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== initial section ==

Is it really true that most mathematicians believe that the diagonal proof was Cantor's first proof of uncountability? I'm no mathematician, but even my topical interest in the matter turned up that fact long before this article was created. Is the misconception really that prevalent? -- [[User:Cyan|Cyan]] 20:51, 3 Nov 2003 (UTC)

I haven't carefully polled mathematicians to ascertain this, but I keep finding it asserted in print, and I've spoken with a number of intelligent mathematicians who were under that impression. [[User:Michael Hardy|Michael Hardy]] 19:53, 4 Nov 2003 (UTC)

I would not have been certain if the diagonal proof was the first one, but my guess (if I would have to bet) would have been that it was, as this is the proof that is most known and famous, so in this sense, I think it's a misconception. Also, mathematicians are pretty sloppy historians (see [[Fermat number]] re: Gauss ''n''-gon construction) so it's best to assume we don't know what we're doing, I think. [[user:revolver|Revolver]]

:I've looked up Cantor's 1874 paper in ''Journal für die Reine und Angewandta Mathematik'', and the argument given in that article is indeed the one given here. See also Joseph Dauben's book about Cantor. This was indeed his first proof of this result. [[User:Michael Hardy|Michael Hardy]] 22:17, 12 Jan 2004 (UTC)

Was it really in 1877 that Cantor discovered the diagonal method, or was it later? I cannot find any proof for this. -- [[User:Zwaardmeester|Zwaardmeester]] 19:52, 15 Jan 2006 (GMT+1)

'''This "proof" is logically flawed'''

The constructivists' counterargument to the preceding "proof" of the "uncountability" of the set of all real numbers hinges on their firm belief that there is no greatest natural number --- hence, there is no last term in the sequence {Xn}. The completed constructibility of the monotone sequences {An} and {Bn} from {Xn} is also dubious for the same reason --- hence, the limit point C is "unreachable" (that is, it perpetually belongs to the elements of R denoted by the 3-dot ellipsis ". . ."). One could state in more familiar terminology that C is an irrational number which does not have a last, say, decimal expansion digit since there is no end in the progression of natural numbers.

For a simple counterargument, even granting arguendo Georg Cantor's own hierarchy of transfinite ordinal numbers, we merely note that the infinite set whose size is "measured" by the ordinal number, say, w+1 = {0,1,2,3,...,w} (here, w is omega)) is also countable. To elaborate, "ordinal numbers" are "order types of well-ordered sets". A well-ordering is an imposition of order on a non-empty set which specifies a first element, an immediate successor for every "non-last element", and an immediate successor for every non-empty subset that does not include the "last element" (if there is one). For examples:

(1) The standard imposition of order on all the non-negative rational numbers:

{ 0, . . ., 1/4, . . ., 1/2, . . ., 3/4, . . ., 1, . . ., 5/4, . . ., 3/2, . . ., 7/4, . . ., 2, . . ., 9/4, . . . }

is not a well-ordering — because, for example, there is no positive rational number immediately following 0.

(2) The following imposition of order on all the non-negative rational numbers is a well-ordering but not a countable ordering or enumeration:

{ 0, 1, 2, 3, . . ., 1/2, 3/2, 5/2, . . ., 1/3, 2/3, 4/3, . . ., 1/4, 3/4, 5/4, . . ., 1/5, 2/5, 3/5, . . . }

(3) The following imposition of order on all the non-negative rational numbers is a well-ordering that is also a countable ordering or enumeration:

{ 0, 1, 1/2, 2, 1/3, 3, 1/4, 2/3, 3/2, 4, 1/5, 5, 1/6, 2/5, 3/4, 4/3, 5/2, 6, 1/7, 3/5, 5/3, 7, . . . }

It must be emphasized that, while the first two representations are not countable ordering or enumeration of the non-negative rational numbers, nevertheless, the set of non-negative rational numbers is countable as the third representation shows.

Rigorously, let us grant arguendo Georg Cantor's claim of completed totality of infinite sets. The sequences {An} and {Bn} were defined to form a sequence of nested closed intervals [An,Bn] that "microscopes" to the limit point C which must be a member of R. The assumption that all the elements of R can be enumerated in the sequence {Xn} carries with it the imposition of a countable ordering --- which deviates from the standard ordering based on the numerical values --- of the members of R. Given an arbitrary countably ordered sequence {Xn}, by the very definition of the construction of {An} and {Bn} stipulated by Georg Cantor, there is only one sequence of nested closed intervals [An,Bn] that can be so constructed --- hence, only one same limit point C for both {An} and {Bn}. In other words, the enumeration scheme of the sequence {Xn} determines exactly the sequences {An} and {Bn} as well as the limit point C. A rearrangement of a finite number of terms of {Xn} is still a countable ordering of the elements of R --- if a rearrangement results in different sequences {An} and {Bn}, then a different limit point C is obtained but, always, there is only one limit point C that is "excluded" in any specified sequence {Xn}. We emphasize that in Cantor's tenet of completed totality of an infinite set, the limit point C of both the sequences {An} and {Bn} is known "all at once" in advance.

