# Zorns Lemma

Zorn's Lemma is a statement about partially ordered sets. It states:

• Every partially ordered set in which every chain has an upper bound contains at least one maximal element.
• (A chain is a totally ordered subset)
This doesn't really sound too controversial. Suppose every chain does have an upper bound. Either that upper bound is a maximal element, or there's something "above" it. Extend the chain, lather, rinse, repeat. Either your chain won't have an upper bound (which is impossible becuase we've assumed every chain has an upper bound, or we must eventually get a maximum.

Well, not so fast. Things get hairy when you have uncountably infinite sized sets, and so things can go wrong.

In fact, Zorn's Lemma is equivalent (using the usual set-theory background) to the Axiom of Choice, and that's not so obvious either.