Hello.
I have nominated Cantor's first uncountability proof for the status of a good article and mentioned that here. Michael Hardy ( talk) 16:41, 20 October 2014 (UTC)
Thank you!
I'm currently working on a French translation of the article, which will have one advantage over the English article: in the footnotes, I use links to the exact page of the 1883 French translation of Cantor's article. (English readers have to look up a translation in a book.) I've previously modified some Wikipédia articles, but have not added an article yet. I would like a native French speaker to check over my work. Do you have any suggestion on how I can locate one to help me. Thanks, -- RJGray ( talk) 15:31, 21 October 2014 (UTC)
Hello. I'm glad to see you're looking at this. The article has been mentioned on math.stackexchange.com a couple of times. As for the amount of attention it gets, there is this: http://stats.grok.se/en/201506/Cantor's%20first%20uncountability%20proof
It averaged almost 43 views per day in June this year and almost 56 per day in May. I'm guessing it's higher during the academic year. Michael Hardy ( talk) 19:10, 15 July 2015 (UTC)
I've moved the draft to Georg Cantor's first set theory article. So far no other articles link to it, so that will be something for everyone concerned to work on. Michael Hardy ( talk) 03:28, 13 February 2016 (UTC)
Hi RJGray, I have moved your discussion about Duolingo's translation system from the admin noticeboard to Wikipedia:Village_pump_(proposals). Regards, De728631 ( talk) 13:17, 21 January 2017 (UTC)
Hi Robert, inspired by your recent edit at Structural induction, I'd like to advertize the view given at Term (logic)#Term structure vs. representation which considers a term as a particular kind of a tree. Similarly, a formula is best viewed as a tree, imho. Best regards - Jochen Burghardt ( talk) 20:19, 20 May 2017 (UTC)
Hi Robert, thank you for your kind reply. Reflecting your arguments, I now think that the distinction is mainly between "terms as strings" (early 20th-century mathematics view) vs. "terms as trees" (later view inherited from computer science), while both views are compatible with structural induction (on sequences vs. trees of symbols). I agree that structural induction is far more familiar to mathematician readers, and helps to avoid introducing tree notation to them. — The description of your new article seems very interesting, so if you send me the link, I'd like to read it. - Jochen Burghardt ( talk) 08:15, 23 May 2017 (UTC)
I just started reading. It is very interesting, but I'm not an expert in this topic (my knowledge is limited about to Halmos' Naive Set Theory book). I boldly inserted an apparently missing 'that', hoping that is ok. If I have less trivial suggestions or comments, I could annotate a copy of your text in my sandbox. Reading through the complete article may take a while, but I'm curious to do it. - Jochen Burghardt ( talk) 21:43, 24 May 2017 (UTC)
I'm using a copy in User:Jochen Burghardt/sandbox1, and started annotating with ((clarify)) requests. clarification needed Hopefully, their contents is displayed when you move the mouse over them - I can't test them, since I don't have a mouse with my tablet. - Jochen Burghardt ( talk) 22:18, 24 May 2017 (UTC)
Looking for a picture, I found File:NGBUonthology.PNG only. It uses Hungarian text. Maybe, you can get some inspiration from it, nevertheless? - Jochen Burghardt ( talk) 05:33, 25 May 2017 (UTC)
I saw your new version of the 'handling paradoxes' paragraph, and found it an improvement. During reading it, I thought about using 'collection' as a neutral notion above both 'set' and 'class'. You then could say e.g. "a paradox arises when a certain collection is too large to be a set. (explain Ord example) If such a collection is made a class instead, the paradox can be avoided". However, having a 3rd notion will probably be confusing. May be if the noun form can be avoided, using the verb 'collect' instead, it is ok? Just a brainstorming-idea... - Jochen Burghardt ( talk) 21:05, 27 May 2017 (UTC)
I saw your new proof layout, and I like it. I wouldn't think the proof should be initially hidden, since the theorem and its proof method seems to be the most essential point of the NBG theory (that is the impression I got from your article; hope I got it right). Maybe, the proof even shouldn't be hidible at all? I'd like to suggest to use cases (x_i IN x_j with i NEQ j; x_i IN x_i; x_i IN C_j) in the base step, too. One might even think about indicationg subcases (i<n; i=n) by the layout, but that may be too much structure.
As another issue, it seems that copying and annotating your text to my sandbox has doubled your workload (a well-known problem in software engineering). If you have any suggestions how to improve our procedures, let me know. For example, you could remove my ((clarify)) requests once you have considered them, I could see this from history and version comparison, and then look at the corresponding text (when it's not obvious, you could point me there by edit message or replacement in the ((clarify)) text) in your sandbox. - Jochen Burghardt ( talk) 05:21, 29 May 2017 (UTC)
Your restated class existence theorem without "parameters" is clearer, imho. I understand that each Ci can be replaced by an arbitrary expression denoting a class (proper or not), and the theorem will still hold. Probably, there never was a problem with your text, but I confused myself - sorry for that. - Jochen Burghardt ( talk) 09:15, 31 May 2017 (UTC)
Hi Robert, now I've read through my copy User:Jochen Burghardt/sandbox1 of your draft User:RJGray/Sandboxcantor1. I stopped when my "++BOOKMARK++" reached the "Notes" section, and didn't look at the later "References", "Talk", "TO DO", etc. I learned a lot about NBG set theory. If you have a new version, I'd like to read it, too. Best regards - Jochen Burghardt ( talk) 13:19, 31 May 2017 (UTC)
Hi Jochen,
Thanks to your insightful comments, I have rewritten quite a bit of the article (see User:RJGray/Sandboxcantor1). My biggest changes were in:
I see that you are a bit confused by the axiom of regularity. I'm not surprised, I was once confused by the axiom, too.
Your suggestion of doesn't work because it only says that for every set , there is a set that doesn't belong to . Consider a set whose only member is , so we have However, the set satisfies Regularity's condition that is needed to prevent Your suggestion of , as you suspected, handles only
I find the best way to get used to the axiom is to realize that it's equivalent (given the axiom of dependent choice) to the non-existence of infinite decreasing membership sequences: The axiom is von Neumann's slick way of using a set to prove the non-existence of these chains.
Assume the axiom of regularity and that we have an infinite descending chain: Now define the set Regularity implies there is a such that However, for some . But which contradicts Therefore, there is no infinite descending membership chain.
The above proof tells us what sets we need for handling and :
The proof that the non-existence of descending chains implies that the axiom of regularity needs the axiom of dependent choice, which is weaker than the axiom of choice. This axiom states: If is a binary relation on such that then there is an infinite sequence such that for all . Similar to the axiom of choice, you don't need the axiom to prove the existence of finite sequences with this property.
We prove the contrapositive. Assume the axiom of regularity is false: which is equivalent to: We now prove that there exists a descending membership chain:
Miscellaneous:
Thanks again for all your help. I look forward to any comments you may have on the revised article. RJGray ( talk) 17:37, 8 July 2017 (UTC)
tabular
environment.Hi Jochen,
Excellent diagrams! I like your page 2 of NBG Evolution.pdf best. However, there is one problem. For example, in Zermelo 1908, you have "elementary sets, …, choice" outside of the box and label them as "Properties". However, they like "extensionality (sets)" are axioms and belong inside the box and the same color as "extensionality (sets)." In the Bernays box, after "Bernays 1931 [letter to Gödel]", the next line could be improved to "1937, 1941 [published]" or "1937, 1941 [axioms published]" with 1941 put under 1931. (I should have thought of this earlier.) Minor points: In von Neumann 1929, move "power " to next line since there's enough room; in Bernays 1931, comma after "von Neumann choice"; also, this box has two lines around it. In Zermelo 1908, if you make the box as wide as the other boxes, then "power set" can be on the first line. It's fine that the Fraenkel 1922, Skolem 1922 box has smaller width since it doesn't refer to an axiom system.
