# Difficulty understanding Zorn's Lemma.

This is a question on chap. 9 of Introduction to Set Theory of Hrbacek and Jech. The book define that the Zorn's Lemma is: If every chain in a partially ordered set has an upper bound, then the partially ordered set has a maximal element.

This is very confused to me. I tried find a upper bound for each chain (is the $\subseteq$-maximal element ?) but I don't know how to try this. Furthermore how I will find the maximal element for the partially ordered set?

• You can't necessarily find the max element. Zorn only tells you that such an element exists. Dec 1, 2014 at 19:09
• You didn't really ask about any equivalences of Zorn's lemma. So I took the liberty of changing the title. Dec 1, 2014 at 19:16

Just like the axiom of choice guarantees that there is a function $f\colon\mathcal P(\Bbb R)\to\Bbb R$ such that for every non-empty set $A$, $f(A)\in A$. It doesn't tell us what is $f(\Bbb R)$ or what is $f(\Bbb{R\setminus Q})$ or anything like that. It exists, move along.
So Zorn's lemma tells us, essentially, that if whenever we pick a chain in $P$, then we can find an element which is larger (or equal) to all the elements in the chain; then we can be sure that there is a maximal element in $P$.