On Dec 16, 7:47 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > Al-Jofar writes: > > 1: "This is a simple Corollary of Cantor's diagonal argument actually. > Proof: Let G be any countable subtree of the infinite binary tree that > is 0 rooted and such that all paths ending by 0-0-0-... or by > 1-1-1-...are among the paths of G. " > > 'Let G be a subtree of the IBBT including all expansions representing > fractions by powers of two (there are two of those for each of those > values).' > > 2: "Let f be a bijective function from the domain N of all > naturals(except 0) to the set of all infinite paths of G. " > > For the countable subtree only containing fractions by powers of two > and their dual representations, that's countable (and not obviously > all the paths). > > 3: "Construct the diagonal path d_f in the following manner: The root > (i.e. the 1st) node of d_f is 0 labeled. Now for n=1,2,3,.. ; The n > +1_th node of d_f is labeled by a label that is opposite to the label > of the n+1_th node of the path of G that f sends n to. " > > Let G be all the paths. In lexicographic order, here as described as > BT _oo (binary tree's breadth-first traversal at infinity), augmented > with .0111... being first to reflect the modification of "the" anti- > diagonal, the generated anti-diagonal path is .0111..., that is the > least element of G in the ordering. >
No, it's not.
> With an un-modified construction of the antidiagonal, the generated > anti-diagonal path is .111... and at the end of any course-of-passage > through the paths in their natural order, i.e., never before the end. > That is, there's a simpler corollary that is the same as the binary > anti-diagonal argument. > > 4: "Now clearly the diagonal d_f (actually the anti-diagonal but it > shall be called the diagonal for short) is a path and clearly it is > labeled in a way that is different from labeling of all paths of G. So > d_f is missing from G. " > > In the course of passage with the natural order of the paths, d_f = > f(0) or here f(1) and is not missing from G. >
> 5: "So any countable subtree of the infinite binary tree, that is 0 > rooted and that has all paths ending with 0-0-0-.. or with 1-1-1.. > among its paths; would be missing a path of the infinite binary tree. > " > > That doesn't follow for the natural ordering of the paths by their > content as expansions. Then though while the paths are totally > ordered, well-ordering them would be as well-ordering the reals. >
> Then I'll happily accept that the anti-diagonal argument is much the > same for the list of expansions or the tree of expansions, then that > EF's antidiagonal is .111... and at the end of the list and BT's > antidiagonal is .111... and rightmost of the tree. > > Thanks, > > Ross Finlayson