One reason I prefer your page 2 is because you show which axioms are removed and which axioms are added in each box. This is an excellent improvement since the reader doesn't have to figure out the differences themselves. I see no need to do it in color since with the arrows it's clear what's being removed and what's being added. If you do want to use colors for conveying meaning, it's good to read Category:Articles with images not understandable by color blind users#Tips for editors.
As for the legend use of "Approach", I can't think of anything better yet. If it wasn't for the Fraenkel 1922, Skolem 1922 box, "Axiom system" could be used.
As for commenting on my rewrite, just put the clarify templates in User:RJGray/Sandboxcantor1. Thanks again for your help, RJGray ( talk) 01:03, 12 July 2017 (UTC)
Hi Robert! Some issues don't fit well into a ((clarify)) template; so I discuss them here:
I'll continue reading, but it will take a few more days than expected; sorry for the delay. Best regards - Jochen Burghardt ( talk) 11:00, 18 July 2017 (UTC)
Thanks for the comments. I used one already. The others need more thought. No need to apologize for the delay—I'm in no rush. It's better to think things through carefully. Also, did you spot my "Reply: " in the first 2 clarifies? Thanks, RJGray ( talk) 22:51, 18 July 2017 (UTC)
Concerning the lead: I copied the "conservative extension" and "Morse-Kelley set theory" material from the original lead. Actually, it was one part of the original lead that I liked. One of the jobs of the lead is to establish context and explain why the topic is notable. "Conservative extension of ZFC" and "Morse-Kelley set theory" is establishing the context in terms of other articles and is establishing notability by its connections with other articles. So I think it belongs in the lead. Here's what WP:Lead says: "The lead should stand on its own as a concise overview of the article's topic. It should identify the topic, establish context, explain why the topic is notable, and summarize the most important points, including any prominent controversies.[2] The notability of the article's subject is usually established in the first few sentences." I did put in your suggestion of the class of all sets. I have mixed feelings about an example being in the lead, but it does show that NBG goes beyond ZFC. Your comments are excellent, to reply I end up learning more about Wikipedia. RJGray ( talk) 22:21, 19 July 2017 (UTC)
My suggestion about the lead wasn't clear enough: I didn't mean to omit the mention of "conservative extension", nor of "MK", but rather to omit the explanation "[cons. ext.] means that ..." and "[the stronger MK] allows ...". These might be considered details which better go to later sections. - Jochen Burghardt ( talk) 06:31, 20 July 2017 (UTC)
Sorry to have misread your suggestion. I agree with what you're saying and have modified the lead. This also lead to a rewrite of the Discussion section, which I broke into two sections ("Discussion" is not a very descriptive name). As you suggested, I also added what I had written in yesterday's lead on MK and put it into the new section "NBG, ZFC, and MK". -- RJGray ( talk) 20:38, 20 July 2017 (UTC)
Based on your comments, I rewrote the explanation of how to use the class existence theorem to produce an NBG proof. It's just 2 sentences and doesn't confuse the reader by talking about the replacement schema.
Thank you for rewriting part of the computer program in a more Pascal form. However, looking it over, I think my more math-like pseudocode is better. The program is meant to be a short example that the reader can read without getting used to different notation. Also, my experience with writing a couple of pure math articles containing pseudocode taught me that quite a few math people don't particularly like computer programs intruding on pure math. I did manage to get one article published in the Mathematical Intelligencer, which is known for publishing articles that take a non-standard approach. Another math journal turned down an article citing that the readership would not like the computer part of the article and also said that a computer person reading the article would find that there was too much math in it. Since this Wikipedia article will be mostly read by math people, I want to minimize the distance my program is from mathematics.
On the function call: Thanks for reformatting it. I really liked your Cpl3 being an prefix operator like , while I was stuck using a raised postfix operator. I remembered that the Complement (set theory) article had several different notations and learned about the Bourbaki notation , which means the same as the operator you invented. So I've reformatted my old function call and also lined it up using Latex spacing to get it similar to your layout without using as much space. Now I have to go back and change the article to use Bourbaki complement notation. Thanks again for your help, RJGray ( talk) 19:44, 21 July 2017 (UTC)
Hi Robert! (I started a new talk section only in order to ease editing here.) At User:Jochen Burghardt/sandbox#NBG, I experimented with implementing all my "clarify" suggestions for User:RJGray/Sandboxcantor1#History, just to see what it would look like. In particular, I agressively italicized all person names and years, and I rewrote anything in past tense or past perfect, even "axiom system was relatively consistent" etc. I feel that the result doesn't read that bad as I had feared. However, I'm not a native English speaker; you may have a look there and decide yourself what suggestions you might adopt. — I also experimented with separating footnotes that contain proper explanations ("[note 1]") from those that just give references ("[1]"); I think it would help the reader to decide whether to look up a footnote. — I found that you use "ZFC" in the main text (except when Cohen's independence proofs are mentioned; BTW: an article, or even a section about it seems to be still missing in wikipedia), but sometimes "ZF" in footnotes; you should probably add an explanation at the first mention of "ZF" (or even intially when "ZFC" is introduced?). Best regards - Jochen Burghardt ( talk) 11:46, 23 July 2017 (UTC)
I uploaded a new version of File:NBG Evolution.pdf which should take your comments into account (my former distinction "axioms"/"properties" was based on a misunderstanding, as was the double surrounding of the "Bernays 1931" box). While different fonts (\rm, \sf, \sl, \bf) should support readbility for colorblind people, I kept some coloring in addition, to give even more reading support to the non-colorblind people. — Comparing the image with User:RJGray/Sandboxcantor1#History, I wonder if "Replacement" should be mentioned in a prominent place in the Neumann 1925/1928 approach?
In the history paragraph starting "Von Neumann approached ...", in the first sentence, I wonder if "the choice axiom: ..." means "Neumann choice" in the diagram (I had to omit the "Von" to save a line)? In this case, better write e.g. "an own version of the choice axiom, viz.: ...". The last sentence of the same paragraph is not very clear: I wonder if both "an axiom system that is closer to ZFC" and "this system" mean the 1929 system? I add these comments here since all ((clarify))s from the History section are dealt with (I hope) and removed in my sandbox version. - Jochen Burghardt ( talk) 13:58, 23 July 2017 (UTC)
An svg version of the image is now available at File:NBG Evolution svg.svg. - Jochen Burghardt ( talk) 11:04, 24 July 2017 (UTC)
Hi Jochen! Excellent diagram! Just a few suggestions:
When I started this article, I considered separating Notes from References as I did in Cantor's first set theory article. However, because of the way I did it in that article, it's very labor-intensive to add new Notes. Since you brought up the subject, I was motivated to look further into this and learned about the "efn" template, which stands for "explanatory footnote". "efn" is as easy to use as "ref". I've changed the Cantor article to use "efn-ua", which uses uppercase alphabetic characters to name the footnotes. In that article, I do have some longer references but only for references with very short notes or references with notes that justify a claim made in the text. I'm not sure if there are any rules about what's a reference versus a note. I find that many math articles don't separate Notes from References, such as the John von Neumann article. It has "Notes" (corresponding to the 2 sections "Notes" and "References" in the Cantor first set theory article) and "References" (corresponding to the "Bibliography" article in that article). So I do plan to separate them in the future.
