User talk:RJGray/The proofs

Excellent work, Jochen! Your diagrams go far beyond my suggestions and have captured the dynamics of the construction with the a₁, a₂, a₃, … Thank you!

I guess I'm now supposed to play critical reader, so here are my suggestions:

1. a_∞ and b_∞ shouldn't have a v-bar, one needs a [ and the other needs a ]
2. The labels in the picture should be the same font size as the print in the "Case" captions. I know this may affect the horizontal size of the picture, but I couldn't read it well with the small font size.
3. The italic "a" in the picture is different from the italic "a" in the text of the proof. Wikipedia italics aren't true italics when it comes to "a". I think Wikipedia is using slanted type rather than italic type. LaTex does support slanted type with its \sl command. Could you try LaTex's slanted type on the variable names and see if it looks similar to the Wikipedia italics in the text?
4. Perhaps the labels could be moved up to be closer to the number line.

Thanks again and I hope you may learn something new from my rewrite. RJGray (talk) 18:29, 6 August 2015 (UTC)

Hi Jochen, I've been thinking about how in Case 2 and Case 3, the x_n gets in the way of putting the "…" between b_n and b₃. I know I was the one who wanted all the labeling below the number line, but now with your excellent pictorial capturing of the construction, I'm thinking that the x_n and c may belong above the line. This way you can fit in the "…". But perhaps more importantly, the reader sees just the dynamics of the construction below the line without any clutter. Then x_n and c sit on the stage for the argument. Thanks, RJGray (talk) 16:09, 7 August 2015 (UTC)

Hi Robert, at last, I implemented your suggestions in a new version. I scaled the image down to 90% (but kept the letter sizes, so that they now appear as 111%); scaling down further would collapse a₂ with a₃ etc. The image bounding box dimensions will have to be adjusted lateron (to remove the surplus white space top+left). Best regards - Jochen Burghardt (talk) 12:09, 18 August 2015 (UTC)

I now adapted the image bounding box dimensions, in order for the downscaling to become effective. Since the labels are still very small, I'll experiment with devoting more horizontal space to the thumbnail images in the article. - Jochen Burghardt (talk) 12:17, 18 August 2015 (UTC)

Are you sure you want \sl fonts? On my browser, wikipedia's a looks like LaTeX's $a$ ($a$, i.e. ordinary math mode, used in previous version of images), while LaTeX's $\sl a$ looks different. - Jochen Burghardt (talk) 12:26, 18 August 2015 (UTC)

Hi Jochen, Excellent Work! Sorry I'm so long in replying -- I'm on vacation and haven't had access to a computer. I'll be back on Friday. Two things: (1) On horizontal space, you can get more by removing a₃ and b₃. Also, I think just a₁ a₂ are enough before the dots and this would simplify the diagrams a bit. On one of my theorems, I use x₁, …, x_g(n). (2) On my browser, wikipedia's italic a looks like a slanted 'a' and not Latex's $a$. If you used Latex's slanted 'a' in the diagrams, on my browser it looks like the wikipedia regular 'a'. So I'm a bit confused on our browser difference (I use Google Chrome) and on why the Latex slanted 'a' doesn't look slanted like the 'a' in the Case descriptions on the diagrams. Thanks again for your excellent work and I'll reply to any comments you have on Friday. RJGray (talk) 01:48, 24 August 2015 (UTC)

Hi Robert, I just uploaded a new version according to your suggestions. I also converted them to svg (while Wikipedia renders pdf images as jpg by default); this usually looks much better (less blurred). Concerning the 'a' renderings, I probably have a too old browser. Best regards - Jochen Burghardt (talk) 10:59, 3 September 2015 (UTC)

Hi Jochen, It looks great, going to svg is a good idea, and I especially like your use of color. About the italic "a" versus slant "a" -- just stick with what you have. I've discovered that my browser renders it a slanted "a", but when I print it, I get an italic "a" ! I never noticed this before, so I suspect that anyone reading math regularly just sees both as an italic "a". So your italic "a" should stay in the diagram. Some other details:

1. I'm thinking of changing c to y since c is usually used as a constant and y as a variable, and we do have an infinity of y's in Cases 1 and 3. Also, using y above the line goes well with the x_n. If you think this is a good idea, please make the change on your next diagram.
2. I noticed that in Cases 2 and 3, the "…" does not always immediately follow the a's and precede the b's like they do in Case 1. In Case 3, there's a space between a and a₁. Also, you might try experimenting with \cdots to see how it looks.
3. I noticed that the a₁, a₂, … and the parentheses on the diagram are hard to see when printed. Is it possible to darken them a bit? I realize that this may require fewer and wider-spaced parentheses when they approach a limit in the diagram.