Therefore, Georg Cantor could have just as well specified the also countable set {X1,X2,X3,...} U {C} = {X1,X2,X3,...,C} [note that there is no real number immediately preceding C] --- instead of the standard enumeration {X1,X2,X3,...} that he assumed in the first sentence of his "proof" as having all of R as its range so that C is clearly not in the sequence {X1,X2,X3,...} but C is in R = {X1,X2,X3,...,C} and no contradiction would be reached.

Furthermore,

{Xn} = {Yn} U {An} U {Bn} U {C} [1]

where the sequence {Yn} consists of all the terms Xi of the sequence {Xn} that were bypassed in the construction of the sequences {An} and {Bn}. If we accept Cantor's reductio ad absurdum argument above, then we deny equality [1] and the easily proved fact that the union set of a finite collection of countable sets is countable.

[e-mail]

: After carefully reading the above post, I conclude that the author is making a mistake similar to those made by many people encountering diagonal-method proofs for the first time. The problem is that if the initial set chosen is {X1,X2,X3, ... C} (with C inserted between two Xi's -- this is not captured by the notation), then the limit value produced by Cantor's argument will not be C, but something else.

:: Even a good high school student would balk at the above objection --- are you saying that a monotone sequence could have a middle-term limit? [[e-mail] (27 Sep 2005)]

:::No. You're saying "For a given proposed enumeration (''x''_''n'') of the reals, Cantor presents a number ''c'' which is not captured. So let's just throw ''c'' in and we have enumerated the reals." The objection to your argument is: once you have thrown in ''c'', Cantor will happily present a ''new'' number (different from ''c'') that is still not covered by your new proposed enumeration. He will always catch you. [[User:AxelBoldt|AxelBoldt]] 18:17, 30 March 2006 (UTC)

:::: This is the same commonest mistake in Cantor's diagonal argument --- once you have thrown in ''c'' and claim a complete list of the real numbers then you're done; a re-application of Cantor's diagonal argument does present a number (different from ''c'') but that new anti-diagonal number was included in the list when ''c'' was the anti-diagonal number. You merely made a new enumeration or re-arranged the row-listed real numbers to get a different anti-diagonal number! → I hope you teach my counter-counterargument to your counterargument in your math classes ... Best regards ... [[e-mail] (16 Oct 2006)]

::::: "...but that new anti-diagonal number was included in the list when ''c'' was the anti-diagonal number." No it wasn't! [[User:Ossi|Ossi]] 19:47, 8 September 2007 (UTC)

:::::: It is wise to understand that any real number is a constant [that is, in place-value positional numeral system like decimal or binary, its every digit is known (or could be computed “given sufficient computing time and space resources” which is a standard axiom in the modern theory of computation)] so they have fixed existence (unlike a variable) and there could not be any algorithm such as Cantor’s anti-diagonalization argument that can produce “new” numbers (that is, not already on the row-list unless it is an extended number such as row-listing all the rationals and having irrational diagonal and anti-diagonal number or row-listing all the real numbers and having complex diagonal and anti-diagonal number) like the 2nd anti-diagonal number, say D, when the first anti-diagonal number C is in the row-list. Any real number is either the anti-diagonal number or in the “actually incomplete” row-listing. If D was not included in the row-listing when C was the anti-diagonal number, then you don’t have to call upon the excluded anti-diagonal number C because your first row-list is already incomplete since it also excludes another real number D. In the sequence of ALL real numbers {X_n} = {Y_n} U {A_n} U {B_n} U {C} of my discussion above, C is an arbitrary real number limit. If the 2nd limit D is not equal to the first limit C, then D is in {Y_n} or in {A_n} or in {B_n} when C is the limit, and vice versa on “C” and “D” — otherwise, {X_n} does not enumerate all the real numbers as presupposed.

