Talk:Cantor's first set theory article

The rationals

Copied from the main article:

(Could someone who understands explain why the set of rational numbers does not have property 4?)

Property 4 says that if you partition the set into two halves, then there must be a boundary point in the set. This is not true for the rationals: take as A the set of all rationals smaller than √2 and as B the set of all rational above √2. Then all rationals are covered, since √2 is irrational, so this is a valid partition. There is no boundary point in the set of rational numbers that separates A from B however. AxelBoldt 02:09, 23 May 2006 (UTC)

• Ok, but could you clarify a little please... in as much as if you have your boundry, and A contains the elements less that that boundry, and B the elements greater than it, the the boundry is not in A or B. Probably missing something here, just can't see what.
  • That isn't a partition. If c is in R, then for {A,B} to be a partition of R, c needs to be in A or in B. Eg, for property 4, c would have to be either the largest member of A or the smallest member B. Aij (talk) 02:13, 15 April 2008 (UTC)

@AxelBoldt - In your example, why can't the boundary point be the largest rational in A or the smallest rational in B? Then, all rationals less than the boundary will be in A and all rationals greater than the boundary will be in B. Or am I misunderstanding the meaning of the word "every point" as "every point in R"? Vijay (talk) 08:36, 11 January 2010 (UTC)

Axel may not be watching anymore — he made that comment in 2006.

Anyway, for A and B as defined, A doesn't have a largest element, and B doesn't have a smallest element. --Trovatore (talk) 09:40, 11 January 2010 (UTC)

There's an explicit exercise in Walter Rudin's Principles of Mathematical Analysis that asks the student to show for any rational number less than √2 how to find a larger rational number that is still less than √2, and similarly for those larger than √2. Michael Hardy (talk) 02:04, 24 January 2010 (UTC)

For $a<{\sqrt {2}}$, one possible choice would be $a+1/\lceil 1/({\sqrt {2}}-a)\rceil$. Paradoctor (talk) 13:27, 16 December 2013 (UTC)

Proposed Changes to Article

I am very happy to see a Wikipedia article about Cantor's first uncountability proof. Since I have studied Cantor's 1874 article and some of his correspondence, I started adding material and making some changes. The result of this work can be found at: Talk:Cantor's first uncountability proof/Temp. I hope you find my revisions interesting and relevant. I'm looking forward to your suggestions, modifications, and feedback. Here's a section-by-section summary of my revisions:

Introduction: Made some changes and mentioned two controversies that have developed around Cantor's article. The "emphasis" controversy ("Why does Cantor's article emphasize the countability of the set of real algebraic numbers?") is already discussed in the current article. The "constructive/non-constructive" controversy concerns Cantor's proof of the existence of transcendental numbers.

Development and Publication: Expanded the current "Publication" section by adding material that comes mostly from Cantor's correspondence. Like the current section, this new section discusses the "emphasis" controversy, but I did add some material here.

The Article: Replaces the current "The theorem" section. Contains statements of the theorems that Cantor proves in his article. Also, uses Cantor's description of his article to bring out the article's structure. This structure is the key to handling the "constructive/non-constructive" controversy.

The Proofs: Contains proofs of Cantor's theorems.

Cantor’s Method of Constructing Transcendental Numbers: Replaces the current "Real algebraic numbers and real transcendental numbers" section. Also, discusses the "constructive/non-constructive" controversy.

I have also added a "Notes" section, and I have added references to the current "References" section.

I highly recommend reading Cantor's original article, which is at: "Über eine Eigenschaft des Ingebriffes aller reelen algebraischen Zahlen". A French translation (which was reviewed and corrected by Cantor) is at: "Sur une propriété du système de tous les nombres algébriques réels". Unfortunately, I have not found an English translation on-line. However, an English translation is in: Volume 2 of Ewald's From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics (ISBN 9780198532712).

Most of the material I added to this Wikipedia article comes from Cantor's article, Cantor's correspondence, Dauben's biography of Cantor (ISBN 0674348710), and the article "Georg Cantor and Transcendental Numbers".

Finally, I wish to thank all the people who have worked on this Wikipedia article. Without the excellent structuring of your article and the topics you chose to cover, I suspect that I would not have written anything. (This is the first time I've written for Wikipedia.) It's much easier to add and revise rather than develop from scratch. RJGray (talk) 23:30, 5 May 2009 (UTC)

Rewrote the section "Cantor’s method of constructing transcendental numbers" and renamed it "Is Cantor’s proof of the existence of transcendentals constructive or non-constructive?" The old section did not explain this constructive/non-constructive controversy. The new section quotes mathematicians on different sides of the controversy, analyzes their versions of "Cantor's proof," looks at relevant letters of Cantor's, mentions some computer programs, and then shows Cantor's diagonal method in a simpler context -- namely, generating the digits of an irrational (rather than the more difficult job of generating the digits of a transcendental).

