Zorn’s lemma, as defined in Wikipedia, is stated as follows:
(Zorn’s lemma) A partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element.
It can be proved that Zorn’s Lemma is actually equivalent to the Axiom of Choice, thus independent from $ZF$ framework. A proof that relies on Zorn’s Lemma usually goes as follows: For a certain partially ordered set with some property, there is a maximal element of it. Any element $x$ greater than the maximal element does not satisfy that property anymore. By choosing the proper $x$ we can run the proof by contradiction.
I found myself still not used to the classical ways of using Zorn’s Lemma after a while, such as the proof of Alexander Subbase Theorem, and the proof of “Nilradical of a commutative ring is the intersection of all prime ideals.” It seems to me that those proofs are very different than the ordinary proofs I learned before, that is, if I get stuck with coming up a proof of a proposition, I will intentionally construct a maximal element and blindly test if I could reach the result closer. Another problem is that I have no clue at all whether a theorem relies on Zorn’s lemma or not from a first look. Can anyone give some suggestions, that I should practice and learn more applications of Zorn’s Lemma, or, in my opinion, to find a book that systematically give motivations for each proof? Any help would be appreciated. Thank you.
Best Answer
I am not really aware of a book like that. There are a few books about the axiom of choice, but they mainly focus on other things, not on "typical applications of Zorn's lemma". These are "Axiom of Choice" by Herrlich and "Axiom of Choice" by Jech, for example.
I would imagine that many books about rings, groups, modules, and other algebraic structures which are infinite, will have a handful of "typical applications of Zorn's lemma".
When I had to explain my students the "typical use", I actually prefer to point them to a different lemma: Teichmüller–Tukey.
Definition. We say that a family of sets $\cal F$ has finite character if the following holds: $A\in\cal F$ if and only if every finite $B\subseteq A$ satisfies $B\in\cal F$.
In other words, $\cal F$ has finite character if in order to verify that $A\in\cal F$, we only need to verify that its finite subsets are in $\cal F$.
Lemma. (Teichmüller–Tukey) If $\cal F$ is a family with finite character, then there is a $\subseteq$-maximal member of $\cal F$.
Note that it is often easy to verify that something has finite character. For example, linearly independent subsets a of a vector space. If a set $A$ is not linearly independent, this property is already given by a finite subset of $A$. So if all finite subsets of $A$ are linearly independent, so must $A$. Now, by the lemma, there is a maximal element, and then it is not hard to prove that such maximal element is a basis.
In the typical use of Zorn's lemma we actually have a family with finite character. So when we take a chain of sets, the union of these sets generates an upper bound (or it is an upper bound in many cases). The reason is that if the union wouldn't be an upper bound, it means that it fails to satisfy the property we are interested in, but by the finite character, this is witnessed by some finitely many elements in the chain, which is a contradiction, since the maximal one of those (in the chain) would indicate otherwise.
Think about being a subgroup, or an ideal, or a chain in a partial order, or so on. These are all properties of a subset which have finite character. Therefore Zorn's lemma or the Teichmüller–Tukey lemma is so useful in proving the existence of maximal sets with these properties.