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EdJohnston (talk | contribs) →Comments: Should not need to give details about the Skolem paradox in the current article |
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:::"The proof of Cantor's second theorem is based on Dedekind's proof, but omits explanation of why a<sub>∞</sub> and b<sub>∞</sub> exist."
::The point is that the reader shouldn't have to look further down to make sense of the bullet point; the explanatory paragraph should just provide extra detail. <span class="nowrap">— '''[[User:Bilorv|Bilorv]]'''<sub>[[Special:Contribs/Bilorv|(c)]][[User talk:Bilorv|('''talk''')]]</sub></span> 01:38, 8 August 2018 (UTC)
*Response to [[User:Bilorv]]’s comment about the Skolem paradox (I put Bilorv’s words in green):
:{{green|“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.)}}
::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)
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