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Groups > sci.logic > #345849 > unrolled thread
| Started by | WM <wolfgang.mueckenheim@tha.de> |
|---|---|
| First post | 2026-05-07 22:48 +0200 |
| Last post | 2026-07-02 16:01 -0700 |
| Articles | 20 on this page of 163 — 8 participants |
Back to article view | Back to sci.logic
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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| From | Moebius <moebius@example.invalid> |
|---|---|
| Date | 2026-06-09 06:35 +0200 |
| Message-ID | <110856o$3ng52$2@dont-email.me> |
| In reply to | #346583 |
Am 09.06.2026 um 06:32 schrieb Moebius: > Am 09.06.2026 um 02:43 schrieb Moebius: >> >> "Every actually infinite set of natural numbers has only finitely many >> elements." (W. Mückenheim, de.sci.mathematik, 07 Feb 2026) >> > In other words, some _infinite_ sets only have _finitely many_ elements > in Mückenheim's world. > > Does that sound SANE? (->Crank) Der Mann hat einfach nicht mehr alle Tassen im Schrank. [That man has simply lost his marbles.] . . . -- Diese E-Mail wurde von Avast-Antivirussoftware auf Viren geprüft. www.avast.com
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| From | Moebius <moebius@example.invalid> |
|---|---|
| Date | 2026-06-02 04:20 +0200 |
| Message-ID | <10vlekv$2l452$3@dont-email.me> |
| In reply to | #346187 |
Am 30.05.2026 um 15:19 schrieb WM: > modern mathematics is inconsistent. Ah ja. ->Wahn ->delusion -- Diese E-Mail wurde von Avast-Antivirussoftware auf Viren geprüft. www.avast.com
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| From | Moebius <moebius@example.invalid> |
|---|---|
| Date | 2026-06-02 04:22 +0200 |
| Message-ID | <10vleoa$2l452$4@dont-email.me> |
| In reply to | #346180 |
Am 30.05.2026 um 09:38 schrieb Mikko: > In mathematics there is no distinction between "visible" and "dark" > numbers. You think? :-P -- Diese E-Mail wurde von Avast-Antivirussoftware auf Viren geprüft. www.avast.com
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| From | Mikko <mikko.levanto@iki.fi> |
|---|---|
| Date | 2026-06-02 10:20 +0300 |
| Message-ID | <10vm07v$2p1kf$2@dont-email.me> |
| In reply to | #346212 |
On 02/06/2026 05:22, Moebius wrote: > Am 30.05.2026 um 09:38 schrieb Mikko: > >> In mathematics there is no distinction between "visible" and "dark" >> numbers. > > You think? :-P Follows from Mückenheim's (partial) definitions of the words. -- Mikko
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| From | wm <wolfgang.mueckenheim@tha.de> |
|---|---|
| Date | 2026-05-22 22:53 +0200 |
| Message-ID | <10uqfor$a1bc$1@solani.org> |
| In reply to | #346048 |
Am 22.05.2026 um 09:23 schrieb Mikko:
> On 21/05/2026 18:24, wm wrote:
>> 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.
>
> Only one of them is the set of natural numbers, and always the same
> one.
>
>>>>>> 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 bijections 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/>
>
> If there is no bijection between two infinite sets then at least one
> of those two sets is uncountable, and if one of them is the the set of
> natural numbers then the other one is uncountable.
All infinite sets are uncountable. Counting requires knowing the
counted. This is impossible for dark numbers. Simplest proof to be
understood even by bad thinkers: Between two rational numbers in natural
order never all rational numbers can be named. Almost all remain not
named.>
> At least there is always a bijection from a set to the same set: the
> identity function.
That is a collective approach, not verified by individual verification.
>> The proof shows that your bujection is not between sets but between
>> potentially infinite collections only.
>
> Both the set of natural numbers and the set of fractions are sets in
> Cantor's sense.
No. See above.>
> In ZF set theory it is possible to identify one set as the set of
> natural numbers. The usual definition is that the set of natural
> numbers is the set that
> - contains the empty set,
> - for each set X it contains also contains the set X ∪ {X}, and
> - is a subset of every other set that satisfies the first two
> conditions.
> This definition agrees with Cantor's construction of natural numbers.
It results in a potentially infinite collection.>
>>>>>>>> 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.
>
> It does not make sense to say that some of them are dark without
> specifying which ones.
Those are dark whih remain unnamed under all circumstanced.>
> Between any two real numbers there are infinitely many other real
> numbers. Of those countably many of those are rational numbers,
> uncountably many are irrational numbers.
