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→Comments: For "Cantor's article is short": added ref; changed "not even" to "less than". |
→Comments: Concerning "The proof of Cantor's second theorem does not state why some limits exist. The proof he was using does." |
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::Excellent. <span class="nowrap">— '''[[User:Bilorv|Bilorv]]'''<sub>[[Special:Contribs/Bilorv|(c)]][[User talk:Bilorv|('''talk''')]]</sub></span> 22:56, 6 August 2018 (UTC)
: Concerning "Cantor's article is short": Added ref to justify use of the word "short". Changed "not even" to "less than"—I feel that a positive phrase here is better than one with a negative in it. I think that having the "short" is important. Readers are more likely to remember "short" than "less than four and a half pages". Also, the fact that Cantor's article is short is one of its characteristics and arises from Weierstrass' desire that Cantor should quickly publish the theorem on the countability of the real algebraic numbers. —[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 18:58, 7 August 2018 (UTC)
: Concerning "The proof of Cantor's second theorem does not state why some limits exist. The proof he was using does.": Please look this over again. The section it is in has the structure:
:* Bullets about facts that historians have discovered about Cantor's article.
:* Then the "lead in": "To explain these facts, historians have pointed to the influence of Cantor's former professors, Karl Weierstrass and Leopold Kronecker."
:* This is followed by explanations of the facts, which goes into detail about the influence of Weierstrass and Kronecker on the bulleted facts.
:The explanation of the fact that you are concerned about is in the following paragraph:
::Kronecker's influence appears in the proof of Cantor's second theorem. Cantor used Dedekind's version of the proof except he left out why the limits ''a''<sub>∞</sub> = lim<sub>''n'' → ∞</sub> ''a<sub>n</sub>'' and ''b''<sub>∞</sub> = lim<sub>''n'' → ∞</sub> ''b<sub>n</sub>'' exist. Dedekind had used his "principle of continuity" to prove they exist. This principle (which is equivalent to the [[least upper bound property]] of the real numbers) comes from Dedekind's construction of the real numbers, a construction Kronecker did not accept.<ref>{{harvnb|Dauben|1979|pp=67–68}}.</ref>
:If my structuring or explanation is not clear to you, please give me suggestions for modifying them. Thanks, — [[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 19:51, 7 August 2018 (UTC)
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