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Extra for Dedekind section[edit]

Cite error: There are <ref> tags on this page without content in them (see the help page). Ferreirós suggests that the reason Cantor did not acknowledge Dedekind's help was his desire to maintain good relations with the Berlin mathematicians, two of which—Kronecker and Ernst Kummer—were angry with Dedekind for publishing his theory of algebraic integers before Kronecker published his equivalent theory (Ferreirós 2007, p. 185). However, two facts seem to contradict Ferreirós' reasoning. In the introduction to his next article, Cantor mentions Dedekind's "beautiful investigations" in algebraic number theory and its use of countably infinite sets (Cantor 1878, p. 243; French translation: Cantor 1883b, p. 312). In 1880, Kronecker proposed that Dedekind be made a member of the Berlin Academy of Sciences for his work in algebraic number theory (Ferreirós 2007, p. 111).

Extra for emphasis section[edit]

Also, at that time, the countability of the real algebraic numbers was the most useful result: Weierstrass used it to construct a function and Cantor used it together with his second theorem to give a new method of constructing transcendentals. The only use the uncountability result had was a non-constructive proof of the existence of transcendentals, which hade bee had two applications

Care must be taken when dealing with Cantor's 1874 remark. For example, Ferreirós suspects that in 1873 Weierstrass might not have accepted the idea that infinite sets can have different sizes. However, Cantor makes no claim about infinite sets having different sizes until his 1878 article where he defines cardinality. All that he has proved in 1874 is that the real algebraic numbers can be written as a sequence, and the real numbers cannot. Weierstrass was amazed by the first result, so it is safe to say that before seeing Cantor's work, Weierstrass probably would have thought that neither set can be written as a sequence. The surprising results are that the first set can and that Cantor can actually prove that the second set cannot. Even with ZFC set theory, we can only really say that the set of real numbers cannot be written as a sequence. Skolem showed that if ZFC is consistent, it has a countable model. This means the set of real numbers can be put into one-to-one correspondence with the positive integers, but this correspondence does not belong to the model.


The following year, Weierstrass "stated that two 'infinitely great magnitudes' are not comparable and can always be regarded as equal."[1] Weierstrass' opinion on infinite sets may have led him to advise Cantor to omit his remark on the essential difference between the collections of real numbers and real algebraic numbers.[2]


Dauben uses examples from Cantor's article to show Kronecker's influence.[3] For example, Cantor did not prove the existence of the limits used in the proof of his second theorem.[4] Cantor did this despite using Dedekind's version of the proof. In his private notes, Dedekind wrote:

… [my] version is carried over almost word-for-word in Cantor's article (Crelle's Journal, 77); of course my use of "the principle of continuity" is avoided at the relevant place …[5]

The "principle of continuity" requires a general theory of the irrationals, such as Cantor's or Dedekind's construction of the real numbers from the rationals. Kronecker accepted neither theory.[6]


It may seem strange to omit a remark and plan to add it to the proofs. Also, Kronecker might read the original article, but probably would not see the modified proofs since only the editor usually sees them.

Carl Borchardt

Ferreirós' strongest statement about the "local circumstances" mentions both Kronecker and Weierstrass: "Had Cantor emphasized it [the uncountability result], as he had in the correspondence with Dedekind, there is no doubt that Kronecker and Weierstrass would have reacted negatively."[7] Ferreirós also mentions another aspect of the local situation: Cantor, thinking of his future career in mathematics, desired to maintain good relations with the Berlin mathematicians.Cite error: A <ref> tag is missing the closing </ref> (see the help page).

With Mr. Weierstrass I had good relations. … Of the conception of enumerability [countability] of which he heard from me at Berlin on Christmas holydays 1873, he became at first quite amazed, but [after] one or two days passed over, it became his own and helped him to an unexpected development of his wonderful theory of functions.[8]
Although I did not yet wish to publish the subject I recently for the first time discussed with you, I have nevertheless unexpectedly been caused to do so. I communicated my results to Mr. Weierstrass on the 22nd; … on the 23rd I had the pleasure of a visit from him, at which I could communicate the proofs to him. He was of the opinion that I must publish the thing at least in so far as it concerns the algebraic numbers. So I wrote a short paper with the title: "On a property of the set of real algebraic numbers," and sent it to Professor Borchardt[9] to be considered for the Journal für Math [Crelle's Journal].[10]
The restriction which I have imposed on the published version of my investigations is caused in part by local [Berlin] circumstances (about which I shall perhaps later speak with you orally) and in part because I believe that it is important to apply my ideas at first to a single case (such as that of the real algebraic numbers) …[10]

