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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 2026-05-20 13:34 +0200
Organization tha
Message-ID <10uk67t$5pbq$1@solani.org> (permalink)
References (19 earlier) <10ueftj$2b1ke$1@dont-email.me> <10uep9g$268t$1@solani.org> <10uh5o5$33rgf$1@dont-email.me> <10ui427$4gbs$1@solani.org> <10ujpn6$3qsm2$1@dont-email.me>

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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.>
>> 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.

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/> 

>>>> 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.

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 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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