Talk:Transcendental number theory

message: I'd love to help get this article into reasonable shape, but it seems inevitable to me that it is going to overlap a lot with the transcendental number article.  I'm definitely not suggesting the two be merged but it'd be useful to get some suggestions on what to put into which article Chenxlee (talk) 20:02, 13 February 2008 (UTC)
replies: text: There also are bibliographic references at Transcendental number that should be migrated over to here. Will Orrick (talk) 22:26, 5 November 2022 (UTC)
thread_title:  Overlap with Transcendental number article 

message: i hate to sound like a retard but does
Other techniques: Cantor and Zilber
Liouville proved in the 1850s that transcendental numbers exist and even gave examples, but his ideas can be used to give only a countable number of transcendental numbers. Indeed, when he published his results the notion of countability and uncountability had not yet been developed. In the 1870s Georg Cantor started to develop set theory and in 1874 published a paper proving that the algebraic numbers could be put in one-to-one correspondence with the set of natural numbers, and thus that the set of transcendental numbers must be uncountable.[15] Later, in 1891, Cantor used his more familiar diagonal argument to prove the same result.[16] While Cantor's result is often quoted as being purely existential and thus unusable for constructing a single transcendental number[17][18], the proofs in both the aforementioned papers give methods to construct transcendental numbers.[19]
mean methods to construct (as in the constructivists sense) all transcendentals?  —Preceding unsigned comment added by Cheat notes (talk • contribs) 04:41, 15 January 2009 (UTC)
replies: text: If you use the method gleaned from Cantor's diagonal argument then you can get every transcendental number, and that's proved in Gray's paper if you can find a copy.  I'm fairly sure that the same holds true if you use his 1874 proof, too. Chenxlee (talk) 14:37, 21 January 2009 (UTC)

text: Added new reference to Gray's paper that gives everyone access to entire article. Also, added reference to Cantor's 1874 article. In Gray's paper, it is proved that every transcendental can be generated by a suitable enumeration of the algebraic real numbers. If you start with a computable transcendental (see computable number), then the proof generates a computable enumeration of the algebraic reals. However, if you start with a highly non-computable transcendental, then the proof generates a highly non-computable enumeration of the algebraic reals. Basically, the enumeration of the algebraic reals will be as complex (or constructive) as the transcendental is (and vice versa). Also, the remark above about using the Cantor's 1874 proof in a similar way is true. I can answer any questions about the article since I wrote it. I'm pleased to see that the article is being discussed. --RJGray (talk) 02:52, 11 August 2009 (UTC)
thread_title:  constructing all the transcendental numbers? 

message: Is number theory really defined broadly enough to include transcendence theory as a branch of it, as the lead sentence here says? I don't get that impression from the article Number theory. 208.50.124.65 (talk) 22:18, 18 June 2014 (UTC)
replies: text: Certainly it is.  See for example Baker's book Transcendental Number Theory.  Deltahedron (talk) 19:53, 19 June 2014 (UTC)
thread_title:  A branch of number theory? 

message: The most recent version of his 'Transcendental Number Theory' was the paperback edition of 1990. I suggest that all citations point to this edition. Scott Tillinghast, Houston TX (talk) 02:02, 17 December 2020 (UTC)
thread_title:  Alan Baker's book