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Re: An afterthought about the Binary Tree

From wm <wolfgang.mueckenheim@tha.de>
Newsgroups sci.logic
Subject Re: An afterthought about the Binary Tree
Date 2026-05-21 17:24 +0200
Organization tha
Message-ID <10un831$7vsp$1@solani.org> (permalink)
References (21 earlier) <10uh5o5$33rgf$1@dont-email.me> <10ui427$4gbs$1@solani.org> <10ujpn6$3qsm2$1@dont-email.me> <10uk67t$5pbq$1@solani.org> <10umc9b$ir8q$1@dont-email.me>

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Am 21.05.2026 um 09:29 schrieb Mikko:
> On 20/05/2026 14:34, wm wrote:
>> Am 20.05.2026 um 10:00 schrieb Mikko:
>>> On 19/05/2026 19:44, wm wrote:
>>>> Set theory requires fixed sets. That is actual infinity. To 
>>>> enumerate the prime numbers, the natural numbers and the fractions 
>>>> however requires a very flexible set.
>>>
>>> There is nothing flexible in any set. There is the set of prime numbers,
>>> whicn is a subset of natural numbers, there is the set of natural
>>> numbers, and there is the set of fractions. Those sets contain what
>>> they contain and don't contain anything else.
> 
>> But these fixed sets are in bijection with ℕ. Therefore ℕ has 
>> sometimes more and sometimes less elements.
> 
> Your "therefore" is false. N always contains all natural numbers and
> nothing else.

But the bijected sets are very different.>
>>>> Of course there are more rational numbers than prime numbers. 1 is 
>>>> rational but not prime.
>>>
>>> There is a simple bijection between rational numbers and integers.
>>> There is also a fairly simple bijection between integers and natural
>>> numbers. There is a less simple bijection between natural numbers
>>> and prime numbers. The existence of these bejections means that the
>>> these sets are equinumerous. Because the set of natural numbers is
>>> one of them they are all countable.
>>
>> Equinumerosity means obviously different numerosity.
> 
> No. Two sets are equinumerous if there is a bijection between them.
> That is the only meaning of the word.

There are no bijrctions between infinite sets, because most of their 
elements are dark, see e.g. 
https://www.reddit.com/r/AspectsOfTheInfinite/comments/1tc6v1l/proof_of_the_existence_of_dark_numbers/> 

>> By the way, the bijection between fractions and natural numbers has 
>> been disproved: https://www.reddit.com/r/AspectsOfTheInfinite/ 
>> comments/1tc6v1l/proof_of_the_existence_of_dark_numbers/
> 
> No. A fallacy does not override a proof. A bijection has been presented
> so it obviously exists. No other "proof" can make impossible what has
> already been done.

The proof shows that your bujection is not between sets but between 
potentially infinite collections only.>
>>>>>> Alas most elements cannot be treated as individuals.
>>>>>
>>>>> For many puroposes those can be that are needed.
>>>>
>>>> Of course. But not for individual treatment.
>>> Those that cannot be handled individually can be for many purposes
>>> handled with quantifiers.
>>
>> At least you accept them. Most set theorists are not even aware of 
>> dark numbers.
> There is no reason to call something dark.

The reasom is this: Choose two fractions as close together as you like. 
Between them there remain infinitely many fractions. Even if you divide 
the interval by 2 or by 10^10000000 this will never change.

Regards, WM



  It is well known that the
> first order logic is not powerful enough to exclude non-standard models
> of natural number arithmetic, some of which are uncontable. But the
> only known completeness theorem is restricte to first order logic. For
> second and higher order logic no complete set of valid inference rules
> is known.
>