Cantor visited Weierstrass in Berlin two weeks after his letter to Dedekind stating his second theorem. At the time, Cantor wanted to discover more results before publishing, but Weierstrass was interested in the countability of the real algebraic numbers and convinced Cantor to write an article that at least contained his results concerning these numbers. Weierstrass also advised Cantor to omit his remark on the difference between the set [a, b] and countable sets such as the real algebraic numbers, but mentioned that it could be added as a marginal note during proofreading. Cantor's remark is weak: it only states that the real numbers in [a, b] cannot be written as a sequence. It does not state that this set forms a larger infinite set than a countably infinite set. Cantor will not define the cardinality of infinite sets and how to compare cardinalities until his next article, which appeared in 1878. Nevertheless, Weierstrass advised omitting this remark until proofreading. Cantor will add his remark at proofreading and it appears as the last sentence in his introduction.

It was Weierstrass who convinced Cantor to write an article. In a letter to Dedekind, Cantor says that he "did not yet wish to publish" his results, but on a trip to Berlin, he had shown his work to Weierstrass who convinced him to publish at least his results about the real algebraic numbers.

Cantor did not plan on publishing his results yet Cantor's remark about the uncountability of an interval of real numbers, which appears at the end of his introductory section, was not part of his submitted article. He added it during his proofreading. This explains the choice of the title "..." since the countability of the real algebraic numbers was the only new result in the submitted article. However, it does not explain why Cantor left the remark out.


  1. Cantor restricted his first theorem to real algebraic numbers even though Dedekind proved that the set of algebraic numbers was countable.
  2. Cantor gave

During the Christmas holidays, Cantor visited Berlin and showed his work to his former professor Karl Weierstrass. On December 25, Cantor wrote to Dedekind about his decision to publish:

Although I did not yet wish to publish the subject I recently for the first time discussed with you, I have nevertheless unexpectedly been caused to do so. I communicated my results to Mr. Weierstrass on the 22nd; … on the 23rd I had the pleasure of a visit from him, at which I could communicate the proofs to him. He was of the opinion that I must publish the thing at least in so far as it concerns the algebraic numbers. So I wrote a short paper with the title: "On a property of the set of real algebraic numbers," and sent it to Professor Borchardt[11] to be considered for the Journal für Math [Crelle's Journal].[10]

In a letter to Philip Jourdain, Cantor provided more details of Weierstrass' reaction:

With Mr. Weierstrass I had good relations. … Of the conception of enumerability [countability] of which he heard from me at Berlin on Christmas holydays 1873, he became at first quite amazed, but [after] one or two days passed over, it became his own and helped him to an unexpected development of his wonderful theory of functions.[12]

Weierstrass probably urged Cantor to publish because he found the countability of the set of algebraic numbers both surprising and useful.[13] On December 27, Cantor told Dedekind more about his article, and mentioned its quick acceptance (only four days after submission):[14]

The restriction which I have imposed on the published version of my investigations is caused in part by local [Berlin] circumstances (about which I shall perhaps later speak with you orally) and in part because I believe that it is important to apply my ideas at first to a single case (such as that of the real algebraic numbers) …[10]
As Mr. Borchardt has already responded to me today, he will have the kindness to include this article soon in the Math. Journal.[15]

Cantor gave two reasons for restricting his article: "local circumstances" and the importance of applying "my ideas at first to a single case." Cantor never told Dedekind what the "local circumstances" were.[16] This has led to a controversy: Who influenced Cantor so that his article emphasizes the countability of the set of algebraic numbers rather than the uncountability of the set of real numbers? This controversy is also fueled by Cantor's earlier letters, which indicate that he was most interested in the set of real numbers.

Cantor biographer Joseph Dauben argues that "local circumstances" refers to the influence of Leopold Kronecker, Weierstrass' colleague at the University of Berlin. Dauben states that publishing in Crelle's Journal could be difficult because Kronecker, a member of the journal's editorial board, had a restricted view of what was acceptable in mathematics.[17] Dauben argues that to avoid publication problems,[18] Cantor wrote his article to emphasize the countability of the set of real algebraic numbers.

