Revision as of 23:20, 27 May 2022 edit GKFX (talk \| contribs) Extended confirmed users 13,147 edits m Remove {{eq}}, which is not going to work after its target becomes a magic word. ← Previous edit		Revision as of 18:37, 1 July 2022 edit undo 2603:6010:b002:3e4:57:9234:1d65:ea1c (talk) 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 Next edit →
Line 363: 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}}

Cantor's first set theory article: Difference between revisions