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

Revision as of 21:23, 8 August 2018

Revision as of 21:37, 8 August 2018

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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]]'''_{[[Special:Contribs/Bilorv|(c)]][[User talk:Bilorv|('''talk''')]]}</span> 15:39, 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 specific records from that time, butthat document the reasons. However, later Harold M. Edwards who has studied Kronecker's work thinksthought 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 that Kroneckerhe had strict views on what is acceptable in mathematics and then Iby gogoing 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)