Talk:Cantor's first set theory article/GA2: Difference between revisions

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Comments: Removed long proof in Examples; replaced with short proof in Case 2 of Second theorem.
Comments: Added something I deleted by accident
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::Can you describe to me how the [[Georg Cantor%27s first set theory article#Example of Cantor's construction|Example of Cantor's construction]] was written? If you mean to say that you claim the entire section falls under [[WP:SCG]], then I'll have to take a closer look and research into what the standard is in this area. I wouldn't describe it as a "simple derivation" or a "routine calculation", not due to complexity but due to the length of the section. The table is trivial, but not the proof of {{nowrap|''x<sub>i</sub>'' ∉ (''a<sub>n</sub>'', ''b<sub>n</sub>'')}}. But then, surely there's a source which covers this step that Cantor omitted. <span class="nowrap">— '''[[User:Bilorv|Bilorv]]'''<sub>[[Special:Contribs/Bilorv|(c)]][[User talk:Bilorv|('''talk''')]]</sub></span> 18:58, 10 August 2018 (UTC)
 
:I came up with a short proof that that for all&nbsp;''n'', (''a''<sub>''n''</sub>,&nbsp;''b''<sub>''n''</sub>) excludes ''x''<sub>1</sub>,&nbsp;…&nbsp;,&nbsp;''x''<sub>2''n''</sub>. This implies Cantor's statement ''x''<sub>''n''</sub> does not belong to the interval (''a''<sub>''n''</sub>,&nbsp;''b''<sub>''n''</sub>). Your comment on looking for a source helped me even though I couldn't find a source that had a proof. Discussions of Cantor's 1874 method seem to be fairly rare; nearly everyone seems to prefer discussing the diagonal method. However, seeing again that Cantor made the unproven observation ''x''<sub>n</sub>&nbsp;&notin;&nbsp;(''a''<sub>n</sub>,&nbsp;''b''<sub>n</sub>) led me to think that he probably left out the proof because it was a simple proof. In his book, Dauben approaches the result differently. He points out, but doesn't prove, that if ''a''<sub>∞</sub>&nbsp;=&nbsp;''x''<sub>''k''</sub>, then for sufficiently large index ''n'', ''x''<sub>k</sub> would be excluded from all intervals nested within (''a''<sub>n</sub>,&nbsp;''b''<sub>n</sub>). I believe that the new proof fits within the [[WP:SCG]] guidelines (it's a simple derivation that makes Cantor's proof more accessible to Wikipedia readers than Cantor's original proof, which was meant for research mathematicians). Also, the trimmed down Example section is now strictly an example.
 
:Your observations have led to this very significant improvement of the article. This together with your other suggestions prove that you've done an excellent job as a Good Article reviewer. In the next few days, I plan to finish working through your remaining suggestions. —[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 23:14, 14 August 2018 (UTC)