:::::: From a different perspective, the following are tautologous diagonalization (not anti-diagonalization like Cantor’s) arguments:

::::::: * a truly complete row-listing with 1st row the positive integers in their respective columns; in row 2, the respective positive integers 2nd powers (squares) ; in row 3, the respective positive integers 3rd powers (cubes); …; in row n, the respective positive integers nth powers; … . Then, clearly, the diagonal sequence of nⁿ is not in the row-list (Leo Zippin in “Uses of Infinity” [MAA, 1962] claims this to be an example of Cantor’s diagonal argument);

::::::: * a truly complete enumeration or row-listing of all the fractional rational numbers would have irrational diagonal and anti-diagonal numbers;

::::::: * a truly complete enumeration or row-listing of all the fractional real numbers &mdash; which consists of all the rational (nonterminating but periodic binary or decimal expansion) and irrational (nonterminating and nonperiodic expansion) fractional numbers with each collection exhausting the positive integer row-indices and the representation 0.b₁b₂b₃… (for the rational numbers, there is no limit in the period-length while for the irrational numbers, each has a nonzero digit at the omegath position due to the 1-1 correspondence, say &pi;-3 in binary, <0.0, 0.00, 0.001, 0.0010, 0.00100, 0.001001, …, &pi;-3> <--> <0, 0, 1, 0, 0, 1, …, 1_&omega;) &mdash; would simply have irrational anti-diagonal (with 1_&omega; if diagonal is rational (without or with 0_&omega;), or vice versa on “ratiobal” and “irrational”;

::::::: * a truly complete enumeration or row-listing of all the fractional algebraic (inclined measurements) real numbers would have transcendental (curved, compounded continuously measurements) real diagonal and anti-diagonal numbers — and vice versa on “algebraic” and “transcendental”;

::::::: * a truly complete row-listing of all the fractional real numbers (complex numbers with no imaginary part &mdash; both algebraic and transcendental) would have complex (with real and imaginary part) diagonal and anti-diagonal numbers;

::::::: * a truly complete enumeration or row-listing of all the fractional complex numbers would have quaternion diagonal and anti-diagonal numbers;

::::::: * etc.

:::::: Even Ludwig Wittgenstein missed my point here -- you might want to read [http://www.math.ucla.edu/%7easl/bsl/0401-toc.htm “An Editor Recalls Some Hopeless Papers”] by Wilfrid Hodges. The very simple flaw in Hodges’ presented variant of Cantor’s anti-diagonal argument which claims to prove the “uncountability” of all the real numbers by “demonstrating” that there could not be any 1-to-1 correspondence between all the natural numbers and all the fractional real numbers is the common false belief that the arbitrary decimal place-value positional numeral system representation 0.d₁d₂d₃…d_n…, where d_n is in {0,1,2,3,4,5,6,7,8,9} for every natural number n, denotes the fractional expansion of either a rational (nonterminating, periodic) or an irrational (nonterminating, nonperiodic) real number between 0 and 1 when indeed just the rational numbers (since they have no period-length limit &mdash; in other words, any finite sequence of digits is a possible period) already exhaust them (on the other hand, the irrational numbers require the representation 0.d₁d₂d₃…d_n…1_&omega; due to the 1-1 correspondence, say &pi; – 3 = 0. 0010010000111... in binary system, <0.0, 0.00, 0.001, 0.0010, 0.00100, 0.001001, …, π - 3> <--> <0, 0, 1, 0, 0, 1, …, 1_&omega;. [[e-mail]] [[User:BenCawaling|BenCawaling]] ([[User talk:BenCawaling|talk]]) 06:25, 26 March 2008 (UTC)

:"Cantor's first proof" is new to me, and I have to say it's delightful. I agree that mathematicians generally believe the diagonal argument to be Cantor's first. However, I'm not completely convinced that this isn't really a diagonal argument in disguise. I need to think about this a bit. [[User:Dmharvey|Dmharvey]] [[User talk:Dmharvey|Talk]] 22:45, 6 Jun 2005 (UTC)

:From the constructionists point view Cantors proof '''is''' "flawed". Let me first say that that I am '''not''' a constructionist, and that I do accept Cantor proof in full. Constructionists don't even consider it proven that there is no greatest number, and counterarguments won't bite. Here is why: Suppose there '''is''' a greatest natural number and call it Max. Then a kid in kindergarten can show that there is a number SuperMax = Max + 1 that is greater than Max. Contradiction right? '''Wrong''', says the constructionist, because you haven't shown Max to exist, and therefore you '''cannot''' use it in any way in a proof of anything. Reductio ad absurdum arguments aren't generally accepted. Constructionists are "more rigorous" than most other mathematicians, probably in a sense that could be quantified exactly if the subset of the aximoms of ZFC that they accept is specified. The Axiom of Infinity is obviously not one of them. By the same token they can "proove fewer theorems" than most other mathematicians. The uncountability of the reals is obviously one of them. [[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 02:20, 30 November 2007 (UTC)

== Simplifications ==

I'm hesitant to make changes to the statement of the theorem and its proof, since we may want to retain historical accuracy, and I don't have access to Cantor's formulation. There's five things I would change:

# '''R''' needs to be non-empty which is currently not required. (but see 2 below)

# if we require that '''R''' have at least ''two'' points, then we can get rid of the endpointlessness requirement. (The first step of the proof would then read "pick the first element of the sequence that's not the largest element of '''R'''"; everything else stays the same.)

