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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

Hi Jochen, I always enjoy hearing from you. Thank you for your email and your link to the Term article. I found it an interesting article. Also, reading the section about formulas made me realized that the Structural Induction article needed some links (in fact, I copied the "Formulas" link from the Term article).

I agree that there are advantages in viewing terms and formulas as trees. For example, I find it very important to think of terms and formulas as trees when I do computer programming. However, I realize that mathematicians tend to think of formulas as built using structural induction. For proofs, there's an advantage of using structural induction—namely, you don't have to do the translation into tree language. As is often the case in mathematics, when you have multiple ways to think about something, your knowledge of it deepens. So it's great that you're visualizing terms and formulas as trees.

In the article rewrite I'm currently working on, there's a proof that uses structural induction. I then capture the induction succinctly as a recursive computer function and supply an example of input and output. I generated the output by thinking in terms of the call tree (formed by the function calls) that was generated by the input. This call tree is generated by the formula's tree. I'm still working on it, but my rewrite is fairly stable now so if you're interested, I can send you the link. --Thanks again for writing, RJGray (talk) 18:53, 22 May 2017 (UTC)[reply]

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)[reply]

Hi Jochen, the link to my rewritten article is User:RJGray/Sandboxcantor1. Comments on the article are very welcome. The proof I mentioned is in the "Class existence theorem" section; the program is at the end of this section. It turns out that the proof does not use structural induction. I wrote my reply to you off the top of my head, and I just think naturally in terms of structural induction. The proof is mostly Gödel's original proof and he does induction on the number of logical symbols in a formula. Of course, he could have used structural induction, but mathematicians, if possible, use ordinary induction, which works fine in this case. I'm thinking of keeping my link to structural induction in the lead because it's a good explanation of how formulas are built and there I'm just giving a general idea of how the construction works. Of course, I welcome feedback on this. Linking to mathematical induction in the lead would be confusing. Of course, I could just drop the link altogether.

The proof is rather long, but the computer program is short. I just use the program to illustrate the proof from a different angle. If you find computer programs in math interesting, I did write an article years ago that used computer programs to prove theorems of analysis: Heine-Borel theorem and Riemann integrability on closed intervals of continuous functions and of bounded functions continuous except on a set of measure zero. The article is just 4 pages and available online, but is unfortunately expensive ($39.95, nearly $10/page!). If you have an interest in it and can tell me how to get scanned images to you, I can send it to you. It's referenced as Gray 1991 in User:RJGray/Sandboxcantor1. —RJGray (talk) 19:10, 24 May 2017 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

Your restated class existence theorem without "parameters" is clearer, imho. I understand that each C_i 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)[reply]

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)[reply]

Hi Jochen, I'm happy that removal of "parameters" to the class existence theorem worked out well even though now you think the old text was fine. The Parameter article is confusing especially in the Parameter#Logic section. If I kept "parameters", I would have to rewrite some of that article. Thanks for all your help. I'm going to take my time going through the rest of your comments. I find your comments extremely helpful. They have challenged me to come up with better ways to express the mathematics. I'll tell you when I have a new version ready. It will be at least a couple of weeks and maybe a bit over a month since I'm doing some traveling in a few weeks.

I'm very pleased that you learned a lot about NBG set theory. This is why I write articles: so people can learn a lot about a subject. I learn a lot by writing them. In fact, this article motivated me to carefully work through much of Gödel's 1940 monograph. If you want to learn more about von Neumann's original set theory and how it relates to NBG, I recently rewrote the Axiom of limitation of size article. Getting through the NBG set theory article is excellent preparation for reading that article. Comments are also welcome on that article. Thanks again for your help, RJGray (talk) 00:44, 1 June 2017 (UTC)[reply]

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:

• Class axioms: I continue giving the informal statement of the class existence theorem from a previous section. I put a border around Example 1 and rewrote parts of it——for example, I explain why intersection handles "and" and domain handles "there exists".

• Class existence theorem: For the proof, I followed your reformatting advice and rewrote parts of the proof following your suggestions. I also rewrote various other parts of this section. In the program, I added a comment to explain c(n). I tried to get wikilinks into the code but it didn't work. I also tried using Pascal syntax, but it doesn't do subscripts, superscripts, and Greek letters well. Anyway, I want the math in the pseudocode to look the same as the math in the proof.