As far as the English present tense, the site Simple Perfect Tense gives two uses that are relevant for the History section:
For example, "Cantor's theory of ordinal numbers could not be developed in Zermelo set theory because it had lacked the axiom of replacement" means to me that Zermelo set theory lacked replacement at that time, but it may no longer lack replacement. For example, Zermelo may have developed his set theory further—of course, this isn't true, but readers may not know this so we would have to tell them. In English, because the lack of replacement is a permanent state (or general truth) of Zermelo set theory, the present tense is needed. So we may have a language difference here between English and German.
You raised a very interesting point on the relation between von Neumann's function notation and Curry's lambda calculus work. So I looked it up and here's what I found: von Neumann's 1925 paper predates Curry's work: Curry started in 1926-27 and first published in 1930; also, it appears that von Neumann's work did not affect Curry's work. But there is Schönfinkel's work who invented combinatory logic and gave a lecture on it in 1920 to Hilbert's group and published it in 1924. The article I got this information from says "We do not know whether von Neumann’s idea came from Schönfinkel's", and goes on to state the evidence for and against. See: History of Lambda-calculus and Combinatory Logic, p. 4-5.
As far as italicizing the names of mathematicians, I find it an interesting experiment. However, I've never seen it done in History sections of articles and I find it distracting. Also, it seems to me that it puts the spotlight on who is doing the work when some readers may be more concerned with the flow of ideas.
Thanks again for all your help. You have given me so many excellent suggestions that I'm having trouble keeping up with them. -- RJGray ( talk) 14:24, 24 July 2017 (UTC)
Hi Robert! I uploaded a new version according to your above comment. It can provide it as svg on Thursday. Concerning tense, I rely on your knowledge as a native English speaker. To be honest, I don't know whether there are rules in German and what they look like; I just followed my feeling, imagining to tell a story from long time ago, and taking that perspective. Name/year italicizing was just a test; probably you are right that it is distracting from the ideas' flow; and without doubt it is unusual. Best regards - Jochen Burghardt ( talk) 20:58, 24 July 2017 (UTC)
Hi Jochen! Thanks for your help with the horizontal spacing problem. My brain got stuck and didn't realize that I could just prefix the two long expressions with and . I also used your idea about shrinking the spacing before and after the EPSILON. In fact, I like the look better since the length of more closely matches the length of . Now the length of the expressions is slightly less than the horizontal length of the computer function. Also, the print size is identical to the print size I get in an article with no use of <math>. One reason I like to work with you so much is that you come up with such good ideas.
You've done an excellent job on diagram! I like the diagram as it is, but I realized something strange about it that I missed when I gave you the information. Von Neumann has no mention of pairing. (Zermelo's pairing is under his Elementary sets.) So I rechecked von Neumann's axioms. They're hard to get to correspond with the others. You can look them over at von Neumann 1928, p. 674-675. He has 5 classes of axioms: Introductory axioms, Arithmetic construction axioms, Logical construction axioms, I-II-objects, and Axioms of infinity. His construction axioms are what I call function existence axioms since they are asserting the existence of functions and are analogous to Bernays' and Gödel's class existence axioms. His I-II-objects include the axiom of limitation of size, and his Axioms of infinity are the axioms of infinity, union, and power set. One of his introductory axioms (extensionality) is already in the diagram. Another (ordered pair operation) is not in the diagram. The other two are less relevant. The first asserts the existence of two I-objects (arguments) A and B. The second asserts the existence of his [f, x] operation and including it would be analogous to including in the other boxes.
So shall we add "Ordered pair operation" to the von Neumann boxes? It would add one line to Von Neumann 1925, 1928.
In other work, I looked into your suggestion of a lemma to simplify the start of the inductive proof. I have figured out one, so I'll be busy doing a rewrite there. Also, I'm planning on rewriting parts of the History section to make it easier to follow. (I figure if you are having trouble with it, lots of others with less experience in math and math history will have trouble, too.) I'm also planning on giving von Neumann, Bernays, and Gödel their own subsections.
So take your time working on the article—I've got a lot of work to do. It's a pleasure working with you, RJGray ( talk) 18:25, 25 July 2017 (UTC)
I just realized that we could probably add "Ordered pair" to the von Neumann boxes without "operation". The pairing axiom in the other boxes is understood to be the unordered pair so we can probably do this without confusing the reader. Also, without "operation", adding "Ordered pair" will probably not add a line to Von Neumann 1925, 1928. Of course, it's best to add it after extensionality, since this is where "Pairing" appears in the other boxes. RJGray ( talk) 00:40, 26 July 2017 (UTC)
Hi Jochen! I've completed my rewrite of the History section (it's in User:RJGray/Sandboxcantor1). My latest work was on the subsection "Von Neumann's 1929 axiom system". I can see why you found my first attempt confusing--hopefully, you won't find the rewrite so confusing. My major remaining work is to rewrite the basis step of the induction in the proof of the class existence theorem using a lemma like you suggested. However, I'm going on vacation tomorrow and won't be around computers much (if at all) for a bit over a week. So take your time working on the article. -- RJGray ( talk) 17:03, 28 July 2017 (UTC)
Hi Jochen! Your diagram looks great! Thanks for all the work you've been doing on it. As for what to call the axiom: the axiom usually goes under the name "Pairing" meaning unordered pairing, but von Neumann uses ordered pairing. I'm thinking that we can handle this the same way we do Extensionality—namely, with parentheses after "Pairing". Then von Neumann's would read "Pairing (ordered)" and the others would read "Pairing (unordered)". That way, in both cases, we would be using the usual names "Extensionality" and "Pairing" and the parenthetical part would handle the difference between von Neumann and the other set theorists.
So I'm back from vacation and I stayed away from computers, but I carried a printout of the article with me and have done quite a bit of rewriting. It will take me several days to get it all in. I'll let you know when it's stable again—I don't want you to waste time reading something that I'm in the process of working on. Your suggestion of a lemma for the class existence theorem works extremely well—I think that readers will be able to understand that part of the proof much better. Thanks again for your work and suggestions -- RJGray ( talk) 00:45, 6 August 2017 (UTC)
I just realized that you might ask: Since we would be putting in and taking out "Pairing (ordered)" and "Pairing (unordered)" using the arrows, would we need to do that for "Extensionality"? I don't think so, Extensionality has the same meaning, it's just on different primitives so it's closely tied to the primitives and doesn't change meaning. However, "Pairing (ordered)" and "Pairing (unordered)" are have very different meanings so they need to be going in and out with the arrows. (Besides, we don't have the room for "Extensionality" with the arrows.) -- RJGray ( talk) 01:07, 6 August 2017 (UTC)
I've been rethink the "Elementary sets" in Zermelo set theory. Basically, there are only 3 sets postulated by the axiom: the empty set; for any set a, the singleton set {a}; for any pairs of sets a and b, the unordered pair {a, b}. I'm thinking it may be a good idea to replace "elementary sets" with "Pairing (unordered)" since Pairing is the only one of importance when comparing the axiom systems (pairing implies the singleton set and the empty set can be proved to exist by the usual assumption in first-order logic that an axiom system implies the existence of at least one object). -- RJGray ( talk) 18:14, 7 August 2017 (UTC)
Hi Jochen, I'll start by congratulating you on achieving the shortest and simplest injection for Cantor's diagonal argument! I and some others suggested injections that involved more work on the reader's part to understand. About my 24 August 2017 Cantor's diagonal argument Talk contribution—sometimes I should just wait a day and think things through rather than defending my proof without fully appreciating another's proof (which was yours). I apologize for doing this—I'll wait a day or until I fully appreciate another's proof the next time a similar situation occurs.
