Talk:Cantor's first set theory article: Difference between revisions

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==The rationals==

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

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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. [[User:AxelBoldt|AxelBoldt]] 02:09, 23 May 2006 (UTC)

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* 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.

|dykentry=... that mathematicians disagree about whether a proof in [[Georg Cantor]]'s '''[[Georg Cantor's first set theory article|first set theory article]]''' actually shows how to construct a [[transcendental number]], or merely proves that such numbers exist?

**That isn't a partition. If c is in R, then for <nowiki>{A,B}</nowiki> 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. [[User:Aij|Aij]] ([[User talk:Aij|talk]]) 02:13, 15 April 2008 (UTC)

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@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''"? [[User:Vijayaraghavan2k5|Vijay]] ([[User talk:Vijayaraghavan2k5|talk]]) 08:36, 11 January 2010 (UTC)

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:Axel may not be watching anymore &mdash; he made that comment in 2006.

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

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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&nbsp;&radic;2 how to find a larger rational number that is still less than&nbsp;&radic;2, and similarly for those larger than&nbsp;&radic;2. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 02:04, 24 January 2010 (UTC)

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: For <math>a<\sqrt2</math>, one possible choice would be <math>a+1/\lceil1/(\sqrt2-a)\rceil</math>. [[User:Paradoctor|Paradoctor]] ([[User talk:Paradoctor|talk]]) 13:27, 16 December 2013 (UTC)

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== Proposed Changes to Article ==

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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:

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'''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.

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'''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: [http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=266194 "Über eine Eigenschaft des Ingebriffes aller reelen algebraischen Zahlen"]. A French translation (which was reviewed and corrected by Cantor) is at: [http://www.springerlink.com/content/37030699752l2573/fulltext.pdf "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 [http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=2907 "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. [[User:RJGray|RJGray]] ([[User talk: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).

--[[User:RJGray|RJGray]] ([[User talk: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. --[[User:RJGray|RJGray]] ([[User talk: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. --[[User:RJGray|RJGray]] ([[User talk: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. [[User:Michael Hardy|Michael Hardy]] ([[User talk: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. " &mdash; 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. &mdash;&nbsp;Carl <small>([[User:CBM|CBM]]&nbsp;·&nbsp;[[User talk:CBM|talk]])</small> 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?

—[[User:RJGray|RJGray]] ([[User talk: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 <math>[a,b] \cap \bigcap_n U_n</math>. 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. &mdash;&nbsp;Carl <small>([[User:CBM|CBM]]&nbsp;·&nbsp;[[User talk:CBM|talk]])</small> 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. &mdash;&nbsp;[[User:RJGray|RJGray]] ([[User talk: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 polynomial]]s (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 - ?

[[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk: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. - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 16:40, 13 December 2013 (UTC)

{| align="right" class="collapsible collapsed" style="text-align:left"

! colspan="10" | '''Cantor's enumeration of algebraic numbers'''

|-

! colspan="10" | '''Height 1:'''

|-

| || || || align="right" | {{color|#c0c0c0|1}}''x'' || {{color|#c0c0c0|+0}} || = 0 || &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; || ''x''₁ = 0

|-

! colspan="10" | '''Height 2:'''

|-

|-

| || || || align="right" | 2''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || || align="right" | {{color|#c0c0c0|1}}''x'' || +1 || = 0 || || ''x''₂ = −1

|-

| || || || align="right" | {{color|#c0c0c0|1}}''x'' || −1 || = 0 || || ''x''₃ = +1

|-

||| || align="right" | {{color|#c0c0c0|1}}''x''² || {{color|#c0c0c0|+0''x''}} || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