# The proof emphasizes "greater than the one considered in the previous step" twice, but that's not needed; just always pick the first member of the sequence that works.

# The proof should make a bit more explicit how the existence of ''c'' follows from the given properties of '''R'''. I.e.: How the sets ''A'' and ''B'' are defined.

# The proof should make explicit how the density property is used.

What do people think about these changes? [[User:AxelBoldt|AxelBoldt]] 17:49, 30 March 2006 (UTC)

== "contrary to what most mathematicians believe" ==

Those six words add nothing to the article but an unverified claim based on the personal experience of wikipedia editors. They're unencyclopedic language and not really appropriate. [[User:Night Gyr|Night Gyr]] 22:15, 29 April 2006 (UTC)

:As mentioned above, it's true that the claim hasn't been formally verified, though anecdotal evidence is strong. I don't see what's wrong with the language. How does removing this statement help our readers? [[User:AxelBoldt|AxelBoldt]] 00:20, 30 April 2006 (UTC)

Because it makes us less reliable as an encyclopedia to include information that can't be relied upon to be anything but the anecdotal experiences of our editors. It's the whole reason behind [[WP:V]]. [[User:Night Gyr|Night Gyr]] 01:13, 30 April 2006 (UTC)

So if I change "most" to "many" and find two sources where mathematicians make the false claim, would you be happy? [[User:AxelBoldt|AxelBoldt]] 14:17, 30 April 2006 (UTC)

I still don't understand why wikipedia needs to state that a misconception exists more prominently than the truth itself. We're here to provide facts, and the fact that a large number mathematicians of mathematicians have a fact incorrect seems less important to the ''first phrase of the entire article'' than ''the fact itself''. [[User:Night Gyr|Night Gyr]] 02:26, 1 May 2006 (UTC)

I agree, it doesn't need to be in the first sentence, or even the first paragraph. But it should be documented nevertheless. [[User:AxelBoldt|AxelBoldt]] 14:42, 3 May 2006 (UTC)

==The rationals==

Line 104 ⟶ 13:

: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)

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)

==Complete is the wrong word==

Technically, the real numbers are not [[complete]]. Of course, the [[extended reals]] are complete. It may be better to remove the completeness requirement and leave the "i.e.". Really, all we need is what Rudin calls the "least-upper-bound property." That is, the least upper bound of any set that is bounded from above exists (and the similar claim about sets bounded from below and infimum). --[[User:TedPavlic|TedPavlic]] 15:32, 7 March 2007 (UTC)

* Additionally, the partition definition only makes sense if the element c is greater than or EQUAL to every element in A and less than or EQUAL TO every element in B. If c is not included in A or B, then (A,B) cannot be a partition of R. --[[User:TedPavlic|TedPavlic]] 21:04, 7 March 2007 (UTC)

**The text seems fine, it does not say that c is greater than every element in A and less than every element in B.--[[User:Patrick|Patrick]] 11:23, 7 June 2007 (UTC)

***I you don't allow for "or equal to" in at least one place i'ts impossible to find such a partition. (Present a counterexample please in case I'm wrong.)[[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 00:40, 30 November 2007 (UTC)

***I correct myself. It is perfectly correct as it stands. Still, I obviously wasn't the only one that fell into the trap. If a wording with "... or equal to" is equivalent to the current wording that would be preferable.[[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 00:51, 30 November 2007 (UTC)

* Note: the term "gapless" may be better than "complete". --[[User:TedPavlic|TedPavlic]] 21:17, 7 March 2007 (UTC)

:Complete is the correct word. From the article you linked: "In order theory and related fields such as lattice and ___domain theory, completeness generally refers to the existence of certain suprema or infima of some partially ordered set. Notable special usages of the term include the concepts of complete Boolean algebra, complete lattice, and complete partial order (cpo). Furthermore, an ordered field is complete if every non-empty subset of it that has an upper bound within the field has a least upper bound within the field, which should be compared to the (slightly different) order-theoretical notion of bounded completeness. Up to isomorphism there is only one complete ordered field: the field of real numbers (but note that this complete ordered field, which is also a lattice, is not a complete lattice)." [[User:Ossi|Ossi]] 19:45, 8 September 2007 (UTC)

