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 people-on-the-undergraduate-level to follow; it certainly did for me. I am currently preparing footnotes intended to help people-on-the-undergraduate-level fill in these gaps. Hopefully, the notes will be ready to submit for review sometime before the end of August of this year. - - --DeaconJohn (talk) 18:10, 26 March 2021 (UTC)

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The sections to be deleted are separated by plus signs (+ + + + +).

The paragraph before the text-to-be-deleted is repeated before the text-to-be-deleted. This makes it easier to find where the text-to-be-deleted used to be in the main body of the text. Those paragraphs are prefaced with a double slash (//) to make them stand out. The paragraphs prefaced with // will not be deleted in the main text.

Prime Mover: Please feel free to delete any of the text on this page that you wish. I will NOT take offense. - - - --DeaconJohn (talk) 18:10, 26 March 2021 (UTC)

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// While Paul Halmos's proof is absolutely beautiful, it skips over quite a few steps. This might make it difficult to follow for people-on-the-undergraduate-level to follow; it certainly did for me. I am currently preparing footnotes intended to help people-on-the-undergraduate-level 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.

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 it 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 $\mathsf{Pr} \infty \mathsf{fWiki}$. 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 $\mathsf{Pr} \infty \mathsf{fWiki}$ 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" ($X$) 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 an email about the "possible typo." In that email, 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. --DeaconJohn (talk) 18:10, 26 March 2021 (UTC)

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 my footnotes on the undergraduate level. - - --DeaconJohn (talk) 18:10, 26 March 2021 (UTC)

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 $\mathsf{Pr} \infty \mathsf{fWiki}$ 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 people-on-the-undergraduate-level 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/she missed. The footnotes are not independent of each other, and this will make it easier to follow their flow.

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)

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// :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)

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- - - - - The text between this line and other the minus-sign-line (in the preceding text) is scheduled to be deleted. - - --DeaconJohn (talk) 18:10, 26 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.

Glad to do it - - - - DeaconJohn (talk) 00:38, 23 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 $\mathsf{Pr} \infty \mathsf{fWiki}$ 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 $\mathsf{Pr} \infty \mathsf{fWiki}$ extensions to $\LaTeX$ which are de rigueur. --prime mover (talk) 09:48, 10 March 2021 (UTC)

Thank you. - - - - DeaconJohn (talk) 00:38, 23 March 2021 (UTC)

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