! colspan="10" | '''Height 3:'''

|-

||| || || align="right" | 3''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || || align="right" | 2''x'' || +1 || = 0 || || ''x''₅ = −1/2

|-

| || || || align="right" | 2''x'' || −1 || = 0 || || ''x''₆ = +1/2

|-

| || || || align="right" | {{color|#c0c0c0|1}}''x'' || +2 || = 0 || || ''x''₄ = −2

|-

| || || || align="right" | {{color|#c0c0c0|1}}''x'' || −2 || = 0 || || ''x''₇ = +2

|-

| || || align="right" | 2''x''² || {{color|#c0c0c0|+0''x''}} || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || +{{color|#c0c0c0|1}}''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || −{{color|#c0c0c0|1}}''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || {{color|#c0c0c0|+0''x''}} || +1 || = 0 || || colspan="2" | no real root

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || {{color|#c0c0c0|+0''x''}} || −1 || = 0 || || colspan="2" | reducible

|-

| || align="right" | {{color|#c0c0c0|1}}''x''³ || {{color|#c0c0c0|+0''x''²}} || {{color|#c0c0c0|+0''x''}} || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

! colspan="10" | '''Height 4:'''

|-

| || || || align="right" | 4''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || || align="right" | 3''x'' || +1 || = 0 || || ''x''₁₃ = −1/3

|-

| || || || align="right" | 3''x'' || −1 || = 0 || || ''x''₁₄ = +1/3

|-

| || || || align="right" | 2''x'' || +2 || = 0 || || colspan="2" | reducible

|-

| || || || align="right" | 2''x'' || −2 || = 0 || || colspan="2" | reducible

|-

| || || || align="right" | {{color|#c0c0c0|1}}''x'' || +3 || = 0 || || ''x''₈ = −3

|-

| || || || align="right" | {{color|#c0c0c0|1}}''x'' || −3 || = 0 || || ''x''₁₉ = +3

|-

| || || align="right" | 3''x''² || {{color|#c0c0c0|+0''x''}} || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || align="right" | 2''x''² || +{{color|#c0c0c0|1}}''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || align="right" | 2''x''² || −{{color|#c0c0c0|1}}''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || align="right" | 2''x''² || {{color|#c0c0c0|+0''x''}} || +1 || = 0 || || colspan="2" | no real root

|-

| || || align="right" | 2''x''² || {{color|#c0c0c0|+0''x''}} || −1 || = 0 || || ''x''₁₆, ''x''₁₁ = ±1/√{{overline|2}}

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || +2''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || −2''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || +{{color|#c0c0c0|1}}''x'' || +1 || = 0 || || colspan="2" | no real root

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || +{{color|#c0c0c0|1}}''x'' || −1 || = 0 || || ''x''₁₅, ''x''₉ = (−1 ± √{{overline|5}}) / 2

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || −{{color|#c0c0c0|1}}''x'' || +1 || = 0 || || colspan="2" | no real root

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || −{{color|#c0c0c0|1}}''x'' || −1 || = 0 || || ''x''₁₈, ''x''₁₂ = (+1 ± √{{overline|5}}) / 2

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || {{color|#c0c0c0|+0''x''}} || +2 || = 0 || || colspan="2" | no real root

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || {{color|#c0c0c0|+0''x''}} || −2 || = 0 || || ''x''₁₇, ''x''₁₀ = ±√{{overline|2}}

|-

| || align="right" | 2''x''³ || {{color|#c0c0c0|+0''x''²}} || {{color|#c0c0c0|+0''x''}} || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || align="right" | {{color|#c0c0c0|1}}''x''³ || +{{color|#c0c0c0|1}}''x''² || {{color|#c0c0c0|+0''x''}} || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || align="right" | {{color|#c0c0c0|1}}''x''³ || −{{color|#c0c0c0|1}}''x''² || {{color|#c0c0c0|+0''x''}} || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || align="right" | {{color|#c0c0c0|1}}''x''³ || {{color|#c0c0c0|+0''x''²}} || +{{color|#c0c0c0|1}}''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || align="right" | {{color|#c0c0c0|1}}''x''³ || {{color|#c0c0c0|+0''x''²}} || −{{color|#c0c0c0|1}}''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || align="right" | {{color|#c0c0c0|1}}''x''³ || {{color|#c0c0c0|+0''x''²}} || {{color|#c0c0c0|+0''x''}} || +1 || = 0 || || colspan="2" | reducible

|-

| || align="right" | {{color|#c0c0c0|1}}''x''³ || {{color|#c0c0c0|+0''x''²}} || {{color|#c0c0c0|+0''x''}} || −1 || = 0 || || colspan="2" | reducible

|-

| align="right" | {{color|#c0c0c0|1}}''x''⁴ || {{color|#c0c0c0|+0''x''³}} || {{color|#c0c0c0|+0''x''²}} || {{color|#c0c0c0|+0''x''}} || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

