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The Order Extension Principle (OEP) is the proposition that every partial order on a set can be extended to a linear order on the same set. The OEP is a theorem of ZFC but not of ZF, and it is among the many choice principles that appear in a book such as Axiom of Choice by Herrlich.

Among notions between those of posets and linearly ordered sets is that of directed sets. One can think of an analogue of the OEP but for directed sets: every partial order on a set can be extended to an order that makes the set directed. Is this a theorem of ZF? (The statement was not on the said book.)

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  • $\begingroup$ I see Zermelo-Fraenkel axioms has something to do with axiom of choice, but how it is different from axiom of choice? $\endgroup$ – Charlie Chang Aug 1 at 4:26
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    $\begingroup$ Just a remark for someone who is trying to solve this: any counterexample must have no maximal elements. So "the obvious" attempts with amorphous sets are doomed to fail. $\endgroup$ – Asaf Karagila Aug 1 at 23:58
  • $\begingroup$ @CharlieChang See the first paragraph here. $\endgroup$ – Alex Kruckman Aug 2 at 1:02
  • $\begingroup$ I think I have a proof that this is not provable in ZF but the details are kind of messy and it will take me a little while to write up. $\endgroup$ – Patrick Lutz Aug 7 at 16:36

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