• Extending the class existence theorem: I reformatted Examples 3 and 4, which you found confusing, and did some rewriting.

• History: I rewrote the first paragraph to go more in depth about von Neumann's use of functions and arguments. Also, I state how von Neumann used his primitives to define sets, classes, and membership. This justifies talking about sets and classes when talking about von Neumann's theory (von Neumann does this himself in his articles).

Concerning your "clarify" statement in the History section about Zermelo's set theory: you figured it out! Halmos is talking about ZFC, which has the axiom of replacement, which Zermelo's 1908 set theory does not have. So other readers don't have to figure it out, I rewrote the paragraph stating right away that Zermelo set theory does not have replacement. Also, I added a reference note where von Neumann points out that although Zermelo had the right definition of ordinal in 1916, he could not prove a key theorem because he lacked the replacement axiom. This together with the von Neumann quote in the paragraph should emphasize the role of replacement enough.

I handled your "clarify" statement about Zermelo thinking that axiom schemas "implicitly involve the concept of natural number" in a reference note.

I changed the link from well-founded to well-founded set, which also brings you to the well-founded relation article, but in such a case, the reader should look for a bold-faced "well-founded set" in the article's lead giving the meaning of well-founded set.

I like your idea of a diagram——the history of NBG is a bit convoluted, so I came up with a diagram (see User:RJGray/Sandboxcantor1#Evolution of NBG (diagram)). Please give me your feedback on the content and layout. Also, since I lack your skill in diagrams, I would greatly appreciate it if you redid it. I only tried an ASCII diagram in one place—putting a border around one of the entries.

• Set axioms: I rewrote the Axiom of replacement and the Axiom of infinity, and also placed them first since they are the set axioms that are different from ZFC's axioms. Because you had questions about the axiom of regularity, I focused my attention on the axiom and realized that I had given the class form of the axiom. Gödel proved later that the set form suffices. So now I'm using the set form, and I added a note about the two forms of this axiom.

One small flaw I left in was my use of the axiom of regularity in the proof of the class existence theorem before I mention it. I'm willing to live with this flaw because I want to emphasize the class existence theorem and want to get to it as early as possible. I copied the French article on this. However, the French article doesn't refer to the axiom of regularity in their proof—it left out that case! I've changed the link to bring you to article's coverage of this axiom rather than link to another article. Since nearly every reader of this article will be familiar with ZFC (which has the same regularity axiom), and because I do supply a link, I don't think I'll get many (if any) objections.

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 ${\displaystyle \forall x\,\exists y\,(\neg y\in x)}$ doesn't work because it only says that for every set ${\displaystyle x}$, there is a set ${\displaystyle y}$ that doesn't belong to ${\displaystyle x}$. Consider a set ${\displaystyle x}$ whose only member is ${\displaystyle x}$, so we have ${\displaystyle x\in x.}$ However, the set ${\displaystyle y=\emptyset }$ satisfies ${\displaystyle \neg y\in x.}$ Regularity's condition that ${\displaystyle x\neq \emptyset \implies (y\in x\land y\cap x=\emptyset )}$ is needed to prevent ${\displaystyle x\in x.}$ Your suggestion of ${\displaystyle \forall x(\neg x\in x)}$, as you suspected, handles only ${\displaystyle x\in x.}$

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: ${\displaystyle \;\;\cdots \in x_{n+1}\in x_{n}\in \cdots \in x_{1}\in x_{0}.}$ The axiom is von Neumann's slick way of using a set ${\displaystyle a}$ to prove the non-existence of these chains.

Assume the axiom of regularity and that we have an infinite descending chain: ${\displaystyle \;\;\cdots \in x_{n+1}\in x_{n}\in \cdots \in x_{1}\in x_{0}.}$ Now define the set ${\displaystyle a=\{\dots ,x_{n+1},x_{n},\dots ,x_{1},x_{0}\}.}$ Regularity implies there is a ${\displaystyle y\in a}$ such that ${\displaystyle y\cap a=\emptyset .}$ However, ${\displaystyle y\in a\implies y=x_{n))$ for some ${\displaystyle n}$. But ${\displaystyle (x_{n+1}\in x_{n}=y)\land x_{n+1}\in a\implies x_{n+1}\in y\cap a,}$ which contradicts ${\displaystyle y\cap a=\emptyset .}$ Therefore, there is no infinite descending membership chain.