--RJGray (talk) 02:55, 5 August 2009 (UTC)

Oops, I forgot to thank Michael Hardy for the feedback that he has given me on my proposed changes. His feedback made me realize that my old section was inadequate. I hope that my new section is more adequate -- I welcome your feedback on it. --RJGray (talk) 03:11, 5 August 2009 (UTC)

Revisions to proposed changes. I have added more material and restructured my proposed changes. The revised text contains the following sections:

• The article

• The proofs

• Is Cantor’s proof of the existence of transcendentals constructive or non-constructive?

• The development of Cantor's ideas

• Why does Cantor's article emphasize the countability of the algebraic numbers?

The biggest changes are the ordering of the sections, and the last two sections. Now the two mathematical sections come first. This was done for several reasons: Since the introduction is about the mathematics, it's natural that the first sections should be mathematical. Also, these two sections prepare the way for the other sections.

The last two sections are a rewrite of the old section: "Development and publication." This rewrite was necessary because I learned of the book: Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought by José Ferreirós. Ferreirós has a different point of view than Joseph Dauben on who influenced Cantor's article. Hence, I felt that Wikipedia's NPOV policy required that I talk about both Dauben's and Ferreirós' opinions.

Finally, various smaller edits appear in the other sections. I welcome your feedback. --RJGray (talk) 01:54, 23 January 2010 (UTC)

I've moved RJGray's draft to the article space and merged its edit history with that of the article as it appeared before. Michael Hardy (talk) 03:54, 23 January 2010 (UTC)

B class

I am going to change the math rating to B class. Here are my specific thoughts about ways the article could be improved:

• There is an obvious relationship between Cantor's proof and the Baire category theorem: the BCT follows immediately by the same proof technique, and the BCT proves Cantor's theorem as a corollary. Somebody must have discussed this in print.
• Is the claim about certain processes requiring sub-exponential time in the source by Gray? I scanned through the reference, but didn't see it.
• In the paragraph beginning "The constructive nature of Cantor's work is most easily demonstrated by using it to construct an irrational number. " — isn't this using the diagonal method rather than the method of Cantor's first proof? Why not make an example that uses the method of the first proof.

I'll read through the article again today to copyedit again. — Carl (CBM · talk) 13:09, 24 January 2010 (UTC)

Thank you very much for your feedback:

• Concerning the relationship between Cantor's proof and the Baire category theorem: I regard the current article as mostly historical and Baire proved his theorem in 1899. Also, the versions of the Baire category theorem as stated at Baire category theorem require some form of the axiom of choice, which Cantor's methods do not need. So I suspect you are talking about a weaker form of the Baire category theorem. Perhaps a note could be added about the relationship between Cantor's 1874 method and the proof of the Baire category theorem if a source could be located.
• Sorry, I left out some references. I have added references to the locations in Gray 1994 where the computer program times are mentioned. (The sub-exponential time is at bottom p. 822 - top p. 823.)
• The diagonal method was used because it is simpler and the idea was just to demonstrate the constructive nature of Cantor's work. In this section, both of Cantor's methods are mentioned so I felt free to use the simplest method. Using Cantor's 1874 method gives the intervals [1/3, 1/2], [2/5, 3/7], [7/17, 5/12], … or in decimals [.33…, .50…], [.400…, 428…], [.4117…, .4166…], … It seems to me that the number generated by the diagonal method is more easily seen to be irrational than the number generated by the 1874 method. I'd like some feedback from other readers before changing methods. Of course, both methods could be illustrated.
• As for the class rating, I'll let the experts on class ratings discuss this. By the way, could you give me a Wiki reference to the definitions of each rating?

—RJGray (talk) 21:34, 24 January 2010 (UTC)

By Baire category theorem I mean: the intersection of a sequence of dense open sets in the real line is dense. This fact does not require the axiom of choice; the proof is completely effective. In particular, if the sequence U_n of dense open sets is computable, then there is a computable function that takes a rational interval [a,b] as input and returns a real in $[a,b]\cap \bigcap _{n}U_{n}$. The axiom of dependent choice is only needed to prove the version of BCT for non-separable complete metric spaces.

A description of the recommendations for math article assessments is at Wikipedia:WikiProject_Mathematics/Wikipedia_1.0/Grading_scheme. However, the "A" class is in limbo right now: there was a system set up to try to review articles before they were rated A class, but that system never caught on, and now it is defunct. — Carl (CBM · talk) 00:23, 25 January 2010 (UTC)

On Feb. 20, I followed your suggestion of having an example of generating an irrational number by using Cantor's 1874 method. This follows the example of generating an irrational number by using Cantor's diagonal method. — RJGray (talk) 01:19, 3 March 2010 (UTC)

Restrict polynomials to irreducible ones in proof of countability of algebraic numbers?

As far as I understood Cantor's 1874 article, he considers in his proof of countability of algebraic numbers only irreducible polynomials (p.258: "und die Gleichung (1.) irreducibel denken" = "and consider equation (1.) to be irreducible"). These are sufficient to get all algebraic numbers, and each of them corresponds to at most one algebraic number, viz. its root (if in ℝ). In this setting it is more clear what it means to "order the real roots of polynomials of the same height by numeric order" (cited from Cantor's first uncountability proof#The proofs). Maybe the article should also restrict polynomials to irreducible ones - ?