I greatly appreciate the excellent work you are doing. As far as my writing is going, I've added my next section. I have three more to do. Thank you, RJGray (talk) 21:35, 3 September 2015 (UTC)

Hi Robert, sorry for the long delay. I updated the pdf images accordingly; I'll update the svg versions in the next days, too, and then again use them. I think \cdots looks better. I hope nobody will notice that there wasn't enough space to place dots between a_N and a_∞ in case 3.

Is the sequence interval color dark enough now? I darkened it from 70% to 60%, where 0% and 100% denotes black and white, respectively. It can easily darkened to another amount; it's just a matter of not letting the sequences disturb the main features of the picture (a, a_N, a_n, a_∞, b, b_N, b_n, b_∞, x_n, y). I could also try to start sequence interval color with (e.g.) black, and fade out into white, as the limit is approached, but that would require some amount of work. Probably it would also destroy the current impression of increasing density of black indicating the limit process, so it might not be a good idea. Best regards - Jochen Burghardt (talk) 10:55, 22 September 2015 (UTC)

Before I forget it again: I had copied the page to User:Jochen Burghardt/sandbox8 on 4 Aug, and had made some edits that I considered helpful to better understand the overall structure. Maybe re-inspecting my edits one by one could give some useful suggestions for your work; but maybe not. Mainly, I've inserted a lot of intermediate (((sub)sub)sub)section titles that were for my own understanding, but not intended to be kept in a final version. I feel that first presenting the bare proof, and thereafter, and explicitly separated, giving the additional remarks might be a good idea. However, this structure might be in conflict with a historically accurate presentation? - Jochen Burghardt (talk) 11:16, 22 September 2015 (UTC)

September 25

Hi Jochen, as always, I'm impressed with your work! Thank you, it looks great! And thank you for changing [c, d] to [y, z] in my text. I want to apologize for making 2 more suggestions, but I know from working with you before that you're like me and want the best for the readers.

On darkening: It's much better, I can now see them when printing them. I think they're almost dark enough. I don't think the markings on the line should be darkened too much more, if at all--they are big enough to be visible. I was wondering if the subscripted letters and dots could be darkened a little more, they are smaller and not as visible. Of course, we don't want a big difference between them and the line markings, but some difference might be okay.

Concerning Case 3: Will anyone notice the lack of dots between a_n and a_∞? Unfortunately, the answer is probably yes. The article will be nominated for Good Article again and if you read Talk:Cantor's first uncountability proof#GA Review, you'll see that the editors seem to notice a lot. I'm using their feedback in my rewrite (one of them noticed a lot of flaws in the old article).

I've come up with a suggestion based on measuring some distances. My measurements are from a printout and may not be the same as in Latex, but you can compute the ratio you need to go from my measurements to Latex.

I measured the distance the dots use from a_n: it's 6 mm. I also measured the distance from a_∞ to y: it's 14 mm. And the distance from y to b_∞ is 8 mm. So if you move a_∞ and b_∞ to the right by 6 mm, it looks to me like you'll have enough room for the dots between a_n and a_∞. This flips the distances from y to a_∞ and b_∞ so it's a mirror reflection that preserves the asymmetry in the diagram.

Now there's not quite enough room for the dots between b_∞ and b_n, but its only off by about 2 mm. On this side of the diagram, there's enough room to move b_n over by 2 mm or even a bit more (also there's room to move x_n if you want move it). My measurements may be a bit off but the basic idea is that there's more room on the right side of the diagram.

Also, thank you for telling me about your edit experiments with the proof. I'm happy to hear that you experiment with proofs. I do it myself. It's a great way to understand a proof.

Right now my plans are to finish the rewrite, then go back to each section, and see they can be improved. I'll be looking at your edits then. I do get ideas from how people react to the proof. For example, the idea of diagrams for the three cases came from a problem an editor mentioned in the old proof. In Case 1, we have a finite interval and have to avoid x_n. The editor wondered if a reader might think that there could be just one point in the finite interval and then you couldn't you avoid it. Another editor suggested that the reader should be reminded that there are infinitely many points in the interval. I wrote this suggestion up, but then realized that geometrically, there's no question that you can easily avoid x_n. This one remark led me to my ASCII diagrams, which you evolved into far better diagrams that even capture the dynamics of the limit process.

You did guess correctly that I'm trying to stay close to Cantor's proof, but I have managed to make two changes already: Using only open intervals, which simplified the theorem's proof and the other proofs in the section. Also, I labeled Cantor's three cases (Cantor broke the argument into two cases, handled the first case, and then broke the second case into two cases; an editor found this confusing).

