It is often said that instead of proving a great theorem a mathematician's fondest dream is to prove a great lemma. Something like Kőnig's tree lemma, or Yoneda's lemma, or really anything from [this list](https://en.wikipedia.org/wiki/List_of_lemmas).


When I was first learning algebra, one of the key lemmas we were taught was [Zorn's lemma](https://en.wikipedia.org/wiki/Zorn%27s_lemma). It was almost magical in its power and utility. However, I can't remember the last time Zorn's lemma appeared in one of my papers (even though I'm an algebraist). In pondering why this is, a few reasons occurred to me, which I'll list below. I don't want to lose my old friend Zorn, and so my question is:



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> What are some reasons to keep (or, perhaps in line with my thoughts, abandon) Zorn's lemma?
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**Edited to add**: One purpose to this question is to know whether or not I should be rewriting my proofs to use Zorn's lemma, instead of my usual practice of using transfinite recursion, if there is a mathematical reason to prefer one over the other. Hopefully this clarifies the mathematical content of this question.




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To motivate the discussion, let me give an example of how I would now teach ungraduates a result that was taught to me using Zorn's lemma.


**Theorem**: Every vector space $V$ has a basis.


*Proof*: First, fix a well-ordering for $V$. We will recursively work our way through the ordering, deciding whether to keep or discard elements of $V$. Suppose we have reached a vector $v$; we keep it if it is linearly independent from the previously kept vectors (equivalently, it is not in their span), otherwise we discard it. If $B$ is the set of kept vectors we see it is a basis as follows. Any vector $v\in V$ is in the span of $B$, because it is either in $B$ or in the span of the vectors previously kept. On the other hand, the elements of $B$ are linearly independent because a nontrivial combination $c\_1 v\_1 + \dotsb +c\_k v\_k=0$, where $v\_1<v\_2<\dotsb<v\_k$ and $c\_k\neq 0$ can be rearranged so $v\_k$ is a linear combination of the previous vectors, so $v\_k$ cannot belong to $B$ after all. $\quad\square$


Here are some of the benefits I see for this type of proof over the usual Zorn's lemma argument.


**1. The use of choice is disentangled from the other parts of the proof.**


When applying Zorn's lemma, it is difficult to see exactly how the axiom of choice is being used to reach the conclusion of a maximal element. One way to visualize its use is that Zorn's lemma lets us recursively build a maximal chain through the poset. This chain must have a greatest element. However, this construction is hidden behind the magic words "Abracadabra Zornify".


Is it a historical artifact that choice is hidden this way?


**2. We can more easily see whether or not to use a choice principle.**


In the proof above, if $V$ is already well-orderable (without AC), then we don't need to ever use the axiom of choice.


**3. Zorn's lemma is no easier than transfinite recursion.**


Each part of transfinite recursion already (implicitly) occurs in most Zorn's lemma arguments. The base case of the recursion corresponds, roughly, to showing that that the poset is nonempty (i.e., has some starting point). The successor ordinal step often occurs at the end; after asserting that some maximal element of the poset exists, we show that this maximal element has some claimed property by working by contradiction, and then passing to a slightly bigger element of the poset (i.e., the next successor). The limit ordinal step occurs when we show that chains have upper bounds.


**4. Zorn's lemma often includes unnecessary complications.**


In the proof I gave above, there is no need to define a complicated set, together with a poset relation. We can use strong induction, to avoid differentiating between the zero, successor, and nonzero limit steps. We don't need to combine the contradiction at the end with any successor step; they are entirely separated.


**5. Transfinite recursion is a more fundamental principle.**


As a matter of pedagogy, shouldn't we teach students about transfinite induction before we teach them a version of it that is also combined with AC, and that requires the construction of a complicated poset?


**6. Transfinite recursion applies to situations where Zorn's lemma does not.**


To give just one example: There are some recursions that continue along all of the ordinals (for a proper class amount of time). Zorn's requires, as a hypothesis, an end.



Zorn's lemma: old friend or historical relic?
I agree with almost everything in your post. But still, I believe I know why people use Zorn's lemma.


**My answer.** Zorn's lemma encapsulates succinctly many of the consequences of AC via transfinite recursion, but without requiring any involvement of the ordinals or knowledge of transfinite recursion to be used.


To those who are deeply familiar with transfinite recursion, of course, every use of Zorn's lemma can be seen as sumblimating the underlying construction, which achieves the maximal elements by a transfinite process that simultanesously explains why they exist. To appeal to Zorn seems to hide this essential explanatory underlying mechanism.


And yet, the alternative perspective is that Zorn's lemma abstracts away from the recursive process, producing in the end a simpler argument that relies only on the core consequences of the recursive process, which do not rely on any explicit engagement with ordinals or recursion. And precisely because of that feature, Zorn's lemma arguments can be undertaken and understood by mathematicians who are unfamiliar with the ordinals and transfinite recursion.


In the vector space example, to show every vector space has a basis, one can mount a transfinite recursive process: you pick an element, and then if it doesn't span, you pick another, and so on transfinitely until you have a basis. (My view of this example is a little different from how you described it, since I view the choice function as more primitive than the well order — I would build the basis by choosing amongst the element not yet in the span — indeed I prefer to view WOP itself as the outcome of recursively choosing elements.)


With Zorn's lemma, however, there is no need for ordinals or transfinite recursion, and the Zorn's lemma argument instead encapsulates abstractly their effects — the partial order consists in effect of partial undertakings of the recursive process. In this sense, the Zorn argument is simpler, abstracting away from the transfinite constructive "process".


I find the situation to be analogous to Martin's axiom and forcing. Martin's axiom is the poor mathematician's forcing, just as Zorn's lemma is the poor mathematician's choice+transfinite recursion.


My personal view is that the ordinals and transfinite recursion are one of the wonders of mathematics, a sublime achievement of the intellect resulting in many beautiful arguments and constructions. I tend to prefer the transfinite recursive arguments as providing a deeply explanatory account of the consequences of Zorn's lemma. (Even the well-order theorem seems fundamentally less mysterious when explained via the transfinite recursion — pick any element as the least element, and now pick a next element, and a next, and so on transfinitely.)


Further, although I recognize that many mathematicians have little involvement or experience with the ordinals and transfinite recursion, I also believe that their mathematical life would be improved by knowing more of them.