Jochen Burghardt (talk) 15:10, 13 December 2013 (UTC)

I made a list of algebraic numbers, ordered by Cantor's rank, as a collapsible table. Maybe it is illustrative to include it (in collapsed form) into the article. At least I my self learned (1) that "irreducible" should mean "cannot be written as product of smaller polynomials with integer coefficients", and (2) an irreducible polynomial in that sense can well have several solutions; two facts that I should have remebered from my school time. - Jochen Burghardt (talk) 16:40, 13 December 2013 (UTC)

Cantor's enumeration of algebraic numbers
Height 1:
			1x	+0	= 0		x1 = 0
Height 2:
			2x	+0	= 0		reducible
			1x	+1	= 0		x2 = −1
			1x	−1	= 0		x3 = +1
		1x2	+0x	+0	= 0		reducible
Height 3:
			3x	+0	= 0		reducible
			2x	+1	= 0		x5 = −1/2
			2x	−1	= 0		x6 = +1/2
			1x	+2	= 0		x4 = −2
			1x	−2	= 0		x7 = +2
		2x2	+0x	+0	= 0		reducible
		1x2	+1x	+0	= 0		reducible
		1x2	−1x	+0	= 0		reducible
		1x2	+0x	+1	= 0		no real root
		1x2	+0x	−1	= 0		reducible
	1x3	+0x2	+0x	+0	= 0		reducible
Height 4:
			4x	+0	= 0		reducible
			3x	+1	= 0		x13 = −1/3
			3x	−1	= 0		x14 = +1/3
			2x	+2	= 0		reducible
			2x	−2	= 0		reducible
			1x	+3	= 0		x8 = −3
			1x	−3	= 0		x19 = +3
		3x2	+0x	+0	= 0		reducible
		2x2	+1x	+0	= 0		reducible
		2x2	−1x	+0	= 0		reducible
		2x2	+0x	+1	= 0		no real root
		2x2	+0x	−1	= 0		x16, x11 = ±1/√2
		1x2	+2x	+0	= 0		reducible
		1x2	−2x	+0	= 0		reducible
		1x2	+1x	+1	= 0		no real root
		1x2	+1x	−1	= 0		x15, x9 = (−1 ± √5) / 2
		1x2	−1x	+1	= 0		no real root
		1x2	−1x	−1	= 0		x18, x12 = (+1 ± √5) / 2
		1x2	+0x	+2	= 0		no real root
		1x2	+0x	−2	= 0		x17, x10 = ±√2
	2x3	+0x2	+0x	+0	= 0		reducible
	1x3	+1x2	+0x	+0	= 0		reducible
	1x3	−1x2	+0x	+0	= 0		reducible
	1x3	+0x2	+1x	+0	= 0		reducible
	1x3	+0x2	−1x	+0	= 0		reducible
	1x3	+0x2	+0x	+1	= 0		reducible
	1x3	+0x2	+0x	−1	= 0		reducible
1x4	+0x3	+0x2	+0x	+0	= 0		reducible
Height 5:
			5x	+0	= 0		reducible
					:

When I wrote the section "The Proofs", my intent was to emphasize the proof of Cantor's second theorem, so I simplified his proof of the countability of algebraic numbers by leaving out "irreducible" so readers wouldn't have to know what an irreducible polynomial is. I'm sorry that you found my method less clear than Cantor's on the ordering of the algebraic numbers of a particular height. Using your enumeration table, the polynomials of height 2 give 0, -1, 1, 0 as roots, so the ordering will be -1, 0, 0, 1. In this enumeration, duplicates often appear within a height and between heights, but Cantor's proof of his second theorem does handle duplicates.

However, you do bring up an excellent question: Shall we mention Cantor's use of irreducible polynomials? I see two ways to mention it: Add it to the text or add a footnote at the end of the paragraph that points out the text's ordering produces duplicates and that Cantor's original enumeration eliminates duplicates by using irreducible polynomials. By the way, the reason for some of the longer footnotes in this article was to explain points in more depth—readers just wanting the main points can skip the footnotes. Which is better in this case? I don't know. Maybe some readers can give us feedback.

