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My Claim:

Let (D,$\le$) be a poset and f a function from the set of linearly ordered subsets of D to D. Assume that for every linearly ordered subset U of D we have $\forall$x $\in$ U, x $\lt$ f(U). Then there is no maximal element in D.

My proof:

Suppose for a contradiction x is a maximal element in D. Clearly, {x} $\subset$ U is linearly ordered. Then x $\lt$ f({x}) $\in$ U. This contradicts with the assumption x being a maximal element in D.

On the other Zorn's Lemma implies that D has a maximal element because every linearly ordered subset U of D is upper bounded by f(U) $\in$ D.

Clearly, there is something wrong here. There are three possibilities that might explain these contradicting results.

  1. My proof of the claim is incorrect.
  2. The use of Zorn's Lemma is incorrect.
  3. There exist no such function f and thus there is no contradiction.

What do you think about this?

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    $\begingroup$ Possibility 3 is the correct one. If the claim is your own invention, then you have nearly rediscovered the proof of Zorn's lemma from the axiom of choice discussed in math.stackexchange.com/questions/4038120/…. $\endgroup$
    – Rob Arthan
    Feb 25 at 14:02
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    $\begingroup$ As far as I can tell your reasoning is correct, so you would have to conclude that no such $f$ can exist. In that case it would be interesting to ask whether such $f$ can exist in a universe without choice. $\endgroup$ Feb 25 at 14:03
  • $\begingroup$ @RobArthan no unfortunately it is a part of an assignment my teacher gave. $\endgroup$
    – Klomanad
    Feb 25 at 14:06
  • $\begingroup$ @MarkKamsma is there a direct way of showing that such f doesn't exist? $\endgroup$
    – Klomanad
    Feb 25 at 14:07
  • $\begingroup$ Well, it depends on what you are assuming... If you assume Zorn's lemma, you have produced yourself the proof that such f does not exist. $\endgroup$ Feb 25 at 14:39
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Working in $\sf ZFC$, every partial order has a maximal chain. Your requirement about $f$ is that $f$ is a strict upper bound, but a maximal chain cannot have a strict upper bound, since adding a strict upper bound to a chain will result in a strictly larger chain.

So assuming $\sf ZFC$, there can be no such $f$. But the same can be said in $\sf ZF$, actually. What $f$ is doing is to choose an upper for every chain, and it is easy to see that it is exactly the choice function needed for the proof of Zorn's lemma, so by assuming it exists we circumvent to appeal to the Axiom of Choice. Consequently, it means that by applying $f$ recursively, we can create a maximal well-ordered chain, which again by maximality cannot have a strict upper bound.

So in fact such $f$ cannot exist, even without assuming the Axiom of Choice. So in either case, your proof is completely vacuous.

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