The above proof tells us what sets ${\displaystyle a}$ we need for handling ${\displaystyle x\in x}$ and ${\displaystyle x\in y\land y\in x}$:

${\displaystyle x\in x}$ generates the infinite chain: ${\displaystyle \;\;\cdots \in x\in x\in \cdots \in x\in x.\,}$ So ${\displaystyle a=\{\dots ,x,x,\dots ,x,x\}=\{x\}.}$

${\displaystyle x\in y\land y\in x}$ generates the infinite chain: ${\displaystyle \;\;\cdots \in x\in y\in x\in \cdots \in x\in y\in x.\,}$ So ${\displaystyle a=\{\dots ,x,y,x,\dots ,x,y,x\}=\{x,y\}.}$

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 ${\displaystyle R}$ is a binary relation on ${\displaystyle a}$ such that ${\displaystyle \forall u\in a\,\exists v\in a(uRv),}$ then there is an infinite sequence ${\displaystyle x_{n))$ such that for all ${\displaystyle n\!:\,x_{n}Rx_{n+1))$. 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: ${\displaystyle \neg \forall a\,[a\neq \emptyset \implies \exists u(u\in a\land u\cap a=\emptyset )],}$ which is equivalent to: ${\displaystyle \exists a[a\neq \emptyset \land \forall u(u\in a\implies u\cap a\neq \emptyset )].}$ We now prove that there exists a descending membership chain:

Define ${\displaystyle uRv}$ to be: ${\displaystyle v\in u\cap a.}$ Since ${\displaystyle \forall u(u\in a\implies u\cap a\neq \emptyset )\implies \forall u[u\in a\implies \exists v(v\in u\cap a)]}$ we have ${\displaystyle \forall u\in a\,\exists v\in a(uRv).}$ By the axiom of dependent choice, there is an infinite sequence ${\displaystyle x_{n))$ such that for all ${\displaystyle n\!:\,x_{n}Rx_{n+1))$. Using the definition of ${\displaystyle R\!:\,x_{n}Rx_{n+1}\iff (x_{n+1}\in x_{n}\cap a)\implies (x_{n+1}\in x_{n}).}$ So ${\displaystyle x_{n))$ is a descending membership chain.

Miscellaneous:

• Classes and sets: Your "clarify" comment: "Order-sorted logic and relation overloading could have avoided some, or even all, of these problems, but it may not have existed in Bernays' days" is correct as far as syntax is concerned. Order-sorted logic and relation overloading seems to me to be "syntactic sugar" that only makes the axioms more pleasing to the eye. Gödel simplified at the deeper, semantic (model-theoretic) level because having all sets being classes made his construction of the constructible universe simpler.

• Discussion: Your comments were helpful and I cleaned up what I could. Some of the material I had just copied over from the original article. Thanks for noticing the confusing sentence towards the end, I copied it over without reading it carefully. Concerning the speed-up results in Pudlak's article (another thing I had just copied over): I located an online copy of the article (see References). However, I don't have the background needed to understand exactly what he is saying, so I can't add any comments about this at this time. (Search for "GB" in Pudlak's article to see his speedup theorems concerning GB [NBG] and ZF [ZFC]).

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)[reply]

Hi Robert,

I didn't have time yet to look at your article, but I just had a glance at your above change notes (anyway thanks for your explanations!)

At least, I could come up with 2 sketches of your diagram, see File:NBG Evolution.pdf (it has 2 pages, one version on each page). I tried to use colors to distinguish approach, primitives, axioms, and properties. I stuck with variants of blue for them, since in the 2nd sketch, I used red/green to indicate properties that were removed/added in the transition from one approach to the next. However, this removed/added information may be unimportant and should be omitted; in that case, it'd be easy to have better-distinguishable colors for the remaining topics.

The box widths should eventally be all equal in each column, I think now.

Also, I guess the main problem will be to exploit the available space in an optimum way, in order to have a readable font, but nevertheless having a sufficiently small image. The 2nd sketch, with the removed/added column omitted, may be the most space-efficient version so far. (As I type, it comes to my mind, that the removed/added information could be given in spite of the omitted column, viz. by coloring the entries immediately below each box appropriately (red/green or neutral dark-yellow for kept properties.)