I like your enumeration table. A few suggestions: Label it "Cantor's enumeration of algebraic numbers". Change "not coprime" to "not irreducible". Coprime refers to a set of two or more integers so it doesn't apply to polynomials such as 2x. The definition of irreducible polynomial states that: "A polynomial with integer coefficients, or, more generally, with coefficients in a unique factorization domain F is said to be irreducible over F if it is not invertible nor zero and cannot be factored into the product of two non-invertible polynomials with coefficients in F." This definition factors: 2x = (2)(x) and it factors: 2x+2 = (2)(x+1). Finally, the exponent 1 in your table always appears in gray and it's well understood that "x" means "x¹". Try leaving out this exponent. I think this might visually simply your table. - RJGray (talk) 01:58, 15 December 2013 (UTC)

When I started this talk section, I had in mind the construction of rationals from integers, and I thought that algebraic numbers could be constructed from rationals in a similar way. The former is done by computing with pairs (p,q) ∈ ℤ×(ℤ\{0}) with the intended meaning p/q; I thought the latter could be done by computing with polynomials, where one polynomial would denote one algebraic number, "viz. its root". Meanwhile I saw that even an irreducible polynomial has several roots, so that there can't be a one-to-one correspondence between polynomials and algebraic numbers, anyway. So I lost my original motivation for asking for irreducibility. Probably the proof is simplest in its current form; maybe a footenote could be added as you suggested.

In the enumeration table, I tried to distinguish several reasons for excluding a polynomial, a non-coprime set of coefficients being one of them, non-irreducibility being another one (admittely subsuming the former); when changing the table to produce duplicates these reasons would disappear, anyway. I used the gray parts to indicate (to myself, in the first place) the systematic way the polynomials are enumerated (nevertheless, I missed all polynomials containing x³ and x⁴; see the new table; I hope it is complete now ...), but you are right: at least the exponent of "x¹" isn't needed for that; I now deleted it. Concerning duplicates: should we have a reason "repetition" (or "duplicate"?) and not assign them a number; or should we assign them a number and mention somewhere that the enumeration is not bijective, but surjective, which suffices for countability? The former case would save some indentation space, since the x⁴ column could be immediately adjacent to the leftmost (number) column, as in each row at least one of them is empty. The latter case wouldn't save much, as "(-1 ± √5) / 2" (to be kept) is about as long as "repetition". - Jochen Burghardt (talk) 12:38, 16 December 2013 (UTC)

I think some readers may find the current text ambiguous on the question of whether duplicates appear in the sequence (of course, it doesn't matter for applying his second theorem). There are two ways to eliminate duplicates and both give the same result. Below is my first attempt at a footnote to clarify the situation and to introduce readers to Cantor's approach and your table:

"Using this ordering and placing only the first occurrence of an real algebraic number in the sequence produces a sequence without duplicates. Cantor obtained the same sequence by using irreducible polynomials: INSERT YOUR TABLE HERE"

Your table is looking better, some more suggestions: remove the "·" in 2·x, etc. In the enumeration, you can use x₁ instead of "1.", etc. (This would connect your table closer to the article where all the sequences are x₁, x₂, ….) Also, in front of the first coefficient, you can leave out the "+" since every polynomial starts with a positive coefficient. Finally, concerning irreducible polynomials versus coprimes, I apologize for not being clearer. I should have quoted the following from "Irreducible polynomial":

"It is helpful to compare irreducible polynomials to prime numbers: prime numbers (together with the corresponding negative numbers of equal magnitude) are the irreducible integers. They exhibit many of the general properties of the concept of 'irreducibility' that equally apply to irreducible polynomials, such as the essentially unique factorization into prime or irreducible factors:"

This means that you factor 6x = (2)(3)(x). Basically, the terms to use when working with factoring polynomials are "reducible" and "irreducible" (they are the counterparts to "composite" and "prime"). I think that you may be generalizing the term coprime to single integers to handle polynomials, such as 3x, when you call this polynomial "not coprime". I've done a Google search and I only found the term "coprime" referring two or more integers. So I think your table would be more accurate and clearer if you used the term "not irreducible". Also, I have the philosophy of placing minimal demands on the reader (whenever possible). By only using the word "irreducible", the reader is not required to understand "coprime".

I hope you don't mind all my suggestions (I can be a bit of a perfectionist when it comes to tables). I think your table is an excellent addition to the article and will definitely help readers understand the ordering. In fact, it motivated me to reread Cantor's article and I noticed a detail that I had forgotten: Cantor gives the number of algebraic reals of heights 1, 2, and 3, which (of course) agree with your table. --RJGray (talk) 18:20, 17 December 2013 (UTC)

I changed the table according to your suggestions (perfectionism in writing optimizes the overall workload, since the table is written only once, but read -hopefully- a lot of times). Maybe the indices like in x₃ should not be in boldface? And: are you sure that no algebraic number may occur as root of two different irreducible polynomials? I've forgotten almost all my algebra knowledge... - Jochen Burghardt (talk) 20:40, 17 December 2013 (UTC)

I like your attitude about perfectionism—I agree, we should think about the reader's workload. I also like the way you nicely simplified the table to have just 2 columns, by putting using "x_n =" with the roots. I think that x₃ is preferable to x₃ because the text doesn't use boldface and it looks better. Some other suggestions: I found double indexing "x_11,16" confusing. Try "x₁₁, x₁₆" or, perhaps better, "x₁₆, x₁₁" to match the way that the + of the ± goes with x₁₆, and the – goes with x₁₁ (or maybe there's a minus-plus symbol with minus on top of the plus). Also, I see no need for the large space between the "x_n =" and the roots at the top of the table. I can see you're lining up with the roots at the bottom of the table, but on a first reading, many users may not go to the bottom of the table and may wonder about the space. Finally, try moving the "…" over a bit at the end of the table.

