TITLE: Zorn's lemma - I think I made a counterproof
QUESTION [7 upvotes]: This is from a textbook on topology:
A subset W of the set Z of integers is said to be closed under addition if given any elements w and w′ of W, w+w′∈W.
Prove that there is a maximal subset of Z which is closed under addition and does not contain 9
My problem is as follows: The set of all multiples of 4 does not contain 9 and is closed under addition, the same holds for all multiples of 5. A maximal set must contain both sets, but if that set is closed under addition, we get that 9 is a member of the set.
EDIT: Thanks for the clarification, on the definition I was really lost. Can anyone give me a sketch of a proof?

REPLY [19 votes]: Maximal does not mean maximum.
Maximal set means that you just cannot add more elements while preserving this property, so indeed a maximal set might include the multiples of $4$ or the multiples of $5$, but certainly not both as you show.
(Recall that if $(P,\leq)$ is a partially ordered set, a maximal element $p\in P$ is such that whenever $p\leq q$ we have that $p=q$; whereas a maximum is an element $p$ such that for every $q$ we have $q\leq p$.)