User talk:RJGray/Sandbox

Rewrite of "Cantor's first …" article is done

Hi Michael,

I've finished my final draft of the article, it's located at User:RJGray/Cantor draft3. It took me longer than expected partly because I keep seeing ways to improve my writing, and partly because I realized that the article had a big flaw: Namely, the article's title was "Cantor's first uncountability proof," but the last four sections had little or nothing to do with the proof. The last three section titles—"The influence of Weierstrass and Kronecker on Cantor's article," "Dedekind's contributions to Cantor's article," "The legacy of Cantor's article"—led me to realize that I've written a fairly comprehensive article about Cantor's article. I also realized that the editors that participate in the Good Article review may also think that the old name doesn't reflect the contents of the new article.

So I've changed its name to "Cantor's first set theory article" and rewrote the lead. Cantor's article is a well-known, often-cited, and much-discussed article so I think it deserves a Wikipedia article. I realize the change of title requires a redirect from the old title. I boldfaced the old article title Cantor's first uncountability proof in the first paragraph of the lead so that redirected readers will know they're at the right place.

The article is ready to post as soon as you are comfortable posting it. Please send me any recommendations you have for improving it. I just have one request about posting. Please tell me when you are going to post it. I have written a Thank You for the people who have helped me with the article. I would like to post it to the article's Talk page soon after you post the article.

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The article rewrite and thanks to all those who helped me

It's been challenging rewriting the "Cantor's first …" article because it's listed in the categories: History of mathematics, Set theory, Real analysis, Georg Cantor. So I had to consider both the math and the math history audiences. I did this by writing the article so all the math in Cantor's article appears in the first two sections, which is followed by a "Development" section that acts as a bridge from the math sections to the math history sections. I changed the title to "Cantor's first set theory article" to reflect its content better; actually, the old article could have used this title. It's a well-known, often-cited, and much-discussed article so I suspect the Wikipedia article will attract a number of readers.

I would like to thank SpinningSpark for his excellent critique of the old article. I really appreciate the time and thought he put into it. The new article owes a lot to SpinningSpark. His detailed section-by-section list of flaws was extremely helpful. I used this list and his suggestions to restructure and rewrite the article. I particularly liked his comment on whether the disagreement about Cantor's proof of the existence of transcendentals "has been a decades long dispute with neither side ever realising that they were not talking about the same proof." The lead now points out this disagreement has been around at least since 1930 and still seems to be unresolved. It was a major flaw of the old article that the longevity of the disagreement was never mentioned. I find it ironic that this disagreement is still around, while most mathematicians now accept transfinite (infinite) sets so the old dispute about the validity of these sets is mostly resolved.

I also thank JohnBlackburn for his comments. His comments that made me realize that I should think of the readers who just want to understand the math in Cantor's proof. This led to the restructuring mentioned above in the first paragraph. His comments also led me to put the long footnotes containing math proofs into the text. I also added some more math to the article.

I thank Jochen Burghardt for his help on the rewrite. He did the case diagrams for the proof of Cantor's second theorem, the subsectioning of "The Proof" section, the calculations in the table "Cantor's enumeration of the real algebraic numbers", and he pointed out places where my writing was unclear. The need for the case diagrams came from reading SpinningSpark's comment on what is now Case 1. I realized that a reader's possible confusion on whether there is point in the finite interval (a_N, b_N) besides x_n could be handled with a diagram. I contacted Jochen with three simple ASCII diagrams. He took my simplistic diagrams and produced diagrams that capture the dynamics of the limiting process.

I thank my daughter Kristen who read a recent draft and made a number of suggestions that improved the writing. Especially important were her suggestions on improving the lead.

I also thank those who edited the old article. I started with a copy of the old article and have kept up with recent edits so that your edits would be preserved (except perhaps in the parts of the article where large changes were made).

Finally, I wish to thank Michael Hardy for his GA nomination for the old article, for giving me the go-ahead for the rewrite, and for his patience with the amount of time it has taken me to do the rewrite. I hope that this rewrite is much closer to GA standards than the old article.

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Have Cantor article translation online, but need guidance with Wikisource

Hi Michael, I would like to have an online English translation of Cantor's article available to readers. I have produced an online English translation of Cantor's article that comes from the 1883 French translation that he reviewed and corrected. It's located at Wikisource:User:RJGray/On a Property of the Set of All Real Algebraic Numbers. I translated the French translation of the article because my German translation was too close to a copyrighted translation (it seems that in translating math, translations are likely to be very similar). Do you know anyone at Wikisource who can help me make it an acceptable translation? Wikisource has a large number of rules including getting scans of the original article, having everything proofread by someone, etc.

