Does the ordered set from Zorn's lemma have a lowest element?

Question

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?

Answer 1

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.