The final version should better be converted to svg, as thumbnail images of pdf are rendered somewhat blurry as jpegs.

Maybe, it would be better to colorize the background, not the foreground (or even both?), but that is difficult to achieve with the LaTeX tabular environment.

After I've read your new article version (I guess, next weekend), and thereby refreshed my memory about NBG history, too, I might have other suggestions w.r.t. the image.

If I still had questions concerning your text, could I add clarify templates (or just <!---markup comments--->?)directly to your User:RJGray/Sandboxcantor1, or would you prefer to append a copy of that page to my User:Jochen Burghardt/sandbox1?

Best regards - Jochen Burghardt (talk) 19:06, 11 July 2017 (UTC)[reply]

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)[reply]

Hi Robert! Some issues don't fit well into a ((clarify)) template; so I discuss them here:

• In the lead's first paragraph, I'd omit the explanation of "conservative extension" and "Morse-Kelley set theory" (and move both to a later place, probably to section "Discussion" / "NBG and ZFC", which could be extended to discuss MK, too). Also, I'd consider (i.e. I'm not sure) omitting "goes beyond ZFC". And I'd mention that NBG allows to speak about the class of all sets. I'm not sure how to order the information to present the most important things first.
• In Example 1, I got the impression that essentially you rewrite the composition as ${\displaystyle G\circ F=Dom(F''\cap G'')=Dom(Permute(...,F')\cap Permute(...,G'))=Dom(Permute(...,Embed(...,F))\cap Permute(...,Embed(...,G)))}$ where "..." indicate the descriptions of permutations and embeddings. I'm not sure, but maybe initially stating the equality ${\displaystyle G\circ F=Dom(Permute(...,Embed(...,F))\cap Permute(...,Embed(...,G)))}$ and then recursively descending along the structure of the right-hand-side term better emphasizes the systematic approach. Most of your current explanations would go into the argument why this equality holds, and to indicate that Embed and Permute are sufficient to handle all constellations of variables in a formula. All that would be (still tedious, but) understandable also on a familiar ZFC background. NBG axioms would come into play only after that, and which are needed would be obvious from the rhs term.
• In section "Axioms for handling language primitives", it took me several times reading until I understood why the uniqueness of E doesn't follow immediately from the extensionality axiom. In retrospect, I must have been blind. Nevertheless, changing the order could possibly make the text easier to grasp, e.g.

"By the axiom of extensionality, class C in the intersection axiom and class B in the complement axiom and the domain axiom are unique. They will be denoted by: ${\displaystyle A\cap B}$, ${\displaystyle A^{\text{c))}$, and ${\displaystyle Dom(A)}$, respectively. On the other hand, extensionality is not applicable to E in the membership axiom since it specifies only those sets in the class that are ordered pairs."

You shouldn't change the presentation order of the axioms, so you can still say "The first three axioms imply ..." in the following paragraph.

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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 ${\displaystyle \neg }$, 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 ${\displaystyle \complement _{V^{3))}$, 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)[reply]

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)[reply]

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)[reply]

An svg version of the image is now available at File:NBG Evolution svg.svg. - Jochen Burghardt (talk) 11:04, 24 July 2017 (UTC)[reply]

Hi Jochen! Excellent diagram! Just a few suggestions:

• In the Bernays box, I would move the "1937," to be next to "1941".
• I think it would be much easier to read the axiom lists without the hyphens "In- finity", etc. This would probably cause an extra line in the von Neumann 1929 box, but my next suggestion would also require this extra line.
• I prefer "von Neumann choice" to "Neumann choice" because it's customary to use the "von". For example, the set theory called "von Neumann–Bernays–Gödel set theory" (the "von" is only dropped in the abbreviation NBG). Also, I changed the History section because you found it confusing. After introducing the choice axiom, in the next sentence, I call it "von Neumann's choice axiom".
• I agree that "Replacement" should be mentioned somewhere in regards to "von Neumann 1925, 1928". I think that the best place may be on the side with the axioms added. Replace "Limitation of size" with "Limitation of size (implies Replacement)". To me, this tells a reader that von Neumann's 1925/1928 axiom system was influenced by the 1922 work of Fraenkel and Skolem, and he made sure that one of his axioms implies Replacement.