Article title

I've looked more into "article" vs. "paper" and am I now neutral on the issue. But we can deal with this later. I do think it would be good to consult the readers to see if they think that a general change of "article" to "paper" throughout the entire Wikipedia article is a good idea.

Let's talk about Wikipedia:Article titles and how your title compares to the existing title. A good Wikipedia article title has the five following characteristics:

• Recognizability – The title is a name or description of the subject that someone familiar with, although not necessarily an expert in, the subject area will recognize. Your suggested title On a Property of the Collection of All Real Algebraic Numbers is unfortunately not recognizable even to students taking set theory unless they have read a historical work that discusses the reason for this strange name. Georg Cantor's first set theory article is recognizable because it's talking about Cantor's work and, in particular, his first article on set theory.
• Naturalness – The title is one that readers are likely to look or search for and that editors would naturally use to link to the article from other articles. Such a title usually conveys what the subject is actually called in English. I don't think that readers will look or search for On a Property of the Collection of All Real Algebraic Numbers while they will search for Cantor's first set theory article since they are likely to start typing "Cantor" and unlikely to start typing "On a Property." Also, since Cantor's first set theory article is a natural shortening of the current title, there is no need to boldface Cantor's first set theory article in the text.
• Precision – The title unambiguously identifies the article's subject and distinguishes it from other subjects. On a Property of the Collection of All Real Algebraic Numbers does not even identify the property that is discussed in Cantor's article. It certainly doesn't capture the Wikipedia article's subject.
• Conciseness – The title is no longer than necessary to identify the article's subject and distinguish it from other subjects. Georg Cantor's first set theory article is shorter than On a Property of the Collection of All Real Algebraic Numbers
• Consistency – The title is consistent with the pattern of similar articles' titles. There are other titles with "Cantor's ..." in it, while your title shares consistency with the Wikipedia article on the Gödel paper.

I also decided to see what effect your suggested title and a redirect would have on the lead (I removed the refs):

On a Property of the Collection of All Real Algebraic Numbers is Georg Cantor's first set theory article. It was published in 1874 and contains the first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is "Cantor's revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the paper, "On a Property of the Collection of All Real Algebraic Numbers," refers to its first theorem: the set of real algebraic numbers is countable.

One problem with your suggested title is that readers who are redirected from Cantor's first uncountability proof may get confused when redirected to a Wikipedia article titled On a Property of the Collection of All Real Algebraic Numbers. The current redirect to Georg Cantor's first set theory article is less confusing because of the "Cantor's first" in the title and because the 2nd sentence in the current article talks about uncountably infinite and the next sentence has Cantor's first uncountability proof in it.

Also, the first sentence in the modified lead is only necessary because of the suggested title change, which also requires the boldfaced Cantor's first set theory article to handle the redirect. I believe quicker leads are better because they entice users to read the article. Also, I wrote the original lead to relegate the obscure title of Cantor's paper to the bottom of the paragraph since the Wikipedia article doesn't devote much space on the countability of the real algebraic numbers. (I deal with the title more in the section "The influence of Weierstrass and Kronecker on Cantor's article.")

I guess my feeling is Cantor got stuck with a poor title for his paper. I don't think we need to get stuck with the same obscure title.

Still prefer

Although I still prefer the descriptive title "Cantor's first set theory article," I'm beginning to understand why some prefer the title On a Property of the Collection of All Real Algebraic Numbers. However, I'm still worried about readers who may find the title confusing, especially those who are redirected from "Cantor's first uncountability proof." I can understand that Wikipedia may have an explicit rule about naming articles after their titles. Could someone direct me to this rule?

I've realized my argument that the descriptive "Cantor's first set theory article" captures the Wikipedia article while the title doesn't is because I get confused. I regarded On a Property of the Collection of All Real Algebraic Numbers as not even capturing Cantor's article so how can it capture the Wikipedia article? I regard "Cantor's first set theory article" as capturing the Wikipedia article because everything in the Wikipedia article grew out of Cantor's article. My problem was that I read Cantor's title literally as being about one theorem in his article. So I wasn't taking it as synonymous to "Cantor's first set theory article".

If the name of the Wikipedia article does change, I felt that it would be nice to have a new first paragraph to handle this. I appreciate Trovatore's point about putting the journal name in the first paragraph. As you can see below, it's in the beginning of the second sentence. The first sentence takes care of the redirect from "Cantor's first set theory article", giving the author's name, and describing the mathematical area of the article, and stating that it's Cantor's first article on set theory. I now regard my leaving out "Crelle's Journal" from my original lead as a oversight on my part. That's one thing I like about Wikipedia, people spot oversights like this.

On a Property of the Collection of All Real Algebraic Numbers is Georg Cantor's first set theory article. It was published in 1874 in Crelle's Journal and contains the first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is "Cantor's revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article refers to its first theorem: the set of real algebraic numbers is countable.

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