In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 24 Mrz., 23:13, "Ross A. Finlayson" <ross.finlay...@gmail.com> > wrote: > > On Mar 24, 2:51 pm, fom <fomJ...@nyms.net> wrote: > > > > > > > > > > > > > On 3/24/2013 4:34 PM, WM wrote: > > > > > > On 24 Mrz., 21:29, Virgil <vir...@ligriv.com> wrote: > > > >> In article > > > >> <729f073f-8948-4eb9-991a-2bd249ac5...@c6g2000yqh.googlegroups.com>, > > > > > >> A binary tree that contains one path of each positive natural number > > > >> length will necessarily also contain exactly one path of infinite > > > >> length. > > > > > > Like the sequence > > > > 0.1 > > > > 0.11 > > > > 0.111 > > > > ... > > > > that necessarily also contains its limit? > > > > A binary tree that contains one path, of all zero-branches, of each > > finite length, will necessarily contain a path of 0-branches of > > infinite length. > > If actual infinity exists. Otherwise it contains nothing more than all > finite paths.
Since "all finite paths", unless some finite maximal length is specified, is like all natural numbers, not a infinite set, WM is once again (and still boringly) wrong!
> But here is a question that is easier to answer and to > decide: Does the Binary Tree that contains all rational paths also > contain all irrational paths?
That depends on what WM means by a "rational path".
If what WM means is a path in an infinite binary tree which has only finitely many branchings in one of the two directions, any tree with all possible such paths will also have "irrational" paths as a sort of "limit path" to monotone sequences of WM's finite paths, just as each irrational strictly between 0 and 1 is a limit of an increasing sequence of binary rationals and also the limit of a decreasing sequence of binary rationals all between 0 and 1. --