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:

1. Permanent state: Jupiter is a very massive planet.
2. General truth: The earth is round.

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)[reply]

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)[reply]

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 ${\displaystyle \phi }$ and ${\displaystyle A}$. 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 ${\displaystyle x_{2}\!\in \!x_{1))$ more closely matches the length of ${\displaystyle E_{2,1,2))$. 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 ${\displaystyle \in }$ 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)[reply]

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)[reply]

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)[reply]

Hi Robert! I placed my final remarks on your article (section "NBG, ZFC, and MK") today. Also, I updated the history diagram according to your suggestions of 26 July. I tried "Unordered pair" instead of "Pairing" to see if that fits, too - it does. If you prefer "Pairing", I'll change that; "Unordered pairing" would fit, too, but give a hyphenation, unless the width is extended. The svg version is now up to date (see User:Jochen Burghardt/sandbox#NBG). Best regards - Jochen Burghardt (talk) 17:47, 5 August 2017 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

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:

• I'm confused by the first half of the clarify on Example 2. Clarify: f [I don't know what "f" refers to] should be given in advance for the example, to be consistent with the above explanation why it can't be inferred from the formula. For convenience, all symbols could be aligned in the following two formulas. (I did the alignment.)

• Old text: "An example explains why the [class existence] theorem states that the free variables are among ${\displaystyle x_{1},\ldots ,x_{n}.}$" Thanks for spotting that "among" is not the best way to state it. I went back to Gödel's proof and now use his terminology (the "among" terminology was from Mendelson): Let ${\displaystyle \phi }$ be a formula that quantifies only over sets and contains no free variables other than ${\displaystyle x_{1},\ldots ,x_{n))$ (not necessarily all of these). I only state "(not necessarily all of these)" on the first occurrence since it's really not needed—it just makes sure that the reader understands what "contains no free variables other than ${\displaystyle x_{1},\ldots ,x_{n))$" means.

• In induction part of proof. Clarify: Maybe, arguments to phi, psi1, psi2 should be written out. Only the descriptions of inductive case 1,2,3 are concerned, so it won't use much extra space (and reading effort). On the other hand, it would ease the reader to follow the proof steps, where arguments *are* written out. Reply: I wrote the arguments out, but when I shrunk the window, "Case", "1:", "${\displaystyle \phi }$", and "${\displaystyle \psi }$" spread some distance from each other when the window size was increased and decreased. I experimented and the best solution so far is to put "Case n" within the "math". Please tell me how it looks, the "Case n:" heading has larger font size than the "Case n:" that I've used in the basis step of the proof.

Set axioms clarify comments:

• Clarify: Naming deviates from common use. Reply: The function definition I give is the same as that of Gödel and Mendelson, and is, as far as I know, commonly used. Perhaps you are thinking of the second definition in Mendelson of ${\displaystyle F:X\rightarrow Y}$ ("function from ${\displaystyle X}$ into ${\displaystyle Y}$" as opposed to the definition of just "function") that requires ${\displaystyle Dom(F)=X}$ and ${\displaystyle Range(F)\subseteq Y.}$ Even in Calculus books, the function definition used by Gödel and Mendelson appears. For example, page 45 of Spivak's Calculus states the definition:

A function is a collection of pairs of numbers with the following property: If (a, b) and (a, c) are both in the collection, then b = c; in other words, the collection must not contain two different pairs with the same first element.

• Your comment: "I guess, this and the following axioms state that something already known to be a class (from the cl,ex.thm.) is even a set." is correct and insightful (I hadn't thought of it in these terms). It turns out that Gödel's presentation handles this. He states the class operations of image, union, and power class are produced by the class existence theorem and points out that the axioms of replacement, union, and power set guarantee that applying these operations to sets produces sets.

• I finally moved the axiom of regularity to before the class existence theorem. I don't know why it took me so long. Thanks for mentioning it a second time. Comments are welcome on its new location.

