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The ordered set in the Zorn's lemma needs to have supremums for all chains, that is, including the empty chain. The supremum of an empty chain is its lowest upper bound. Since any element is an upper bound for the empty chain, we conclude that the set contains a lowest element (in particular, it cannot be empty). Is that right?

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  • $\begingroup$ The only thing you need is some element in the set and chains just need to have upper bounds, not supremums. $\endgroup$ – user2345215 Nov 29 '14 at 11:03
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No, that is not Zorn's lemma. Zorn's lemma says that if every chain has an upper bound, then there is a maximal element.

Not all upper bounds are suprema. And in fact, there lies the crux of choice. You need to choose which upper bound to use when you are attempting to reach the maximal element. Since a supremum is unique, if we had a supremum, we didn't have to make any choices at limit steps.

Since every element is an upper bound of the empty chain, there isn't necessarily a least element. Take for example "The set of all infinite subsets of $X$", when $X$ is infinite.

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  • $\begingroup$ Sure, I misunderstood the statement of the lemma, thank you! We still can conclude that the set is not empty, can we? $\endgroup$ – lisyarus Nov 29 '14 at 11:06
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    $\begingroup$ In some places partially ordered sets are by definition non-empty (there, by convention, chains are usually non-empty as well); when empty sets are allowed to be partially ordered sets, then the empty chain does not have an upper bound. $\endgroup$ – Asaf Karagila Nov 29 '14 at 11:08
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    $\begingroup$ Zorn's lemma says "If bla bla bla, then blah". If the empty chain is considered as a chain, then any element is an upper bound of it - as I remarked in my answer. The only problem arises when considering the empty set as a partial order, but then there is a chain without an upper bound. Since "bla bla bla" is not satisfied we don't care anymore. Zorn's lemma is not the statement "Every chain in every partial order has an upper bound". $\endgroup$ – Asaf Karagila Nov 29 '14 at 11:13
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    $\begingroup$ You are not reading my comments. Since Zorn's lemma is a conditional, either every element is an upper bound of the empty chain (in which case the empty chain does in fact have an upper bound), or we are in the case where the partial order is empty, in which case the empty chain does not have an upper bound and there is no maximal element either. I don't know what else to add on that, since you keep asking the same question, and seem to expect a different answer. $\endgroup$ – Asaf Karagila Nov 29 '14 at 11:17
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    $\begingroup$ Glad to be of service. $\endgroup$ – Asaf Karagila Nov 29 '14 at 11:20

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