Dauben uses examples from Cantor's article to show Kronecker's influence.[19] For example, Cantor did not prove the existence of the limits used in the proof of his second theorem.[20] Cantor did this despite using Dedekind's version of the proof. In his private notes, Dedekind wrote:

… [my] version is carried over almost word-for-word in Cantor's article (Crelle's Journal, 77); of course my use of "the principle of continuity" is avoided at the relevant place …[21]

The "principle of continuity" requires a general theory of the irrationals, such as Cantor's or Dedekind's construction of the real numbers from the rationals. Kronecker accepted neither theory.[22]

In his history of set theory, José Ferreirós analyzes the situation in Berlin and arrives at a different conclusion. Ferreirós' strongest statement about the "local circumstances" mentions both Kronecker and Weierstrass: "Had Cantor emphasized it [the uncountability result], as he had in the correspondence with Dedekind, there is no doubt that Kronecker and Weierstrass would have reacted negatively."[23]

Ferreirós emphasizes Weierstrass' influence: Weierstrass was interested in the countability of the set of real algebraic numbers because he could use it to build interesting functions.[24] Also, Ferreirós suspects that in 1873 Weierstrass might not have accepted the idea that infinite sets can have different sizes. The following year, Weierstrass "stated that two 'infinitely great magnitudes' are not comparable and can always be regarded as equal."[25] Weierstrass' opinion on infinite sets may have led him to advise Cantor to omit his remark on the essential difference between the collections of real numbers and real algebraic numbers.[26] (This remark appears above in "The article.") Cantor mentions Weierstrass' advice in his December 27 letter:

The remark on the essential difference of the collections, which I could have very well included, was omitted on the advice of Mr. Weierstrass; but [he also advised that I] could add it later as a marginal note during proofreading.[27]

Ferreirós' strongest statement about the "local circumstances" mentions both Kronecker and Weierstrass: "Had Cantor emphasized it [the uncountability result], as he had in the correspondence with Dedekind, there is no doubt that Kronecker and Weierstrass would have reacted negatively."[28] Ferreirós also mentions another aspect of the local situation: Cantor, thinking of his future career in mathematics, desired to maintain good relations with the Berlin mathematicians.[29] This desire could have motivated Cantor to create an article that appealed to Weierstrass' interests, and did not antagonize Kronecker.[30]

Dirichlet and existence proofs[edit]

History and Philosophy of Modern Mathematics, vol. 11, p. 249

Generating an irrational using Cantor's construction
Interval Interval (decimal) Rational numbers outside interval


Extra[edit]

Steward 2015, p. 285: "Meanwhile Georg Cantor, in 1874, had produced a revolutionary proof of the existence of transcendental numbers, without actually constructing any."

Pudlák 2013, p. 259: "Cantor's proof shows that almost every number is transcendental, but it does not provide a single concrete example of a transcendental number.

Chowdhary 2015, p. 19: "A consequence of this is that there exists a multitude of transcendental numbers, even though the proof, by contradiction, does not produce a single specific example."

Frazer Jarvis, Algebraic Number Theory NC, p. 18 2014

Springer, 978-3-319-07544-0 "...the simplest way to prove the existence of t.n. is due to Cantor himself (1874) although it does not give any way to construct them."

Ian Stewart, David Tall 2015 The Foundations of Mathematics, 2nd ed. Oxford, 978-0-19-870644-1 p. 333


This disagreement shows no signs of being resolved. Since 2014, at least two books appeared stating that Cantor's proof is constructive, and at least four appeared stating that his proof does not construct any (or a single) transcendental.[31]

  • Grattan-Guinness, Ivor (1971), "The Correspondence between Georg Cantor and Philip Jourdain", Jahresbericht der Deutschen Mathematiker-Vereinigung, 73: 111–130.
  • Havil, Julian (2012), The Irrationals, Princeton University Press, ISBN 978-0-691-14342-2.
  • Kanamori, Akihiro (2012), "Set Theory from Cantor to Cohen" (PDF), in Gabbay, Dov M.; Kanamori, Akihiro; Woods, John (eds.), Handbook of the History of Logic, Volume 6: Sets and Extensions in the Twentieth Century, Elsevier, pp. 1–71, ISBN 978-0-444-51555-1 {{citation}}: External link in |title= (help).
  • Pudlák, Pavel (2013), Logical Foundations of Mathematics and Computational Complexity, Springer, ISBN 978-3-319-00119-7.

