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”.)

yesbut I don't really know... $\endgroup$ – Tim Campion♦ Apr 18 at 20:22