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)[reply]

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:

• Example 2 [It took me a while to remember what I'd meant]: In the version of 5 Aug 2017, f was used in the preceding Bound variable renaming rule as the number of free variables; it is n in the current version. In the earlier section Class existence theorem, you explained that the number of free variables that may occur need to be specified (it cannot be obtained from the formula). To be consistent with that, I suggested to say in Example 2 something like "... The bound variable renaming rule is applied to a formula in one free vaqriable that defines ...". — In some situation, the renaming rule could be applied to the very same formula, but considered to have free variables among x1,x2,x3; another renaming would result. Therefore I think, f, resp. n should be given as input to the renaming procedure, and hence stated before its result is shown.

• In induction part of proof: At first glance, I didn't notice the different sizes of "Case n". But now that I've searched for it, after reading your explanation, it appears somewhat strange. Did you also try to avoid <math> and ((math))? Except for the complement "${\displaystyle \complement }$", all symbols should be available in Unicode, too. (There was a time when using many <math>s caused rendering time in the minutes range, and I then converted some long/slow articles to Unicode, usually without problems.) Another alternative could be to use "\mathsf" for "Case n" in the whole proof. — In any case, I'd still be in favor of keeping the arguments to φ and ψ, since you manipulate them during the proof; using x→ and C→ could be an alternative, except that the former is broken down into its components in the ∃ case. But of course the decision is up to you.

Set axioms clarify comments:

• Clarify: Naming deviates from common use: Oops, I thought that "${\displaystyle F\subseteq V^{2}\land \forall x\,\forall y\,\forall z\,[(x,y)\in F\,\land \,(x,z)\in F\implies y=z]}$" defines a "partial function"; if additionally "${\displaystyle \forall x\exists y[(x,y)\in F]}$" is required, then I'd call it a "function" (or "total function", if emphasis is needed). At least Function (mathematics), Partial function, Halmos "Naive Mengenlehre = Naive Set Theory" (Ch.8) agree with that (Kamke doesn't mention relations or functions at all). I wonder if there are different communities that use the two different notions of "function". Anyway, I think it is useful if you add a footnote stating the difference between Gödel / Mendelson / Spivak / NBG set theory vs. Function (mathematics) / Partial function; preferably when one of the latter is first wikilinked.

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)[reply]

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)[reply]

Thank you. I'll be ready to deal with any feedback generated by the review. RJGray (talk) 19:50, 1 June 2018 (UTC)[reply]

See Talk:Georg Cantor's first set theory article/GA2.

I am told the following:

The article Georg Cantor's first set theory article you nominated as a good article has been placed on hold Symbol wait.svg. The article is close to meeting the good article criteria, but there are some minor changes or clarifications needing to be addressed. If these are fixed within 7 days, the article will pass; otherwise it may fail.

Michael Hardy (talk) 23:27, 30 July 2018 (UTC)[reply]

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)[reply]

Hello, RJGray. You have new messages at Template:Did you know nominations/Georg Cantor's first set theory article.
Message added 09:44, 21 October 2018 (UTC). You can remove this notice at any time by removing the ((Talkback)) or ((Tb)) template.

Narutolovehinata5 ^tccsdnew 09:44, 21 October 2018 (UTC)[reply]

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, ), 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)[reply]

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:

• The use of bold face is very much regulated and I believe that the bold face used in footnote E should be replaced by some other form of emphasis, e.g. italic, or none at all. Regarding the bold face lettering for the set of reals I think it should be fine (seeing the article on the real numbers) but just in case you could use ${\displaystyle \mathbb {R} }$ instead, which is even better math-wise anyway.
  • I've changed bold face to italics in Footnote E. On the choice of symbols for the set of real numbers: R versus ℝ, I prefer R because on my screen ℝ is not very visible compare to R. In fact, I suspect that the contributors to Real number use R throughout most of their article for the same reason. Also, the article Rational number uses ℚ once but elsewhere uses Q.
• Why aren't ref 3, 4, 6 12, 32 and 47, given as footnotes ?
  • Because they are references with an annotation: see WP:Citing sources#Additional annotation. The harvnb and the sfn templates support these annotations with their "ps=" parameter. Also, there are featured articles using references with an annotation: see Hubble Space Telescope (ref 65, 66, 87, 1120) and History of evolutionary thought (ref 89). Refs 65 and 66 use "ps=", the rest don't. However, there is a problem with my additional annotations: I placed them at the start of the reference instead of at the end. I have fixed this. Because I eliminated most of the refs with an annotation, there is just one remaining ref with an annotation that now uses "ps=".
• Publisher location is lacking in all the books. Such info is always given on https://www.worldcat.org/ . I know it is a nitty-gritty detail but believe me this is checked at FAC.
  • Publisher location has been added for all books. Thank you for pointing out that I could get the information from the Worldcat site.
• Any reference that is not in English should have its language stated. For example "Fraenkel, Abraham (1930), "Georg Cantor", Jahresbericht der Deutschen Mathematiker-Vereinigung, 39: 189–266." is in German. Just add the "language = German" field in the cite template.
  • "language = German" has been added where needed.
• All pictures must have an alt text that describes the picture in a few words for people who cannot see the pic. Add a "alt= Description here" field in the file template "|thumb|upright|alt=..."
  • They already have an "alt=" field. It turns out that the last field without an "alt=", which I use, works the same as having the "alt=". This is commonly done: for example, see the featured articles Georg Cantor and General relativity.
• Some paragraphs have no references at all. This will be raised at FAC, as editors tend to look at possible issues first by spotting locations with no citations. This is e.g. the case of the first paragraph of the section "The proofs / Second theorem" and also "Example of Cantor's construction". There are more examples.
  • Fixed the ones you mentioned and others. But there are a few that have a reference before the end of the paragraph which is followed by a sentence of two where I'm making clarifying remarks: For example, in User:RJGray/Sandbox100#Dedekind's contribution to Cantor's article: look at the last 2 sentences of the 3rd paragraph and the last sentence of the 5th paragraph. Is there anything wrong with these clarifying remarks?
• The lead may need some reworking to be made more precise. For example " In addition, they have looked at the article's legacy, which includes the impact that the uncountability theorem and the concept of countability have had on mathematics. " is a sentence that has nearly no informational content other than "people have considered the legacy of something". Here instead you need to state what this legacy is in more concrete terms, saying e.g. "Historians of mathematics have since recognised the key role played by Cantor's discovery in the subsequent development of measure theory, set theory, integration and logic" (this might not be the best sentence but it says more than the one currently given and does summarise to some extend the last section.
  • Thank you for your advice. I've used it my rewrite of the lead.
• As a MOS rule, the lead should have no references, except for statements that are highly contentious. This is not the case here so all refs of the lead should be removed. This would be raised at FAC immediately.
  • Wikipedia:Manual of Style/Lead section#Citations states that:
    • "The lead must conform to verifiability, biographies of living persons, and other policies. The verifiability policy advises that material that is challenged or likely to be challenged, and direct quotations, should be supported by an inline citation."
  • This covers my use of the direct quotation "Cantor's revolutionary discovery". As for the other inline citations, I've deleted them but I will put back any that get challenges on the statement that it protects.
• I am not sure but some pictures might need a US-PD tag and not just a PS-old one. This would be checked at FAC and can be fixed then.
• Bold face at the begining of the article should refer to the article name or a close variant of it and should be absent after that. Thus I suggest starting this article by "Georg Cantor's first set theory article was published in 1874. In it, Georg Cantor presents the first theorems of transfinite [...]". I suggest removing bold face from the "Cantor's first uncountability proof," in the lead and the one after as well.
  • Thank you for the suggestion on starting the article. I've used it in my rewrite of the lead. As for the use of boldface at the beginning of an article, MOS states:
    • "The most common use of boldface is to highlight the first occurrence of the title word/phrase of the article in the lead section. This is also done at the first occurrence of a term (commonly a synonym in the lead) that redirects to the article or one of its subsections, whether the term appears in the lead or not. These applications of boldface are done in the majority of articles, but are not a requirement. It will not be helpful in a case where a large number of terms redirect to a single article, e.g. a plant species with dozens of vernacular names."
    • For example: the featured article Laplace–Runge–Lenz vector uses boldface for 3 redirects in the last paragraph of the lead.
  • The two uses of boldface are that you are concerned about are both for redirects. In math articles, redirects to the article or one of its subsections are always (as far as I've seen) in boldface in the lead.
• All books cited should have an ISBN. If they don't have an ISBN, they should have an OCLC, which necessarily exists. OCLC numbers are available on https://www.worldcat.org/ . Just add the | oclc = something in the cite book template. For example, Oskar Perron's Irrationalzahlen which you cite, has its full info here, which include the oclc 4636376 and the publisher location Leipzig, Berlin.
  • Publication location has been added for all books. Also, added OCLC for books with no ISBN. Thank you for pointing out that I could get the information from the Worldcat site.
• Could we not include some wikilink to Epistemology somewhere in there ? Because some of the historians of mathematics cited here would refer to themselves as epistemologists.
  • The focus of this Wikipedia article is on Cantor's first set theory article, which is just a short math article, partly because of Weierstrass' influence (he wanted to use the correspondence between the integers and the real algebraic numbers in his work) and partly because Cantor only had two results. So I see no good place to bring epistemology into the discussion. On the other hand, things change when writing an article on Cantor's 1883 Grundlagen article, which someday I hope to do. This is the first article in which Cantor uses both mathematics and philosophy to justify his handling of the infinite. For more on Cantor's philosophy of the infinite, see Georg Cantor#Philosophy, religion, literature and Cantor's mathematics (perhaps you could put in a wikilink to Epistemology.