Your question about the possibility of an algebraic number occurring as the root of two different irreducible polynomials is very relevant. At the site: Algebraic Number (Encyclopedia of Math), you can read about the minimal polynomial of an algebraic number. This minimal polynomial is the polynomial of least degree that has α as a root, has rational coefficients, and first coefficient 1. It is irreducible. By multiplying by the least common denominator of all its coefficients, you obtain α's irreducible polynomial with integer coefficients that Cantor uses. The minimal polynomial Φ(x) of the algebraic number α can be easily shown to be a factor of any polynomial p(x) with rational coefficients that has root α. You start by dividing p(x) by Φ(x) using long division. This gives: p(x) = q(x) Φ(x) + r(x) where deg(r(x)) < deg(Φ(x)). Assume r(x) ≠ 0. Since p(α) = Φ(α) = 0, we then have r(α) = 0 which contradicts the fact that the minimal polynomial Φ(x) is the polynomial of least degree with root α. So r(x) must be 0. Therefore: p(x) = q(x) Φ(x) so the minimal polynomial is a factor of p(x). --RJGray (talk) 20:32, 18 December 2013 (UTC)

I didn't have web access during xmas holidays, but now I updated the table according to your recent suggestions. There is a "∓" symbol, but I think it looks unusual in an expression, so I instead changed the order of the lhs variables. I moved the final dots into the "=" column and simulated vertical dots by a colon, as I couldn't find an appropriate symbol or template.

I like your suggestion for a footnote containing our table. As you are currently editing the article anyway, would you insert your footnote and move the table? Maybe it is best to remove it from the talk page, to avoid confusion about where to do possible later table edits.

Last not least: Thank you for your explanation why there is only one minimal irreducible polynomial for an algebraic number; it helped me to bring back my memories about algebra. - Jochen Burghardt (talk) 14:09, 27 December 2013 (UTC)

Sorry to be so slow in getting back to you. I've been busy and haven't watching my Watchlist. I see that you've already made the necessary changes, which is great--you deserve the credit. I think that the way you improved your table is much better than my suggestion. Keep up your excellent Wikipedia work! --RJGray (talk) 18:36, 8 April 2014 (UTC)

Contrast 2nd theorem with sequence of rational numbers?

Cantor's 2nd theorem seems obvious at first glance to many people, as we usually are unable to imagine a sequence that could completely fill a whole interval. However, there are sequences (like that of all positive rational numbers) whose set of accumulation points equals a whole interval (or even whole ℝ₊; cf. the picture there). Mentioning this in the article might prevent novice readers from thinking "Mathematicians make a big fuzz proving things that are obvious, anyway", and might generally help to sharpen one's intuition about what a sequence can do in relation to an interval and what it cannot. It would require, however, to explain the notion of an accumulation point (which is poorly represented in English Wikipedia in general). - Jochen Burghardt (talk) 11:52, 17 December 2013 (UTC)

Title containing "article"

This looks like a fascinating (Wikipedia) article, and I'm looking forward to reading it in detail.

I'm not too convinced by the title, though. I think it's more usual to refer to such contributions as "papers" rather than "articles". To me "article" sounds like something you find in a magazine, not a journal. Also, as I alluded to above, it's a bit problematic that "article" also is used to refer to Wikipedia articles, and there's possible interference from that, for editors discussing the Wikipedia article, but more importantly also for readers.

We could move it to Georg Cantor's first set theory paper, but to be honest I would rather move it to the actual title, On a Property of the Collection of All Real Algebraic Numbers, and put it in italics. I think in general we write articles on notable papers by their titles. See for example On Formally Undecidable Propositions of Principia Mathematica and Related Systems (not sure why it's not in italics; I think it should be). --Trovatore (talk) 03:38, 13 February 2016 (UTC)

Just an update on the italics issue — I went ahead and added {{italic title}} to the other article. But now I'm having second thoughts; there may be an argument for preserving a distinction between book titles, which are italicized, and titles of papers, which are not. I don't know. I think it looks better with italics; it makes it clearer that it's a title of a published work. --Trovatore (talk) 03:59, 13 February 2016 (UTC)

The article rewrite and thanks to all those who helped me

It's been challenging rewriting the "Cantor's first uncountability proof" 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. --RJGray (talk) 15:26, 13 February 2016 (UTC)

No need to merge article with Cantor's first uncountability proof -- already contains its content

There is no need to merge this article with Cantor's first uncountability proof because this is a rewrite of that article with guidance from the GA Review of that article. What is needed is a redirect from Cantor's first uncountability proof to this new article. Note that Cantor's first uncountability proof appears in boldface in first paragraph so readers will know that this article contains the content of that article. --RJGray (talk) 15:37, 13 February 2016 (UTC)

