We have the following statement:
Zorn's lemma is equivalent to the following statement: every partially ordered ser $\langle A,<_R\rangle$ has a chain which contains is own upper bounds
I have proved this first part quite easily, following an argument similar to the one that is used while proving the equivalence between Zorn's lemma and Hausdorff's maximal principle.
However, the corresponding exercise has a second statement:
If in the previous statement we add that the chain is well ordered, does the equivalence still hold?
I think that is not the case, but I don't know how to prove it.
It would be pretty interesting to have a partially ordered set that satisfies the Zorn's lemma hypothesis, but is such that every subset of it is not well-ordered with the restriction of the original partial order. That would make it, however, is such a thing possible? If not, how can we prove the equivalence does not hold? I am completely clueless at this point.
Thanks in advance for your interest.