On 22 Jun., 01:32, Virgil <Vir...@home.esc> wrote: > In article > <e0b8cbfa-70a9-4be3-a08f-117e0af7f...@q12g2000yqj.googlegroups.com>, > > > > > > WM <mueck...@rz.fh-augsburg.de> wrote: > > On 21 Jun., 22:17, Virgil <Vir...@home.esc> wrote: > > > > > But the set of all diagonals constructed according to the scheme given > > > > above is countable but cannot be listed. > > > > Whatever do yu mean by "the set of all diagonals"? > > > If you mean one "diagonal" for each possible list, that set of diagonals > > > is not countable. > > > If you mean the set of all "diagonals" for a single list, that is not > > > countable either, since each of uncountably many permutations of the > > > list produces a different "diagonal". > > > Take a list of all rationals. Construct, accordingto a given > > substitution rule the antidiagonal. Add it at first position to the > > list. Construct, according to the given rule the antidiagonal, add > > it ... and so on. > > You are saying given a list, create its anti-diagonal and prepend it to > that list. Repeat with the new list ad infinitum. > > > > > The number of antidiagonal is countable. Nevertheless it cannot be > > listed. > > I see no problem in listing the set of such anti-diagonals.
Including all rational numbers!
> They can > even be listed in the order in which each anti-diagonal is to be created. > > Note that the list of anti-diagonals so far created is finite so long as > the number of iterations is finite, and only becomes infinite when the > process achieves infinitely many iterations, and thus it is countable. > > Which such anti-diagonals does WM claim remain unlisted?
That of the intended list of all those antidiagonals and all rational numbers. >
> > The complete infinite binary tree contains, by definition, all real > > numbers between 0 and 1 as infinite paths, i. e., as infinite > > sequences { 0, 1 }^N of bits. > > > 0, > > / \ > > 0 1 > > / \ / \ > > 0 1 0 1 > > / > > 0 ... > > > The set { K_k | k in N } of nodes K_k of the tree is countable. > > > K_0 > > / \ > > K_1 K_2 > > / \ / \ > > K_3 K_4 K_5 K_6 > > / > > K_7 ... > > > The tree is constructed by extending the configurations B_j as > > explained below: > > > _________________ > > B_0 = > > > K_0 > > _________________ > > B_1 = > > > K_0 > > / > > K_1 > > _________________ > > B_2 = > > > K_0 > > / \ > > K_1 K_2 > > _________________ > > B_3 = > > > K_0 > > / \ > > K_1 K_2 > > / > > K_3 > > _________________ > > > B_4 = > > > K_0 > > / \ > > K_1 K_2 > > / \ > > K_3 K_4 > > _________________ > > ... > > _________________ > > B_j = > > > K_0 > > / \ > > K_1 K_2 > > / \ > > K_3 K_4 ... > > ... > > ... K_j > > _________________ > > ... > > _________________ > > > The complete infinite binary tree (including all those infinite paths > > which consist of nodes and edges only) is constructed by a countable > > number of steps. In no step more than one infinite paths is extended. > > Hence there are not more than countably many infinite paths. > > I have no idea what sort of definition of "countability" of infinite > sets that WM is using,
I know that you cannot read the above text. But perhaps somewhat else can.
