Cantor's first set theory article: Difference between revisions

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{{Short description|First article on transfinite set theory}}

{{good article}}

[[File:Georg Cantor3.jpg|thumb|alt=refer to caption|Georg Cantor, {{spaces|4|hair}}{{circa}} 1870]]

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Cantor's article also contains a proof of the existence of [[transcendental number]]s. Both [[constructive proof|constructive and non-constructive proofs]] have been presented as "Cantor's proof." The popularity of presenting a non-constructive proof has led to a misconception that Cantor's arguments are non-constructive. Since the proof that Cantor published either constructs transcendental numbers or does not, an analysis of his article can determine whether or not this proof is constructive.<ref>{{harvnb|Gray|1994|pp=819&ndash;821}}.</ref> Cantor's correspondence with [[Richard Dedekind]] shows the development of his ideas and reveals that he had a choice between two proofs: a non-constructive proof that uses the uncountability of the real numbers and a constructive proof that does not use uncountability.

Historians of mathematics have examined Cantor's article and the circumstances in which it was written. For example, they have discovered that Cantor was advised to leave out his uncountability theorem in the article he submitted{{space|hair|4}} —{{space|hair|4}} he added it during [[Galley proof|proofreading]]. They have traced this and other facts about the article to the influence of [[Karl Weierstrass]] and [[Leopold Kronecker]]. Historians have also studied Dedekind's contributions to the article, including his contributions to the theorem on the countability of the real algebraic numbers. In addition, they have recognized the role played by the uncountability theorem and the concept of countability in the development of set theory, [[measure theory]], and the [[Lebesgue integral]].

==The article==

Cantor's article is short, less than four and a half pages.{{efn-ua|In letter to Dedekind dated December 25, 1873, Cantor states that he has written and submitted "a short paper" titled ''On a Property of the Set of All Real Algebraic Numbers''.

({{harvnb|Noether|Cavaillès|1937|p=17}}; English translation: {{harvnb|Ewald|1996|p=847}}.)}} It begins with a discussion of the real [[algebraic number]]s and a statement of his first theorem: The set of real algebraic numbers can be put into [[one-to-one correspondence]] with the set of positive integers.<ref name=Cantor1874>{{harvnb|Cantor|1874}}. English translation: {{harvnb|Ewald|1996|pp=840&ndash;843}}.</ref> Cantor restates this theorem in terms more familiar to mathematicians of his time: "The set of real algebraic numbers can be written as an infinite [[sequence]] in which each number appears only once."<ref name=Gray828>{{harvnb|Gray|1994|p=828}}.</ref>

Cantor's second theorem works with a [[closed interval]] [''a'',&nbsp;''b''], which is the set of real numbers ≥&nbsp;''a'' and ≤&nbsp;''b''. The theorem states: Given any sequence of real numbers ''x''₁, ''x''₂, ''x''₃, ... and any interval [''a'',&nbsp;''b''], there is a number in [''a'',&nbsp;''b''] that is not contained in the given sequence. Hence, there are infinitely many such numbers.<ref name=Ewald840_841>{{harvnb|Cantor|1874|p=259}}. English translation: {{harvnb|Ewald|1996|pp=840&ndash;841}}.</ref>

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Either the number of intervals generated is finite or infinite. If finite, let (''a''_''L'',&nbsp;''b''_''L'') be the last interval. If infinite, take the [[Limit of a sequence|limit]]s ''a''_∞&nbsp;=&nbsp;lim_{''n''&nbsp;→&nbsp;∞}&nbsp;''a''_''n'' and ''b''_∞&nbsp;=&nbsp;lim_{''n''&nbsp;→&nbsp;∞}&nbsp;''b''_''n''. Since ''a''_''n''&nbsp;<&nbsp;''b''_''n'' for all ''n'', either ''a''_∞&nbsp;=&nbsp;''b''_∞ or ''a''_∞&nbsp;<&nbsp;''b''_∞. Thus, there are three cases to consider:

{{Anchor|Case1}}[[File:Cantor's first uncountability proof Case 1 svg.svg|thumb|350px|alt=Illustration of case 1. [[Real line]] containing closed interval [''a'',&nbsp;''b''] that contains nested open intervals (''a''_''n'',&nbsp;''b''_''n'') for ''n''&nbsp;= 1 to ''L''. Two distinct numbers ''y'' and one ''x''_''n'' are in (''a''_''L'',&nbsp;''b''_''L'').|Case 1: Last interval (''a''_''L'', ''b''_''L'')]]

:Case 1: There is a last interval (''a''_''L'',&nbsp;''b''_''L''). Since at most one ''x''_''n'' can be in this interval, every ''y'' in this interval except ''x''_''n'' (if it exists) is not contained in the given sequence.

