In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 25 Mrz., 15:48, "Ross A. Finlayson" <ross.finlay...@gmail.com> > wrote: > > On Mar 25, 4:45 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > Does the Binary Tree that contains all rational paths also > > > contain all irrational paths? > > > Given that, for R[0,1]: > > > > a) each irrational has a unique infinite expansion as path > > That is the question. If so, why has never anybody written it using > digits or bits? Everyone who tried died of boredom before finishing.
Why don't you try, WM? > > > > b) each initial segment of the expansion is the initial segment of a > > rational > > > > c) every rational's path is in the tree > > That is the question too. Why has never anybody written the complete > decimal- or binary expansion of a periodic rational?
Everyone who tried died of boredom before finishing. Why don't you try, WM? > > > > d) the union of finite initial segments of the expansion as tree > > contains the expansion as path > > > > e) thus each irrational's expansion is a path in the tree of rationals > > > > then, yes, that appears to be so. > > I agree with your conclusion but not with the premises. > > Remember: Never has anybody written an infinite sequence other than by > using the symbolic method: "1/9" or "1/pi" or "1/(SUM 1/n!)". These > however are only names to identify or formulas to construct infinite > paths - not paths that belong to the Binary Tree.
No one has ever written down an actual number rather than the name of, or numeral for, that number, so according to WM's thesis, we might as well ignore all numbers entirely, as we never actually deal with them directly. --