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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 Mikko <mikko.levanto@iki.fi> - 2026-05-13 12:39 +0300
                Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-05-13 22:39 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-14 11:46 +0300
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-05-14 16:52 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-15 08:50 +0300
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-05-15 18:39 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-16 12:43 +0300
                Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-05-17 16:15 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-18 10:42 +0300
                Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-05-18 12:22 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-19 11:07 +0300
                Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-05-19 18:44 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-20 11:00 +0300
                Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-05-20 13:34 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-21 10:29 +0300
                Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-05-21 17:24 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-22 10:23 +0300
                Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-05-22 22:53 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-23 09:31 +0300
                Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-05-23 23:09 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-03 12:35 +0300
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-05-23 22:40 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-24 12:03 +0300
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-05-24 12:59 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-25 11:33 +0300
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-05-25 18:54 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-26 11:43 +0300
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-05-26 12:43 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-27 10:37 +0300
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-05-27 14:19 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-28 09:34 +0300
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-05-28 14:46 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-29 09:52 +0300
                Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-05-29 16:36 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-30 10:38 +0300
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-05-30 15:19 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-05-31 12:14 +0300
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-05-31 16:28 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-01 10:51 +0300
                Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-06-01 17:17 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-02 10:18 +0300
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-06-02 16:00 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-03 11:38 +0300
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-06-03 22:49 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-04 11:05 +0300
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-06-04 22:47 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-05 11:28 +0300
                Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-06-05 16:29 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-06 11:51 +0300
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-06 22:25 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-07 11:25 +0300
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-07 20:25 +0200
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-06-06 22:52 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-07 11:33 +0300
                Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-06-07 21:47 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-08 09:29 +0300
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-06-08 14:19 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-09 11:22 +0300
                Re: An afterthought about the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-08 08:21 -0700
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-08 10:11 +0300
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-06-08 14:42 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-09 11:25 +0300
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-09 15:22 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-09 02:43 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-09 02:46 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-09 02:57 +0200
                Re: An afterthought about the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-09 07:52 -0700
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-09 06:31 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-09 06:32 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-09 06:35 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-02 04:20 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-02 04:22 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-02 10:20 +0300
                Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-05-22 22:53 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-03 12:40 +0300
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-03 13:42 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-04 11:11 +0300
                Re: An afterthought about the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-03 09:43 -0700
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-06-03 23:07 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-04 11:13 +0300
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-04 16:06 +0200
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-06-04 22:52 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-05 11:44 +0300
                Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-06-05 16:37 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-06 12:19 +0300
                Re: An afterthought about the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-06-06 13:35 -0700
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-07 11:37 +0300
                Re: An afterthought about the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-06-08 13:04 -0700
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-08 22:21 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-08 22:36 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-08 22:56 +0200
                Re: An afterthought about the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-09 07:51 -0700
                Re: An afterthought about the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-06-09 14:54 -0700
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-09 23:56 +0200
                Re: An afterthought about the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-06-09 15:12 -0700
                Re: An afterthought about the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-06-09 15:33 -0700
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-10 00:59 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-10 01:03 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-06 22:41 +0200
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-06-06 22:45 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-07 11:58 +0300
                Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-06-07 21:41 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-08 09:36 +0300
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-06-08 14:29 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-09 11:51 +0300
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-09 17:58 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-09 18:18 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-08 17:42 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-10 06:45 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-06 12:03 +0300
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-06 22:28 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-07 12:04 +0300
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-07 21:02 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-09 02:33 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-09 11:52 +0300
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-06 22:29 +0200
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-06-06 22:49 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-07 12:09 +0300
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-07 17:02 +0200
                Re: An afterthought about the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-07 17:34 -0700
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-06-07 21:33 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-08 10:05 +0300
                Re: An afterthought about the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2026-06-08 14:34 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-09 12:10 +0300
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-10 06:32 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-10 13:30 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-10 13:33 +0200
                Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-06-18 22:15 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-08 17:53 +0200
                Re: An afterthought about the Binary Tree Mikko <mikko.levanto@iki.fi> - 2026-06-09 12:04 +0300
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-09 18:39 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-09 18:50 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-09 23:05 +0200
                Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-09 23:05 +0200
                Re: An afterthought about the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-05-20 18:07 -0700
                Re: An afterthought about the Binary Tree wm <wolfgang.mueckenheim@tha.de> - 2026-05-21 17:13 +0200
      Re: An afterthought about the Binary Tree Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2026-07-02 13:52 +0100
        Re: An afterthought about the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-07-02 08:50 -0700
    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
  Re: An afterthought about the Binary Tree Moebius <moebius@example.invalid> - 2026-06-06 04:35 +0200
  Re: An afterthought about the Binary Tree Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2026-07-02 13:12 +0100
    Re: An afterthought about the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-07-02 09:01 -0700
      Re: An afterthought about the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-07-02 11:53 -0700
        Re: An afterthought about the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-07-02 13:32 -0700
          Re: An afterthought about the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-07-02 13:46 -0700
            Re: An afterthought about the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-07-02 15:36 -0700
              Re: An afterthought about the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-07-02 16:01 -0700

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