http://www.academia.edu/6160599/Eric_Temple_Bells_Men_of_Mathematics_From_Influential_to_Infamous Eric Temple Bell's Men of Mathematics: From Influential to Infamous V. Frederick Rickey Published 1937 Continuously in print Libraries have 4300 copies

grouped references will help structure the article!!



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Proof

The proof uses Farey sequences and continued fractions. The Farey sequence Fn is the increasing sequence of completely reduced fractions whose denominators are ≤ n. If a/b and c/d are adjacent in a Farey sequence, the lowest denominator fraction between them is their mediant (a + c)/(b + d). This mediant is adjacent to both a/b and c/d in the Farey sequence Fb + d . (LeVeque 2002, pp. 154–155.) Cantor's construction produces mediants because we sequenced rational numbers by increasing denominator. The first interval in the table is [1/3, 1/2]. Since 1/3 and 1/2 are adjacent in F3, their mediant 2/5 is the first rational in the sequence between 1/3 and 1/2. Since 2/5 is adjacent to both 1/3 and 1/2 in F5, the next fraction is 3/7, the mediant of 2/5 and 1/2. Since we use mediants, we have: 1/3 < 2/5 < 3/7 < 1/2. So the second interval is [2/5, 3/7].

We will prove that the endpoints of the intervals converge to the continued fraction [0, 2, 2,…]. This continued fraction is the limit of its convergents pn/qn = [0, 2,… 2] (n 2's). The pn and qn sequences satisfy the equations: p0 = 0, p1 = 1; q0 = 1, q1 = 2; pn + 1 = 2pn + pn − 1 and qn + 1 = 2qn + qn − 1 for n ≥ 1. (LeVeque 2002, pp. 174–175.)

First, we prove by induction that the n-th interval in the table is ((pn + pn − 1)/(qn + qn − 1), pn/qn) for odd n. For even n, the endpoints are reversed.

This is true for the first interval since (1/3, 1/2) = [(p1 + p0)/(q1 + q0), p1/q1].

Assume that the inductive hypothesis is true for the k-th interval. If k is odd, this interval is [(pk + pk − 1)/(qk + qk − 1), pk/qk]. The mediant of its endpoints, (2pk + pk − 1)/(2qk + qk − 1) = pk + 1/qk + 1, is the first fraction between these endpoints. The second fraction is (pk + 1 + pk)/(qk + 1 + qk), the mediant of pk + 1/qk + 1 and pk/qk. Since we use mediants, we have: (pk + pk − 1)/(qk + qk − 1) < pk + 1/qk + 1 < (pk + 1 + pk)/(qk + 1 + qk) < pk/qk. So the (k + 1)-st interval is [pk + 1/qk + 1, (pk + 1 + pk)/(qk + 1 + qk)]. Note that pk + 1/qk + 1 is the left endpoint, which is required since k + 1 is even. Thus, the inductive hypothesis is true for the (k + 1)-st interval. For even k, the proof is similar.

Since the left endpoints of the intervals are increasing and every other endpoint is p2n/q2n, their limit equals limn → ∞ pn/qn. The right endpoints have the same limit because they are decreasing and every other endpoint is p2n − 1/q2n − 1. As mentioned above, this limit is the continued fraction [0, 2, 2,…], which equals √2 − 1. (Weisstein 2003, p. 541.)


Sometimes, a mathematician may not produce a constructive proof quickly and publish the On the other hand, Hilbert published the first widely-acknowledged non-constructive proof in 1890 solving a problem of Gordon's in invariant theory. Some authors claim that Gordon's reaction was. However, even Hilbert (who strongly supported non-constructive proofs) would publish a constructive proof two years later.

Old ASCII Case Diagrams[edit]

——(—————|——–|———)————
aN c xn bN
Case 1: Last interval (aN, bN)

Either the number of intervals generated is finite or infinite. Case 1: if finite, let (aN, bN) be the last interval. Since aN < bN and at most one xn can be in (aN, bN), any c in this interval besides xn (if it exists) does not belong to the given sequence.

