In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 22 Feb., 14:27, fom <fomJ...@nyms.net> wrote: > > > > > what you are describing here is the structure of the > > natural numbers as a directed set. > > > > There is actual mathematics that describes this. > > > > There are logics in which your reasoning can be > > formalized. > > That is not necessary for pople who can think without crutches.
But it is painfully obvious to everyone BUT WM that WM needs far more crutches than he will allow himself to use. > > > > And the fact that this structure remains as absolute > > infinity relative to Cantor's transfinite arithmetic > > does not make that arithmetical calculus not > > mathematics. > > Cantor has been disproved.
By whom? Cantor has not been disproved within ZF or ZFC or NBG or any standard set theory, so in what axiom system does WM claim it has been disproved?
And where has that disproof been posted?
> Try to understand the Binary Tree. Then you > will understand that not more than countably many paths can be > distinguished.
But uncountably many of them can be shown to exist, like the power set of |N being larger than |N everywhere outside WMytheology.
While in WMytheology 'distinguishability' and 'existence' may be cognates, in mathematics they are not. > > 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.
You really should learn to use mathematical terms correctly, WM, before pontificating, the way you do.
> 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.
So again, WM's ignorance of CIBTs, trips him up! --