! colspan="10" | '''Height 5:'''

|-

| || || || align="right" | 5''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | 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 2''x''. 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 [[unit (ring theory)|invertible]] nor zero and cannot be factored into the product of two non-invertible polynomials with coefficients in ''F''." This definition factors: 2''x'' = (2)(''x'') and it factors: 2''x''+2 = (2)(''x''+1). Finally, the exponent 1 in your table always appears in gray and it's well understood that "''x''" means "''x''¹". Try leaving out this exponent. I think this might visually simply your table. - [[User:RJGray|RJGray]] ([[User talk: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 ''x''³ and ''x''⁴; see the new table; I hope it is complete now ...), but you are right: at least the exponent of "''x''¹" 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 ''x''⁴ 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". - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk: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 polynomial]]s: INSERT YOUR TABLE HERE"

::Your table is looking better, some more suggestions: remove the "·" in 2·''x'', etc. In the enumeration, you can use ''x''₁ instead of "1.", etc. (This would connect your table closer to the article where all the sequences are ''x''₁, ''x''₂, ….) 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 number]]s: prime numbers (together with the corresponding negative numbers of equal magnitude) are the irreducible [[integer]]s. 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 6''x'' = (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 3''x'', 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. --[[User:RJGray|RJGray]] ([[User talk: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 ''x''_'''3''' 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... - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk: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 ''x''₃ is preferable to ''x''_'''3''' because the text doesn't use boldface and it looks better. Some other suggestions: I found double indexing "''x''_11,16" confusing. Try "''x''₁₁, ''x''₁₆" or, perhaps better, "''x''₁₆, ''x''₁₁" to match the way that the + of the ± goes with ''x''₁₆, and the – goes with ''x''₁₁ (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: [http://www.encyclopediaofmath.org/index.php/Algebraic_number 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)''. --[[User:RJGray|RJGray]] ([[User talk: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. - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk: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! --[[User:RJGray|RJGray]] ([[User talk: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 point]]s equals a whole interval (or even whole ℝ₊; cf. the picture [[accumulation point|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). - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk: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). --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 03:38, 13 February 2016 (UTC)

:<small>Just an update on the italics issue &mdash; I went ahead and added {{tl|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. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 03:59, 13 February 2016 (UTC) </small>

== 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. --[[User:RJGray|RJGray]] ([[User talk: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. --[[User:RJGray|RJGray]] ([[User talk: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. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 21:41, 13 February 2016 (UTC)

== Requested move 13 February 2016 ==

Line 364 ⟶ 121:

* '''Oppose''' this complicated proposal. The present title seems fine. [[User:Dicklyon|Dicklyon]] ([[User talk:Dicklyon|talk]]) 05:07, 4 April 2016 (UTC)

----

:''The above discussion is preserved as an archive of a [[Wikipedia:Requested moves|requested move]]. <span style="color:red">'''Please do not modify it.'''</span> Subsequent comments should be made in a new section on this talk page or in a [[Wikipedia:Move review|move review]]. No further edits should be made to this section.''</div><!-- Template:RM bottom -->

== 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 [[User:William M. Connolley|William M. Connolley]] ([[User talk: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. &mdash;&nbsp;Carl <small>([[User:CBM|CBM]]&nbsp;·&nbsp;[[User talk:CBM|talk]])</small> 15:05, 3 April 2016 (UTC)

== A replacement for the removed sentence ==

Line 407 ⟶ 158:

Actually, I've got them working with &amp;nbsp; except in the one case of a subscript 2''n''&nbsp;–&nbsp;1 where I had to use a nowrap within the <nowiki> _… </nowiki>. So I've managed to work around the problem. I've went into Internet Explorer and the nowrap in my example above does work there so it might just be a Chrome bug. Also, the sfrac problem is in both Chrome and Internet Explorer so that problem seems to be a Wikipedia bug. [[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 20:25, 4 May 2016 (UTC)

== Problems with "&amp;nbsp;" in Internet Explorer and with "nowrap" in Chrome ==

Thank you David for telling me that I may be seeing a bug. I've done some more experimenting and found a sentence that I can get to wrap properly in either Chrome and Internet Explorer, but not both.

The following sentence which uses "&amp;nbsp;" wraps properly in Chrome, but not in Internet Explorer. For example, by shrinking your window, the 3 occurrences of "''g''(" can wrap before the "(".

A one-to-one correspondence between ''T'' and '''R''' is given by the function: ''g''(''t'')&nbsp;<nowiki>=</nowiki>&nbsp;''t'' if ''t''&nbsp;∈&nbsp;''T''₀, ''g''(''t''_{{{nowrap|2''n'' – 1}}})&nbsp;<nowiki>=</nowiki>&nbsp;''t_n'', and ''g''(''t''_2''n'')&nbsp;<nowiki>=</nowiki>&nbsp;''a_n''.

The next sentence which uses "nowrap" wraps properly in Internet Explorer, but not in Chrome. The 1st occurrence works for Chrome, but Chrome can wrap within the subscripts in the next 2 occurrences.

A one-to-one correspondence between ''T'' and '''R''' is given by the function: {{nowrap|''g''(''t'') <nowiki>=</nowiki> ''t''}} if ''t''&nbsp;∈&nbsp;''T''₀, {{nowrap|''g''(''t''_{2''n'' – 1}) <nowiki>=</nowiki> ''t_n''}}, and {{nowrap|''g''(''t''_2''n'') <nowiki>=</nowiki> ''a_n''}}.

I've just fixed the 1st occurrence so that at least it works in both browsers. In many cases "&amp;nbsp;" works in both browsers, which is why I've been switching over to it.

I find these examples very interesting. I am specifying the same no-wrap regions in two different ways. Since I'm getting two different behaviors in Chrome or Internet Explorer, it seems unclear whether the bug is in a browser or in Wikipedia code. However, since I can get the proper no-wrapping behavior by using different text, the problem can be fixed in Wikipedia. Actually, I could do it myself with we had a browser template so I could write: <nowiki> {{browser | Chrome | … }} {{browser | Internet Explorer | … }} {{browser | default | … }} </nowiki>. Of course, it would be preferable for this to be done by the Wikipedia people who maintain "&amp;nbsp;" and "nowrap". Can my examples be communicated to them? Thanks, [[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 14:31, 6 May 2016 (UTC)

:: {{ping|RJGray}} Is this still an issue now? There is a system for reporting bugs, but it's changed since last time I did that, so I'd have to look into how it's done all over again. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 21:14, 14 September 2018 (UTC)

== Collapsed content ==

Can someone explain how the initially-hidden content of this article (e.g. the list of algebraic numbers, proof of the value for the method for generating an irrational, etc) is in compliance with [[MOS:DONTHIDE]]? Because it doesn't seem to be to me. (Also the square roots in the list of algebraic numbers are horribly formatted in android mobile view.) —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 05:44, 2 July 2018 (UTC)

:I still think the table of algebraic numbers is merely supplementary, similar to the past-years statistics example given in [[MOS:DONTHIDE]]. The enumeration proof for algbraic numbers can be understood without uncollapsing the table, but it gives some impression about how the enumeration looks like. Similarly, as far as I remember, the value ot the constructed irrational number is just supplementary; its irrationality has been proven before. Probably {{u|RJGray}} knows the details. &mdash;

