> > 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. 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. 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.