Now on to Von Neumann–Bernays–Gödel set theory: Thanks for your insightful comments; they have been extremely helpful and have led to a number of improvements. Your suggestion of a lemma to handle the basis step of the proof of the class existence theorem was a great suggestion—it does an excellent job of simplifying that part of the proof.
It's taken me awhile to work through all of your suggestions, but I'm finally done with the rewrite (see User:RJGray/Sandboxcantor1). Here are my notes on the clarify's I had some trouble with:
Class existence theorem clarify comments:
Set axioms clarify comments:
By the way, part of the reason that this rewrite took so long is that when I first write an article, I just grab the information from a variety of sources. At some point, I go over all the definitions, axioms, theorems, and proofs and make sure I have the historically correct ones. For example, I realized that the tuple-handling axioms that I got from French Wikipédia differed from the ones in Gödel and Mendelson, so I now use theirs. I also realized that the product by V axiom that I'm using is Bernays' original axiom, which Gödel changed, so I mention this. Also, I've added "efn" notes that explain how the proof in this Wikipedia article differs from Gödel's proof, so a reader who reads the article is better prepared to read Gödel's original proof.
Looking forward to your comments. Take your time—I'm in no rush to post the article. Thanks again for all the great suggestions you've made, RJGray ( talk) 19:25, 5 September 2017 (UTC)
Hi Robert, thanks for your compliments, but the diagonal proof originates from Kamke, or -probably- earlier, I didn't do more than cite it. For now, let me just reply on your clarify comments.
Class existence theorem clarify comments:
Set axioms clarify comments:
In the next days, I'll read through the article, and insert my comments, if any, using "clarify" as before. Best regards - Jochen Burghardt ( talk) 12:28, 8 September 2017 (UTC)
I've submitted the nomination for "Good article" status again: Talk:Georg_Cantor's_first_set_theory_article/GA2. Michael Hardy ( talk) 19:06, 1 June 2018 (UTC)
Thank you. I'll be ready to deal with any feedback generated by the review. RJGray ( talk) 19:50, 1 June 2018 (UTC)
See Talk:Georg Cantor's first set theory article/GA2.
I am told the following:
Michael Hardy ( talk) 23:27, 30 July 2018 (UTC)
I have just returned from a vacation that started on July 28 and I wasn't around computers. I see that you have already made some changes. What remains to be done and can we get a time extension? RJGray ( talk) 20:03, 5 August 2018 (UTC)
Narutolovehinata5 t c csd new 09:44, 21 October 2018 (UTC)
On 7 December 2018, Did you know was updated with a fact from the article Georg Cantor's first set theory article, which you recently created, substantially expanded, or brought to good article status. The fact was ... that mathematicians disagree about whether a proof in Georg Cantor's first set theory article actually shows how to construct a transcendental number, or merely proves that such numbers exist? The nomination discussion and review may be seen at Template:Did you know nominations/Georg Cantor's first set theory article. You are welcome to check how many page hits the article got while on the front page ( here's how, Georg Cantor's first set theory article), and it may be added to the statistics page if the total is over 5,000. Finally, if you know of an interesting fact from another recently created article, then please feel free to suggest it on the Did you know talk page.
Mifter ( talk) 00:02, 7 December 2018 (UTC)
I would be happy to be your mentor for " Georg Cantor's first set theory article" as we lack good quality maths articles on wikipedia. I see that the article is already in great shape plus it turns out that I recently taught a course where there was a presentation of Cantor's set as a set of reals with no 1 when written in base 3. Before I dwelve into the math of the article, a couple of minor things caught my attention in terms of formatting that would be raised at FAC:
More to come. My advises may look daunting but this is a very fine article. I am convinced it will succeed at FAC if we fix the details. Iry-Hor ( talk) 17:28, 14 November 2019 (UTC)
Here are a few more details for the article:
This is a very good article really, bravo. I will continue to look into this in details but it will be a glorious FA. Then I suggest that we make it so that it appears as the "Today's Featured Article" on the main page at some point (after FA)! Iry-Hor ( talk) 09:06, 15 November 2019 (UTC)
Hi Robert!
I see you are experimenting with various versions of captions for the 3 case diagrams.
In my opinion, the constellations in case 1 to 3 are completely described in the proof text; the diagrams' only purpose can be to help the reader's brain to construct a graphical image of each constellation.
As a consequence, every attempt to describe the diagrams verbally in alt=
text at best can amount to duplicate the proof text.
Therefore, I'd be in favor of just keeping the old captions ("Illustration of case 1", etc.), and to add an empty alt=
text; improvements of the descriptions, if necessary at all, should better be devoted to the proof text itself.
Alternatively, I thought of a naive, proof-unrelated, alt
description like "Real number line with distinct points, left to right: a, a1 to aL, y, xn, bL to b1, b" for case 1.
Best regards -
Jochen Burghardt (
talk)
18:53, 27 January 2020 (UTC)
Addendum: I also thought of "Real number line with relations a < a1 < ... <aL < y < xn < bL < ... < b1 < b"; this is shorter to write, but supposedly longer to hear (when read by a screen reader). - An advantage of keeping caption and alt
separate: when not using a screen reader, the user wouldn't have to read long captions that duplicate the proof text. - All this are just some immature thoughts; feel free to ignore any of them. -
Jochen Burghardt (
talk)
09:41, 28 January 2020 (UTC)
Hi Jochen! Thanks for pointing out that I was just duplicating the proof. I got a bit carried away with my experiments. Also, the old way was better because the old diagrams just barely touched the next section while the new diagrams went several lines deep into the next section. Concerning alt text: I've experimented with it by loading a copy of NVDA, which is a widely-used and free screen reader, and found that a blank alt text reads the file name. I think this may be why WP:ALT recommends using "refer to caption". I've also changed my "Visual representation of case" to your "Illustration of case". As for using the "< relations", I've decided to stick with nested intervals. I explain what they are in the second paragraph of the section and readers can also click on the link. Also, at least for me, as soon as I hear "nested intervals", I immediately imagine an illustration of them. The < relations requires me to think through the dot-dot-dot. Also, I've discovered that NVDA just skips over "...", you have to use ". . . ." Then it only ignores the first ".". By the way, I just tried NVDA on your use of "< relations" above—it doesn't do a good job. There's additional software for reading math that can be added, but I don't have the time to try it yet so that software may fix it. Also, thanks for getting me involved with this section again. I discovered that I can't stack the 3 " File:..." because NVDA reads all 3 files before it goes into the proof texts. I just needed to place each " File:..." before the proof text. Thanks again for your help, RJGray ( talk) 14:09, 28 January 2020 (UTC)
Hello.
