In article <email@example.com>, WM <firstname.lastname@example.org> 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.
In ZF, and most other set theories in which the set of all naturals is allowed, the uncountability of the set of all subsets of the set of naturals is provable.
> 2) An uncountable set has (infinitely many) more elements than a > countable set.
Quite so, at least outside WMytheology.
> 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).
But nothing guarantees that one can always find such a decimal or binary representation.
> 4) In many cases the string cannot be defined by a finite word.
That some reals are inaccessible is well known.
> 5) Without loss of information the first bits of two strings, if > equal, need not be written twice.
Without loss of information the FISs of uncountably many binary strings, if equal, need not be written twice.
> 6) Application of this rule leads to the Binary Tree.
Since WM's misunderstandings of of a Complete Infinite Binary Tree are pervasive and apparently incorrectable, from here on WM's stuff is even more junky than usual. --