On Sat, 15 Jul 2006 15:43:28 -0600, Virgil <email@example.com> wrote:
>In article <firstname.lastname@example.org>, > Lester Zick <DontBother@nowhere.net> wrote: > > >> Well I don't have Turing's paper with me but as I recollect it was >> entitled something like "On the Computability of Numbers". What I >> don't know was exactly how exhaustive the paper was in that regard. >> But I'm not sure I'd agree that all the reals are computable in this >> sense. Certainly the rationals are but irrationals and transcendentals >> not obviously so. > > >That seems to imply that irrationals and transcendentals are disjoint. >In fact the set of transcendentals is a proper subset of the set of >irrationals.
Well, Virgil, no disrespect but I've been over and over this issue on various venues. The conventional definition you refer to is based on non repeating fractional rational approximations first, I believe, enunciated by Euler.I consider this definitional regression incorrect, that rationals and irrationals are defined on straight line segments and transcendentals on curves.