On 22 Feb., 23:23, Virgil <vir...@ligriv.com> wrote:
> > Here is a summary of the argument concerning the Binary Tree: > > > 1) The set of all real numbers of the unit interval is (said to be) > > uncountable. > > 2) An uncountable set has (infinitely many) more elements than a > > countable set. > > 3) Every real number has at least one unique representation as an > > infinite binary string (some rationals have even two representations > > but that's peanuts). > > 4) In many cases the string cannot be defined by a finite word. > > 5) Without loss of information the first bits of two strings, if > > equal, need not be written twice. > > 6) Application of this rule leads to the Binary Tree. > > 7) The binary strings of the unit interval are isomorphic to the paths > > of the Binary Tree. > > If WM means they are of equal cardinality or biject with each other , > true, but to establish an isomorphism, as WM is claiming, one must > specify the structure that is being preserved by the bijection, which WM > has NOT done.
The mapping is bijective and linear.
> > 8) It is not possible to distinguish more than countably many paths by > > their nodes. > > The set of paths of a CIBT is easily bijected with the set of all > subsets of |N (the path generates the set of naturals corresponding to > the levels at which that path branches left rather than right) which > allows us easily to distinguish any path from any other by the diffences > in their corresponding sets of naturals.
This shows a contradiction - at least in case someone accepts Hessenberg's trick as part of mathematics.