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)[reply]

Here are a few more details for the article:

• Algebraic number should be wikilinked the first time it appears in the text. As a general rule, wikilinks in the leads do not count, so something linked in the lead should be relinked the first time it appears in the text and not after that.
  • Done.
• "Only the first part of Cantor's second theorem needs to be proved." might be contentious or curious to some readers unacquainted to math. Perhaps the sentence should be modified or the logical connection between the first and second parts of the theorem be explained in a footnote? For mathematicians it is obvious but the article is supposed to be accessible to all.
  • Thank you for pointing this out. I proved the first part of Cantor's second theorem implies the second part in the preceding "The article" section but the proof really belongs to the "First theorem" section. So I moved it, expanded it, and put it in as a footnote.
• I think that Controversy over Cantor's theory should be wikilinked somewhere in the article, either in "The disagreement about Cantor's existence proof" or in "The legacy of Cantor's article" section. I don't know if you know but you don't have to use the exact article title to wikilink something, rather write [ [ Exact article name | what appears in the text ] ], which produces things like this.
  • The Wikipedia article Controversy over Cantor's theory is addressing the controversy about set theory that occurs in regard to Cantor's later work. The reason why I want to distinguish Cantor's 1874 article from his later controversial work is explained in the article Georg Cantor and Transcendental Numbers. On page 828, this article asks: "Why do some mathematicians misinterpret Cantor's [1874] article?" It goes on to state: "We will only discuss how these misinterpretations are encouraged by some commonly-held views about mathematics and its history. One such view states that Cantor's set theory was initially attacked by many mathematicians of his time. The problem with this view is that it fails to distinguish the parts of Cantor's work that were attacked, from those that were not." The article goes on to tell how Kronecker did not even delay publication of Cantor's article although he had previously delayed publication of an article written Heine, one of Cantor's colleagues. On page 829, the article concludes: "So the popular view of the history of set theory needs to be refined. Criticism of Cantor's theory did not begin with the publication of his 1874 article. It began with his 1878 article, which contains arguments that require the use of infinite sets. [With this article, there was a publication delay that Cantor blamed on Kronecker and he stopped sending articles to Crelle's Journal.] Criticism increased as Cantor introduced new concepts involving the infinite."
  • So my reason for not wanting a reference to Controversy over Cantor's theory is that I don't want to contribute to the view that Cantor's 1874 article was attacked. Writing my reason down has made me realize that I should rewrite more of the "Controversy over Cantor's theory" article. I had previously rewritten the mathematical part of it.
• In the references: Saunders Mac Lane, Eric Temple Bell, Paul Cohen, Oskar Perron, G. H. Hardy and E. M. Wright should all be wikilinked once, irrespectively of whether or not they are already wikilinked in the text. EDIT: actually I am not sure there is something in the MOS on this, so you can leave the article as it is on this point.

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)[reply]

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 a_L, y, x_n, b_L to b₁, b" for case 1. Best regards - Jochen Burghardt (talk) 18:53, 27 January 2020 (UTC)[reply]

Addendum: I also thought of "Real number line with relations a < a₁ < ... <a_L < y < x_n < b_L < ... < 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)[reply]

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)[reply]

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