Ok, but in that case we needed to merge the histories to make it clearer that much of the content was rewritten from the older version. I have completed a history merge, so now the histories of both articles are in one place, and removed the merge tags. —David Eppstein (talk) 21:41, 13 February 2016 (UTC)

Requested move 13 February 2016

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: No move. This RM has been open for nearly two months, and it seems clear that there isn't consensus for the move. Cúchullain ^t/c 15:11, 4 April 2016 (UTC)

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Georg Cantor's first set theory article → On a Property of the Collection of All Real Algebraic NumbersOn a Property of the Collection of All Real Algebraic Numbers – Article is about this paper, so name it after the paper. See more elaborate remarks at the section #Title_containing_.22article.22 above. Trovatore (talk) 20:20, 13 February 2016 (UTC)

A few points to think about:

• The title of Cantor's article "On a Property of the Collection of All Real Algebraic Numbers" does not capture what the article is famous for--namely, the uncountability of the set of real numbers. Gödel's article "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I" does capture what the article is famous for. The reason for Cantor's choice of name is covered in the Wikipedia article.
• If we drop the "Georg" in the Wikipedia article's title, a reader who is interested in Cantor can type "Cantor" into the Search box and see the Wikipedia article along with other articles about Cantor's work. Being a reader who uses this feature of the Search box when I'm curious about a mathematician's or scientist's work, I think "Cantor's first set theory article" would be more reader-friendly than "On a Property of the Collection of All Real Algebraic Numbers."
• As far as "article" vs. "paper". See Difference Between Research Article and Research Paper. Two items of interest: "• There is a trend to refer to term papers and academic papers written by students in colleges as research papers whereas articles submitted by scholars and scientists with their groundbreaking research are termed as research articles. • Research articles are published in renowned scientific journals whereas papers written by students do not go to journals."--RJGray (talk) 21:35, 13 February 2016 (UTC)

1. As to your first point: Yes, that's true, but that's an interesting story in itself, which as far as I can see (but I haven't read the article carefully yet), the article does not currently discuss, but should probably be added. As I'm sure you know, Cantor is thought to have chosen this (somewhat inferior) title deliberately, to avoid a confrontation with Kronecker. Dauben goes into some detail on this point, I believe (don't have him to hand unfortunately). In any case I don't see that this should determine what this article should be called. This article is about the paper, and the paper has that name, and I really think that ought to settle it. Update — it looks like you say that the point about the reason for the title of Cantor's paper is covered in the Wikipedia article. I took a quick look, and I still don't see it; can you point me to it? I would expect it to be more prominent.
2. As to typing in the search box, it works for redirects too. Try it. There is in any case going to be a redirect from the current title, so I don't really see a problem.
3. I still strongly prefer "paper". To me a "paper" is more academic; an "article" is more likely to be for mass consumption. Also a "paper" is more likely to be a primary source, whereas an "article" is probably a secondary or tertiary source. Also the interference issue is real; the word "article" invites confusion with Wikipedia articles, whereas "paper" does not. But in any case we don't need to decide that in this RM as the proposed title does not contain either word. --Trovatore (talk) 21:46, 13 February 2016 (UTC)

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. If they do, I'd be happy to make the change.

Let's talk about Wikipedia:Article titles and how your title compares to the existing title. Wikipedia says that: 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. Here's the modified 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 Georg 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.--RJGray (talk) 00:43, 15 February 2016 (UTC)

The article is about a published work, so we go with the title of that work. I don't know any exception to that. If you had made it about the content, then there would be more options, but you made it about the paper itself, so I think there is really only one choice. --Trovatore (talk) 05:37, 15 February 2016 (UTC)

I am inclined to keep the present title. @Trovatore: Have you attempted to find out which articles on published works exist on Wikipedia and examine them to see whether some Wikipedia article titles differ from the titles of the published works? Michael Hardy (talk) 01:40, 16 February 2016 (UTC)

I'm not sure how I would really do that except by serially looking through articles on published works. I'm not aware of any exception.

Can we agree that, as article titles, descriptions are inferior to names, assuming a canonical name exists? I would think that's kind of obvious, actually. Sometimes there is no agreed name, and you have to fall back to a description, but that's an unfortunate necessity. But pretty much every published work has a name, namely its title, so I don't see any justification for titling this article with a description. --Trovatore (talk) 19:17, 16 February 2016 (UTC)

On suggested move:

On Formally Undecidable Propositions of Principia Mathematica and Related Systems

This article is about the paper. For theorems proved in this paper, see Gödel's incompleteness theorems.

I'm not sure what you're trying to say here. Are you saying there's a disanalogy because this article is (currently) more about the content than about the paper per se? If so, then the title should refer to the content, and not to the published work; it currently refers to the published work.

