Talk:Cantor's first uncountability proof

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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? -- 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. 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. 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. 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. -- 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.

BenCawaling@Yahoo.com

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? [BenCawaling@Yahoo.com (27 Sep 2005)]
No. You're saying "For a given proposed enumeration (xn) 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. 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 ... [BenCawaling@Yahoo.com (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! 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 {Xn} = {Yn} U {An} U {Bn} 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 {Yn} or in {An} or in {Bn} when C is the limit, and vice versa on “C” and “D” — otherwise, {Xn} 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 — 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 π-3 in binary, <0.0, 0.00, 0.001, 0.0010, 0.00100, 0.001001, …, π-3> <--> <0, 0, 1, 0, 0, 1, …, 1ω) — would simply have irrational anti-diagonal (with 1ω if diagonal is rational (without or with 0ω), 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 — 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 “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…dn…, where dn 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 — 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…dn…1ω due to the 1-1 correspondence, say π – 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, …, 1ω. [BenCawaling@Yahoo.com] 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. Dmharvey Image:User_dmharvey_sig.png 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. YohanN7 (talk) 02:20, 30 November 2007 (UTC)

Contents

[edit] 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:

  1. R needs to be non-empty which is currently not required. (but see 2 below)
  2. 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.)
  3. 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.
  4. 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.
  5. The proof should make explicit how the density property is used.

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

[edit] "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. 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? 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. 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? 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. 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. AxelBoldt 14:42, 3 May 2006 (UTC)

[edit] The rationals

Copied from the main article:

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

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

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

[edit] 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). --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. --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.--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.)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.YohanN7 (talk) 00:51, 30 November 2007 (UTC)
  • Note: the term "gapless" may be better than "complete". --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)." Ossi 19:45, 8 September 2007 (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. — Carl (CBM · talk) 13:15, 23 July 2007 (UTC)