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
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.


 * It took me some time to work out what you meant. The main page of Proof 1 contains no such request. I did find some discussion in Talk:Axiom of Choice Implies Zorn's Lemma where various points are discussed, that may be what you are talking about.

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.


 * I confess I could not find any gaps. If you follow the proof carefully, you find that everything follows on from everything else. If you can identify such gaps, you are invited to invoke an instance of the Explain template to suggest what needs to be explained. --prime mover (talk) 09:48, 10 March 2021 (UTC)

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."


 * I am disinclined to use the word "dimension", as this has specific connotations which will add all sorts of confusion into the mind of the reader.


 * It may be instructive to divide the proof into sections with subheadings, if you consider that a useful way to go, and even extract whatever self-contained sections as can be identified into separate lemmata which can be transcluded accordingly as appropriate. --prime mover (talk) 09:48, 10 March 2021 (UTC)

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.


 * I would need to be convinced of the wisdom of including such discursion on (and at the very least, on this particular page). The philosophy of this site is that a proof page contains the proof, and nothing else. Digressions, implications and philosophical discussions about the proof are housed off into separate pages, transcluded if appropriate. We occasionally have a section titled "Motivation", and we also have plenty of "Historical Notes" when such information is available and instructive.


 * I for one find Halmos's style difficult: while he is usually clear and unambiguous in his exposition, a) he tends to be terse to the point of gnomic, and b) he often runs strings of statements together without allowing the reader to pause for breath, which makes him very hard work to comprehend.

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.


 * I am not sure exactly what you are referring to here. How much of this is relevant to the proof of this result? --prime mover (talk) 09:48, 10 March 2021 (UTC)

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.


 * Such material is explained on other pages. See Hausdorff Maximal Principle. There is a wealth of material on containing proofs on this subject. It's just that we do not take the approach of putting everything on the same page.


 * Treatment of Morse-Kelley set theory has yet to be started. It's on my to-do list, but I'm currently working on other things, and I have yet to be inspired to do a comprehensive review of Kelley. (Such is my intellectual laziness, I find hard work to be hard work.) --prime mover (talk) 09:48, 10 March 2021 (UTC)

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?


 * We have a standard structure on for the inclusion of examples that illustrate the use of definitions or results. We could easily do what you suggest using that structure. --prime mover (talk) 09:48, 10 March 2021 (UTC)

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.


 * Again, I had to hunt around before I could find this. I believe you mean Proof 2.


 * The "explain" tag here is to alert the person writing the page that they need to explain the notation on that page. This is conventionally done by providing a link to the appropriate page, with a note along the lines "where $f \sqbrk \lambda$ denotes the image under $f$ of $\lambda$" and is well understood -- just that the contributor in question, when active, expressed distaste for the tedium of explaining the notation (his personal philosophy differed from that of, he was much cleverer than everybody else).


 * Hence the note was an attempt to guide him towards completing the page and bringing it up to house standards.


 * It's also worth noting that Halmos himself did not use the notation $f \sqbrk A$ to mean $\set {y \in \Img f: \exists x \in A: \map f x = y}$; this notation is discussed in some detail in Definition:Image (Set Theory)/Mapping/Subset/Notation, and IMO it's an important point.

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


 * You are encouraged to develop the habit of always enclosing mathematical elements of your communications between dollar signs. For example, in your paragraph above starting "A Much More Minor Point", all of those instances of $f$ and $A$ and so on would be presented so. In this way we differ significantly with Wikipedia, whose approach to presentation of mathematics relies on using a combination of invocation of italic script, special symbols and direct html markup, which results in mathematical exposition (when presented as in-line text) which is difficult to maintain and even harder to comprehend. Our approach is to present everything mathematical, even if just a one-letter variable appearing in an in-line sentence, between $\LaTeX$ "dollar" tags. The difference in font style then makes it immediately apparent as to what is text and what is mathematics -- and has the enhanced side-effect of making similarly presented characters absolutely unambiguous. Compare "Let I(l) denote the identity mapping on l " with "Let $\map I l$ denote the identity mapping on $l$" to see what I mean.


 * For a comprehensive list of $\LaTeX$ commands, you are invited to browse Symbols:LaTeX Commands which is fairly comprehensive, and contains links to a website which explains in further detail. In order to contribute actively and effectively to it is of considerable importance to have the more common aspects of these commands under your fingertips. I also direct you to Symbols:LaTeX Commands/ProofWiki Specific which contains  extensions to $\LaTeX$ which are de rigueur. --prime mover (talk) 09:48, 10 March 2021 (UTC)

Any Comments?

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