>
> In mathematics proofs matter, opinions don't.
ZF "proofs" are rubbish. What matters is that something can be done. To
name all rational numbers between two given numbers cannot be done.
Regards, WM>
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| From | Mikko <mikko.levanto@iki.fi> |
|---|---|
| Date | 2026-06-03 12:40 +0300 |
| Message-ID | <10vosqo$3hl43$1@dont-email.me> |
| In reply to | #346058 |
On 22/05/2026 23:53, wm wrote: > Am 22.05.2026 um 09:23 schrieb Mikko: >> On 21/05/2026 18:24, wm wrote: >>> 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. >> >> Only one of them is the set of natural numbers, and always the same >> one. >> >>>>>>> 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 bijections 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/> >> >> If there is no bijection between two infinite sets then at least one >> of those two sets is uncountable, and if one of them is the the set of >> natural numbers then the other one is uncountable. > > All infinite sets are uncountable. The set of the natural numbers is countable per definition. A set that can be bijectively mapped to the set of the natural numbers is also countable by definition. Also by definition, no other set is countable. If you don't respect the definitions you can't talk about any mathmatical topic. -- Mikko
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| From | Moebius <moebius@example.invalid> |
|---|---|
| Date | 2026-06-03 13:42 +0200 |
| Message-ID | <10vp3um$3jv6q$1@dont-email.me> |
| In reply to | #346251 |
Am 03.06.2026 um 11:40 schrieb Mikko: > On 22/05/2026 23:53, wm wrote: >> All infinite sets are uncountable. [WM] Holy shit! As I told you: mad as a hatter. > The set of the natural numbers is countable per definition. A set > that can be bijectively mapped to the set of the natural numbers > is also countable by definition. Also by definition, no other set > is countable. Indeed! > If you don't respect the definitions you can't talk about any > mathmatical topic. You think? "[WM's] conclusions are based on the sloppiness of his notions, his inability of giving precise definitions, his fundamental misunderstanding of elementary mathematical concepts, and sometimes, as the late Dik Winter remarked [...], on nothing at all." -- Franz Lemmermeyer -- Diese E-Mail wurde von Avast-Antivirussoftware auf Viren geprüft. www.avast.com
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| From | Mikko <mikko.levanto@iki.fi> |
|---|---|
| Date | 2026-06-04 11:11 +0300 |
| Message-ID | <10vrbup$899v$1@dont-email.me> |
| In reply to | #346255 |
On 03/06/2026 14:42, Moebius wrote: > Am 03.06.2026 um 11:40 schrieb Mikko: >> On 22/05/2026 23:53, wm wrote: > >>> All infinite sets are uncountable. [WM] > > Holy shit! > > As I told you: mad as a hatter. > >> The set of the natural numbers is countable per definition. A set >> that can be bijectively mapped to the set of the natural numbers >> is also countable by definition. Also by definition, no other set >> is countable. > > Indeed! > >> If you don't respect the definitions you can't talk about any >> mathmatical topic. > > You think? There is nothing is mathematics except definitions and consequences of those definitions. > "[WM's] conclusions are based on the sloppiness of his notions, his > inability of giving > precise definitions, his fundamental misunderstanding of elementary > mathematical > concepts, and sometimes, as the late Dik Winter remarked [...], on > nothing at all." > > -- Franz Lemmermeyer The result of that may look like mathematics but it isnt'. -- Mikko
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| From | Ross Finlayson <ross.a.finlayson@gmail.com> |
|---|---|
| Date | 2026-06-03 09:43 -0700 |
| Message-ID | <N8acndgM38gyxL33nZ2dnZfqnPudnZ2d@giganews.com> |
| In reply to | #346251 |
On 06/03/2026 02:40 AM, Mikko wrote: > On 22/05/2026 23:53, wm wrote: >> Am 22.05.2026 um 09:23 schrieb Mikko: >>> On 21/05/2026 18:24, wm wrote: >>>> 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. >>> >>> Only one of them is the set of natural numbers, and always the same >>> one. >>> >>>>>>>> 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 bijections 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/> >>> >>> If there is no bijection between two infinite sets then at least one >>> of those two sets is uncountable, and if one of them is the the set of >>> natural numbers then the other one is uncountable. >> >> All infinite sets are uncountable. > > The set of the natural numbers is countable per definition. A set > that can be bijectively mapped to the set of the natural numbers > is also countable by definition. Also by definition, no other set > is countable. > > If you don't respect the definitions you can't talk about any > mathmatical topic. > Then, a usual idea is that the definition of "function" or "mapping" or "functional" or "distribution" or "non-Cartesian function" as with regards to set theory's usual naive account of "domains" and "ranges" and "images" and "co-images", has that there's no such thing as a "function" in set theory: only "sets" their relations that happen to fulfill being the structure and a _model_ of "functions" in the theory. It's like Jordan measure and for accounts of Dirichlet measure vis-a-vis usual accounts of Riemann/Lebesgue measure and the sigma-algebras for everybody. Anyways Jordan measure fulfills being "measure" yet since it's so readily demonstrable the contradictions with Lebesgue "measure", then they'll say something like "Jordan content", when really both of them are to be accounts of "measure", since it _broke_ their language, that they had failed to incorporate the implicits to dis-ambiguate the distinctions. So, you can't "respect" (and observe, and honor) the definitions if you don't even know what they are and how they are. Examples of finite-state-machines are simple and complete. The theory of all finite-state-machines isn't. So, "function" and "topology" are about the most under-defined usual accounts of terms in definitions in "mathematics".