:Concerning the root formatting: I remember I had used a unicode "√" followed by (e.g.) "<nowiki>{{overbar|5}}</nowiki>" to generate "√{{overbar|5}}". Subsequently, sombody has changed this to "<nowiki>{{radic|5}}</nowiki>", which generates "{{radic|5}}".Both look the same on my PC; I can't compare them in android mobile view. Anyway, it appears to be a problem of the "<nowiki>{{radic}}</nowiki>" template now. - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 11:19, 2 July 2018 (UTC)

{{Did you know nominations/Georg Cantor's first set theory article}}

== Reply to "Why doesn't Cantor's second theorem apply to a world of purely algebraic numbers?" ==

Ipsic asks the following: Explain what seems to me to be a subtle point. Why doesn't Cantor's second theorem apply to a world of purely algebraic numbers?

Here's the text that came with this request: (This text was in the section "Second theorem", just after the paragraph starting with "The proof is complete since ...")

Note, that under cases 1 and 3, above, the real number in [''a'',&nbsp;''b''] that is not a contained in the sequence may be chosen to be any of an infinite number of algebraic numbers that are contained within the intervals (''a''_''N'',&nbsp;''b''_''N'') or [''a''_∞,&nbsp;''b''_∞], respectively. However, in case 2 where ''a''_∞&nbsp;=&nbsp;''b''_∞, there is no interval from which an arbitrary algebraic number may be chosen. The value of ''a''_∞ must not be algebraic, because asserting that it is algebraic leads to a contradiction with the first theorem.

I agree that this can be a subtle point. The important point is to distinguish what Cantor's theorem does in general from what it does in a particular application, such as being applied to the sequence of all algebraic numbers in an interval. In general, Cantor's theorem only guarantees that the real number obtained from his construction is not in the given sequence. In many applications, such as applying it to the sequence of rationals in (0, 1) as in the section "Example of Cantor's construction", you are not guaranteed to obtain a transcendental number but in this case you are guaranteed to obtain an number that is not rational. Applying the theorem to this case, you obtain <math>\sqrt{2}-1</math>, which is a non-rational algebraic number.

However, if you apply Cantor's theorem to the sequence of all algebraic numbers in an interval (''a'',&nbsp;''b''), you are guaranteed to obtain a non-algebraic number in the interval.

Since you brought up that you can hit an algebraic number in Case 1 or Case 3, I wish to point out that the section "Dense sequences" proves that a dense sequence—such as, the sequence of algebraic numbers in an interval—never ends up in Case 1 or Case 3, so your argument fails here. To obtain a transcendental number, Cantor is using such a dense sequence.

If you still find this confusing, please let me know. By the way, if you have questions about an article, the questions really belong on the Talk pages and not in the article's text. --[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 02:00, 13 September 2018 (UTC)

== "Disagreement" on constructivity ==

The article currently says:

:''Books as recent as 2014 and 2015 indicate that this disagreement has not been resolved.''

There is a cite, which quotes Sheppard:

:''Cantor's proof of the existence of transcendental numbers is not just an existence proof. It can, at least in principle, be used to construct an explicit transcendental number.''

and Stewart:

:''Meanwhile Georg Cantor, in 1874, had produced a revolutionary proof of the existence of transcendental numbers, without actually constructing any.''

But these two statements are not actually in conflict, and we have no evidence that Sheppard and Stewart actually "disagree" in the slightest. Sheppard correctly notes that Cantor's method can be applied to find a particular transcendental; Stewart correctly notes that Cantor did not in fact ''do'' that.

I'm not sure how to fix this, but I don't think it can stand as is. There is no modern "disagreement" on the question, not phrased this way. (I'm not sure whether intuitionists consider the proof constructive, because it might use excluded middle (?) but that's a bit of a different issue.) --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 07:05, 13 September 2018 (UTC)

Thank you for your feedback. I agree with you that "Sheppard correctly notes that Cantor's method can be applied to find a particular transcendental." However, I disagree that "Stewart correctly notes that Cantor did not in fact ''do'' that."