{{Anchor|Case2}}[[File:Cantor's first uncountability proof Case 2 svg.svg|thumb|350px|alt=Illustration of case 2. Real line containing interval [''a'',&nbsp;''b''] that contains nested intervals (''a''_''n'', ''b''_''n'') for ''n''&nbsp;=&nbsp;1 to ∞. These intervals converge to ''a''_∞.|Case 2: ''a''_∞ = ''b''_∞]]

:Case 2: ''a''_∞&nbsp;=&nbsp;''b''_∞. Then ''a''_∞ is not contained in the given sequence since for all ''n''{{space|hair}}: ''a''_∞ belongsis toin the interval (''a''_''n'',&nbsp;''b''_''n'') but ''x''_''n'' does not belong to (''a''_''n'',&nbsp;''b''_''n''). In symbols: ''a''_∞&nbsp;[[∈]]&nbsp;(''a''_''n'',&nbsp;''b''_''n'') but ''x''_''n''&nbsp;[[∉]]&nbsp;(''a''_''n'',&nbsp;''b''_''n'').

{{Anchor|x_n not in}}

:{| class="wikitable collapsible collapsed"

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! style="background: f5f5f5;" |'''Proof that the number generated is {{sqrt|2}}&nbsp;&minus;&nbsp;1'''

|- style="text-align: left; vertical-align: top; background: white"

| style="padding-left: 1em; padding-right: 1em" |The proof uses [[Farey sequence]]s and [[simple continued fractionsfraction]]s. The Farey sequence <math>F_n</math> is the increasing sequence of [[completely reduced fraction]]s whose denominators are <math>\leq n.</math> If <math>\frac{a}{b}</math> and <math>\frac{c}{d}</math> are adjacent in a Farey sequence, the lowest denominator fraction between them is their [[mediant (mathematics)|mediant]] <math>\frac{a+c}{b+d}.</math> This mediant is adjacent to both <math>\frac{a}{b}</math> and <math>\frac{c}{d}</math> in the Farey sequence <math>F_{b+d}.</math><ref>{{harvnb|LeVeque|1956|pp=154&ndash;155}}.</ref>

Cantor's construction produces mediants because the rational numbers were sequenced by increasing denominator. The first interval in the table is <math>(\frac{1}{3}, \frac{1}{2}).</math> Since <math>\frac{1}{3}</math> and <math>\frac{1}{2}</math> are adjacent in <math>F_3,</math> their mediant <math>\frac{2}{5}</math> is the first fraction in the sequence between <math>\frac{1}{3}</math> and <math>\frac{1}{2}.</math> Hence, <math>\frac{1}{3} < \frac{2}{5} < \frac{1}{2}.</math> In this inequality, <math>\frac{1}{2}</math> has the smallest denominator, so the second fraction is the mediant of <math>\frac{2}{5}</math> and <math>\frac{1}{2},</math> which equals <math>\frac{3}{7}.</math> This implies: <math>\frac{1}{3} < \frac{2}{5} < \frac{3}{7} < \frac{1}{2}.</math> Therefore, the next interval is <math>(\frac{2}{5}, \frac{3}{7}).</math>

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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,&nbsp;β], then the remarkable case can happen that ''every'' interval [γ,&nbsp;δ] contained in [α,&nbsp;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,&nbsp;β].{{efn-ua|Translated from {{harvnb|Cantor|1879|p=2}}: {{lang-langx|de|Liegt ''P'' theilweise oder ganz im Intervalle (α&nbsp;.&nbsp;.&nbsp;.&nbsp;β), so kann der bemerkenswerthe Fall eintreten, dass ''jedes noch so kleine'' in (α&nbsp;.&nbsp;.&nbsp;.&nbsp;β) enthaltene Intervall (γ&nbsp;.&nbsp;.&nbsp;.&nbsp;δ) Punkte von ''P'' enthält. In einem solchen Falle wollen wir sagen, dass ''P'' ''im Intervalle'' (α&nbsp;.&nbsp;.&nbsp;.&nbsp;β) ''überall-dicht'' sei.|label=none|italic=unset}}}}