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| From | WM <wolfgang.mueckenheim@tha.de> |
|---|---|
| Date | 2026-06-03 23:07 +0200 |
| Message-ID | <10vq525$3vs9m$1@dont-email.me> |
| In reply to | #346251 |
Am 03.06.2026 um 11:40 schrieb Mikko: > On 22/05/2026 23:53, wm wrote: >> All infinite sets are uncountable. > > The set of the natural numbers is countable per definition. But it is not conuntable in fact. Because by counting you will never exhaust it. But without counting you can exhaust it colleczively. > If you don't respect the definitions you can't talk about any > mathmatical topic. The definitions are mistaken. Countable sets are claimed to be exhaustible by counting but they are not. They are uncountable. Regards, WM
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| From | Mikko <mikko.levanto@iki.fi> |
|---|---|
| Date | 2026-06-04 11:13 +0300 |
| Message-ID | <10vrc2m$899v$2@dont-email.me> |
| In reply to | #346289 |
On 04/06/2026 00:07, WM wrote: > Am 03.06.2026 um 11:40 schrieb Mikko: >> On 22/05/2026 23:53, wm wrote: > >>> All infinite sets are uncountable. >> >> The set of the natural numbers is countable per definition. > > But it is not conuntable in fact. If you don't respect the mathematical definitions you can't talk about a mathematical topic. Non-mathematical facts are not relevant to mathematics. -- Mikko
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| From | Moebius <moebius@example.invalid> |
|---|---|
| Date | 2026-06-04 16:06 +0200 |
| Message-ID | <10vs0ou$e20q$1@dont-email.me> |
| In reply to | #346308 |
Am 04.06.2026 um 10:13 schrieb Mikko: > On 04/06/2026 00:07, WM wrote: >> Am 03.06.2026 um 11:40 schrieb Mikko: >>> On 22/05/2026 23:53, wm wrote: >>>> >>>> All infinite sets are uncountable. >>>> >>> The set of the natural numbers is countable per definition. >> >> But it is not conuntable in fact. Learn some logic, learn some math, Mückenheim! https://en.wikipedia.org/wiki/Countable_set > If you don't respect the mathematical definitions you can't talk > about a mathematical topic. > > Non-mathematical facts are not relevant to mathematics. > -- Diese E-Mail wurde von Avast-Antivirussoftware auf Viren geprüft. www.avast.com
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| From | WM <wolfgang.mueckenheim@tha.de> |
|---|---|
| Date | 2026-06-04 22:52 +0200 |
| Message-ID | <10vsoic$lu02$1@dont-email.me> |
| In reply to | #346308 |
Am 04.06.2026 um 10:13 schrieb Mikko:
> On 04/06/2026 00:07, WM wrote:
>> Am 03.06.2026 um 11:40 schrieb Mikko:
>>> On 22/05/2026 23:53, wm wrote:
>>
>>>> All infinite sets are uncountable.
>>>
>>> The set of the natural numbers is countable per definition.
>>
>> But it is not conuntable in fact.
>
> If you don't respect the mathematical definitions you can't talk
> about a mathematical topic.
I can prove that thesae definitions make fools.Simplest by showing the
nonsense of Cantor's bijections.
According to Cantor all positive fractions
1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
...
can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 + m
which attaches the index k to the fraction m/n in Cantor's sequence
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2,
5/1, 1/6, 2/5, 3/4, ... .
