A few facts: Zorn's lemma is an axiom (which has a few equivalent reformulations such as each set can be well-ordered, or the cartesian product of any family of sets is not void ('axiom of choice')) which is independent of the axioms of Fraenkel-Zermelo set theory (the most popular axiom system of set theory). So, from a logical point of view you can accept it or refuse it. If you refuse it you get rid of paradoxical phenomena such as the Banach-Tarski paradox.

If you accept it (as most mathematical textbooks do) you get for free a lot of mostly useless theorems.

Here is an concrete example: Zorn's lemma allows an easy proof of: Every linear space has a Hamel basis. This proof gives no executable method how to get such a base. Even for the simple case of R (real numbers) linear space over Q (rational numbers), no Hamel basis is known, and nobody misses it. For the existence of the much more usefull Schauder bases of topological linear spaces no Zorn's lemma is needed. Good mathematical texts tell where Zorn's lemma is required (it is often used in proofs for laziness even if it is not required).

Sometimes Zorn's lemma is regarded as an axiom, sometimes as a theorem, it depends on the accepted axiom system. If it is regarded as an axiom, every system in which Zorn's lemma does not fit in, simply must be built upon other axiomatic basis.

It is my opinion that logics must be an abstraction of real world. For instance, if some contradiction were noticed in group theory or other theory else, I do not be sure if we must change the theory or the underlying logics ruling it. However, if the considered theory connects with real world, then I prefer changing logics. Real world is always my choice instead of a dogmatic system.

A few facts: Zorn's lemma is an axiom (which has a few equivalent reformulations such as each set can be well-ordered, or the cartesian product of any family of sets is not void ('axiom of choice')) which is independent of the axioms of Fraenkel-Zermelo set theory (the most popular axiom system of set theory). So, from a logical point of view you can accept it or refuse it. If you refuse it you get rid of paradoxical phenomena such as the Banach-Tarski paradox.

If you accept it (as most mathematical textbooks do) you get for free a lot of mostly useless theorems.

Here is an concrete example: Zorn's lemma allows an easy proof of: Every linear space has a Hamel basis. This proof gives no executable method how to get such a base. Even for the simple case of R (real numbers) linear space over Q (rational numbers), no Hamel basis is known, and nobody misses it. For the existence of the much more usefull Schauder bases of topological linear spaces no Zorn's lemma is needed. Good mathematical texts tell where Zorn's lemma is required (it is often used in proofs for laziness even if it is not required).

As Ulrich explain, Zorn's lemma is independent form ZF and therefore cannot be incorrect. Bishop-style constructivism rejects Zorn's lemma (among others) and therefore proides an example of mathematics without Zorn.

Zorn's lemma is equivalent to the Axiom of Choice which has been proven to be independent on ZF in 1963 by Cohen using forcing. The following paper: http://arxiv.org/abs/0712.1320 is a nice introduction, but uses forcing to prove the independence of the continuum hypothesis on ZF. Mind you, the assumption is the consistency of ZF. If ZF were not to be consistent, the result is not necessarily true. Let me mention a few reasons why I like the axiom of choice (but still like to think about the use) is Hahn-Banach. This is strictly weaker and implied by the ultrafilter lemma. Additionally, Hahn-Banach already gives us a non-measurable set. So, many strange things, like the Banach-Tarski paradox, arise from the assumption of ZFC, but many nice things get lost if we don't. Especially analysis is virtually non-existent without AC. Of course, there are schools that study that, but the analysis that is applicable in applied mathematics is certainly not one of those :-). There is a model of the reals due to Solovay that makes all the subsets of the reals Lebesgue measurable (iirc). That sounds nice right? However, it also gives very contradictory things like boundedness of basically all linear functions. Summarizing, I wanted to add that the assumption of Zorn's lemma not only makes life easier, it also makes life possible. And, as you know, life is not fair, even not in mathematics. Cheers! Jonas

@William: Did you intend to add to anwering the question? For me this reads as bla, bla. 'Mathematics is only a model' ... if model at all, than a family of models, at least one with and without Zorn's lemma.

For followers of this thread, the following article may be of interest:

J.S. Russell, Indefinite divisibility:

http://phiblog.files.wordpress.com/2009/02/extensibility2.pdf

See page 7, where the author writes:

The reason for the apparent failure of Zorn’s Lemma is that the lemma (like the comprehension scheme) involves an ambiguous quantifier (twice over, in stating the existence of a chain, and in stating that it is maximal). While we can truly say there is_1 a chain that is maximal_1 (i.e. no_1 chain properly contains it), we may yet go on to use a more inclusive quantifier “there is_2 ”, with respect to which that chain is not maximal_2 after all—for there is_2 a chain that properly contains it. This is just what we should expect in the presence of indefinite extensibility.

James; thank you for pointing this excellent source

http://phiblog.files.wordpress.com/2009/02/extensibility2.pdf

Zorn's lemma is equivalent with the Axiom of Choice (Ac). These cannot be true or false since there are independent. You have to choices: To accept the axiom of choice and do classical mathematics or accept the negation of AC and then you are in non-Cantorian mathematics. 

A remark on the AC: Intuitively, AC imposes a level of reality, so that when AC holds, then it is as we accept the abstraction of "rigid bodies" which give white or black entities. If you accept the negetion of AC you are essentially in moving entities like particles.  To see all these one needs to go to examples that describe all these.

Axioms are satisfied by some objects, while other do not satisfy them. For instance, commutativity is only satisfied by abelian groups. Not every group need to satisfy commutativity property.

Likewise, Axiom of Choice is satisfied by some sets. Accordingly Zorn's lemma is valid only for some sets. This is why, in some contexts it is regarded as an axiom instead of a lemma.

Summarizing, if you apply Zorn's lemma, working with some set that does not satisfy the Axiom of Choice, then it is a wrong procedure, but not a lack of accuracy of algebraic theories.

@Costas

please note that a "rigid body" has an invariable shape but is well allowed to move in space. There is a huge body of knowledge termed "rigid body dynamics" dealing with moving rigid bodies. This spoils your metaphor which obviously was intended to suggest some kind of contradistinction.