In this discussion of Cantor's proof: ''a'',&nbsp;''b'',&nbsp;''c'',&nbsp;''d'' are used instead of α,&nbsp;β,&nbsp;γ,&nbsp;δ. Also, Cantor only uses his interval notation if the first endpoint is less than the second. For this discussion, this means that (''a'',&nbsp;''b'') implies ''a''&nbsp;<&nbsp;''b''.

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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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{{space|12}}ω₁,&nbsp;ω₂,&nbsp;.&nbsp;.&nbsp;.&nbsp;,&nbsp;ω_ν,&nbsp;.&nbsp;.&nbsp;.

|{{lang-langx|de|[Seite 5]<br>

&nbsp;.&nbsp;.&nbsp;.&nbsp;Dem widerspricht aber ein sehr allgemeiner Satz, welchen wir in Borchardt's Journal, Bd. 77, pag. 260, mit aller Strenge bewiesen haben, nämlich der folgende Satz:

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{{space|5}}Da in unserer Reihe:<br>

{{space|12}}ω₁,&nbsp;ω₂,&nbsp;.&nbsp;.&nbsp;.&nbsp;,&nbsp;ω_ν,&nbsp;.&nbsp;.&nbsp;.

<br>|label=none|italic=unset}}

|-

|[Page 6]<br>

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{{space|5}}''That if ν is an arbitrary whole number, the [real] quantity ω_ν lies outside the interval [α^(ν)&nbsp;.&nbsp;.&nbsp;.&nbsp;β^(ν)].''

|{{lang-langx|de|[Seite 6]<br>

sicher Zahlen ''innerhalb'' des Intervalls (α&nbsp;.&nbsp;.&nbsp;.&nbsp;β) vorkommen, so muss eine von diesen Zahlen den ''kleinsten Index'' haben, sie sei ω_κ₁, und eine andere: ω_κ₂ mit dem nächst grösseren Index behaftet sein.

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|[Page 7]<br>

{{space|5}}Since the numbers α', α<nowiki>''</nowiki>, α<nowiki>'''</nowiki>,&nbsp;.&nbsp;.&nbsp;., α^(ν),&nbsp;.&nbsp;.&nbsp;. are continually increasing by value while simultaneously being enclosed in the interval [α,&nbsp;β], they have, by a well-known fundamental theorem of the theory of magnitudes [see note 2 below], a limit that we denote by A, so that:<br>

{{space|12}}{{nowrap|A {{eq=}} Lim α^(ν) for ν {{eq=}} ∞.}}

{{space|5}}The same applies to the numbers β', β<nowiki>''</nowiki>, β<nowiki>'''</nowiki>,&nbsp;.&nbsp;.&nbsp;., β^(ν),&nbsp;.&nbsp;.&nbsp;., which are continually decreasing and likewise lying in the interval [α,&nbsp;β]. We call their limit B, so that:<br>

{{space|12}}{{nowrap|B {{eq=}} Lim β^(ν) for ν {{eq=}} ∞.}}

{{space|5}}Obviously, one has:<br>

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to the assumption.

{{space|5}}Thus, there only remains the case A&nbsp;{{eq=}}&nbsp;B and now it is demonstrated that the number:<br>

{{space|12}}η&nbsp;{{eq=}}&nbsp;A&nbsp;{{eq=}}&nbsp;B<br>

does ''not'' occur in our sequence (ω).

{{space|5}}If it were a member of our sequence, such as the ν^th, then one would have: η {{eq=}} ω_ν.