Its terms can be represented by matrices. When we attach all indeXes k =
1, 2, 3, ..., for clarity represented by X, to the integer fractions m/1
and indicate missing indexes by hOles O, then we get the matrix M(0) as
starting position:
XOOO... XXOO... XXOO... XXXO...
XOOO... OOOO... XOOO... XOOO...
XOOO... XOOO... OOOO... OOOO...
XOOO... XOOO... XOOO... OOOO...
... ... ... ...
M(0) M(2) M(3) M(4) ...
M(1) is the same as M(0) because index 1 remains at 1/1. In M(2) index 2
from 2/1 has been attached to 1/2. In M(3) index 3 from 3/1 has been
attached to 2/1. In M(4) index 4 from 4/1 has been attached to 1/3.
Successively all fractions of the sequence get indexed. In the limit,
denoted by M(∞), we see no fraction without index remaining. Note that
the only difference to Cantor's enumeration is that Cantor does not
render account for the source of the indices.
Every X, representing the index k, when taken from its present fraction
m/n, is replaced by the O taken from the fraction to be indexed by this
k. Its last carrier m/n will be indexed later by another index.
Important is that, when continuing, no O can leave the matrix as long as
any index X blocks the only possible drain, i.e., the first column. And
if leaving, where should it settle?
As long as indexes are in the drain, no O has left. The presence of all
O indicates that almost all fractions are not indexed. And after all
indexes have been issued and the drain has become free, no indexes are
available which could index the remaining matrix elements, yet covered by O.
It should go without saying that by rearranging the X of M(0) never a
complete covering can be realized. Lossless transpositions cannot suffer
losses. The limit matrix M(∞) only shows what should have happened when
all fractions were indexed. Logic proves that this cannot have happened
by exchanges. The only explanation for finally seeing M(∞) is that there
are invisible matrix positions, existing already at the start. Obviously
by exchanging O and X no O can leave the matrix, but the O can disappear
by moving without end, from visible to invisible positions.
The number of not indexed fractions remains |ℕ|*(|ℕ|-1) for all
definable terms of the sequence 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3,
3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ... .
Hence |ℕ|*(|ℕ|-1) fractions cannot be indexed by definable indices.
Regards, WM
Regards, WM
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| From | Mikko <mikko.levanto@iki.fi> |
|---|---|
| Date | 2026-06-05 11:44 +0300 |
| Message-ID | <10vu29t$10kme$1@dont-email.me> |
| In reply to | #346386 |
On 04/06/2026 23:52, WM wrote: > Am 04.06.2026 um 10:13 schrieb Mikko: >> On 04/06/2026 00:07, WM wrote: >>> Am 03.06.2026 um 11:40 schrieb Mikko: >>>> On 22/05/2026 23:53, wm wrote: >>> >>>>> All infinite sets are uncountable. >>>> >>>> The set of the natural numbers is countable per definition. >>> >>> But it is not conuntable in fact. >> >> If you don't respect the mathematical definitions you can't talk >> about a mathematical topic. > > I can prove that thesae definitions make fools. No need to prove the obvious. They made you a fool. That is convincing enough. > Simplest by showing the nonsense of Cantor's bijections. > > According to Cantor all positive fractions > > 1/1, 1/2, 1/3, 1/4, ... > 2/1, 2/2, 2/3, 2/4, ... > 3/1, 3/2, 3/3, 3/4, ... > 4/1, 4/2, 4/3, 4/4, ... > ... > > can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 + m > which attaches the index k to the fraction m/n in Cantor's sequence > > 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, > 5/1, 1/6, 2/5, 3/4, ... . Obviously, what Cantor did is doable. > Its terms can be represented by matrices. When we attach all indeXes k = > 1, 2, 3, ..., for clarity represented by X, to the integer fractions m/1 > and indicate missing indexes by hOles O, then we get the matrix M(0) as > starting position: > > XOOO... XXOO... XXOO... XXXO... > XOOO... OOOO... XOOO... XOOO... > XOOO... XOOO... OOOO... OOOO... > XOOO... XOOO... XOOO... OOOO... > ... ... ... ... > M(0) M(2) M(3) M(4) ... The description of the process is not clear. What does "missing index" mean? > M(1) is the same as M(0) because index 1 remains at 1/1. Before explaining whay M(1) is what it is you should tell us what it is. And the same about M(2), M(3), and all others. -- Mikko
