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A proof by refutation (e.g., that assuming the real numbers are countable leads to a contradiction) is not a proof by contradiction. A proof by contradiction proceeds by refuting the *negation* of the original conclusion, thereby implying the conclusion itself. Cantor's diagonal argument is *not* a proof by contradiction. Tag: Reverted |
Undid revision 1096011298 by 2603:6010:B002:3E4:57:9234:1D65:EA1C (talk) Cantor's proof starts by assuming the negation of the thing to be proved, and proceeds as in any proof by contradiction. |
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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=14–15}}. English translation: {{harvnb|Ewald|1996|pp=845–846}}.</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 7th 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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