On Mar 25, 8:09 am, WM <mueck...@rz.fh-augsburg.de> 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? > > > > > 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? > > > > > 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. > > Regards, WM
Two different paths, finite or infinite, have at least one node not in common.
And they have at most countably many nodes not in common, there are only countably many nodes.
And, there are only countably many nodes a distinct path, from a path, could have. Then, there are only that many distinct paths, as for each there are only countably many others.
Yet, mapping the paths onto 2^w and thus to P(w), that would be a contradiction to Cantor's theorem.
So, are there uncountably many nodes? Because, there's a rational for each node, and the rationals are countable. There are only countably many different paths, from a path. The paths are rooted.
Seems rather muddled.
Infinite sequences are written out as their specification. No, there are not infinite digital resources to emit each (element of an infinite sequence), but, a rule to emit each suffices for many purposes.
Then in as to whether Eudoxus/Dedekind/Cauchy expansions are sufficient to represent each element of the linear continuum of what would be the real numbers, is a separate notion. (And, no, they aren't.)