But really I don't think the content of the paper is a very natural topic for an article, given the divergent character of the two results. I think we should have an article about the paper, and I think it should be named after the paper, and it should spend more time on the paper per se than it currently does. (For example, currently, the article doesn't even seem to give the journal in which the paper was published, which I believe was Crelle's Journal.) --Trovatore (talk) 21:59, 16 February 2016 (UTC)

Sorry, I thought I was writing in one of my private files. I didn't realize until later that I wasn't. But you got part of the gist of what I planned to write about. I'm thinking that there should be two Wikipedia articles, similar to the way Gödel's work is divided into two parts. The article on Gödel's paper just covers material on the paper itself. I'd be happy to write a second article that would be similar in format to On Formally Undecidable Propositions of Principia Mathematica and Related Systems and would be called On a Property of the Collection of All Real Algebraic Numbers. It would include an outline of the Cantor's article, which by the way also contains an improvement of a theorem of Minnigerode that most coverage of the article leaves out (Dauben does mention it). It would also cover translations of the article, including the 1883 French translation and how that came about (and, of course, Ewald's English translation). It would also give the German name of the article and the journal in which it was published. By the way, in the current article, the section Georg Cantor's first set theory article#The influence of Weierstrass and Kronecker on Cantor's article contains the sentence: "Cantor would submit his article to Crelle's Journal."

I think one problem we are facing is that the current Wikipedia article goes far beyond Cantor's article: Starting with the Development section, it mentions 16 other mathematicians. So titling the article On a Property of the Collection of All Real Algebraic Numbers doesn't capture the article's content but would capture the content of the 2nd article I'm offering to write. The math history section of the current Wikipedia article covers a slice of math history that came about because of Cantor's article. For everyone reading this: I welcome suggestions as what to call the current Wikipedia article that would accurately capture both its math and math history content (I'm working on this myself). In the absence of title suggestions: What's wrong with having two articles, one titled On a Property of the Collection of All Real Algebraic Numbers and one titled Georg Cantor's first set theory article? RJGray (talk) 22:50, 16 February 2016 (UTC)

I believe strongly that a Wikipedia article should be about one clear thing, and that thing should be named in the title. If the title refers to the paper, then the article should be about the paper, and I think should be named after the paper. (We wouldn't put the article about Gone With the Wind at Margaret Mitchell's most famous novel, say.)

You seem to be saying this article should be about the content of the paper. I don't think that's a natural topic for an article, the two results being so divergent. I think the article is fine, but it should say more about the paper per se, and be named after the paper.

I did find the reference to Crelle's Journal, later, but it's too buried. If the article is about the paper, which I think it should be, then the journal and publication date should be named in the first paragraph, probably in the first sentence. --Trovatore (talk) 22:59, 16 February 2016 (UTC)

I, for one, support the move. As WP:TITLE says, "In Wikipedia, an article title is a natural language word or expression that indicates the subject of the article: as such the article title is usually the name of the person, or of the place, or of whatever else the topic of the article is." If the topic of the present article is Georg Cantor's first set theory article, then the article title almost surely should be the title of that article.

If that doesn't capture the scope of the article, then the article should be revised in one way or another. We can either abandon the idea that the article is about one of Cantor's papers, or we can move some of the material less relevant to Cantor's paper to, say, History of set theory. I don't see any need for the latter article to spring fully-formed from our foreheads; it could be started and left incomplete for now. Or, if the result would be too short, then I think it's fine for the present article to overgrow its proper scope. Eventually the relevant material can be reorganized. Ozob (talk) 00:25, 17 February 2016 (UTC)

Several more points:

• I can find no explicit Wikipedia rule that requires the name of an article be used. In fact, I've already discovered three examples where it is not used:
  • Wikipedia article title: Grothendieck's Tôhoku paper. Paper's name: "Sur quelques points d'algèbre homologique." (Mathematics)
  • Wikipedia article title: Alpher–Bethe–Gamow paper. Paper's name: "The Origin of Chemical Elements." (Physics)
  • Wikipedia article title: Lighthill report. Paper's name: "Artificial Intelligence: A General Survey." (Computer Science)
• WP:TITLE states: "A good Wikipedia article title has the five following characteristics: Recognizability … Naturalness … Precision … Conciseness … Consistency. I discussed these above and compared the current Wikipedia article title with the proposal to use the title of Cantor's article. No one has refuted my claim that the current title is better than the proposed title in the first four characteristics and in the last characteristic they are tied.
• Concerning the example: "We wouldn't put the article about Gone With the Wind at Margaret Mitchell's most famous novel, say." For me, Margaret Mitchell's most famous novel doesn't work because of Recognizability. I've seen the movie Gone With the Wind so I would recognize it. However, I've never read the book so I wouldn't recognize the author's name. In the case of On a Property of the Collection of All Real Algebraic Numbers: many (most?) mathematicians and students of mathematics would not recognize this title, but nearly all of them would recognize the author's name in Georg Cantor's first set theory article. They would also recognize the area of "set theory" and a good number could tell you his first article's most significant result: the uncountability of the set of real numbers. These are two of the problems with the proposed title: (1) It fails the WP:TITLE Recognizability characteristic: "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." (2) The title is unexpected and confusing since it doesn't mention the theorem the article is well-known for. (By the way, Margaret Mitchell's most famous novel is not an acceptable Wikipedia title because it uses the peacock word "famous"--see WP:PEACOCK).
• Concerning Wikipedia article content: I regard there as being two approaches to article content: A narrow article that only talks about what is in the math article versus a more comprehensive article that does this and puts it into historical context. In mathematics and the sciences, it is often of interest to understand what led to an article and what an article has led to. In the case of Gödel's article, the Wikipedia article "Gödel's incompleteness theorems" is the comprehensive article and On Formally Undecidable Propositions of Principia Mathematica and Related Systems is the narrow article (it only contains publication info, outline of the paper, and a section on translations of the paper). I was thinking of having the same division with current Wikipedia article title being the more comprehensive article and another Wikipedia article (whose title would be the title of Cantor's article) that would be very narrow. It was a compromise measure I was proposing, but it obviously got nowhere and I only confused people. I take it that everyone wants just one Wikipedia article (which I think is the best way to go). --RJGray (talk) 17:12, 19 February 2016 (UTC)

