Cantor's first set theory article: Difference between revisions

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== Cantor's 1879 uncountability proof ==

=== Everywhere dense ===

In 1879, Cantor published a new uncountability proof that modifies his 1874 proof. He first defines the [[topological]] notion of a point set ''P'' being "everywhere [[dense set|dense]] in an interval":{{efn-ua|Cantor was not the first to define "everywhere dense" but his terminology was adopted with or without the "everywhere" (everywhere dense: {{harvnb|Arkhangel'skii|Fedorchuk|1990|p=15}}; dense: {{harvnb|Kelley|1991|p=49}}). In 1870, [[Hermann Hankel]] had defined this concept using different terminology: "a multitude of points …... ''fill the segment'' if no interval, however small, can be given within the segment in which one does not find at least one point of that multitude" ({{harvnb|Ferreirós|2007|p=155}}). Hankel was building on [[Peter Gustav Lejeune Dirichlet]]'s 1829 article that contains the [[Dirichlet function]], a non-([[Riemann integral|Riemann]]) [[integrable function]] whose value is 0 for [[rational number]]s and 1 for [[irrational number]]s. ({{harvnb|Ferreirós|2007|p=149}}.)}}

:If ''P'' lies partially or completely in the interval [α, β], then the remarkable case can happen that ''every'' interval [γ, δ] contained in [α, β], ''no matter how small,'' contains points of ''P''. In such a case, we will say that ''P'' is ''everywhere dense in the interval'' [α, β].{{efn-ua|Translated from {{harvnb|Cantor|1879|p=2}}: {{lang-de|Liegt ''P'' theilweise oder ganz im Intervalle (α . . . β), so kann der bemerkenswerthe Fall eintreten, dass ''jedes noch so kleine'' in (α . . . β) enthaltene Intervall (γ . . . δ) Punkte von ''P'' enthält. In einem solchen Falle wollen wir sagen, dass ''P'' ''im Intervalle'' (α . . . β) ''überall-dicht'' sei.|label=none|italic=unset}}}}

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=== Cantor's 1879 proof ===

Cantor modified his 1874 proof with a new proof of its [[#Second theorem|second theorem]]: Given any sequence ''P'' of real numbers ''x''₁, ''x''₂, ''x''₃,&nbsp;... and any interval [''a'',&nbsp;''b''], there is a number in [''a'',&nbsp;''b''] that is not contained in ''P''. Cantor's new proof has only two cases. First, it handles the case of ''P'' not being dense in the interval, then it deals with the more difficult case of ''P'' being dense in the interval. This division into cases not only indicates which sequences are more difficult to handle, but it also reveals the important role denseness plays in the proof.<ref group=proof name=p>Since Cantor's proof has not been published in English, an English translation is given alongside the original German text, which is from {{harvnb|Cantor|1879|pp=5–7}}. The translation starts one sentence before the proof because this sentence mentions Cantor's 1874 proof. Cantor states it was printed in Borchardt's Journal. Crelle’sCrelle's Journal was also called Borchardt’sBorchardt's Journal from 1856-1880 when [[Carl Wilhelm Borchardt]] edited the journal ({{harvnb|Audin|2011|p=80}}). Square brackets are used to identify this mention of Cantor's earlier proof, to clarify the translation, and to provide page numbers. Also, "{{lang|de|Mannichfaltigkeit|italic=no}}" (manifold) has been translated to "set" and Cantor's notation for closed sets (α&nbsp;.&nbsp;.&nbsp;.&nbsp;β) has been translated to [α,&nbsp;β]. Cantor changed his terminology from {{lang|de|Mannichfaltigkeit|italic=no}} to {{lang|de|Menge|italic=no}} (set) in his 1883 article, which introduced sets of [[ordinal number]]s ({{harvnb|Kanamori|2012|p=5}}). Currently in mathematics, a [[manifold]] is type of [[topological space]].

