Talk:Cantor's first set theory article/Archive 1: Difference between revisions

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{{GA nominee|16:34, 20 October 2014 (UTC)|nominator=[[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]])|page=1|subtopic=Mathematics and mathematicians|status=onreview|note=}}

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Older discussion is archived at [[/Archive1]]

==The rationals==

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There's an explicit exercise in [[Walter Rudin]]'s ''Principles of Mathematical Analysis'' that asks the student to show for any rational number less than √2 how to find a larger rational number that is still less than √2, and similarly for those larger than √2. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 02:04, 24 January 2010 (UTC)

: For <math>a\lt<\sqrt2</math>, one possible choice would be <math>a+1/\lceil1/(\sqrt2-a)\rceil</math>. [[User:Paradoctor|Paradoctor]] ([[User talk:Paradoctor|talk]]) 13:27, 16 December 2013 (UTC)

== Proposed Changes to Article ==

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I have also added a "Notes" section, and I have added references to the current "References" section.

I highly recommend reading Cantor's original article, which is at: [http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=266194 "Über eine Eigenschaft des Ingebriffes aller reelen algebraischen Zahlen"]. A French translation (which was reviewed and corrected by Cantor) is at: [http://www.springerlink.com/content/37030699752l2573/fulltext.pdf "Sur une propriété du système de tous les nombres algébriques réels"]. Unfortunately, I have not found an English translation on-line. However, an English translation is in: Volume 2 of Ewald's ''From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics'' ({{ISBN |9780198532712}}).

Most of the material I added to this Wikipedia article comes from Cantor's article, Cantor's correspondence, Dauben's biography of Cantor ({{ISBN |0674348710}}), and the article [http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=2907 "Georg Cantor and Transcendental Numbers"].

Finally, I wish to thank all the people who have worked on this Wikipedia article. Without the excellent structuring of your article and the topics you chose to cover, I suspect that I would not have written anything. (This is the first time I've written for Wikipedia.) It's much easier to add and revise rather than develop from scratch. [[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 23:30, 5 May 2009 (UTC)

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== Contrast 2nd theorem with sequence of rational numbers? ==

Cantor's 2nd theorem seems obvious at first glance to many people, as we usually are unable to imagine a sequence that could completely fill a whole interval. However, there are sequences (like that of all positive rational numbers) whose set of [[accumulation point]]s equals a whole interval (or even whole ℝ₊; cf. the picture [[accumulation point|there]]). Mentioning this in the article might prevent novice readers from thinking "''Mathematicians make a big fuzz proving things that are obvious, anyway''", and might generally help to sharpen one's intuition about what a sequence ''can'' do in relation to an interval and what it ''cannot''. It would require, however, to explain the notion of an ''accumulation point'' (which is poorly represented in English Wikipedia in general). - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 11:52, 17 December 2013 (UTC)

{{Talk:Georg Cantor's first uncountabilityset theory proofarticle/GA1}}