Talk:Axiom of Choice implies Zorn's Lemma/Proof 1

Refactor
Refactoring is badly, badly needed. This is a long rambling essay which goes from Zorn to what I believe Kelley calls the maximal principle, to a maximal chain condition I believe is sometimes called Kuratowski's.... It's a mess. --Dfeuer (talk) 20:23, 1 July 2013 (UTC)


 * It is as presented in Halmos. As such it stands as is. --prime mover (talk) 20:26, 1 July 2013 (UTC)

Notes in preparation for an upgrade in Proof 1 of Zorn's Lemma
Hello, I am John Aiken. PhD in mathematical physics (von Neumann Algebras) from Louisiana State University, Baton Rouge, La, in 1972. About half a dozen publications in professional math, quantum chemistry, and computer science journals and conference papers. Spent my career working in applied math, electrical engineering, and computer science. Finally ended up at MITRE - "M.I.T. Research Engineering," a government consulting firm that spun out of M.I.T. in the 1950's.

Am 74 years old, happily married, have 14 grandchildren. Live in Winston-Salem, North Carolina.

Never quite understood how the Axiom of Choice implies Zorn's Lemma. The Axiom of Choice seems so intuitively clear (at least to me), and Zorn's Lemma seems quite mysterious. Being retired, I now have the time to look into the matter more deeply.

While Paul Halmos's proof, a variant of Zorn's original proof, is absolutely beautiful, it skips over quite a few steps. This might make it difficult to follow for undergraduates; it certainly did for me. I am currently preparing notes intended to fill in these gaps. Hopefully, the notes will be ready to submit for your review before the end of July of this year.

Currently posted on the main page of Proof 1 is a request that somebody please explain Zorn's lemma and its proof. Here are my "off the top of my head" responses to that request.

1) Hopefully the notes that are in preparation will explain how to get from one statement to the next in the proof. Filling in each gap is not particularly difficult, once you look at the problem from the right viewpoint, but there are so many gaps that it can become confusing.

2) Another approach is to explain the overall structure of the proof. Maybe something like this:

"The proof itself is 3-dimensional. In the first part, one looks at "chains" in the partially ordered set "X". That is the first dimension. In the second part, one looks at chains that are built on the chains constructed in the first part. That is the second dimension. The third (and longest) part of the proof looks at 'towers' that are built on the chains of the second part. That is the third dimension. An amazing thing is that at the end of the first part of the proof, the partial order on X can be completely ignored, and the problem can be reformulated in terms of the partial order from the second part of the proof."

3) On the page preceding his proof, Halmos explains what makes Zorn's Lemma so mysterious. He explains why Zorn's Lemma does not look at all obvious, even though the Axiom of Choice seems to be intuitively clear. In other words, a choice function certainly "should" exist, while it is not at all clear that the a maximal element should exist.  More precisely, it is not clear how to prove that a maximal element exists without running into Russel's paradox.

One could invoke "classes" and "von Neumann ordinals" to explain how to avoid Russel's paradox, or one could use the existence of "well-orderings" on arbitrary sets, but such a choice seems like "using a sledge hammer to kill a fly." Not to mention that such a choice inserts many unnecessary steps -- such a pity, if one is only interested in understanding how the Axiom of Choice implies Zorn's Lemma.

4) Another aspect of "explaining" Zorn's lemma might be to put it the context of the other "axioms" that are equivalent to the Axiom of Choice. John Kelly has a nice start on this in his book "General Topology." But, there are several stumbling blocks in his approach. The main stumbling block is that he does not include the equivalence of the Hausdorff Maximality Principle in his list of equivalent axioms. In any case, running through equivalent axioms does not enhance my understanding of how the Axiom of Choice implies Zorn's lemma. Another stumbling block is that Kelly does not use Zermelo-Fraenkel set theory, the standard formulation of axiomatic set theory.

5) Yet another way to help "explain" Zorn's lemma is to give a "canonical" application. An application that is illuminating.  An application that is simple without being trivial.  An application that makes Zorn's lemma look like it "should" be true.

A standard example in set theory is the equivalence relation on the real line where x is equivalent to y if and only if y-x is a rational number. Each equivalence class is countable, but the set of all of these equivalence classes has the same cardinality as the real numbers themselves (the cardinality of the continuum). This example is simple without being trivial, but does it have anything to do with Zorn's lemma?

A Much More Minor Point: Currently posted on the ProofWiki page of Proof 1 is the question, "What is f[A]?" Answer: f[A] is the image of the set "A" under the function "f". In other words, if a function "f" maps a set A into a set B, (that is, f:A-->B), then the set "f[A]" is the set of all y in B such that y = f(x) for some x in A. "B" is the range of "f." f[A] is a subset of B.

Finally, I am looking forward to learning LaTex. I've used LaTex before, but I never really learned it.

Any Comments?

--DeaconJohn (talk) 06:53, 10 March 2021 (UTC)