{| class="wikitable collapsible collapsed"

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The development leading to Cantor's 1874 article appears in the correspondence between Cantor and [[Richard Dedekind]]. On November 29, 1873, Cantor asked Dedekind whether the collection of positive integers and the collection of positive real numbers "can be corresponded so that each individual of one collection corresponds to one and only one individual of the other?" Cantor added that collections having such a correspondence include the collection of positive rational numbers, and collections of the form (''a''_{''n''₁,&nbsp;''n''₂,&nbsp;.&nbsp;.&nbsp;.&nbsp;,&nbsp;''n''_''ν''}) where ''n''₁, ''n''₂, .&nbsp;.&nbsp;. , ''n''_''ν'', and ''ν'' are positive integers.<ref>{{harvnb|Noether|Cavaillès|1937|pp=12&ndash;13}}. English translation: {{harvnb|Gray|1994|p=827}}; {{harvnb|Ewald|1996|p=844}}.</ref>

Dedekind replied that he was unable to answer Cantor's question, and said that it "did not deserve too much effort because it has no particular practical interest.". Dedekind also sent Cantor a proof that the set of algebraic numbers is countable.<ref name=Noether18>{{harvnb|Noether|Cavaillès|1937|p=18}}. English translation: {{harvnb|Ewald|1996|p=848}}.</ref>

{{Anchor|Cantor's December 2nd letter}}

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[[File:Oskar Perron.jpg|thumb|upright=0.93|alt=Oskar Perron reading a book while standing in front of a blackboard containing equations|Oskar Perron, {{spaces|4|hair}}c. 1948]]

The correspondence containing Cantor's non-constructive reasoning was published in 1937. By then, other mathematicians had rediscovered his non-constructive, reverse-order proof. As early as 1921, this proof was called "Cantor's proof" and criticized for not producing any transcendental numbers.<ref>{{harvnb|Gray|1994|pp=827&ndash;828}}.</ref> In that year, [[Oskar Perron]] gave the reverse-order proof and then stated: "…... Cantor's proof for the existence of transcendental numbers has, along with its simplicity and elegance, the great disadvantage that it is only an existence proof; it does not enable us to actually specify even a single transcendental number."<ref>{{harvnb|Perron|1921|p=162}}</ref>{{efn-ua|By "Cantor's proof," Perron does not mean that it is a proof published by Cantor. Rather, he means that the proof only uses arguments that Cantor published. For example, to obtain a real not in a given sequence, Perron follows Cantor's 1874 proof except for one modification: he uses Cantor's 1891 diagonal argument instead of his 1874 nested intervals argument to obtain a real. Cantor never used his diagonal argument to reprove his theorem. In this case, both Cantor's proof and Perron's proof are constructive, so no misconception can arise here. Then, Perron modifies Cantor's proof of the existence of a transcendental by giving the reverse-order proof. This converts Cantor's 1874 constructive proof into a non-constructive proof which leads to the misconception about Cantor's work.}}

[[File:Adolf Abraham Halevi Fraenkel.jpg|thumb|upright=0.93|alt=refer to caption|Abraham Fraenkel, between 1939 and 1949]]

As early as 1930, some mathematicians have attempted to correct this misconception of Cantor's work. In that year, the set theorist [[Abraham Fraenkel]] stated that Cantor's method is "…... a method that incidentally, contrary to a widespread interpretation, is fundamentally constructive and not merely existential."<ref>{{harvnb|Fraenkel|1930|p=237}}. English translation: {{harvnb|Gray|1994|p=823}}.</ref> In 1972, [[Irving Kaplansky]] wrote: "It is often said that Cantor's proof is not 'constructive,' and so does not yield a tangible transcendental number. This remark is not justified. If we set up a definite listing of all algebraic numbers …... and then apply the [[Cantor's diagonal argument|diagonal procedure]] …..., we get a perfectly definite transcendental number (it could be computed to any number of decimal places)."<ref>{{harvnb|Kaplansky|1972|p=25}}.</ref>{{efn-ua|This proof is the same as Cantor's 1874 proof except for one modification: it uses his 1891 diagonal argument instead of his 1874 nested intervals argument to obtain a real.}} Cantor's proof is not only constructive, it is also simpler than Perron's proof, which requires the detour of first proving that the set of all reals is uncountable.<ref>{{harvnb|Gray|1994|pp=829&ndash;830}}.</ref>

Cantor's diagonal argument has often replaced his 1874 construction in expositions of his proof. The diagonal argument is constructive and produces a more efficient computer program than his 1874 construction. Using it, a computer program has been written that computes the digits of a transcendental number in [[polynomial time]]. The program that uses Cantor's 1874 construction requires at least [[sub-exponential time]].<ref>{{harvnb|Gray|1994|pp=821&ndash;824}}.</ref>{{efn-ua|The program using the diagonal method produces <math>n</math> digits in [[Big O notation#Use in computer science|<math>{\color{Blue}O}(n^2 \log^2 n \log \log n)</math>]] steps, while the program using the 1874 method requires at least <math>O(2^{\sqrt[3]{n}})</math> steps to produce <math>n</math> digits. ({{harvnb|Gray|1994|pp=822&ndash;823}}.)}}

