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===The disagreement about Cantor's existence proof===
*In the third paragraph, the "exact" asymptotics <math>O(2^{\sqrt[3]{n}})</math> and <math>O(n^2 \log^2 n \log \log n)</math> might as well be mentioned.
*<s>"The proof of Cantor's second theorem does not state why some limits exist. The proof he was using does." — I don't quite understand either of these sentences. My understanding from reading the paper is that Cantor implicitly invokes what I would describe as the [[Monotone convergence theorem|Monotonic Sequence Theorem]] to deduce that <math>\alpha^\infty</math> and <math>\beta^\infty</math> exist. Does this bullet point mean that Cantor is using this without statement, but requires it for his claims? Or that Cantor didn't explain this bit but the editor added in an explanation? Or something completely different?</s>
*<s>Is the 1870 Heine article relevant to Cantor or his work (e.g. a close friend / publication on a similar topic)? Is the point here that Kronecker is delaying publication in the journal because of his finitist views? (If so, add a few words to spell this out.)</s>
*<s>"Cantor chose "On a Property of the Collection of All Real Algebraic Numbers,"" — I would put the German title here and then the English in parentheticals (pedantic, but the German name is what Cantor was really choosing).</s>
===The legacy of Cantor's article===
*<s>"In 1922, Thoralf Skolem proved that if the axioms of set theory are consistent" — Which axioms of set theory exactly? Is this referring to a specific collection of axioms (e.g. ZFC), or saying generally "given any set of consistent axioms ..."? (In the latter case, the definite article "'''the''' axioms of set theory" is misleading.)</s>
*<s>"[[Skolem's paradox|this does not contradict Cantor's uncountability theorem]]" is an [[WP:EASTER|Easter egg]] link; mention the phrase "Skolem's paradox" in prose.</s>
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::In my opinion, the current article does not need to give a full explanation of the Skolem paradox, so long as the reader can easily follow the links to a longer discussion. An answer to Bilorv's specific question can be seen in [http://boole.stanford.edu/skolem/ “Skolem’s paradox up close and personal”, by Vaughan Pratt]. (The Skolem paradox follows from the Zermelo axioms alone and doesn't require the full ZFC set of axioms). The modern statement of the [[Löwenheim–Skolem theorem]] as presented in our article speaks of 'any countable first-order theory' so the paradox holds even for a variety of axiomatizations. It appears that the first-orderness is what causes the paradox. The lead of [[Löwenheim–Skolem theorem]] says "In general, the Löwenheim–Skolem theorem does not hold in stronger logics such as second-order logic." But these refinements come from later. The purpose of mentioning Skolem's paradox in the current article is surely to show what happened to the notion of 'countability' as employed by Cantor in the work of later mathematicians. [[User:EdJohnston|EdJohnston]] ([[User talk:EdJohnston|talk]]) 02:26, 8 August 2018 (UTC)
:::Right, but I wasn't suggesting that a full explanation should be given. The problem was the ambiguity. I'm happy if "the axioms" is replaced with "standard axioms" or "Zermelo axioms" or "ZFC axioms" or "certain axioms (e.g. ZFC)" etc. But the paragraph needs to be meaningful and fully accurate when read in isolation, so I still think something needs to be changed. <span class="nowrap">— '''[[User:Bilorv|Bilorv]]'''<sub>[[Special:Contribs/Bilorv|(c)]][[User talk:Bilorv|('''talk''')]]</sub></span> 15:39, 8 August 2018 (UTC)
::::Oops, I forgot that "the axioms" was changed to "conventional axioms" [[Special:Diff/852871054|a while ago]]. That addresses my concern. <span class="nowrap">— '''[[User:Bilorv|Bilorv]]'''<sub>[[Special:Contribs/Bilorv|(c)]][[User talk:Bilorv|('''talk''')]]</sub></span> 21:42, 8 August 2018 (UTC)
: Concerning "Is the 1870 Heine article relevant to Cantor or his work": Put in that Heine was one of Cantor's colleagues. Being a colleague of someone who has had problems with Kronecker would have made Cantor cautious. As for Kronecker's exact reasons for the delay of Heine's article, I know of no records from that time that document the reasons. However, later Harold M. Edwards who has studied Kronecker's work thought it could be due to Heine's study of ''arbitrary'' trigonometric series. Instead of speculating on Kronecker's motives for Heine's article delay, I prepare the reader at the end of the last section for Kronecker by stating he had strict views on what is acceptable in mathematics and by going into more detail in a footnote. This should give the reader some idea of Kronecker's thinking. —[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 21:23, 8 August 2018 (UTC)
::Yes, this makes sense. <span class="nowrap">— '''[[User:Bilorv|Bilorv]]'''<sub>[[Special:Contribs/Bilorv|(c)]][[User talk:Bilorv|('''talk''')]]</sub></span> 21:42, 8 August 2018 (UTC)
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