{{space|5}}But the latter equation is not possible for any value of ν because η is in the ''interior'' of the interval [α^(ν),&nbsp;β^(ν)], but ω_ν lies ''outside'' of it.

|{{lang-langx|de|[Seite 7]<br>

{{space|5}}Da die Zahlen α', α<nowiki>''</nowiki>, α<nowiki>'''</nowiki>, &nbsp;.&nbsp;.&nbsp;., α^(ν), &nbsp;.&nbsp;.&nbsp;. ihrer Grösse nach fortwährend wachsen, dabei jedoch im Intervalle (α&nbsp;.&nbsp;.&nbsp;.&nbsp;β) eingeschlossen sind, so haben sie, nach einem bekannten Fundamentalsatze der Grössenlehre, eine Grenze, die wir mit A bezeichnen, so dass:<br>

{{space|12}}{{nowrap|A {{eq=}} Lim α^(ν) für ν {{eq=}} ∞.}}

{{space|5}}Ein Gleiches gilt für die Zahlen β', β<nowiki>''</nowiki>, β<nowiki>'''</nowiki>,&nbsp;.&nbsp;.&nbsp;., β^(ν),&nbsp;.&nbsp;.&nbsp;. welche fortwährend abnehmen und dabei ebenfalls im Intervalle (α&nbsp;.&nbsp;.&nbsp;.&nbsp;β) liegen; wir nennen ihre Grenze B, so dass:<br>

{{space|12}}{{nowrap|B {{eq=}} Lim β^(ν) für ν {{eq=}} ∞.}}

{{space|5}}Man hat offenbar:<br>

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{{space|5}}Es ist aber leicht zu sehen, dass der Fall A&nbsp;<&nbsp;B hier ''nicht'' vorkommen kann; da sonst jede Zahl ω_ν, unserer Reihe ''ausserhalb'' des Intervalles (A&nbsp;.&nbsp;.&nbsp;.&nbsp;B) liegen würde, indem ω_ν, ausserhalb des Intervalls (α^(ν) . . . β^(ν)) gelegen ist; unsere Reihe (ω) wäre im Intervall (α&nbsp;.&nbsp;.&nbsp;.&nbsp;β) ''nicht überalldicht,'' gegen die Voraussetzung.

{{space|5}}Es bleibt daher nur der Fall A&nbsp;{{eq=}}&nbsp;B übrig und es zeigt sich nun, dass die Zahl:<br>

{{space|12}}{{nowrap|η {{eq=}} A {{eq=}} B}}<br>

in unserer Reihe (ω) ''nicht'' vorkommt.

{{space|5}}Denn, würde sie ein Glied unserer Reihe sein, etwa das ν^te, so hätte man: η {{eq=}} ω_ν.

{{space|5}}Die letztere Gleichung ist aber für keinen Werth von v möglich, weil η im ''Innern'' des Intervalls [α^(ν),&nbsp;β^(ν)], ω_ν aber ''ausserhalb'' desselben liegt.|label=none|italic=unset}}

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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}}

On December 2, Cantor responded that his question does have interest: "It would be nice if it could be answered; for example, provided that it could be answered ''no'', one would have a new proof of [[Liouville number|Liouville's theorem]] that there are transcendental numbers."<ref>{{harvnb|Noether|Cavaillès|1937|p=13}}. English translation: {{harvnb|Gray|1994|p=827}}.</ref>

On December 7, Cantor sent Dedekind a [[proof by contradiction]] that the set of real numbers is uncountable. Cantor starts by assuming that the real numbers in <math>[0,1]</math> can be written as a sequence. Then, he applies a construction to this sequence to produce a number in <math>[0,1]</math> that is not in the sequence, thus contradicting his assumption.<ref name=Dec7letter>{{harvnb|Noether|Cavaillès|1937|pp=14&ndash;15}}. English translation: {{harvnb|Ewald|1996|pp=845&ndash;846}}.</ref> Together, the letters of December 2 and 7 provide a non-constructive proof of the existence of transcendental numbers.<ref>{{harvnb|Gray|1994|p=827}}</ref> Also, the proof in Cantor's December 7th7 letter shows some of the reasoning that led to his discovery that the real numbers form an uncountable set.<ref>{{harvnb|Dauben|1979|p=51}}.</ref>

{{Anchor|Cantor's December 7, 1873 proof}}

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One sees that it is possible to form an infinite sequence of nested intervals <math>[p, q] \supsetneq [p_1, q_1] \supsetneq [p_2, q_2] \supsetneq \ldots</math> such that:<br>

