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By convention a Lemma is a technical intermediate step which has no standing as an independent result. Lemmas are only used to chop big proofs into handy pieces. (Quoted from here)

I am wondering why Zorn's Lemma is called a lemma at all. Is it a "handy piece" of some proof of an important result?

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Zuriel
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  • It is usually used to show the Hahn-Banach theorem. – Eff Feb 03 '15 at 11:49
  • @Eff, thank you for the comment! Then why it is called a lemma and the Axiom of choice is called an axiom? It is a bit weird that a lemma is equivalent to an axiom. – Zuriel Feb 03 '15 at 11:52
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    https://en.wikipedia.org/wiki/Zorn%27s_lemma#History – Najib Idrissi Feb 03 '15 at 12:09
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    Mathematics is not always that congruent with naming, symbols and stuff. Most people I know would consider a "lemma" to be a provable result of lesser importance within some given frame, and from this point of view Zorn's lemma is so not a lemma, in fact. Some others consider a lemma to be a result helpful to prove other results, yet still the other results are more important, usually, than the lemma. From all this I'd rather call it Zorn's Supertheorem, yet the name "lemma" is so deeply rooted that it could be misleading to change it now. – Timbuc Feb 03 '15 at 12:09
  • Because Zorn was so clever, what the common man calls a theorem, he calls a lemma. :) – goblin GONE Feb 03 '15 at 12:12
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    @goblin: Does that rhyme on purpose? – Nikolaj-K Feb 03 '15 at 12:26
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    So that people could make jokes like "What's yellow and equivalent to the axiom of choice - Zorn's Lemon. What's fluffy and equivalent to the axiom of choice - Zorn's Llama, etc." – Tom Collinge Feb 03 '15 at 12:28
  • @TomCollinge, What's purple and commutes? What is yellow, linear, normed and complete? – Zuriel Feb 03 '15 at 12:30
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    Rhyme, you say? Why yes, it does, @NikolajK. – goblin GONE Feb 03 '15 at 12:32
  • @Zuriel , the first one I know (I think): an abelian grape. For the second I only think of a rather lame Bananach space...? – Timbuc Feb 03 '15 at 13:01
  • @Timbuc, you are exactly correct! – Zuriel Feb 03 '15 at 13:06
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    In the original paper by Zorn, it served as a lemma for the theorem he really wanted to prove. Another example is Fatou's Lemma: in Fatou's paper it was a lemma for his main theorem on boundary values. – GEdgar Feb 03 '15 at 13:37
  • There are many famous lemmas in math whose fame renders their lemma status dubious: http://en.wikipedia.org/wiki/List_of_lemmas. – Quinn Culver Feb 03 '15 at 15:08

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When you want to construct something very large, or just handle something very large, you can do it using the axiom of choice and/or the well-ordering theorem coupled with transfinite recursion and induction.

It's not very difficult, but one has to understand the structure of the ordinals in order to use the ordinals properly. In particular one has to know a bit more about well-orders in order to use transfinite induction.

If someone wants to prove that every vector space has a basis; or if every unital commutative ring has a maximal ideal; or whatever. They might not be very interested in well-orders, ordinals and transfinite recursions.

Zorn's lemma offers a way out. It offers working with partial orders, often we use subsets or some other objects that we know sufficiently well.

And of course, it has plenty of easy uses, since partial orders where every chain has an upper bound are abundant in mathematics (see two easy examples above). It is a lemma since we're not generally interested in the partial order or in finding a maximal element there. We are, however, interested in specific partial orders (the set of linearly independent sets ordered by inclusion, for example). So the term "lemma" is very fitting, since it tells you some nontrivial information about a general context that is applicable to your case.

Asaf Karagila
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From Proofs from the Book, Second Edition, Martin Aigner and Günter* M. Ziegler, p. 149:

The essence of mathematics is proving theorems -- and so, that is what mathematicians do: they prove theorems. But to tell the truth, what they really want to prove, once in their lifetime, is a Lemma, like the one by Fatou in analysis, the Lemma of Gauss in number theory, or the Burnside-Frobenius Lemma in combinatorics.

Now what makes a mathematica statement a true Lemma? First, it should be applicable to a wide variety of instances, even seemingly unrelated problems. Secondly, the statement should, once you have seen it, be completely obvious. The reaction of the reader might well be one of faint envy: Why haven't I noticed this before? And thirdly, on an esthetic level, the Lemma -- including it's proof -- should be beautiful!

Edit: I was so excited to drop that quote that I didn't carefully read the question. Here's a link to Timothy Gowers's Weblog where he describes "How to use Zorn’s lemma" which hopefully more directly answers the question, including:

Quick description: If you are building a mathematical object in stages and find that (i) you have not finished even after infinitely many stages, and (ii) there seems to be nothing to stop you continuing to build, then Zorn’s lemma may well be able to help you.

*Cut-and-pasted the "ü" in Günter per suggestion in the comment -- LaTeX encoding didn't seem to work.

joeA
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