Cantor's first set theory article: Difference between revisions

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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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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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== 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/>