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| From | wm <wolfgang.mueckenheim@tha.de> |
|---|---|
| Date | 2026-06-05 16:37 +0200 |
| Message-ID | <10vumvr$df7t$1@solani.org> |
| In reply to | #346414 |
Am 05.06.2026 um 10:44 schrieb Mikko: > On 04/06/2026 23:52, WM wrote: >> Simplest by showing the nonsense of Cantor's bijections. >> >> According to Cantor all positive fractions >> >> 1/1, 1/2, 1/3, 1/4, ... >> 2/1, 2/2, 2/3, 2/4, ... >> 3/1, 3/2, 3/3, 3/4, ... >> 4/1, 4/2, 4/3, 4/4, ... >> ... >> >> can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 + m >> which attaches the index k to the fraction m/n in Cantor's sequence >> >> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, >> 5/1, 1/6, 2/5, 3/4, ... . > > Obviously, what Cantor did is doable. But what he believed to have done is not.> >> Its terms can be represented by matrices. When we attach all indeXes k >> = 1, 2, 3, ..., for clarity represented by X, to the integer fractions >> m/1 and indicate missing indexes by hOles O, then we get the matrix >> M(0) as starting position: >> >> XOOO... XXOO... XXOO... XXXO... >> XOOO... OOOO... XOOO... XOOO... >> XOOO... XOOO... OOOO... OOOO... >> XOOO... XOOO... XOOO... OOOO... >> ... ... ... ... >> M(0) M(2) M(3) M(4) ... > > The description of the process is not clear. What does "missing index" > mean? "Missing index" is said of a fraction that has no index.> >> M(1) is the same as M(0) because index 1 remains at 1/1. > > Before explaining why M(1) is what it is you should tell us what it is. I did above. But again: All indices 1, 2, 3, ... are used to index all unit fractions in the first column. No.one can claim that there are too few indices in the matrix. Then these indices are distributed according to Cantor's description. Most critics of this proof claim that the limit has other properties as the sequence.(https://www.reddit.com/r/AspectsOfTheInfinite/comments/1tc6v1l/proof_of_the_existence_of_dark_numbers/) but there is no limit in Cantor's theory but only the complete sequence. Regards, WM
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| From | Mikko <mikko.levanto@iki.fi> |
|---|---|
| Date | 2026-06-06 12:19 +0300 |
| Message-ID | <1100on1$1o4cb$1@dont-email.me> |
| In reply to | #346422 |
On 05/06/2026 17:37, wm wrote: > Am 05.06.2026 um 10:44 schrieb Mikko: >> On 04/06/2026 23:52, WM wrote: > >>> Simplest by showing the nonsense of Cantor's bijections. >>> >>> According to Cantor all positive fractions >>> >>> 1/1, 1/2, 1/3, 1/4, ... >>> 2/1, 2/2, 2/3, 2/4, ... >>> 3/1, 3/2, 3/3, 3/4, ... >>> 4/1, 4/2, 4/3, 4/4, ... >>> ... >>> >>> can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 + >>> m which attaches the index k to the fraction m/n in Cantor's sequence >>> >>> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, >>> 5/1, 1/6, 2/5, 3/4, ... . >> >> Obviously, what Cantor did is doable. > > But what he believed to have done is not. Cantor's theological opinons are not relevant to mathematics. >>> Its terms can be represented by matrices. When we attach all indeXes >>> k = 1, 2, 3, ..., for clarity represented by X, to the integer >>> fractions m/1 and indicate missing indexes by hOles O, then we get >>> the matrix M(0) as starting position: >>> >>> XOOO... XXOO... XXOO... XXXO... >>> XOOO... OOOO... XOOO... XOOO... >>> XOOO... XOOO... OOOO... OOOO... >>> XOOO... XOOO... XOOO... OOOO... >>> ... ... ... ... >>> M(0) M(2) M(3) M(4) ... >> >> The description of the process is not clear. What does "missing index" >> mean? > > "Missing index" is said of a fraction that has no index.> >>> M(1) is the same as M(0) because index 1 remains at 1/1. >> >> Before explaining why M(1) is what it is you should tell us what it is. > > I did above. No, you did not. You only told what M(0) is. > But again: All indices 1, 2, 3, ... are used to index all > unit fractions in the first column. No.one can claim that there are too > few indices in the matrix. Then these indices are distributed according > to Cantor's description. Most critics of this proof claim that the limit > has other properties as the sequence.(https://www.reddit.com/r/ > AspectsOfTheInfinite/comments/1tc6v1l/ > proof_of_the_existence_of_dark_numbers/) but there is no limit in > Cantor's theory but only the complete sequence. Indeed, no concept of limit is needed for Cantor's proof. All that matters is that every positive rational number is covered by at least one positive integer and that positive integers are a subset of positive rationals. It is possible to start with a bijection between a finite set of positive fractions and a finite set of positive integers and then to note that when the sizes of those sets are increased beyond any limit so that there always is a bijuction between then tnen every fraction and every integer is included, sooner or later. But that is not the way Cantor did it. Anyway, the conclusion is the same. -- Mikko