I have nominated Cantor's first uncountability proof for the status of a good article and mentioned that here. Michael Hardy ( talk) 16:41, 20 October 2014 (UTC)
Thank you!
I'm currently working on a French translation of the article, which will have one advantage over the English article: in the footnotes, I use links to the exact page of the 1883 French translation of Cantor's article. (English readers have to look up a translation in a book.) I've previously modified some Wikipédia articles, but have not added an article yet. I would like a native French speaker to check over my work. Do you have any suggestion on how I can locate one to help me. Thanks, -- RJGray ( talk) 15:31, 21 October 2014 (UTC)
Hello. I'm glad to see you're looking at this. The article has been mentioned on math.stackexchange.com a couple of times. As for the amount of attention it gets, there is this: http://stats.grok.se/en/201506/Cantor's%20first%20uncountability%20proof
It averaged almost 43 views per day in June this year and almost 56 per day in May. I'm guessing it's higher during the academic year. Michael Hardy ( talk) 19:10, 15 July 2015 (UTC)
I've moved the draft to Georg Cantor's first set theory article. So far no other articles link to it, so that will be something for everyone concerned to work on. Michael Hardy ( talk) 03:28, 13 February 2016 (UTC)
Hi RJGray, I have moved your discussion about Duolingo's translation system from the admin noticeboard to Wikipedia:Village_pump_(proposals). Regards, De728631 ( talk) 13:17, 21 January 2017 (UTC)
Hi Robert, inspired by your recent edit at Structural induction, I'd like to advertize the view given at Term (logic)#Term structure vs. representation which considers a term as a particular kind of a tree. Similarly, a formula is best viewed as a tree, imho. Best regards - Jochen Burghardt ( talk) 20:19, 20 May 2017 (UTC)
Hi Robert, thank you for your kind reply. Reflecting your arguments, I now think that the distinction is mainly between "terms as strings" (early 20th-century mathematics view) vs. "terms as trees" (later view inherited from computer science), while both views are compatible with structural induction (on sequences vs. trees of symbols). I agree that structural induction is far more familiar to mathematician readers, and helps to avoid introducing tree notation to them. — The description of your new article seems very interesting, so if you send me the link, I'd like to read it. - Jochen Burghardt ( talk) 08:15, 23 May 2017 (UTC)
I just started reading. It is very interesting, but I'm not an expert in this topic (my knowledge is limited about to Halmos' Naive Set Theory book). I boldly inserted an apparently missing 'that', hoping that is ok. If I have less trivial suggestions or comments, I could annotate a copy of your text in my sandbox. Reading through the complete article may take a while, but I'm curious to do it. - Jochen Burghardt ( talk) 21:43, 24 May 2017 (UTC)
I'm using a copy in User:Jochen Burghardt/sandbox1, and started annotating with ((clarify)) requests. clarification needed Hopefully, their contents is displayed when you move the mouse over them - I can't test them, since I don't have a mouse with my tablet. - Jochen Burghardt ( talk) 22:18, 24 May 2017 (UTC)
Looking for a picture, I found File:NGBUonthology.PNG only. It uses Hungarian text. Maybe, you can get some inspiration from it, nevertheless? - Jochen Burghardt ( talk) 05:33, 25 May 2017 (UTC)
I saw your new version of the 'handling paradoxes' paragraph, and found it an improvement. During reading it, I thought about using 'collection' as a neutral notion above both 'set' and 'class'. You then could say e.g. "a paradox arises when a certain collection is too large to be a set. (explain Ord example) If such a collection is made a class instead, the paradox can be avoided". However, having a 3rd notion will probably be confusing. May be if the noun form can be avoided, using the verb 'collect' instead, it is ok? Just a brainstorming-idea... - Jochen Burghardt ( talk) 21:05, 27 May 2017 (UTC)
I saw your new proof layout, and I like it. I wouldn't think the proof should be initially hidden, since the theorem and its proof method seems to be the most essential point of the NBG theory (that is the impression I got from your article; hope I got it right). Maybe, the proof even shouldn't be hidible at all? I'd like to suggest to use cases (x_i IN x_j with i NEQ j; x_i IN x_i; x_i IN C_j) in the base step, too. One might even think about indicationg subcases (i<n; i=n) by the layout, but that may be too much structure.
As another issue, it seems that copying and annotating your text to my sandbox has doubled your workload (a well-known problem in software engineering). If you have any suggestions how to improve our procedures, let me know. For example, you could remove my ((clarify)) requests once you have considered them, I could see this from history and version comparison, and then look at the corresponding text (when it's not obvious, you could point me there by edit message or replacement in the ((clarify)) text) in your sandbox. - Jochen Burghardt ( talk) 05:21, 29 May 2017 (UTC)
Your restated class existence theorem without "parameters" is clearer, imho. I understand that each Ci can be replaced by an arbitrary expression denoting a class (proper or not), and the theorem will still hold. Probably, there never was a problem with your text, but I confused myself - sorry for that. - Jochen Burghardt ( talk) 09:15, 31 May 2017 (UTC)
Hi Robert, now I've read through my copy User:Jochen Burghardt/sandbox1 of your draft User:RJGray/Sandboxcantor1. I stopped when my "++BOOKMARK++" reached the "Notes" section, and didn't look at the later "References", "Talk", "TO DO", etc. I learned a lot about NBG set theory. If you have a new version, I'd like to read it, too. Best regards - Jochen Burghardt ( talk) 13:19, 31 May 2017 (UTC)
Hi Jochen,
Thanks to your insightful comments, I have rewritten quite a bit of the article (see User:RJGray/Sandboxcantor1). My biggest changes were in:
I see that you are a bit confused by the axiom of regularity. I'm not surprised, I was once confused by the axiom, too.
Your suggestion of doesn't work because it only says that for every set , there is a set that doesn't belong to . Consider a set whose only member is , so we have However, the set satisfies Regularity's condition that is needed to prevent Your suggestion of , as you suspected, handles only
I find the best way to get used to the axiom is to realize that it's equivalent (given the axiom of dependent choice) to the non-existence of infinite decreasing membership sequences: The axiom is von Neumann's slick way of using a set to prove the non-existence of these chains.
Assume the axiom of regularity and that we have an infinite descending chain: Now define the set Regularity implies there is a such that However, for some . But which contradicts Therefore, there is no infinite descending membership chain.
The above proof tells us what sets we need for handling and :
The proof that the non-existence of descending chains implies that the axiom of regularity needs the axiom of dependent choice, which is weaker than the axiom of choice. This axiom states: If is a binary relation on such that then there is an infinite sequence such that for all . Similar to the axiom of choice, you don't need the axiom to prove the existence of finite sequences with this property.
We prove the contrapositive. Assume the axiom of regularity is false: which is equivalent to: We now prove that there exists a descending membership chain:
Miscellaneous:
Thanks again for all your help. I look forward to any comments you may have on the revised article. RJGray ( talk) 17:37, 8 July 2017 (UTC)
tabular
environment.Hi Jochen,
Excellent diagrams! I like your page 2 of NBG Evolution.pdf best. However, there is one problem. For example, in Zermelo 1908, you have "elementary sets, …, choice" outside of the box and label them as "Properties". However, they like "extensionality (sets)" are axioms and belong inside the box and the same color as "extensionality (sets)." In the Bernays box, after "Bernays 1931 [letter to Gödel]", the next line could be improved to "1937, 1941 [published]" or "1937, 1941 [axioms published]" with 1941 put under 1931. (I should have thought of this earlier.) Minor points: In von Neumann 1929, move "power " to next line since there's enough room; in Bernays 1931, comma after "von Neumann choice"; also, this box has two lines around it. In Zermelo 1908, if you make the box as wide as the other boxes, then "power set" can be on the first line. It's fine that the Fraenkel 1922, Skolem 1922 box has smaller width since it doesn't refer to an axiom system.