I really don't think there's a choice here. If the article is about the paper, which I think it should be, then it should be named after the paper. --Trovatore (talk) 17:49, 19 February 2016 (UTC)

I looked back and saw your examples. That is a point. Still, no one treats "Cantor's first set theory article" as a name of the paper (whereas for Alpher-Beta-Gamow they arguably do). The current title is a description; that's what offends me the most about it. Descriptions are the last choice for WP article titles. --Trovatore (talk) 17:52, 19 February 2016 (UTC)

A better example than the Alpher-Beta-Gamow paper is the Grothendieck paper. The title Grothendieck's Tôhoku paper is as descriptive as Cantor's first set theory article. The former is saying the Wikipedia article is about Grothendieck's paper that appeared in the Tôhoku Mathematical Journal, while the latter is saying the Wikipedia article is about Cantor's first article on set theory.

I think that descriptions for WP article titles should only be used if there are good reasons for their use. Usually, I would avoid using a description for a WP article title. However, as I pointed out above we are in an unusual case where Cantor was coaxed (or pressured) into choosing a title that has nothing to do with the revolutionary result the article is famous for. Hence, the descriptive title easily beats the title of Cantor's article as measured by the five characteristics of a good Wikipedia article title (WP:TITLE). We also need to think of readers who come to the Wikipedia article via various links; for example, at least 20 articles have links to "Cantor's first uncountability proof." Not all these readers will be well-versed in math since some of the links are in an easier section of an article. These readers will find themselves at an article with a long name — "On a Property of the Collection of All Real Algebraic Numbers" — that's talking about some unspecified property of the collection of real algebraic numbers. I don't find this reader-friendly at all and I wouldn't be surprised if some readers think they're not at the correct article and just leave.

Also, I like to think of the first sentence of an article as a welcome mat. Starting an article with "Georg Cantor's first set theory article" is far more welcoming.

Finally, some books completely avoid the name of Cantor's paper and use a description similar to the current Wikipedia article title. Here's a couple of examples: Hao Wang, Popular Lectures on Mathematical Logic, p. 119: "Cantor's first paper on set theory was published in 1874." Gerard Buskes and Arnoud van Rooij, Topological Spaces: From Distance to Neighborhood, page 112: "Cantor's first set theory paper was his 1874 paper on algebraic numbers." It seems that they think the article is about algebraic numbers (perhaps the title confused them), but they do go on to cover the article's theorem about the uncountability of the real numbers.

I do like the respectful tone of our discussion and it has challenged me to think about why I have the opinion I do. Perhaps the next discussion we're both involved in will have us in agreement. RJGray (talk) 02:56, 20 February 2016 (UTC)

The article Grothendieck's Tôhoku paper has the advantage that people do in fact refer to that paper as "Grothendieck's Tôhoku paper" both in conversation and in writing. I don't ever recall having a conversation with anyone in which Cantor's first set theory article was discussed (but maybe that's because I work in algebraic geometry and not set theory or logic) so I don't know how people refer to it.

Neither the present title nor "Grothendieck's Tôhoku paper" strike me as being appropriate. After all, the papers have names, and I continue to recoil at the idea of calling them something other than their names. Though I'm reminded of Haddocks' Eyes. Ozob (talk) 04:28, 20 February 2016 (UTC)

• Oppose this complicated proposal. The present title seems fine. Dicklyon (talk) 05:07, 4 April 2016 (UTC)

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The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

Constructive?

The lede text on constructiveness seems a bit confused. There's not clear evidence for a controversy; the only "against" is "Stewart, 2015" but that's not enough to identify who Stewart is. Or who Sheppard 2014 might be. Fraenkel is a heavyweight though so you'd need a good reason to disagree with him William M. Connolley (talk) 22:46, 2 April 2016 (UTC)

Those are standard Harvard citations and the sources are listed at the end, so it is clear enough to me what the references point to. There seems to be an entire section of the article related to this issue, with numerous sources. — Carl (CBM · talk) 15:05, 3 April 2016 (UTC)