TITLE: Cofinal chains in directed sets
QUESTION [9 upvotes]: Let $(D,\prec) $ be a directed set. Is there a "cofinal chain" in $D $? By chain I mean a $D'\subset D $ such that for all $\lambda,\gamma \in D'$, then $\lambda\prec \gamma $ or $\gamma\prec \lambda $. By cofinal chain I  mean a chain $D'$ for which, for every $\gamma\in D $, there is some $\lambda \in D'$ such that $\gamma\prec \lambda $.

REPLY [5 votes]: Take $X$ to be your favorite uncountable set. Now take $D$ to be the finite subsets of $X$, ordered by inclusion.
Easily, this set is directed. But there are no chains with order type longer than $\omega$, and so no chain can have an uncountable union, and in particular, no chain is cofinal.
(If you choose $X$ wisely, then no choice is needed either. And by wisely I just mean not a countable union of finite sets.)