On 2/4/2013 2:15 AM, WM wrote: > On 3 Feb., 23:04, fom <fomJ...@nyms.net> wrote: > >>> There is no sensible way of saying that 0.111... is more than every >>> FIS. And every FIS is in a line. >> >> Do you mean "is in some line"? > > What is s? In decimal it is 1/9. In binary it is 1. In paths or > strings of bits or decimals it does not exist! >> >> As in "there exists a line containing a given FIS"- > > There is not more than every FIS of 0.111... Not even all FIS. Here > you may see why: > 1) The set of all real numbers of the unit interval is (said to be) > uncountable.
So, one takes "uncountable" as negatively defined in distinction with the usual definition concerning correspondence with the smallest completed infinity containing every natural number.
...compared with, say, "a set of vectors is linearly independent if it is not linearly dependent"
> 2) An uncountable set has (infinitely many) more elements than a > countable set.
By "more," you mean that the construction of a new name may be accomplished and by "infinitely many" you mean that consecutive constructions can always be performed sequentially without end from any initial finite configuration of names.
> 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).
By "uniqueness", you mean there is a strategy for constructing names that always allows you to differentiate a single object from a plurality on the basis of "naming"
> 4) In many cases the string cannot be defined by a finite word.
What would be the limitation here? Is it the negative logic of "since there are more numbers than names..."?
> 5) Without loss of information the first bits of two strings, if > equal, need not be written twice.
This starts to become a little problematic. Now, your numbers are turning into classes of numbers. And, your names are turning into the names for canonical representatives of those classes if the partition is viewed as an equivalence partition.
> 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. > 8) It is not possible to distinguish more than countably many paths by > their nodes. > 9) This is proven by constructing the Binary Tree node by node. > 10) Further this is proven by colouring all edges and nodes and paths > of the complete Binary Tree by countably many paths.
This last part is not problematic. I do not consider the debate you have been having as reflecting the actual sense of the Cantor space (by which I mean "topological"). You are describing the well-founded k-equivalent parts of a tree structure in a discrete sense.
In trying to understand your position, I hope my paraphrasing has been reasonably accurate.