——(——[———|——]———)———|—
an a∞ c b∞ bn xn
Case 3: a∞ < b∞
——(———————|———–)———|—
an a∞ bn xn
Case 2: a∞ = b∞

Cantor presents his results cautiously. CITE AKI'S article. He emphasizes countability and only states that there is a "clear difference" between the set of reals and a countable set, he does not claim that the set of reals form a set of larger infinity than a countable set. He will not define how to compare the size of infinite sets until his 1878 article. Cantor's 1874 article only introduces the concept of countability, provides an example of a countable dense set. Some of Cantor's later work did generate some controversy, but there was also interest in his work. Most of his articles up to 1883 were translated by French mathematicians and these translations were published that year and the next? in Acta Mathematica. These articles include his work on trigonometric series, his 1874 article, his 1878 article in which he first introduced , his series of articles on what we now call point-set topology, and his 1883 article on transfinite ordinal numbers.

Borel and Lebesque were "semi-intuitionists" who criticized parts of set theory, but found other parts of Cantor's theory very useful. Borel even used Cantor's countable ordinals (which are a transfinite extension of the natural numbers) in his first proof of the Heine-Borel theorem.

  1. ^ Ferreirós 2007, p. 184, footnote 3. However, by the mid-1880s, Weierstrass accepted Cantor's discovery that infinite sets can have different cardinalities. (Ferreirós 2007, p. 185.)
  2. ^ Ferreirós 2007, p. 184.
  3. ^ Dauben 1979, pp. 66–70.
  4. ^ These are the limits: a∞ =  limn → ∞ an  and  b∞ =  limn → ∞ bn (see "The proofs").
  5. ^ Dauben 1979, p. 67. Dedekind's principle of continuity appears in his 1872 construction of the real numbers, which uses Dedekind cuts of rational numbers. The principle of continuity states: For every cut in the real numbers, there is a unique real number that produces this cut. (A cut separates the real numbers into two sets A and B such that every member A is less than every member of B. A real number produces a cut if it is either the greatest member of A or the least member of B.) Dedekind's principle of continuity is equivalent to the least upper bound property: Every set of real numbers that is bounded above has a least upper bound. (Dedekind 1901, p. 9.)
  6. ^ Dauben 1979, p. 69.
  7. ^ Ferreirós 2007, p. 185.
  8. ^ Grattan-Guinness 1971, p. 124. (The letter was written in 1905.)
  9. ^ The editor of Crelle's Journal, which was called Borchardt's Journal at the time.
  10. ^ a b c d Noether and Cavaillès 1937, p. 17. English translation: Ewald 1996, pp. 847–848.
  11. ^ The editor of Crelle's Journal, which was called Borchardt's Journal at the time.
  12. ^ Grattan-Guinness 1971, p. 124. (The letter was written in 1905.)
  13. ^ Cantor realized that his fellow mathematicians might be initially surprised by this result. In his article, he states: "The real algebraic numbers form in their totality a set of numbers, which shall be designated by (ω); as is readily seen, (ω) has the property that in every neighbourhood of any given number α there are infinitely many numbers from (ω). So it ought at first glance to be all the more striking that one can correlate the set (ω) one-to-one with the set … of all positive integers …" (Cantor 1874, p. 258, Ewald 1996, p. 840).
  14. ^ The submission date, December 23, 1873, appears at the end of the article—see Cantor 1874, p. 262.
  15. ^ Dugac 1976, p. 226. Cantor's article was published within weeks after its submission (Dauben 1993, p. 5).
  16. ^ Noether and Cavaillès 1937, p. 20. English translation: Ewald 1996, p. 850.
  17. ^ Dauben 1993 (p. 5) states: "By the early 1870s, Kronecker was already vocal in his opposition to the Bolzano-Weierstrass theorem, upper and lower limits, and to irrational numbers in general." For more on Kronecker's views: see Dauben 1979, pp. 66–70, and Dauben 1993, p. 5–7.
  18. ^ Cantor's colleague, Eduard Heine, had recently experienced a publication delay because of Kronecker. (Dauben 1979, p. 67; Dauben 1993, p. 5.)