: 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)

The article linked to complete lattices, and you're right the reals are not a complete lattice. They are a complete metric space, but that is not the subject here. I reworded it a little. I don't think "gapless" is better - we should try to explain things, but not to the point of inventing our own terminology. &mdash;&nbsp;Carl <small>([[User:CBM|CBM]]&nbsp;·&nbsp;[[User talk:CBM|talk]])</small> 13:15, 23 July 2007 (UTC)

I added an External Link from this page to [http://www.theoremoftheday.org/Theorems.html#19 Theorem of the Day, no. 19]. It seemed an innocent enough thing to do. Theorem of the Day is an academic project which tries to bring beautiful mathematics to a wider public without sacrificing rigour. It has over 100 theorems on offer now and is reasonably widely known. I am new to Wikipedia, however, so if this is what is regarded as spam, then maybe I rethink how I publicise my project? [[User:Charleswallingford|Charleswallingford]] ([[User talk:Charleswallingford|talk]]) 22:56, 24 June 2008 (UTC)

== Cantor Anti-Diagonal Argument &mdash; Clarifying Determinateness and Consistency in Knowledgeful Mathematical Discourse ==

Perhaps my unfinished manuscript [http://www.geocities.com/bencawaling/cantor_anti-diagonal_argument.pdf "Cantor Anti-Diagonal Argument -- Clarifying Determinateness and Consistency in Knowledgeful Mathematical Discourse"] would be useful now to those interested in understanding Cantor anti-diagonal argument. I was hoping to submit it to the Bulletin of Symbolic Logic this year. Unfortunately, since 1 January 2008, I have been suffering from recurring extremely blurred vision due to frequent “exploding optical nerves” brought on by my diabetes (I can’t afford laser eye surgery) and I had only about 20 productive days in the last 8 months. At this rate, it would take me a long while to finish my paper or may not be able to complete it if I go permanently blind soon. I just hope my endeavors to clarify mathematical infinity and modern logic would reach the next (if not the present) generations of mathematicians, philosophers, and logicians. [[e-mail]] [[User:BenCawaling|BenCawaling]] ([[User talk:BenCawaling|talk]]) 08:08, 4 September 2008 (UTC)

== Proposed Changes to Article ==

Line 138 ⟶ 33:

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)

Line 150 ⟶ 45:

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:

== I suggest scrapping the article. ==

* 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.

The four properties say nothing about countability. One could reword the entire theorem to state:

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)

A set G composed of linearly ordered sets A and B with all members of A < B contains a member c that is indeterminate.

: 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)

This is just another flavour of Dedekind's definition of real number. Defining a real number does not make it 'countable', whatever countable means for indeed it is not defined in any way. What is being defined first here - countability or a real number? The Diagonal Argument is very different to this inconsequential theorem. Both the theorem and diagonal argument are flawed. [[Special:Contributions/91.105.179.213|91.105.179.213]] ([[User talk:91.105.179.213|talk]]) 10:21, 19 January 2010 (UTC)

== B class ==

: You're confused. The set G you refer to DOES NOT generally contain a member between A and B; in particular, if G is countable then it can be partitioned into such sets A and B such that G contains no such element. And you haven't defined "indeterminate". Countability is used in order to reach the conclusion. If you missed that, you haven't read carefully. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 14:29, 19 January 2010 (UTC)

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

OK, more long windedly:

* 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.

* "A set G composed of linearly ordered sets A and B with all members of A < B"

* 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.

No, that's not right. What is written about is a '''linearly ordered set ''G'''''. If is not enough for ''A'' and ''B'' '''separately''' to be linearly ordered; the whole set ''G'' is supposed linearly ordered.

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)

* "with all members of A < B"

:

I presume this means every member of ''A'' is less than every member of ''B''; it should have said so.

* "contains a member c that is indeterminate"

:

What does "indeterminate" mean? The article claims

:*''c'' is '''not a member of either ''A'' or ''B''''', and

:*'''every member of ''A'' is less than ''c''''', and

:*'''''c'' is less than every member of ''B'''''.

:

Moreover, the article says ''c'' is '''not a member of G'''! If ''c'' is '''not''' a member of ''G'', then ''G'' does '''not''' "contain"&nbsp;''c''! Yet, you've set "contains a member that is...".

:

Is it true that if a linearly ordered set ''G'' is partitioned into subsets ''A'' and ''B'' with every member of ''A'' less than every member of ''B'', then such an element ''c'' exists? '''No'''. It is '''not''' true. For example, let ''G'' be the set of all real numbers, let ''A'' be the set of all negative numbers, and let ''B'' be the set of all non-negative numbers, i.e. ''B'' contains all positive numbers and&nbsp;0. Does ''G'' contain any member ''c'' that is greater than every member of ''A'' and less than every member of ''B''? '''No, it does not.'''

