Groups | Search | Server Info | Login | Register

Groups > sci.logic > #345866

Re: An afterthought about the Binary Tree

Show all headers | View raw

On 08/05/2026 15:46, wm wrote:
> 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?

In Common Language the word "go" has a different meaning when the
subject is "path" (or something else of similar nature). ALthough
the mathematical meaning of "path" is not the same as in Common
Language it is similar enough that the word "go" can be understood
even in a mathematical context. Likewise the words "from", "through",
and "to". Of course, in an actual mathematical presentation all
these words must be defined if used.
>> anywhere other
>> tnan to nodes of the tree. Unless at least some nodes are real numbers

> No. Nodes are points.

They needn't be anything other that nodes. But they can be anyting.

> They can be defined by natural numbers.

Nodes can even be natural numbers. That is the simplest way to relate
nodes to natural numbers.

> Real numbers are (represented by) paths. But obviously only countably many 
> can be distinguished by nodes.

That is not obvious and not true. And if the nodes are defined as the
set of nodes they go through there is no need to any distinction than
that onother path is a different set.

>> the paths don't go to any real number
> 
> They are (representing) real numbers.

Without a mapping from paths to real numbers a path cannot represent
a real number. Without presenting the mapping it is not possible to
determine whether there are real numbers that are represented by
several paths or none.

>>> 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???

Every rational number can be represented by a pair of integer numbers
where the send one is greater than zero. Every integer can be
represented as a finite string of characters of a finite alphabeth.
Therefore every rational number can be represented as a finite string.
You may wtite whatever humbug you want but that does not change the
facts.

-- 
Mikko

Back to sci.logic | Previous | Next — Previous in thread | Next in thread | Find similar

Thread

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

csiph-web