# About the hypothesis of Zorn's lemma

The proofs I know of Zorn's lemma give the following refinement:

Let $(X,<)$ be a partially ordered set such that every well-ordered subset of $X$ has an upperbound. Then $X$ has a maximal element.

In fact, Zorn's lemma is sometimes stated as such. In comparison, the usual statement asks that every totally ordered subset of $X$ has an upper bound.

Does this refinement have some application?

EDIT (after comments). Your comments make me realize that I did not think enough of my question, which is what the voters-to-close probably guessed. To help future readers, let me sum up the comments.

• If $(X,<)$ satisfies the hypothesis of the refined Zorn lemma (RZL), then (modulo AC), it satisfies the hypothesis of classical Zorn lemma as well, since any totally ordered set contains a cofinal ordered subset. (Comment of Ramiro de la Vega).

• ZF+RZL implies ZF+AC (Noah S).

• In Cohen's first model of ZF, there exists a set which satisfies the hypothesis of RZL but not that of ZL (Asaf Karagila).

• Since that seemingly weaker version of Zorn implies AC, they are in fact equivalent. (To see that this version implies AC: given a set $A$ consider the poset of partial well-orderings of $A$.) Oct 16 '14 at 16:22
• @NoahS: I believe the question asks for examples where checking only well-ordered chains is much easier than checking all chains. Oct 16 '14 at 16:25
• One interesting way to interpret the question is to ask for a model of ZF in which AC fails, which has a partial order in which all well-ordered chains have upper bounds, but some totally ordered chains do not. This would show, if indeed there are such models, that the equivalence of the two hypotheses was itself a kind of choice principle. Oct 16 '14 at 17:37
• @Noah: Cohen's first model satisfies that. The Dedekind-finite set of reals is unbounded, but every well-ordered chain is finite, so it has a maximal element. Oct 16 '14 at 18:15
• Can one of the voters-to-close explain why they're voting to close? Oct 16 '14 at 18:26