:

The element ''c'' is explicitly stated '''not''' to be a member of ''G'', yet you say ''G'' "contains a member that..." etc.

:

Countability was used in reaching the conclusion. The conclusion was not that the countable set ''G'' contains the number ''c'', but rather that the set of ''real numbers'' contains the number ''c'', and ''G'' '''does not''' contain the number&nbsp;''c''.

:

You need to pay attention. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 14:45, 19 January 2010 (UTC)

Thank you very much for your feedback:

'''Is it I or you who needs to pay attention?''' The article states:

* 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.

''Property 4 has the following least upper bound property: For any partition of R into two nonempty sets A and B such that every member of A is less than every member of B, there is a boundary point c (in R), so that every point less than c is in A and every point greater than c is in B.

''

: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)

Just replace R with G and you have exactly what I claimed:

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)

'''''A set R composed of linearly ordered sets A and B with all members of A < B contains a member c that is indeterminate.

'''''

Okay. So why is c indeterminate? Because your article states it is so: '''c is a boundary point in R.'''

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

Mr. Hardy, I object to your patronizing tone. Do not talk down to me! [[Special:Contributions/91.105.179.213|91.105.179.213]] ([[User talk:91.105.179.213|talk]]) 20:57, 19 January 2010 (UTC)

:Please refrain from this sort of argument on article talk pages. See [[WP:TALK]]. 91, if you want to discuss this sort of issue with ''Dr'' Hardy, I suggest you contact him directly. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 21:05, 19 January 2010 (UTC)

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 - ?

::Excuse me? And who might you be? I don't give a hoot whether he is a Mr. or a Dr or whatever. He better not be talking down to me! Trovatore - I suggest you '''pay attention!''' Before you start blurting out baseless accusations. Look Travo, just stay out of this conversation. Hardy does not need your help. I certainly have not violated any WP:TALK rules. You should be directing your comments at him! Besides, this is the discussion page - outside of this page, I have nothing to say to Hardy. [[Special:Contributions/91.105.179.213|91.105.179.213]] ([[User talk:91.105.179.213|talk]]) 22:06, 19 January 2010 (UTC)

:::Actually you have gone against talk page guidelines &mdash; talk pages are not for general discussion of the subject matter. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 01:27, 20 January 2010 (UTC)

[[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 15:10, 13 December 2013 (UTC)

"91.105.179.213", I do think you need to pay closer attention. You wrote about

: "A set G composed of linearly ordered sets A and B"

:

The fact is, the least upper bound property was attributed here to the linearly ordered set of real numbers, '''not''' to just any linearly ordered set. And the proof shows that any '''countable''' subset lacks that property. Essential use of countability was made in the argument. Therefore this does indeed have to do with countability. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 00:21, 20 January 2010 (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)

....OK, a bit more long-windedly again:

* There are '''some''' linearly ordered sets that when partitioned into subsets every member of one of which is less than every member of the other, contain a boundary point (which must of course be a member of one of those two subsets or the other). The set of real numbers in their usual order is such a case.

* There are '''some''' linearly ordered sets ''G'' that can be partitioned into subsets every member of one of which is less than every member of the other such that there is no such boundary point within&nbsp;''G''. Such a boundary point will be found within a completion of ''G'', of which ''G'' itself is a subset. The set of all rational numbers is an example.

:

Your statement about an "indeterminate point" seems to say that that indeterminate point must be '''within'''&nbsp;''G''. But the example of the rational numbers shows that that is not true.

:

At any rate, it's clear that you haven't understood the argument. You also haven't said what you mean by "indeterminate". [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 00:40, 20 January 2010 (UTC)

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

:: That I haven't understood the argument is correct. However, I think the problem is not with my understanding but rather with the argument. The article states 4 properties that are not equivalent to the same requirement for rational numbers to be considered countable. In other words, there is nothing that demonstrates on any level that the real numbers can be placed into a one to one correspondence with the natural numbers '''or not'''. Well, if this is what was written in Cantor's original work, it does not matter whether it is correct or not as far as Wikipedia is concerned. After all, the goal is to report the argument, not to debate it. Try not to be so arrogant Hardy. [[Special:Contributions/91.105.179.213|91.105.179.213]] ([[User talk:91.105.179.213|talk]]) 05:23, 20 January 2010 (UTC)

! 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

|-

| || || || || || ''':'''

|}

The four requirements have nothing to do with the rationals being countable. The four requirements are satisfied by the '''real''' numbers in their usual order, not the rational numbers. The claim is that any linearly ordered set satisfying those four requirements is uncountable. The proof shows that any countable subset must fail to contain every member of the linearly ordered set. The way it is proved is by showing that regardless of which countable set you pick, there will be some elements of the initially considered linearly ordered set that fail to be contained within it.