One reason I prefer your page 2 is because you show which axioms are removed and which axioms are added in each box. This is an excellent improvement since the reader doesn't have to figure out the differences themselves. I see no need to do it in color since with the arrows it's clear what's being removed and what's being added. If you do want to use colors for conveying meaning, it's good to read Category:Articles with images not understandable by color blind users#Tips for editors.
As for the legend use of "Approach", I can't think of anything better yet. If it wasn't for the Fraenkel 1922, Skolem 1922 box, "Axiom system" could be used.
As for commenting on my rewrite, just put the clarify templates in User:RJGray/Sandboxcantor1. Thanks again for your help, RJGray ( talk) 01:03, 12 July 2017 (UTC)
Hi Robert! Some issues don't fit well into a ((clarify)) template; so I discuss them here:
I'll continue reading, but it will take a few more days than expected; sorry for the delay. Best regards - Jochen Burghardt ( talk) 11:00, 18 July 2017 (UTC)
Thanks for the comments. I used one already. The others need more thought. No need to apologize for the delay—I'm in no rush. It's better to think things through carefully. Also, did you spot my "Reply: " in the first 2 clarifies? Thanks, RJGray ( talk) 22:51, 18 July 2017 (UTC)
Concerning the lead: I copied the "conservative extension" and "Morse-Kelley set theory" material from the original lead. Actually, it was one part of the original lead that I liked. One of the jobs of the lead is to establish context and explain why the topic is notable. "Conservative extension of ZFC" and "Morse-Kelley set theory" is establishing the context in terms of other articles and is establishing notability by its connections with other articles. So I think it belongs in the lead. Here's what WP:Lead says: "The lead should stand on its own as a concise overview of the article's topic. It should identify the topic, establish context, explain why the topic is notable, and summarize the most important points, including any prominent controversies.[2] The notability of the article's subject is usually established in the first few sentences." I did put in your suggestion of the class of all sets. I have mixed feelings about an example being in the lead, but it does show that NBG goes beyond ZFC. Your comments are excellent, to reply I end up learning more about Wikipedia. RJGray ( talk) 22:21, 19 July 2017 (UTC)
My suggestion about the lead wasn't clear enough: I didn't mean to omit the mention of "conservative extension", nor of "MK", but rather to omit the explanation "[cons. ext.] means that ..." and "[the stronger MK] allows ...". These might be considered details which better go to later sections. - Jochen Burghardt ( talk) 06:31, 20 July 2017 (UTC)
Sorry to have misread your suggestion. I agree with what you're saying and have modified the lead. This also lead to a rewrite of the Discussion section, which I broke into two sections ("Discussion" is not a very descriptive name). As you suggested, I also added what I had written in yesterday's lead on MK and put it into the new section "NBG, ZFC, and MK". -- RJGray ( talk) 20:38, 20 July 2017 (UTC)
Based on your comments, I rewrote the explanation of how to use the class existence theorem to produce an NBG proof. It's just 2 sentences and doesn't confuse the reader by talking about the replacement schema.
Thank you for rewriting part of the computer program in a more Pascal form. However, looking it over, I think my more math-like pseudocode is better. The program is meant to be a short example that the reader can read without getting used to different notation. Also, my experience with writing a couple of pure math articles containing pseudocode taught me that quite a few math people don't particularly like computer programs intruding on pure math. I did manage to get one article published in the Mathematical Intelligencer, which is known for publishing articles that take a non-standard approach. Another math journal turned down an article citing that the readership would not like the computer part of the article and also said that a computer person reading the article would find that there was too much math in it. Since this Wikipedia article will be mostly read by math people, I want to minimize the distance my program is from mathematics.
On the function call: Thanks for reformatting it. I really liked your Cpl3 being an prefix operator like , while I was stuck using a raised postfix operator. I remembered that the Complement (set theory) article had several different notations and learned about the Bourbaki notation , which means the same as the operator you invented. So I've reformatted my old function call and also lined it up using Latex spacing to get it similar to your layout without using as much space. Now I have to go back and change the article to use Bourbaki complement notation. Thanks again for your help, RJGray ( talk) 19:44, 21 July 2017 (UTC)
Hi Robert! (I started a new talk section only in order to ease editing here.) At User:Jochen Burghardt/sandbox#NBG, I experimented with implementing all my "clarify" suggestions for User:RJGray/Sandboxcantor1#History, just to see what it would look like. In particular, I agressively italicized all person names and years, and I rewrote anything in past tense or past perfect, even "axiom system was relatively consistent" etc. I feel that the result doesn't read that bad as I had feared. However, I'm not a native English speaker; you may have a look there and decide yourself what suggestions you might adopt. — I also experimented with separating footnotes that contain proper explanations ("[note 1]") from those that just give references ("[1]"); I think it would help the reader to decide whether to look up a footnote. — I found that you use "ZFC" in the main text (except when Cohen's independence proofs are mentioned; BTW: an article, or even a section about it seems to be still missing in wikipedia), but sometimes "ZF" in footnotes; you should probably add an explanation at the first mention of "ZF" (or even intially when "ZFC" is introduced?). Best regards - Jochen Burghardt ( talk) 11:46, 23 July 2017 (UTC)
I uploaded a new version of File:NBG Evolution.pdf which should take your comments into account (my former distinction "axioms"/"properties" was based on a misunderstanding, as was the double surrounding of the "Bernays 1931" box). While different fonts (\rm, \sf, \sl, \bf) should support readbility for colorblind people, I kept some coloring in addition, to give even more reading support to the non-colorblind people. — Comparing the image with User:RJGray/Sandboxcantor1#History, I wonder if "Replacement" should be mentioned in a prominent place in the Neumann 1925/1928 approach?
In the history paragraph starting "Von Neumann approached ...", in the first sentence, I wonder if "the choice axiom: ..." means "Neumann choice" in the diagram (I had to omit the "Von" to save a line)? In this case, better write e.g. "an own version of the choice axiom, viz.: ...". The last sentence of the same paragraph is not very clear: I wonder if both "an axiom system that is closer to ZFC" and "this system" mean the 1929 system? I add these comments here since all ((clarify))s from the History section are dealt with (I hope) and removed in my sandbox version. - Jochen Burghardt ( talk) 13:58, 23 July 2017 (UTC)
An svg version of the image is now available at File:NBG Evolution svg.svg. - Jochen Burghardt ( talk) 11:04, 24 July 2017 (UTC)
Hi Jochen! Excellent diagram! Just a few suggestions:
When I started this article, I considered separating Notes from References as I did in Cantor's first set theory article. However, because of the way I did it in that article, it's very labor-intensive to add new Notes. Since you brought up the subject, I was motivated to look further into this and learned about the "efn" template, which stands for "explanatory footnote". "efn" is as easy to use as "ref". I've changed the Cantor article to use "efn-ua", which uses uppercase alphabetic characters to name the footnotes. In that article, I do have some longer references but only for references with very short notes or references with notes that justify a claim made in the text. I'm not sure if there are any rules about what's a reference versus a note. I find that many math articles don't separate Notes from References, such as the John von Neumann article. It has "Notes" (corresponding to the 2 sections "Notes" and "References" in the Cantor first set theory article) and "References" (corresponding to the "Bibliography" article in that article). So I do plan to separate them in the future.
