Cantor's First Uncountability Proof - Why Does Cantor's Article Emphasize The Countability of The Algebraic Numbers?

Why Does Cantor's Article Emphasize The Countability of The Algebraic Numbers?

During the Christmas holidays, Cantor visited Berlin and showed his work to his former professor Karl Weierstrass. On December 25, Cantor wrote 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 to be considered for the Journal für Math .

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 of which he heard from me at Berlin on Christmas holydays 1873, he became at first quite amazed, but 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 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:

… 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 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 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, 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.