{{space|6}}the members of the <math>1^\text{st}, 2^\text{nd}, \ldots, (k - 1)^\text{st}</math> sequence lie outside <math>[p, q];</math><br>

{{space|6}}the members of the <math>k^\text{th}, \ldots, (k_1 - 1)^\text{st}</math> sequence lie outside <math>[p_1, q_1];</math><br>

{{space|6}}the members of the <math>(k_1)^\text{th}, \ldots, (k_2 - 1)^\text{st}</math> sequence lie outside <math>[p_2, q_2];</math><br>

{{space|6}}<math>\ldots;</math><ref name=Dec7letter />

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|}

Dedekind received Cantor's proof on December 8. On that same day, Dedekind simplified the proof and mailed his proof to Cantor. Cantor used Dedekind's proof in his article.<ref>{{harvnb|Noether|Cavaillès|1937|p=19}}. English translation: {{harvnb|Ewald|1996|p=849}}.</ref> The letter containing Cantor's December 7th7 proof was not published until 1937.<ref>{{harvnb|Ewald|1996|p=843}}.</ref>

On December 9, Cantor announced the theorem that allowed him to construct transcendental numbers as well as prove the uncountability of the set of real numbers:

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The non-constructive proof uses two proofs by contradiction:

# The proof by contradiction used to prove the uncountability theorem (see [[#Proof of Cantor's uncountability theorem|Proof of Cantor's uncountability theorem]]).

# The proof by contradiction used to prove the existence of transcendental numbers from the countability of the real algebraic numbers and the uncountability of real numbers. [[#Cantor's December 2nd letter|Cantor's December 2nd letter]] mentions this existence proof but does not contain it. Here is a proof: Assume that there are no transcendental numbers in [''a'',&nbsp;''b'']. Then all the numbers in [''a'',&nbsp;''b''] are algebraic. This implies that they form a [[subsequence]] of the sequence of all real algebraic numbers, which contradicts Cantor's uncountability theorem. Thus, the assumption that there are no transcendental numbers in [''a'',&nbsp;''b''] is false. Therefore, there is a transcendental number in [''a'',&nbsp;''b''].{{efn-ua|The beginning of this proof is derived from the proof below by restricting its numbers to the interval [''a'',&nbsp;''b''] and by using a subsequence since Cantor was using sequences in his 1873 work on countability.<br>''German text:'' {{lang-langx|de|''Satz 68. Es gibt transzendente Zahlen.''<br>Gäbe es nämlich keine transzendenten Zahlen, so wären alle Zahlen algebraisch, das Kontinuum also identisch mit der Menge aller algebraischen Zahlen. Das ist aber unmöglich, weil die Menge aller algebraischen Zahlen abzählbar ist, das Kontinuum aber nicht.|label=none|italic=unset}}<ref>{{harvnb|Perron|1921|p=162}}.</ref><br>''Translation: Theorem 68. There are transcendental numbers.''<br> If there were no transcendental numbers, then all numbers would be algebraic. Hence, the [[continuum (set theory)|continuum]] would be identical to the set of all algebraic numbers. However, this is impossible because the set of all algebraic numbers is countable, but the continuum is not.}}

Cantor chose to publish the constructive proof, which not only produces a transcendental number but is also shorter and avoids two proofs by contradiction. The non-constructive proof from Cantor's correspondence is simpler than the one above because it works with all the real numbers rather than the interval [''a'',&nbsp;''b'']. This eliminates the subsequence step and all occurrences of [''a'',&nbsp;''b''] in the second proof by contradiction.<ref name=Ewald840_841 />

== A misconception about Cantor's work ==

[[Akihiro Kanamori]], who specializes in set theory, stated that "Accounts of Cantor's work have mostly reversed the order for deducing the existence of transcendental numbers, establishing first the uncountability of the reals and only then drawing the existence conclusion from the countability of the algebraic numbers. In textbooks the inversion may be inevitable, but this has promoted the misconception that Cantor's arguments are non-constructive."<ref name=Kanamori4>{{harvnb|Kanamori|2012|p=4}}.</ref>