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To explain these facts, historians have pointed to the influence of Cantor's former professors, [[Karl Weierstrass]] and Leopold Kronecker. Cantor discussed his results with Weierstrass on December 23, 1873.<ref name=Noether16_17>{{harvnb|Noether|Cavaillès|1937|pp=16&ndash;17}}. English translation: {{harvnb|Ewald|1996|p=847}}.</ref> Weierstrass was first amazed by the concept of countability, but then found the countability of the set of real algebraic numbers useful.<ref>{{harvnb|Grattan-Guinness|1971|p=124}}.</ref> Cantor did not want to publish yet, but Weierstrass felt that he must publish at least his results concerning the algebraic numbers.<ref name=Noether16_17 />

From his correspondence, it appears that Cantor only discussed his article with Weierstrass. However, Cantor told Dedekind: "The restriction which I have imposed on the published version of my investigations is caused in part by local circumstances …..."<ref name=Noether16_17 /> Cantor biographer [[Joseph Dauben]] believes that "local circumstances" refers to Kronecker who, as a member of the editorial board of ''[[Crelle's Journal]]'', had delayed publication of an 1870 article by [[Eduard Heine]], one of Cantor's colleagues. Cantor would submit his article to ''Crelle's Journal''.<ref>{{harvnb|Dauben|1979|pp=67, 308&ndash;309}}.</ref>

Weierstrass advised Cantor to leave his uncountability theorem out of the article he submitted, but Weierstrass also told Cantor that he could add it as a marginal note during proofreading, which he did.<ref name=Ferreiros184 /> It appears in a

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''b''_∞&nbsp;=&nbsp;lim_{''n''&nbsp;→&nbsp;∞}&nbsp;''b_n'' exist. Dedekind had used his "principle of continuity" to prove they exist. This principle (which is equivalent to the [[least upper bound property]] of the real numbers) comes from Dedekind's construction of the real numbers, a construction Kronecker did not accept.<ref>{{harvnb|Dauben|1979|pp=67&ndash;68}}.</ref>

Cantor restricted his first theorem to the set of real algebraic numbers even though Dedekind had sent him a proof that handled all algebraic numbers.<ref name=Noether18 /> Cantor did this for expository reasons and because of "local circumstances.".<ref>{{harvnb|Ferreirós|2007|p=183}}.</ref> This restriction simplifies the article because the second theorem works with real sequences. Hence, the construction in the second theorem can be applied directly to the enumeration of the real algebraic numbers to produce "an effective procedure for the calculation of transcendental numbers.". This procedure would be acceptable to Weierstrass.<ref>{{harvnb|Ferreirós|2007|p=185}}.</ref>

==Dedekind's contributions to Cantor's article==

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Dedekind's second contribution is his proof of Cantor's second theorem. Dedekind sent this proof in reply to Cantor's letter that contained the uncountability theorem, which [[#Cantor's December 7, 1873 proof|Cantor proved]] using infinitely many sequences. Cantor next wrote that he had found a simpler proof that did not use infinitely many sequences.<ref>{{harvnb|Noether|Cavaillès|1937|pp=14&ndash;16, 19}}. English translation: {{harvnb|Ewald|1996|pp=845&ndash;847, 849}}.</ref> So Cantor had a choice of proofs and chose to publish Dedekind's.<ref>{{harvnb|Ferreirós|1993|pp=358&ndash;359}}.</ref>

Cantor thanked Dedekind privately for his help: "…... your comments (which I value highly) and your manner of putting some of the points were of great assistance to me."<ref name=Noether16_17 /> However, he did not mention Dedekind's help in his article. In previous articles, he had acknowledged help received from Kronecker, Weierstrass, Heine, and [[Hermann Schwarz]]. Cantor's failure to mention Dedekind's contributions damaged his relationship with Dedekind. Dedekind stopped replying to his letters and did not resume the correspondence until October 1876.<ref>{{harvnb|Ferreirós|1993|p=350}}.</ref>{{efn-ua|Ferreirós has analyzed the relations between Cantor and Dedekind. He explains why "Relations between both mathematicians were difficult after 1874, when they underwent an interruption…interruption..." ({{harvnb|Ferreirós|1993|pp=344, 348&ndash;352.}})}}

==The legacy of Cantor's article==