:

If I try to explain the argument to you, does that make me "arrogant"? [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 08:01, 20 January 2010 (UTC)

::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.

::Talking down to someone by saying that person '''needs to pay attention''' is disdainful, arrogant and unnecessary.

::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.

::The four requirements have nothing to do with the rational numbers being countable – Yes, that would be correct. As I stated, the 4 properties are Dedekind's definition of a real number. So, what definition of countability does Cantor assume in his definition (''Then R is not '''countable''''') that the real numbers are '''uncountable'''? A few circular references here do not quite add up. Do you know where one can find the original text on the internet? The wiki link to this text is broken - Geocities no longer exists. I read German and am curious to see exactly how Cantor phrased it. [[Special:Contributions/91.105.179.213|91.105.179.213]] ([[User talk:91.105.179.213|talk]]) 10:00, 20 January 2010 (UTC)

::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)

The four properties are ''consequences of'' Dedekind's definition. Those four properties do not apply '''only''' to real numbers, but also to some other linearly ordered sets.

:

The definition of "countable" is just that a countable set is one whose members are in a sequence: a first one, a second one, a third one, etc., so that each is reached after only finitely many steps. I don't know what '''you''' mean by the word "definition"? You say "his definition (''Then R is not '''countable''''')". That's not a '''definition'''; that's a '''conclusion'''. And you write of "his definition[...]that the real numbers are '''uncountable'''". Again: that's '''not''' a '''definition'''; that's a '''conclusion'''. Using words in odd ways like this that are different from what everybody else uses does not help people understand what you're saying. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 17:59, 20 January 2010 (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.

::The four properties '''are''' Dedekind's definition of real number. These are not '''consequences of''' as you claim. However this is immaterial to my understanding. The problem lies with the formulation of the theorem itself.

:::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:

::Let me see if I can explain this more simply:

::"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"

::Article states: Suppose a set R has the following '''4 properties''' then R is not '''countable'''.

::This logic is equivalent to stating that proposition P implies NOT proposition C. P can be the statement of the 4 properties. But what is C? Countability, one would imagine. If so, where is countability defined? If this Cantor's first countability proof, then where did he define countability seeing the proof uses the word '''countable''' as in '''R is not countable'''. Make sense to you now? [[Special:Contributions/91.105.179.213|91.105.179.213]] ([[User talk:91.105.179.213|talk]]) 18:39, 20 January 2010 (UTC)

::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]]":

Dedekind's definition of real number identified a real number with a Dedekind cut in the set of rational numbers. There are other linearly ordered sets than the rationals with which one can do the same thing, and get different linearly ordered sets satisfying the same properties, but those are not the set of real numbers.

:

I believe the way Cantor wrote the argument originally is simply that he set no sequence contains all real numbers, understanding ''sequence'' to mean something that has a first element, a second, a third, and so on, each element being reached after some finite number of steps. He also wrote in the same paper that there ''is'' a sequence containing all real algebraic numbers, and showed how to construct it.

:

I looked for the paper in pdf format and was surprised that I couldn't find it quickly. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 06:01, 21 January 2010 (UTC)

::"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:"

::I can't comment about Cantor's sequence containing all real algebraic numbers. My first reaction is it does not sound correct. Regarding your comments on linearly ordered sets, again, property 4 of the theorem makes it clear Cantor is dealing with '''real''' numbers (least upper bound property implies c is real number).

::You could not find the paper ''quickly'' - does this imply you found it? If so, please provide a link to it. [[Special:Contributions/91.105.179.213|91.105.179.213]] ([[User talk:91.105.179.213|talk]]) 07:42, 21 January 2010 (UTC)

::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 have not found it on the web. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 21:30, 21 January 2010 (UTC)

::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)

Saying that the four conditions mentioned in the article are Dedekind's definition of "real number" seems quite confused. Dedekind identified a real number with Dedekind cut within the linearly ordered set of rational numbers. Such a cut is a partition of the rationals into two sets ''A'',&nbsp;''B'', with every element of ''A'' less than every element of ''B''. It is '''not''' the case that the rationals contain a boundary point between those two sets, which would have to be a member of one or the other of the two sets. If that were true, what Dedekind was doing wouldn't have worked. It is true that if you partition the '''real''' numbers in that way, then the set of reals contains such a boundary point. But that is certainly not Dedekind's definition of the reals. Nor are the reals the only linearly ordered set with that property; there are others, essentially different from the reals in the sense of not being isomorphic as an ordered set. Hence the four conditions fall short of fully characterizing the reals as a linearly ordered set. Stating the four conditions is simpler than what Dedekind did. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 21:38, 21 January 2010 (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)