As far as the English present tense, the site Simple Perfect Tense gives two uses that are relevant for the History section:
For example, "Cantor's theory of ordinal numbers could not be developed in Zermelo set theory because it had lacked the axiom of replacement" means to me that Zermelo set theory lacked replacement at that time, but it may no longer lack replacement. For example, Zermelo may have developed his set theory further—of course, this isn't true, but readers may not know this so we would have to tell them. In English, because the lack of replacement is a permanent state (or general truth) of Zermelo set theory, the present tense is needed. So we may have a language difference here between English and German.
You raised a very interesting point on the relation between von Neumann's function notation and Curry's lambda calculus work. So I looked it up and here's what I found: von Neumann's 1925 paper predates Curry's work: Curry started in 1926-27 and first published in 1930; also, it appears that von Neumann's work did not affect Curry's work. But there is Schönfinkel's work who invented combinatory logic and gave a lecture on it in 1920 to Hilbert's group and published it in 1924. The article I got this information from says "We do not know whether von Neumann’s idea came from Schönfinkel's", and goes on to state the evidence for and against. See: History of Lambda-calculus and Combinatory Logic, p. 4-5.
As far as italicizing the names of mathematicians, I find it an interesting experiment. However, I've never seen it done in History sections of articles and I find it distracting. Also, it seems to me that it puts the spotlight on who is doing the work when some readers may be more concerned with the flow of ideas.
Thanks again for all your help. You have given me so many excellent suggestions that I'm having trouble keeping up with them. -- RJGray ( talk) 14:24, 24 July 2017 (UTC)
Hi Robert! I uploaded a new version according to your above comment. It can provide it as svg on Thursday. Concerning tense, I rely on your knowledge as a native English speaker. To be honest, I don't know whether there are rules in German and what they look like; I just followed my feeling, imagining to tell a story from long time ago, and taking that perspective. Name/year italicizing was just a test; probably you are right that it is distracting from the ideas' flow; and without doubt it is unusual. Best regards - Jochen Burghardt ( talk) 20:58, 24 July 2017 (UTC)
Hi Jochen! Thanks for your help with the horizontal spacing problem. My brain got stuck and didn't realize that I could just prefix the two long expressions with and . I also used your idea about shrinking the spacing before and after the EPSILON. In fact, I like the look better since the length of more closely matches the length of . Now the length of the expressions is slightly less than the horizontal length of the computer function. Also, the print size is identical to the print size I get in an article with no use of <math>. One reason I like to work with you so much is that you come up with such good ideas.
You've done an excellent job on diagram! I like the diagram as it is, but I realized something strange about it that I missed when I gave you the information. Von Neumann has no mention of pairing. (Zermelo's pairing is under his Elementary sets.) So I rechecked von Neumann's axioms. They're hard to get to correspond with the others. You can look them over at von Neumann 1928, p. 674-675. He has 5 classes of axioms: Introductory axioms, Arithmetic construction axioms, Logical construction axioms, I-II-objects, and Axioms of infinity. His construction axioms are what I call function existence axioms since they are asserting the existence of functions and are analogous to Bernays' and Gödel's class existence axioms. His I-II-objects include the axiom of limitation of size, and his Axioms of infinity are the axioms of infinity, union, and power set. One of his introductory axioms (extensionality) is already in the diagram. Another (ordered pair operation) is not in the diagram. The other two are less relevant. The first asserts the existence of two I-objects (arguments) A and B. The second asserts the existence of his [f, x] operation and including it would be analogous to including in the other boxes.
So shall we add "Ordered pair operation" to the von Neumann boxes? It would add one line to Von Neumann 1925, 1928.
In other work, I looked into your suggestion of a lemma to simplify the start of the inductive proof. I have figured out one, so I'll be busy doing a rewrite there. Also, I'm planning on rewriting parts of the History section to make it easier to follow. (I figure if you are having trouble with it, lots of others with less experience in math and math history will have trouble, too.) I'm also planning on giving von Neumann, Bernays, and Gödel their own subsections.
So take your time working on the article—I've got a lot of work to do. It's a pleasure working with you, RJGray ( talk) 18:25, 25 July 2017 (UTC)
I just realized that we could probably add "Ordered pair" to the von Neumann boxes without "operation". The pairing axiom in the other boxes is understood to be the unordered pair so we can probably do this without confusing the reader. Also, without "operation", adding "Ordered pair" will probably not add a line to Von Neumann 1925, 1928. Of course, it's best to add it after extensionality, since this is where "Pairing" appears in the other boxes. RJGray ( talk) 00:40, 26 July 2017 (UTC)
Hi Jochen! I've completed my rewrite of the History section (it's in User:RJGray/Sandboxcantor1). My latest work was on the subsection "Von Neumann's 1929 axiom system". I can see why you found my first attempt confusing--hopefully, you won't find the rewrite so confusing. My major remaining work is to rewrite the basis step of the induction in the proof of the class existence theorem using a lemma like you suggested. However, I'm going on vacation tomorrow and won't be around computers much (if at all) for a bit over a week. So take your time working on the article. -- RJGray ( talk) 17:03, 28 July 2017 (UTC)
Hi Jochen! Your diagram looks great! Thanks for all the work you've been doing on it. As for what to call the axiom: the axiom usually goes under the name "Pairing" meaning unordered pairing, but von Neumann uses ordered pairing. I'm thinking that we can handle this the same way we do Extensionality—namely, with parentheses after "Pairing". Then von Neumann's would read "Pairing (ordered)" and the others would read "Pairing (unordered)". That way, in both cases, we would be using the usual names "Extensionality" and "Pairing" and the parenthetical part would handle the difference between von Neumann and the other set theorists.
So I'm back from vacation and I stayed away from computers, but I carried a printout of the article with me and have done quite a bit of rewriting. It will take me several days to get it all in. I'll let you know when it's stable again—I don't want you to waste time reading something that I'm in the process of working on. Your suggestion of a lemma for the class existence theorem works extremely well—I think that readers will be able to understand that part of the proof much better. Thanks again for your work and suggestions -- RJGray ( talk) 00:45, 6 August 2017 (UTC)
I just realized that you might ask: Since we would be putting in and taking out "Pairing (ordered)" and "Pairing (unordered)" using the arrows, would we need to do that for "Extensionality"? I don't think so, Extensionality has the same meaning, it's just on different primitives so it's closely tied to the primitives and doesn't change meaning. However, "Pairing (ordered)" and "Pairing (unordered)" are have very different meanings so they need to be going in and out with the arrows. (Besides, we don't have the room for "Extensionality" with the arrows.) -- RJGray ( talk) 01:07, 6 August 2017 (UTC)
I've been rethink the "Elementary sets" in Zermelo set theory. Basically, there are only 3 sets postulated by the axiom: the empty set; for any set a, the singleton set {a}; for any pairs of sets a and b, the unordered pair {a, b}. I'm thinking it may be a good idea to replace "elementary sets" with "Pairing (unordered)" since Pairing is the only one of importance when comparing the axiom systems (pairing implies the singleton set and the empty set can be proved to exist by the usual assumption in first-order logic that an axiom system implies the existence of at least one object). -- RJGray ( talk) 18:14, 7 August 2017 (UTC)
Hi Jochen, I'll start by congratulating you on achieving the shortest and simplest injection for Cantor's diagonal argument! I and some others suggested injections that involved more work on the reader's part to understand. About my 24 August 2017 Cantor's diagonal argument Talk contribution—sometimes I should just wait a day and think things through rather than defending my proof without fully appreciating another's proof (which was yours). I apologize for doing this—I'll wait a day or until I fully appreciate another's proof the next time a similar situation occurs.