  19. ^ Dauben 1979, pp. 66–70. Gray 1994 (p. 828) also gives an example: "… [Cantor's] theorems only deal with sequences that are ordered by a 'law.'" Gray states that "Cantor may have inserted this restriction to avoid problems with Kronecker." However, Ferreirós 2007 (p. 263) corrects Gray: "Like all other authors in the early period of the history of sets, he [Cantor] tended to think of them [sets] as given by a concept or a law, indeed he emphasized this aspect more than other contemporaries." Ferreirós provides examples from Cantor's articles dating from 1872 to 1883.
  20. ^ These are the limits: a∞ =  limn → ∞ an  and  b∞ =  limn → ∞ bn (see "The proofs").
  21. ^ Dauben 1979, p. 67. Dedekind's principle of continuity appears in his 1872 construction of the real numbers, which uses Dedekind cuts of rational numbers. The principle of continuity states: For every cut in the real numbers, there is a unique real number that produces this cut. (A cut separates the real numbers into two sets A and B such that every member A is less than every member of B. A real number produces a cut if it is either the greatest member of A or the least member of B.) Dedekind's principle of continuity is equivalent to the least upper bound property: Every set of real numbers that is bounded above has a least upper bound. (Dedekind 1901, p. 9.)
  22. ^ Dauben 1979, p. 69.
  23. ^ Ferreirós 2007, p. 185.
  24. ^ For example, Weierstrass used it to build a function that is continuous everywhere but differentiable only at transcendental numbers. Ferreirós 2007, p. 184.
  25. ^ Ferreirós 2007, p. 184, footnote 3. However, by the mid-1880s, Weierstrass accepted Cantor's discovery that infinite sets can have different cardinalities. (Ferreirós 2007, p. 185.)
  26. ^ Ferreirós 2007, p. 184.
  27. ^ Dugac 1976, p. 226; Ferreirós 2007, p. 184. Cantor did add his remark during proofreading.
  28. ^ Ferreirós 2007, p. 185.
  29. ^ Ferreirós 2007 (p. 185–186) mentions this subject as one possible reason why Cantor did not acknowledge Dedekind's help in his article. Cantor acknowledged the help of Kronecker, Weierstrass, Schwarz, and Heine in earlier articles. He did thank Dedekind privately in his December 25 letter: "… your comments (which I value highly) and your manner of putting some of the points were of great assistance to me." Dedekind's private notes on his correspondence reveal how great this assistance was. Dedekind wrote: "… [I] stated and fully proved the theorem that even the totality of all algebraic numbers can be correlated in the stated manner with the totality (n) of all natural numbers. (Shortly thereafter, this theorem and proof appeared almost word-for-word in Cantor's paper in Crelle, Vol. 77, …)" Dedekind's proof of Cantor's second theorem also appears in the article (see quotation above in discussion of Dauben's views). Ferreirós suspects that one reason for Cantor not acknowledging Dedekind's help was his desire to maintain good relations with the Berlin mathematicians. Two of these mathematicians—Kronecker and Kummer—were angry with Dedekind for publishing his theory of algebraic integers before Kronecker published his equivalent theory. Dedekind published his theory in 1871. Kronecker's theory dates back to 1858, but he published it in 1883. For more details on Dedekind's contributions to Cantor's article, see Ferreirós 2007, pp. 177–186. The quotations above are from: Noether and Cavaillès 1937, pp. 17–19; English translation: Ewald 1996, pp. 847–849.
  30. ^ In his December 27 letter, Cantor mentions Weierstrass' interests when he states "it is important to apply my ideas at first to a single case (such as that of the real algebraic numbers)." His desire to avoid antagonizing Kronecker appears in Dauben's analysis, and in Ferreirós' statement "that Kronecker and Weierstrass would have reacted negatively" if Cantor had emphasized his uncountability result.
  31. ^ Proof is constructive: Havil 2012, p. 206-209; Kanamori 2012, pp. 3-5; Dasgupta 2014, p. 107; Sheppard 2014, pp. 131-132. Proof is non-constructive: Pudlák 2013, pp. 258-259; Jarvis 2014, p. 18; Chowdhary 2015, p. 19; Steward 2015, p. 285.
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