Cantor's published proof and the reverse-order proof both use the theorem: Given a sequence of reals, a real can be found that is not in the sequence. By applying this theorem to the sequence of real algebraic numbers, Cantor produced a transcendental number. He then proved that the reals are uncountable: Assume that there is a sequence containing all the reals. Applying the theorem to this sequence produces a real not in the sequence, contradicting the assumption that the sequence contains all the reals. Hence, the reals are uncountable.<ref name=Ewald840_841/> The reverse-order proof starts by first proving the reals are uncountable. It then proves that transcendental numbers exist: If there were no transcendental numbers, all the reals would be algebraic and hence countable, which contradicts what was just proved. This contradiction proves that transcendental numbers exist without constructing any.<ref name=Kanamori4/>

[[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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The presentation of the non-constructive proof without mentioning Cantor's constructive proof appears in some books that were quite successful as measured by the length of time new editions or reprints appeared—for example: Oskar Perron's Irrationalzahlen (1921; 1960, 4th edition), [[Eric Temple Bell|Eric Temple Bell's]] ''[[Men of Mathematics]]'' (1937; still being reprinted), [[Godfrey Hardy]] and [[E. M. Wright|E. M. Wright's]] ''An Introduction to the [[Theory of numbers|Theory of Numbers]]'' (1938; 2008 6th edition), [[Garrett Birkhoff]] and [[Saunders Mac Lane|Saunders Mac Lane's]] ''A Survey of [[Modern Algebra]]'' (1941; 1997 5th edition), and [[Michael Spivak|Michael Spivak's]] ''[[Calculus]]'' (1967; 2008 4th edition).<ref>{{harvnb|Bell|1937|pp=568&ndash;569}}; {{harvnb|Hardy|Wright|1938|p=159}} (6th ed., pp. 205&ndash;206); {{harvnb|Birkhoff|Mac Lane|1941|p=392}}, (5th ed., pp. 436&ndash;437); {{harvnb|Spivak|1967|pp=369&ndash;370}} (4th ed., pp. 448&ndash;449).</ref>{{efn-ua|Starting with Hardy and Wright's book, these books are linked to Perron's book via their bibliographies: Perron's book is mentioned in the bibliography of Hardy and Wright's book, which in turn is mentioned in the bibliography of Birkhoff and Mac Lane's book and in the bibliography of Spivak's book. ({{harvnb|Hardy|Wright|1938|p=400}}; {{harvnb|Birkhoff|Mac Lane|1941|p=441}}; {{harvnb|Spivak|1967|p=515}}.)}} Since 2014, at least two books have appeared stating that Cantor's proof is constructive,<ref>{{harvnb|Dasgupta|2014|p=107}}; {{harvnb|Sheppard|2014|pp=131&ndash;132}}.</ref> and at least four have appeared stating that his proof does not construct any (or a single) transcendental.<ref>{{harvnb|Jarvis|2014|p=18}}; {{harvnb|Chowdhary|2015|p=19}}; {{harvnb|Stewart|2015|p=285}}; {{harvnb|Stewart|Tall|2015|p=333}}.</ref>

Asserting that Cantor gave a non-constructive argument without mentioning the constructive proof he published can lead to erroneous statements about the [[history of mathematics]]. In ''A Survey of Modern Algebra,'' Birkhoff and Mac Lane state: "Cantor's argument for this result [Not every real number is algebraic] was at first rejected by many mathematicians, since it did not exhibit any specific transcendental number." <ref>{{harvnb|Birkhoff|Mac Lane|1941|p=392}}, (5th ed., pp. 436&ndash;437).</ref> The proof that Cantor published produces transcendental numbers, and there appears to be no evidence that his argument was rejected. Even [[Leopold Kronecker]], who had strict views on what is acceptable in mathematics and who could have delayed publication of Cantor's article, did not delay it.<ref name=Gray828/> In fact, applying Cantor's construction to the sequence of real algebraic numbers produces a limiting process that Kronecker accepted—namely, it determines a number to any required degree of accuracy.{{efn-ua|Kronecker's opinion was: "Definitions must contain the means of reaching a decision in a finite number of steps, and existence proofs must be conducted so that the quantity in question can be calculated with any required degree of accuracy."<ref>{{harvnb|Burton|1995|p=595}}.</ref> So Kronecker would accept Cantor's argument as a valid existence proof, but he would not accept its conclusion that transcendental numbers exist. For Kronecker, they do not exist because their definition contains no means for deciding in a finite number of steps whether or not a given number is transcendental.<ref>{{harvnb|Dauben|1979|p=69}}.</ref> Cantor's 1874 construction calculates numbers to any required degree of accuracy because: Given a ''k'', an ''n'' can be computed such that {{nowrap|''b''_''n'' – ''a_n'' ≤ {{sfrac|1|''k''}}}} where (''a_n'',&nbsp;''b_n'') is the {{nowrap|''n''-th}} interval of Cantor's construction. An example of how to prove this is given in {{harvnb|Gray|1994|p=822}}. Cantor's diagonal argument provides an accuracy of 10^−''n'' after ''n'' real algebraic numbers have been calculated because each of these numbers generates one digit of the transcendental number.<ref>{{harvnb|Gray|1994|p=824}}.</ref>}}

