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Re: An afterthought about the Binary Tree

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On 07/05/2026 23:48, WM wrote:

> Meanwhile I know three mathematicians [1, 2, 3] who deny that the Binary 
> Tree can produce the paths belonging to single real numbers.

Mathematical objecs like a binary tree don't produce. They just are.
There are paths in a binary tree but they don't go anywhere other
tnan to nodes of the tree. Unless at least some nodes are real numbers
the paths don't go to any real number and in any case they don't go
to any other real number.

> There 
> remain sheaves or bunches of paths, each one containing uncountably many 
> paths which are not further distinguishable in the infinite Binary Tree.

Every path is distinguished from every other path by any one of the
nodes that one of them contains and the other does not.

>   /\
> /\/\
> ...
> 
> In my opinion this forbids the complete digit sequence of any real 
> number because a path in the Binary Tree is nothing else than a sequence 
> of bits. On the other hand Cantor's diagonal argument produces a 
> complete digit sequence (in the original version [4] a complete bit 
> sequence, using the symbols W M) of a real number, namely the famous 
> diagonal number.

A bit sequence is useful for proving that the power set of a
countable set is not countable. For uncountablility of reals there is
the problem that bit sequences with only finitely many zeros are
different from bit sequences with only finitely many ones but denote
the same real numbers. This problem is avoided with base 3 or higher.

> How can this contradiction be resolved?

The most effective way is to stick to formal proofs that are verified
with a good simple proof checker.

-- 
Mikko

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An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-05-07 22:48 +0200
  Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-08 10:58 +0300
    Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-05-08 14:46 +0200
      Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-09 10:59 +0300
        Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-05-09 23:20 +0200
          Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-10 10:25 +0300
            Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-05-10 15:56 +0200
              Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-11 10:51 +0300
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-05-11 13:42 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-12 10:50 +0300
                Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-05-12 13:36 +0200
    Re: An afterthought about the Binary Tree Moebius <invalid@example.invalid> - 2026-05-08 14:57 +0200
      Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-05-08 15:16 +0200
      Re: An afterthought about the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-05-08 10:16 -0700
        Re: An afterthought about the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-05-09 11:02 -0700
    Re: An afterthought about the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-05-08 09:58 -0700
      Re: An afterthought about the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-05-08 10:47 -0700

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