::Read property 3 very carefully. Can you show me any other linearly ordered set (besides the rational numbers) of which it can be said: between any two members there is always another? Allow me to answer this question. '''NO'''. Property 3 cannot be 'digested' - such an assumption requires a huge leap of faith. So, since no other linearly ordered set meets this requirement, the matter is settled - property 3 can be reworded as "'''Assume R contains rational numbers'''". Property 4 becomes a bygone conclusion and Cantor's proof is just another flavour of Dedekind's ideas. [[Special:Contributions/91.105.179.213|91.105.179.213]] ([[User talk:91.105.179.213|talk]]) 22:03, 21 January 2010 (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.

:::In my previous post I was assuming the real numbers are '''not well-defined'''. Yes, yes. I know you think the real numbers are well-defined. So I too will assume they are well-defined. Even so, I still see many problems with the theorem.

:::1. Stating the 4 properties, says that R contains real numbers.

:::2. The theorem states that if R meets these conditions, then R is not countable without defining countable. If one accepts that one can first define '''not countable''' and then determine '''countable''' as its '''compliment''', then it might appear that Cantor's first ideas had nothing to do with the notion of a '''one-to-one correspondence of a subset with its proper subset''' as the definition of countable.

:::3. The theorem could be stated as follows: '''Any set that has at least these 4 properties is not countable'''. But to state it this way, is the same as saying the real numbers are not countable by virtue of their '''design/architecture'''.

:::4. In set theory, if a proper subset contains a certain property, then it implies the superset must also posses that property by '''inclusion''' of the subset. So if the rational numbers are countable, this would imply the real numbers must therefore also be countable. [[Special:Contributions/91.105.179.213|91.105.179.213]] ([[User talk:91.105.179.213|talk]]) 05:36, 22 January 2010 (UTC)

::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)

Sorry, you're quite wrong. You wrote:

:: Can you show me any other linearly ordered set (besides the rational numbers) of which it can be said: between any two members there is always another? Allow me to answer this question. '''NO'''.

:

That is nonsense. I can easily show you many such sets. One of them is the real numbers. Another is the real algebraic numbers. Another is rational functions in one variable with real coefficients with a certain natural ordering. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 05:42, 22 January 2010 (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.

You wrote:

:::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.

:: In set theory, if a proper subset contains a certain property, then it implies the superset must also posses that property by '''inclusion''' of the subset.

:::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)

:

That is nonsense. The set {1,&nbsp;2,&nbsp;3} has the property of finiteness. The set {1,&nbsp;2,&nbsp;3,&nbsp;...} does not. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 05:47, 22 January 2010 (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)

You wrote:

: property 3 can be reworded as "Assume R contains rational numbers". Property 4 becomes a bygone conclusion

:

That is nonsense. The set of all real algrebraic numbers contains the set of all rational numbers, but does not satisfy property 4 at all. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 05:51, 22 January 2010 (UTC)

== Contrast 2nd theorem with sequence of rational numbers? ==

::It's quite amazing to me how you picked out what you felt like answering and ignored the big elephant in the room. So, I respond as follows: I posted a second comment that clarified my first assumption. Your example of a finite and infinite set is irrelevant - all the sets we are discussing are infinite: naturals, rationals and reals. So how about addressing the real issues -

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)

{{Talk:Georg Cantor's first set theory article/GA1}}

::1. Stating the 4 properties, says that R contains real numbers.

:::2. The theorem states that if R meets these conditions, then R is not countable without defining countable. If one accepts that one can first define '''not countable''' and then determine '''countable''' as its '''compliment''', then it might appear that Cantor's first ideas had nothing to do with the notion of a '''one-to-one correspondence of a subset with its proper subset''' as the definition of countable.

:::3. The theorem could be stated as follows: '''Any set that has at least these 4 properties is not countable'''. But to state it this way, is the same as saying the real numbers are not countable by virtue of their '''design/architecture'''.

:::4. In set theory, if a proper subset contains a certain property, then it implies the superset must also posses that property by '''inclusion''' of the subset. So if the rational numbers are countable, this would imply the real numbers must therefore also be countable. [[Special:Contributions/91.105.179.213|91.105.179.213]] ([[User talk:91.105.179.213|talk]]) 09:14, 22 January 2010 (UTC)