== The influence of Weierstrass and Kronecker on Cantor's article ==

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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==

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| publisher = Simon & Schuster

| ___location = New York

| year = 1937}}. Reprinted, 1984, {{isbn|978-0-671-62818-5}}.

| isbn = 978-0-671-62818-5}}.

* {{Citation | first1 = Garrett

| last1 = Birkhoff

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| publisher = Macmillan

| ___location = New York

| year = 1941}}. Reprinted, Taylor & Francis, 1997, {{isbn|978-1-56881-068-3}}.

| year = 1941

| isbn = 978-1-56881-068-3}}.

* {{Citation | last = Burton

| first = David M.

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| journal = Journal für die Reine und Angewandte Mathematik

| year = 1878| doi = 10.1515/crll.1878.84.242

| doi-broken-date = 11 July 2025

}}.

* {{Citation | first = Georg

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| title = Fundamentals of Discrete Mathematical Structures

| publisher = PHI Learning

| ___location = DehliDelhi, India

| year = 2015

| edition = 3rd

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| pmid=16578557

| pmc=221287| bibcode = 1963PNAS...50.1143C

| doi-access = free

}}.

* {{Citation | last = Dasgupta

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| pages = 343–363

| year = 1993

| doi = 10.1006/hmat.1993.1030}}.| doi-access = free

}}.

* {{Citation | last = Ferreirós

| first = José

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| pages = 111–130

| year = 1971}}.

* {{Citation
| first = Robert

|last | last = Gray

|title | title = Georg Cantor and Transcendental Numbers

|url | url = http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gray819-832.pdf

|journal | journal = [[American Mathematical Monthly]]

|volume | volume = 101

|issue | issue = 9

|pages | pages = 819–832

|year | year = 1994

|doi | doi= 10.2307/2975129

|mr | mr= 1300488

|zbl = 0827.01004

| zbl=0827.01004| jstor = 2975129}}.

|jstor = 2975129

|access-date = 2016-02-13

|archive-date = 2022-01-21

|archive-url = https://web.archive.org/web/20220121155859/https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gray819-832.pdf

|url-status = dead

}}.

* {{Citation | last1 = Hardy

| first1 = Godfrey

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| ___location = Oxford

| year = 1938

|<!-- isbn = 978-0-19-921985-8 6th edition, not 1938 edition -->}}.

* {{Citation | last = Havil

| first = Julian

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|volume = I

|publisher = Addison-Wesley

|___location = Reading, Mass.Massachusetts

|year = 1956

}}. ([https://archive.org/details/topicsinnumberth0000leve Reprinted by Dover Publications], 2002, {{isbn|978-0-486-42539-9}}.)

|isbn = 978-0-486-42539-9

|url-access = registration

|url = https://archive.org/details/topicsinnumberth0000leve

}}. (Reprinted by Dover Publications, 2002.)

* {{Citation | editor-last = Noether

| editor-first = Emmy

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| isbn = 978-1-58488-347-0}}.

{{Mathematical logic}}

[[Category:SetGeorg theoryCantor]]

[[Category:History of mathematics]]

[[Category:Set theory]]

[[Category:Real analysis]]

[[Category:GeorgSet Cantortheory]]