the following statements in $\newcommand{\ZF}{\sf (ZF)}\ZF$:

  • (Inj) If $A, B$ are sets, then there is an injection $\iota:A\to B$, or an injection $\iota:B\to A$, or both.
  • (Surj) If $A, B$ are non-empty sets, then there is a surjection $s:A\to B$, or a surjection $s:B\to A$, or both.

Clearly, (Inj) implies (Surj). Does (Inj) imply (AC)? Is (Surj) strictly weaker than (AC) or (Inj)?

Dominic van der Zypen
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    What you write as Inj is Bernstein's Theorem (given any two sets $X$ and $Y$, there is either a bijection between $X$ and a subset of $Y$, or a bijection between a $Y$ and a subset of $X$). it is equivalent to the Axiom of Choice. You can see a sketch of the proof that Bernstein's Theorem implies the Well Ordering Principle in this George Bergman Handout. – Arturo Magidin Feb 16 '24 at 17:20
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    @ArturoMagidin I know. I had to switch to a computer, searching for dupes on my phone can be taxing at time. Especially when walking on an uneven sidewalk. – Asaf Karagila Feb 16 '24 at 17:23
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    Thanks for adding the second one. – Arturo Magidin Feb 16 '24 at 18:02

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