To understand what Cantor did, I'll quote from the article [http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gray819-832.pdf Georg Cantor and Transcendental Numbers ]. From the bottom of page 819 to page 820, it states:

:Cantor begins his article by defining the algebraic reals and introducing the notation: (ω) for the collection of all algebraic reals, and (ν) for the collection of all natural numbers. Next he states the property mentioned in the article's title; namely, that the collection (ω) can be placed into a one-to-one correspondence with the collection (ν), or equivalently:

::... the collection (ω) can be thought of in the form of an infinite sequence: (2.) ω₁, ω₂, ..., ω_η, ... which is ordered by a law and in which all individuals of (ω) appear, each of them being located at a fixed place in (2.) that is given by the accompanying index.

:Cantor states that this property of the algebraic reals will be proved in Section 1 of his article, and then he outlines the rest of the article:

::To give an application of this property of the collection of all real algebraic numbers, I supplement Section I with Section 2, in which I show that when given an arbitrary sequence of real numbers of the form (2.), one can determine [Note: By this Cantor means that he can "construct"] in any given interval (α ··· β), numbers that are not contained in (2.). Combining the contents of both sections thus gives a new proof of the theorem first demonstrated by Liouville: In every given interval (α ··· β), there are infinitely many transcendentals, that is, numbers that are not algebraic reals. Furthermore, the theorem in Section 2 presents itself as the reason why collections of real numbers forming a so-called continuum (such as, all the real numbers which are ≥ 0 and ≤ 1), cannot correspond one-to-one with the collection (v); thus I have found the clear difference between a so-called continuum and a collection like the totality of all real algebraic numbers.

:To appreciate the structure of Cantor's article, we number his theorems and corollaries:

:Theorem 1. The collection of all algebraic reals can be written as an infinite sequence.

:Theorem 2. Given any sequence of real numbers and any interval [α, β], one can determine a number η in [α, β] that does not belong to the sequence. Hence, one can determine infinitely many such numbers η in [α, β]. (We have used the modern notation [α, β] rather than Cantor's notation (α ··· β).)

:Corollary 1. In any given interval [α, β], there are infinitely many transcendental reals.

:Corollary 2. The real numbers cannot be written as an infinite sequence. That is, they cannot be put into a one-to-one correspondence with the natural numbers.

:Observe the flow of reasoning: Cantor's second theorem holds for any sequence of reals. By applying his theorem to the sequence of algebraic reals, Cantor obtains transcendentals. By applying it to any sequence that allegedly enumerates the reals, he obtains a contradiction—so no such enumerating sequence can exist. Kac and Ulam reason differently [20, p. 12-13]. They prove Theorem 1 and then Corollary 2. By combining these results, they obtain a non-constructive proof of the existence of transcendentals.

So Cantor does give a method of constructing transcendental numbers. By the way, Kac and Ulam present the same non-constructive proof that Stewart uses. <!-- Template:Unsigned --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:RJGray|RJGray]] ([[User talk:RJGray#top|talk]] • [[Special:Contributions/RJGray|contribs]]) 01:56, 14 September 2018 (UTC)</small>

::OK, well, I haven't read Stewart's article; I don't know exactly what Stewart claims. I was making an inference based on what the citation said.

::If Stewart does not claim that Cantor didn't find a particular transcendental, then there's even less support for the claim of "disagreement". --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 06:06, 14 September 2018 (UTC)

If this doesn't qualify as actual disagreement, then maybe the DYK hook should get rephrased. But if it's not actual disagreement, it may still be accurate to say many actual mathematicians have misunderstood the matter in published writings. Many mathematicians including Dirichlet (who may be the originator of the error) have written in books and papers that Euclid's proof of the infinitude of primes is by contradiction. Most of that may be just following what they've read rather than substantially disagreeing. But some have written, erroneously, that Euclid's proof is non-constructive. "Disagreement"? Or (moderately) widespread error? If the latter, it could be a DYK hook if sufficiently supported. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 22:05, 14 September 2018 (UTC)

Trovatore, concerning your statement:

::If Stewart does not claim that Cantor didn't find a particular transcendental, then there's even less support for the claim of "disagreement".

