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Extra for Dedekind section
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
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." 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.

Dauben uses examples from Cantor's article to show Kronecker's influence. For example, Cantor did not prove the existence of the limits used in the proof of his second theorem. 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 …

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.

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." 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. Ferreirós 2007 (p. 185–186)

Cantor biographer Joseph Dauben argues that "local circumstances" refers to 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.[15] Dauben argues that to avoid publication problems,[16] Cantor wrote his article to emphasize the countability of the set of real algebraic numbers.

advised him to omit his remark containing the uncountability of the set of real numbers, but also mentioned that he could add it during proofreading, which he did. Omitting this remark meant that the only significant new result was the countability of the set of real algebraic numbers, which explains the article's title. Further advice from Weierstrass probably explains the other facts listed above.

In a letter to Dedekind, Cantor gives 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.


 * 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.


 * 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  to be considered for the Journal für Math [ Crelle's Journal ] .


 * 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) …

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.

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:
 * 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


 * 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  to be considered for the Journal für Math [ Crelle's Journal ] .

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.

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


 * 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) …


 * As Mr. Borchardt has already responded to me today, he will have the kindness to include this article soon in the Math. Journal.

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. 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. Dauben argues that to avoid publication problems, 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. For example, Cantor did not prove the existence of the limits used in the proof of his second theorem. 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 …

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.

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."

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. 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." 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. (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.

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." 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. This desire could have motivated Cantor to create an article that appealed to Weierstrass' interests, and did not antagonize Kronecker.

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

Extra
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.



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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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
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.

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.