Concerning the rarity of provably transcendental real numbers - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T19:49:34Z http://mathoverflow.net/feeds/question/59638 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59638/concerning-the-rarity-of-provably-transcendental-real-numbers Concerning the rarity of provably transcendental real numbers David Feldman 2011-03-26T07:44:48Z 2012-04-18T03:12:30Z <p>Does there exist any rubric where <em>provably</em> transcendental real numbers emerge, in a meaningful way, as rare among <em>all</em> the transcendental numbers?</p> <p>Here are some of the things I'm worried about:</p> <p>1) To talk about provably transcendental numbers, it seems only fair to consider them as a subset of some sort of set of definable real numbers (relative to some appropriate language). If the language is countable, that means comparing two countable sets, so measure-theoretic language doesn't seem to help.</p> <p>2) Some transcendentality proofs naturally apply to all the numbers in a definable uncountable set (which of course contains many undefinable numbers). Small variations of Liouville's famous original construction yield uncountable sets of transcendentals. So a countable language that can only encode a countable number of proofs can still establish the transcendentality of more than countably many numbers.</p> <p>Perhaps something like this: relative to a fixed language one can define a complexity for definable transcendental reals by the length of their shortest defining formula. Among those of a given complexity, some fraction admit transcendentality proofs. Perhaps this fraction must go to 0 with the complexity for any reasonable theory? (This seems to me a meaningful question despite that attendant undecibilities concerning whether a formula defines a numbers, whether two formulas define the same number, etc.)</p> http://mathoverflow.net/questions/59638/concerning-the-rarity-of-provably-transcendental-real-numbers/94350#94350 Answer by Eyal Goren for Concerning the rarity of provably transcendental real numbers Eyal Goren 2012-04-18T03:12:30Z 2012-04-18T03:12:30Z <p>I am not sure this is what your question aims at, but: \$e^{1/n}\$ (see Wikipedia) for natural n> 0, are transcendental and have very simple continued fraction expansion. If memory serves, Hurwitz has theorems generalizing this phenomenon (and perhaps Baker, much later). Sorry to be so shaky on the details here... The characterization, if memory serves, is again in terms of continued fractions where the pattern of the coefficients follows some arithmetic series. In that sense, it is a countable set of provably transcendental real numbers, that is nonetheless very rare among all transcendental numbers. </p>