Now on to Von Neumann–Bernays–Gödel set theory: Thanks for your insightful comments; they have been extremely helpful and have led to a number of improvements. Your suggestion of a lemma to handle the basis step of the proof of the class existence theorem was a great suggestion—it does an excellent job of simplifying that part of the proof.
It's taken me awhile to work through all of your suggestions, but I'm finally done with the rewrite (see User:RJGray/Sandboxcantor1). Here are my notes on the clarify's I had some trouble with:
Class existence theorem clarify comments:
Set axioms clarify comments:
By the way, part of the reason that this rewrite took so long is that when I first write an article, I just grab the information from a variety of sources. At some point, I go over all the definitions, axioms, theorems, and proofs and make sure I have the historically correct ones. For example, I realized that the tuple-handling axioms that I got from French Wikipédia differed from the ones in Gödel and Mendelson, so I now use theirs. I also realized that the product by V axiom that I'm using is Bernays' original axiom, which Gödel changed, so I mention this. Also, I've added "efn" notes that explain how the proof in this Wikipedia article differs from Gödel's proof, so a reader who reads the article is better prepared to read Gödel's original proof.
Looking forward to your comments. Take your time—I'm in no rush to post the article. Thanks again for all the great suggestions you've made, RJGray ( talk) 19:25, 5 September 2017 (UTC)
Hi Robert, thanks for your compliments, but the diagonal proof originates from Kamke, or -probably- earlier, I didn't do more than cite it. For now, let me just reply on your clarify comments.
Class existence theorem clarify comments:
Set axioms clarify comments:
In the next days, I'll read through the article, and insert my comments, if any, using "clarify" as before. Best regards - Jochen Burghardt ( talk) 12:28, 8 September 2017 (UTC)
I've submitted the nomination for "Good article" status again: Talk:Georg_Cantor's_first_set_theory_article/GA2. Michael Hardy ( talk) 19:06, 1 June 2018 (UTC)
Thank you. I'll be ready to deal with any feedback generated by the review. RJGray ( talk) 19:50, 1 June 2018 (UTC)
See Talk:Georg Cantor's first set theory article/GA2.
I am told the following:
Michael Hardy ( talk) 23:27, 30 July 2018 (UTC)
I have just returned from a vacation that started on July 28 and I wasn't around computers. I see that you have already made some changes. What remains to be done and can we get a time extension? RJGray ( talk) 20:03, 5 August 2018 (UTC)
Narutolovehinata5 t c csd new 09:44, 21 October 2018 (UTC)
On 7 December 2018, Did you know was updated with a fact from the article Georg Cantor's first set theory article, which you recently created, substantially expanded, or brought to good article status. The fact was ... that mathematicians disagree about whether a proof in Georg Cantor's first set theory article actually shows how to construct a transcendental number, or merely proves that such numbers exist? The nomination discussion and review may be seen at Template:Did you know nominations/Georg Cantor's first set theory article. You are welcome to check how many page hits the article got while on the front page ( here's how, Georg Cantor's first set theory article), and it may be added to the statistics page if the total is over 5,000. Finally, if you know of an interesting fact from another recently created article, then please feel free to suggest it on the Did you know talk page.
Mifter ( talk) 00:02, 7 December 2018 (UTC)
I would be happy to be your mentor for " Georg Cantor's first set theory article" as we lack good quality maths articles on wikipedia. I see that the article is already in great shape plus it turns out that I recently taught a course where there was a presentation of Cantor's set as a set of reals with no 1 when written in base 3. Before I dwelve into the math of the article, a couple of minor things caught my attention in terms of formatting that would be raised at FAC:
More to come. My advises may look daunting but this is a very fine article. I am convinced it will succeed at FAC if we fix the details. Iry-Hor ( talk) 17:28, 14 November 2019 (UTC)
Here are a few more details for the article:
This is a very good article really, bravo. I will continue to look into this in details but it will be a glorious FA. Then I suggest that we make it so that it appears as the "Today's Featured Article" on the main page at some point (after FA)! Iry-Hor ( talk) 09:06, 15 November 2019 (UTC)
Hi Robert!
I see you are experimenting with various versions of captions for the 3 case diagrams.
In my opinion, the constellations in case 1 to 3 are completely described in the proof text; the diagrams' only purpose can be to help the reader's brain to construct a graphical image of each constellation.
As a consequence, every attempt to describe the diagrams verbally in alt=
text at best can amount to duplicate the proof text.
Therefore, I'd be in favor of just keeping the old captions ("Illustration of case 1", etc.), and to add an empty alt=
text; improvements of the descriptions, if necessary at all, should better be devoted to the proof text itself.
Alternatively, I thought of a naive, proof-unrelated, alt
description like "Real number line with distinct points, left to right: a, a1 to aL, y, xn, bL to b1, b" for case 1.
Best regards -
Jochen Burghardt (
talk)
18:53, 27 January 2020 (UTC)
Addendum: I also thought of "Real number line with relations a < a1 < ... <aL < y < xn < bL < ... < b1 < b"; this is shorter to write, but supposedly longer to hear (when read by a screen reader). - An advantage of keeping caption and alt
separate: when not using a screen reader, the user wouldn't have to read long captions that duplicate the proof text. - All this are just some immature thoughts; feel free to ignore any of them. -
Jochen Burghardt (
talk)
09:41, 28 January 2020 (UTC)
Hi Jochen! Thanks for pointing out that I was just duplicating the proof. I got a bit carried away with my experiments. Also, the old way was better because the old diagrams just barely touched the next section while the new diagrams went several lines deep into the next section. Concerning alt text: I've experimented with it by loading a copy of NVDA, which is a widely-used and free screen reader, and found that a blank alt text reads the file name. I think this may be why WP:ALT recommends using "refer to caption". I've also changed my "Visual representation of case" to your "Illustration of case". As for using the "< relations", I've decided to stick with nested intervals. I explain what they are in the second paragraph of the section and readers can also click on the link. Also, at least for me, as soon as I hear "nested intervals", I immediately imagine an illustration of them. The < relations requires me to think through the dot-dot-dot. Also, I've discovered that NVDA just skips over "...", you have to use ". . . ." Then it only ignores the first ".". By the way, I just tried NVDA on your use of "< relations" above—it doesn't do a good job. There's additional software for reading math that can be added, but I don't have the time to try it yet so that software may fix it. Also, thanks for getting me involved with this section again. I discovered that I can't stack the 3 " File:..." because NVDA reads all 3 files before it goes into the proof texts. I just needed to place each " File:..." before the proof text. Thanks again for your help, RJGray ( talk) 14:09, 28 January 2020 (UTC)