Zorn’s Lemma applies to posets in which every chain has an upper bound. However, in all applications I know, the poset is also evidently chain-complete — chains have least upper bounds. A few classic such applications:
- AC, using a poset of “partial choice functions”
- the Well-Ordering Principle, using a poset of “partial well-orderings” (ordered by end-extension, not simply $\subseteq$)
- Hahn–Banach, using a poset of “partial functionals defined on subspaces”
- extension of filters to ultrafilters, using a poset of filters.
Are there any natural applications of Zorn’s Lemma where the poset isn’t chain-complete, or where chain-completeness is less obvious than existence of upper bounds?
(Indeed, all examples I know are equally obviously directed-complete. Under AC, this is equivalent to chain-complete, but constructively, directed-completeness is stronger. For those wondering why I’m caring about use of AC while considering Zorn’s Lemma, note that constructively, ZL does not imply AC — this is shown in John Bell’s very nice analysis Zorn's lemma and complete Boolean algebras in intuitionistic type theories, JSL 1997, https://doi.org/10.2307/2275642. His surprising (to me) insight is that ZL itself doesn’t imply LEM or AC — most applications, including the classical proof of ZL=>AC, use LEM for the final step “a maximal partial object must be total”.)