Stewart said:

:''Meanwhile Georg Cantor, in 1874, had produced a revolutionary proof of the existence of transcendental numbers, ''without actually constructing any.''

I believe that most (nearly all?) people reading ''without actually constructing any'' would come away believing that Cantor's proof is a pure existence proof that does not construct transcendentals.

Cantor's constructive proof allows him to construct a transcendental number on any closed interval that is provided. Cantor was writing a research article and could expect that his readers would see this without him providing an example. In fact, he wrote at a time when non-constructive proofs were rare (this started to change in 1890 when Hilbert gave a non-constructive proof of his Basis Theorem). As pointed out in the Wikipedia article, even Kronecker would accept that Cantor's construction applied to the sequence of algebraic numbers produces a definite real number. This may be one reason why Cantor's article was accepted so quickly—only four days after submission. His next article suffered a long delay which he blamed on Kronecker.

By the way, you bring up an important point—would intuitionists accept Cantor's proof? I believe the answer is no because his proof of his Theorem 2 uses the fact that an increasing or decreasing bounded sequence of reals has a limit. Intuitionists like limiting procedures that are coming from below and above. However, the footnote on page 821 in [http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gray819-832.pdf Georg Cantor and Transcendental Numbers] says that page 27 in Bishop and Bridges ''Constructive Analysis'' (1985) has a proof of Theorem 2 that meets the demands of constructive mathematicians (and probably also intuitionists).

Concerning the word "disagree", I chose it to replace the word "controversy" in my first rewrite of the Wikipedia article. The word "controversy" has two problems: It's a "peacock" term and it's inaccurate because controversy implies that the mathematicians who are stating the proof is constructive or non-constructive are aware of the choice they are making. Disagreement simply means that what an article or book says disagrees with at least one source in the literature. I'm not particularly attached to the word "disagree". However, changing it could take some time (depending of course on the particular term chosen) and the GA review had no trouble with "disagree".

Most sources seem to say that Cantor's proof is non-constructive. Ivor Grattan-Guinness in his two-sentence 1995 review of "Georg Cantor and Transcendental Numbers" stated: "It is commonly believed that Cantor's proof of the existence of transcendental numbers, published in 1874, merely proves an existence theorem. The author refutes this view by using a computer program to determine such a number". Working on this Cantor article rewrite, I still found more sources stating his proof is non-constructive, but it was nice to find a few saying the proof is constructive.

As far as the DYK hook: Michael, I think you are working on this. I'd be happy to help. Just give me your latest and best ideas.

Thank you, Michael and Trovatore, for the work you are doing on this. Now I have to work on adding a few references requested by the DYK review. --[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 19:02, 15 September 2018 (UTC)

:{{U|RJGray}}: There is not a disagreement. There is not a controversy. It's such a simple question that ''everyone agrees''. They just ''phrase'' it differently. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 23:20, 1 May 2020 (UTC)

== Proof of Lemma needs a change ==

The lemma required for the second case in the proof states that x(n+1) and x(n+2) are the end points for the interval (a(n+1), b(n+1)), however these are just the first 2 candidates as end points, we do not know if they lie inside or outside the interval (a(n), b(n)). This is indeed enough to satisfy the criterion that x(n+1) and x(n+2) are either larger or best case are indeed the end-points.

The simplest modification would be to state that these two points are at best the end points. If not the induction actually runs better. [[Special:Contributions/2001:1C02:1203:8500:49CD:D219:D882:A34|2001:1C02:1203:8500:49CD:D219:D882:A34]] ([[User talk:2001:1C02:1203:8500:49CD:D219:D882:A34|talk]]) 13:32, 26 January 2023 (UTC)

== visited? ==

"...he second column lists the terms visited during the search for the first two terms..."

what do you mean by "terms _visited_"? [[Special:Contributions/217.149.171.204|217.149.171.204]] ([[User talk:217.149.171.204|talk]]) 08:12, 5 July 2023 (UTC)