Cantor's first set theory article: Difference between revisions

Revision as of 14:32, 15 March 2022

Revision as of 23:20, 27 May 2022

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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==

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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}}&#61; Lim α^(ν) for ν {{eq}}&#61; ∞.}}

{{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}}&#61; Lim β^(ν) for ν {{eq}}&#61; ∞.}}

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

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to the assumption.

{{space|5}}Thus, there only remains the case A&nbsp;{{eq}}&#61;&nbsp;B and now it is demonstrated that the number:<br>

{{space|12}}η&nbsp;{{eq}}&#61;&nbsp;A&nbsp;{{eq}}&#61;&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}}&#61; 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.

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|{{lang-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}}&#61; Lim α^(ν) für ν {{eq}}&#61; ∞.}}

{{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}}&#61; Lim β^(ν) für ν {{eq}}&#61; ∞.}}

{{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}}&#61;&nbsp;B übrig und es zeigt sich nun, dass die Zahl:<br>

{{space|12}}{{nowrap|η {{eq}}&#61; A {{eq}}&#61; 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}}&#61; 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}}