Talk:Cantor's first set theory article/Archive 1

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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)

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)