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Re: An afterthought about the Binary Tree

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Am 08.05.2026 um 09:58 schrieb Mikko:
> 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.

That is a matter of taste. It is possible to describe what happens when 
we go through a mathematical object. Then a sequence can decrease, a sum 
or series can grow and a node can produce a sheaf.

  They just are.
> There are paths in a binary tree but they don't go

In fact, they go? Fast or slow?

> anywhere other
> tnan to nodes of the tree. Unless at least some nodes are real numbers

No. Nodes are points. They can be defined by natural numbers. Real 
numbers are (represented by) paths. But obviously only countably many 
can be distinguished by nodes.

> the paths don't go to any real number

They are (representing) real numbers.
> 
>> 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.

But it is obvious that the current formalism is nonsense since it 
assumes or even proves that every rational number can be finitely 
defined (disproved above) and that uncountably many paths differing by 
nodes are existing in the Binary Tree (contradiction accepted by 
yourself). So why should anybody depend on that???

Regards, WM

Regards, WM>

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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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