Cantor's first set theory article: Difference between revisions

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! style="background: f5f5f5;" |'''Proof that the number generated is {{sqrt|2}} − 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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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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{{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}}''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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{{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 {{=}} Lim α^(ν) für ν {{=}} ∞.}}

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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 />

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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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| ___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