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

Revision as of 15:28, 9 August 2018

Revision as of 16:13, 10 August 2018

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: 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'', ''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 (leaving it unchanged or using Liouville's theorem in both places) or do you have another option? —[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 15:05, 9 August 2018 (UTC)

::Yes, I'd rather have Liouville's theorem used to describe both (and I think the linking you just used is better than the article's "[[Liouville number#Liouville numbers and transcendence|Liouville's theorem that there are transcendental numbers]]", which is too long a phrase to link). <span class="nowrap">— '''[[User:Bilorv|Bilorv]]'''_{[[Special:Contribs/Bilorv|(c)]][[User talk:Bilorv|('''talk''')]]}</span> 15:28, 9 August 2018 (UTC)

:Concerning "The proofs": My approach was to stay within the guidelines of [[WP:Scientific citation guidelines#Examples, derivations and restatements]] whose first paragraph states:

:"Wikipedia is neither a textbook nor a journal. Nonetheless, in mathematics and the mathematical sciences, it is frequently helpful to quote theorems, include simple derivations, and provide illustrative examples. For reasons of notation, clarity, consistency, or simplicity it is often necessary to state things in a slightly different way than they are stated in the references, to provide a different derivation, or to provide an example. This is standard practice in journals, and does not make any claim of novelty.[1] In Wikipedia articles this does not constitute original research and is perfectly permissible – in fact, encouraged – provided that a reader who reads and understands the references can easily see how the material in the Wikipedia article can be inferred. Furthermore, copying extensively from a source with only minor modifications is not normally permitted by copyright law, unless the source has a free license."

:In particular, I wrote "The proofs" knowing that providing a different derivation or an example is "perfectly permissible" and is "in fact, encouraged" as long as a reader knowledgeable about the references (in this case, Cantor's 1874 article) can easily see how the Wikipedia derivation can be inferred.

:The table in the first proof is an "illustrative example". This example has the additional feature of being able to be verified by doing calculations to generate the polynomials with height ≤ 4, identify the irreducible polynomials, and compute the roots.

:The closed interval simplification in the second proof is a slightly "different derivation" that "a reader who reads and understands the references can easily see how the material in the Wikipedia article can be inferred." In fact, I help the reader by describing the major difference between Cantor's derivation and the one using the closed interval simplification.

:My reason for the closed interval simplification is for "clarity" and "simplicity" so the Wikipedia proof is more accessible to Wikipedia readers than Cantor's original proof, which was meant for research mathematicians. For example, the closed interval simplification is crucial to the case diagrams, which handle some criticism from the GA review of an older version of this Wikipedia article. Cantor's method of bouncing between closed intervals and their interiors would lead very cluttered diagrams. By the way, it was natural for Cantor to only work with closed intervals—he only had notation for them. He later defined "closed set". Open sets came much later—they were first defined in print by Baire in 1899. (See Gregory Moore's [https://www.sciencedirect.com/science/article/pii/S0315086008000050 The emergence of open sets, closed sets, and limit points in analysis and topology].)

:The example of Cantor's construction is another "illustrative example" with the additional feature of being able to be verified by doing some calculations. — [[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 16:12, 10 August 2018 (UTC)