In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 29 Mrz., 19:34, Virgil <vir...@ligriv.com> wrote: > > > > > > So we have established the fact that an irrational number has no node > > > of its own. > > > > No number in any infinite binary tree has any node "of its own", as > > every node has two child nodes belonging to necessarily different > > numbers. > > That is correct, but only establishes the fact that no actually > infinite path can be distinguished from all rational paths as should > be possible in a Cantor-list - but is not.
Every rational path in ANY Complete Infinite Binary Tree is an actually infinite path, as that is the only sort of paths such a tree is allowed to have.
Thus every binary-rational path in a Complete Infinite Binary Tree is an actually infinite path. Such binary-rational paths are the ones that are eventually all left-child nodes or all right-child nodes from some node onward. > > > > The first bit causes that the anti- > > > diagonal differs from 2^-1 of all entries and agrees with 2^-1, the > > > second bit leaves 2^2 entries and the n-th bits leaves 2-^n entries of > > > the list agreeing with the anti-diagonal. There is no bit that would > > > leave zero entries agreeing with the diagonal, since there is no bit > > > at position infinity. > > > > Thus there is always at least one bit of any listed entry disagreeing > > with the antidiagonanl, just as the Cantor proof requires. > > In a list containing every rational: Is there always, i.e., up to > every digit, an infinite set of paths identical with the anti- > diagonal? Yes or no?
The set of paths in any Complete Infinite Binary Tree which agree with any particular path up to its nth node is equinumerous with the set of all paths in the entire tree i.e., is uncountably infinite.
Note that in Wolkenmuekenheim, at least according to WM's standard constraints there, no such thing as a Complete Infinite Binary Tree can exist.
And WM's ability to think about things outside Wolkenmuekenheim is seriosly inhibited by his delusions about how things work outside of Wolkenmuekenheim, . --