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

Revision as of 21:42, 8 August 2018

Revision as of 15:05, 9 August 2018

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: 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]]'''_{[[Special:Contribs/Bilorv|(c)]][[User talk:Bilorv|('''talk''')]]}</span> 21:42, 8 August 2018 (UTC)

: Concerning "Liouville's theorem that there are transcendental numbers": I wrote both parts almost identical to the way it appears in Cantor's article and letter. If I understand you correctly, you would like the part in the article to read: "Cantor observes that combining his two theorems yields a new proof of [[Liouville number|Liouville's theorem]] that every interval [''a'',&nbsp;''b''] contains infinitely many [[transcendental number]]s." The part in the letter would be unchanged: "It would be nice if it could be answered; for example, provided that it could be answered ''no'', one would have a new proof of [[Liouville number|Liouville's theorem]] that there are transcendental numbers." The only problem here is that Liouville's theorem is expressed two different ways: one asserting infinitely many transcendental numbers and the other asserting the existence of transcendental numbers. The first implies the second, and the second implies the first as soon as one realizes that the existence of one transcendental implies the existence of infinitely many (which goes back to my earlier mention of generating infinity many by adding rationals to the transcendental that was proved to exist). We are stuck with these choices because Cantor knows that if you proved one statement, the other statement is an easy consequence, so he feels comfortable informally using the two differing statements as if they were equivalent. Which option do you prefer or do you have another option? —[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 15:05, 9 August 2018 (UTC)