I really enjoy working with you and didn't mind the delay. My optimistic time frame has me working for at least 3-4 weeks more (it will most likely take me longer). Then I'll try to get an editor or two to look it over before the rewrite gets posted for everyone to see. So there's plenty of time. RJGray (talk) 20:36, 25 September 2015 (UTC)

October 21

Hi Jochen, Well I've finally finished my first version of my first draft I've updated "The Proofs" page. Any comments are welcome. I'm now going to start looking over the article from the beginning and look over your proof suggestions. But first, I'm going to take a week's break from the computer. Thanks for all your work, --RJGray (talk) 15:52, 21 October 2015 (UTC)

Hi Robert, I adapted the pictures in the pdf version; the svg version will follow on Monday. Sorry for the new delay; I didn't find time earlier. Best regards, Jochen Burghardt (talk) 12:34, 6 November 2015 (UTC)

Adapted svg version, too. - Jochen Burghardt (talk) 09:55, 9 November 2015 (UTC)

November 21

Hi Jochen, The new pictures look great! I really appreciate all the work you've done on them. I'm now finally done with my second draft. I looked over your work on my proof and the comments following it, and it has helped. The proof now has 3 parts: Generating the nested intervals; dividing into cases; handling the cases. Cantor had mixed the last two together, which was fine for his audience of research mathematicians. But Wikipedia articles need to have clearer proofs. Your pictures are a big help here. I made the ends of the proofs clearer: Both the second theorem proof and the dense sequence proof end in the paragraph after the case handling. I also took out some of my comments—too many comments can be confusing for the reader. It's as if I'm saying: "Look at this, look at that, etc."

As for time delays, they are fine with me. As you can see, it took me longer than expected to get this draft out. I'm not that far from my final draft; I'm sending this one to an editor for feedback. Also, any feedback you have is appreciated. --RJGray (talk) 02:44, 22 November 2015 (UTC)

Hi Robert, I've put a copy into User:Jochen Burghardt/sandbox1 and started inserting suggestions and comments (by abuse of the ((clarify)) template) as I read it. I stopped right before section "The disagreement about Cantor's proof", where I intend to continue in the next days. Also, I didn't read all details in the parts "(Appropriate heading1)" and "(Appropriate heading2)" (subsectioning suggestions made by me) in section "The proofs". Maybe, some comments or suggestions are useful; but feel free to ignore any of them, as I just wrote down what came to my mind, without thinking too much about it. - Jochen Burghardt (talk) 22:18, 22 November 2015 (UTC)

Today, I've complete reading the article. Again, my comments are just brainstorming results. - Jochen Burghardt (talk) 21:08, 3 December 2015 (UTC)

December 13

Hi Jochen, Thank you for all your suggestions. Your subsection suggestion was great! I only had to move the paragraph that mentions the computer program that uses the two proofs. I haven't went through all your suggestions yet because I got bogged down in a rewrite of the example in "The Proofs" section. You can see that I simplified the table using color, which is an idea I got from your work on the case diagrams. I rewrote the proof that x_n ∉ (a_n,b_n) about 5 different ways until I found an approach that gets both the best g(n) function and has the shortest math induction proof so I could remove the hide/show box. I'll be looking at your other suggestions in a while. I'm getting a bit tired of working on the article and need a short break. I now think it may take me until January to finish it. Thank you! --RJGray (talk) 23:30, 13 December 2015 (UTC)

December 19

Hi Jochen, Decided to delay my short break. I wanted to see if somehow I could move your work on Cantor's enumeration of the real algebraic numbers into the First Theorem subsection. Just taking away the <ref> and </ref> and "showing" it caused several pages of table to appear and spill over into other subsections, which I suspected would not please the Good Article editors.

So I tried to shorten it. I wanted readers to get a feel for the enumeration with the shortest possible table. It seemed to me that the essence of the enumeration is the 19 real algebraic numbers you calculated, their associated polynomials, and the polynomial heights. After several attempts in different formats, I chose a wikitable because it cleanly expands the subsection without going into other subsections. Also, I was able to keep it to a bit less than one page on my screen (I use a medium size font). I hope you find my work acceptable.

With the smaller wikitable, readers may spot patterns in the enumeration. For example, there are mirror symmetries with a sign change for the height 3 numbers (reflect at the x₅–x₆ boundary), and for the height 4 numbers if you reflect at the x₁₃–x₁₄ boundary. Of course, the height 2 numbers also have this symmetry. I was able to prove this always happens, and I can prove how the polynomial associated with the number will change. I first attempted a complicated approach to prove it, but it turned out the proof is fairly simple. I also took care of a few more of your "clarifications needed". Thanks for taking the time to put them in.RJGray (talk) 16:59, 19 December 2015 (UTC)

January 11

Hi Jochen, I'm finally happy enough with the article to send it to the editor I'm working with. He'll post it when he considers it ready. I've changed the name of the article since it goes beyond Cantor's first uncountability proof. It's now "Cantor's first set theory" article, which will require a redirect from its old name. Thanks again for all your help. I'll be acknowledging your help in a Thank you that will appear in the Talk section of the article. Thanks, --RJGray (talk) 15:38, 11 January 2016 (UTC)