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 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 footnotes intended to help undergraduates fill in these gaps. Hopefully, the notes will be ready to submit for your review before the end of July of this year.

Currently (or at least recently) 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.


 * Yes.


 * I apologize for having wasted your time. Hopefully that will happen less often as I get more familiar with ProofWiki. -- --DeaconJohn (talk) 05:21, 22 March 2021 (UTC)

1) Hopefully the footnotes 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)


 * Every time Arjun Jain uses the word "Lemma" in his arXiv.org paper https://arxiv.org/pdf/1207.6698v1.pdf he is filling in one of the holes in Halmos' proof. There are 8 Lemmas; that is 8 holes. I found and filled in Jain's first 5 holes myself, but I got stuck on the sixth hole. I have not actually verified that Lemmas #7 and #8 do indeed correspond holes, but, I strongly suspect that they do.


 * I'm looking through the paper myself, to see what he has to add.


 * First thing I notice is his "Lemma 1. A necessary and sufficient condition for $\overline s (x) \subset \overline s(y)$ is $x \le y$." This is already covered in . We have done exactly the same as Jain did, we linked to this result which is a basic standard result in set theory, proved (for example) by Devlin (1993) and linked to by us in our Ordering is Equivalent to Subset Relation.


 * We use a number of such subsidiary results in our exposition. It is how is structured.


 * It may be worth revisiting this proof to see just how much of the detail of Jain's expansion of this proof is similarly hidden behind such blue links to established results.


 * I believe it may be a retrogressive step to extract the reasoning from all such results and present the exposition of each one onto this page here.


 * Hence my contention that there are actually no gaps in this presentation. (Again, I beg to be proven wrong.) I am not going to go into a full analysis of this against Jain's expansion now, though, as I have other plans this morning. I will leave it up to you to take a look. --prime mover (talk) 06:56, 22 March 2021 (UTC)


 * Beware, though, I believe that Jain's Lemma #6 contains a serious typo: he uses "Z" every place "script X" should appear (IMHO). Also, it appears that he is writing to people who teach Axiomatic Set Theory on the graduate level, and to their students.  In other words, his exposition seems to me to be way-above the undergraduate level.  I have written him and email about the "possible typo." I mentioned that Lemma 6 was of great help to me in filling in the hole corresponding to that lemma.  I also promised to give him full credit when I publish my footnotes.


 * What would you think about adding a template for referencing papers in arXiv.org? Of course, I don't really need to provide a link, since a reference in text would be adequate.  But, your help page does say that references to online scholarly papers are acceptable.  If you don't want to have a template for arXiv.org, what is the syntax for linking to on-line papers outside of ProofWiki?  I would like to have provided such a link for you personally in this paragraph.


 * I am a slow and careful worker. Even so, I make a lot of mistakes, and that slows me down even more. It would take me too much time to carefully craft explanations of the holes to fit into the Explain template format on the undergraduate level.  After all, the holes were not apparent to even you, and you are way past the level of a typical undergraduate.


 * I'm not. As my teachers always screamed at me constantly, I'm a lazy slacker who doesn't deserve a job as a janitor, with a slapdash and casual attitude that deserves to me the torments of Dante's inner hell. It is because of that I barely scraped my Summa cum Laude MMath, and that I was told I was going to fail all my exams at school.


 * And as you say, I can't see the holes as such, but maybe that's because I transcribed this proof onto this page in the first place, and (bearing in mind what I said further up under your comments on Jain's lemmas) I was fairly sure the train of thought was indeed covered. I did make a mistake at one point, but it was pointed out to me and I corrected it. --prime mover (talk) 06:56, 22 March 2021 (UTC)


 * I suspect that there are holes that Jain did not address. He is writing to graduate students, and their teachers, who are already active in Axiomatic Set Theory, whereas I am writing on the undergraduate level.


 * The only difference between undergraduate and graduate is the arbitrary line caused by particular terminology and techniques not having been covered. Bearing in mind my previous comments above concerning the details already being extracted and proved in separately established results, it is the intention of that every result is ultimately accessible to all readers who have the intellectual commitment to address it. But ultimately the reason this is considered a "postgraduate" result is not so much because the techniques are not covered at undergraduate level, as the fact that the result itself is highly challenging.


 * Neither Zorn's Lemma nor the Axiom of Choice were even mentioned in my formal studies (which, as I say, got no further than the substandard summa-cum-laude MMath I scraped some 16 years back), but then again I have never made a good fist of formal studies because of personality deficiencies, and so most of what I have now comes from deliberately un-learning everything I learned formally and re-learning from working through various published materials that can be found either in bookshops or online.

I intend to begin each footnote with an explanation of why there is a gap.

At the beginning of the proof of Zorn's Lemma, I intend to advise undergraduates to try to figure out why there is a gap each time they see a footnote, and then try to fill in the gap before referring to the footnotes. If they don't do this, they will not really understand the proof. Many of the crucial steps in the proof are contained in the gaps, IMHO.

Having all the footnotes at the end of the page, will allow the advanced student to read through the proof and then read the footnotes to see what he missed. The footnotes are not independent of each other, and this will make it easier to follow their flow. -- --DeaconJohn (talk) 05:21, 22 March 2021 (UTC)


 * I've not been a fan of the footnote approach, because it compromises the convenience of the transclusion process. As I say, the standard house style of structuring of such exposition is to make the statement on the page, including the detail behind the blue links. But I would be interested to see how you expedite your plan. --prime mover (talk) 06:56, 22 March 2021 (UTC)

2) Another approach is to explain the overall structure of the proof.


 * 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)


 * That sounds like a good idea. Let's revisit this later, if necessary. --DeaconJohn (talk) 05:21, 22 March 2021 (UTC)


 * Prime Mover: Please delete the paragraphs between the two lines of asterisks unless you make a further comments. ********************* --DeaconJohn (talk) 05:21, 22 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 discussion 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.


 * Got it.

3) 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. He does not "close the loop."  His exposition does not prove that the Axiom of Choice implies Zorn's Lemma. Even if he die, 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.


 * Got it.


 * 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)


 * I advise against including anything on Morse-Kelley set theory. It has been replaced by ZFC, as far a I know. If Morse-Kelly makes a "come-back" then, maybe.


 * It was ever the aim of to document anything and everything coming our way. The Morse-Kelley approach is historical and pedagogically interesting in its own right, and deserves documenting on its own intrinsic merits. We (collectively) enjoy the concept of the historical expansion of mathematical concepts, and the metaphorical fumbling-in-the-dark that went on in this subject over the decades before Zermelo's and Fraenkel's lucidity affords ample scope for allowing us to expose the inner workings of the development of a mathematical enlightenment. --prime mover (talk) 06:56, 22 March 2021 (UTC)

3) 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)


 * Got it.

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.


 * Again, I apologize.


 * 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.


 * Got it.


 * Prime Mover: Please delete everything between the two lines of asterisks unless you make a further comments. ********************* --DeaconJohn (talk) 05:21, 22 March 2021 (UTC)

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.


 * Too late for me to do that this evening, but I will get to it later. --DeaconJohn (talk) 05:21, 22 March 2021 (UTC)


 * 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)

Thank you, Prime Mover for all the time you have spent introducing me to ProofWiki. -- --DeaconJohn (talk) 05:21, 22 March 2021 (UTC)

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