In article <email@example.com>, Lester Zick <DontBother@nowhere.net> wrote:
> On Sat, 15 Jul 2006 15:43:28 -0600, Virgil <firstname.lastname@example.org> wrote: > > >In article <email@example.com>, > > 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.
That may be your definition, but it is no one else's.
According to the Harper Collins Dictionary of mathematics, for example,
irrational: not expressible as a ratio of integers,
transcendental: not the root of any polynomial equation with rational coefficients.
Unless you can show that your "definitions" are at least compatible with these, yours are wrong.