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| From | "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> |
|---|---|
| Date | 2026-06-06 13:35 -0700 |
| Message-ID | <11020b7$239o1$2@dont-email.me> |
| In reply to | #346464 |
On 6/6/2026 2:19 AM, Mikko wrote: > On 05/06/2026 17:37, wm wrote: >> Am 05.06.2026 um 10:44 schrieb Mikko: >>> On 04/06/2026 23:52, WM wrote: >> >>>> Simplest by showing the nonsense of Cantor's bijections. >>>> >>>> According to Cantor all positive fractions >>>> >>>> 1/1, 1/2, 1/3, 1/4, ... >>>> 2/1, 2/2, 2/3, 2/4, ... >>>> 3/1, 3/2, 3/3, 3/4, ... >>>> 4/1, 4/2, 4/3, 4/4, ... >>>> ... >>>> >>>> can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 + >>>> m which attaches the index k to the fraction m/n in Cantor's sequence >>>> >>>> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, >>>> 4/2, 5/1, 1/6, 2/5, 3/4, ... . >>> >>> Obviously, what Cantor did is doable. >> >> But what he believed to have done is not. > > Cantor's theological opinons are not relevant to mathematics. > >>>> Its terms can be represented by matrices. When we attach all indeXes >>>> k = 1, 2, 3, ..., for clarity represented by X, to the integer >>>> fractions m/1 and indicate missing indexes by hOles O, then we get >>>> the matrix M(0) as starting position: >>>> >>>> XOOO... XXOO... XXOO... XXXO... >>>> XOOO... OOOO... XOOO... XOOO... >>>> XOOO... XOOO... OOOO... OOOO... >>>> XOOO... XOOO... XOOO... OOOO... >>>> ... ... ... ... >>>> M(0) M(2) M(3) M(4) ... >>> >>> The description of the process is not clear. What does "missing index" >>> mean? >> >> "Missing index" is said of a fraction that has no index.> >>>> M(1) is the same as M(0) because index 1 remains at 1/1. >>> >>> Before explaining why M(1) is what it is you should tell us what it is. >> >> I did above. > > No, you did not. You only told what M(0) is. > >> But again: All indices 1, 2, 3, ... are used to index all unit >> fractions in the first column. No.one can claim that there are too few >> indices in the matrix. Then these indices are distributed according to >> Cantor's description. Most critics of this proof claim that the limit >> has other properties as the sequence.(https://www.reddit.com/r/ >> AspectsOfTheInfinite/comments/1tc6v1l/ >> proof_of_the_existence_of_dark_numbers/) but there is no limit in >> Cantor's theory but only the complete sequence. > > Indeed, no concept of limit is needed for Cantor's proof. All that > matters is that every positive rational number is covered by at least > one positive integer and that positive integers are a subset of positive > rationals. > > It is possible to start with a bijection between a finite set of > positive fractions and a finite set of positive integers and then > to note that when the sizes of those sets are increased beyond any > limit so that there always is a bijuction between then tnen every > fraction and every integer is included, sooner or later. But that > is not the way Cantor did it. Anyway, the conclusion is the same. > Basically wrt unsigned integers, its (Cantor pairing) a lossless bijection between a single X to a unique pair (A, B), and back from (A, B) to X...
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| From | Mikko <mikko.levanto@iki.fi> |
|---|---|
| Date | 2026-06-07 11:37 +0300 |
| Message-ID | <1103ake$2cvsk$1@dont-email.me> |
| In reply to | #346469 |
On 06/06/2026 23:35, Chris M. Thomasson wrote: > On 6/6/2026 2:19 AM, Mikko wrote: >> On 05/06/2026 17:37, wm wrote: >>> Am 05.06.2026 um 10:44 schrieb Mikko: >>>> On 04/06/2026 23:52, WM wrote: >>> >>>>> Simplest by showing the nonsense of Cantor's bijections. >>>>> >>>>> According to Cantor all positive fractions >>>>> >>>>> 1/1, 1/2, 1/3, 1/4, ... >>>>> 2/1, 2/2, 2/3, 2/4, ... >>>>> 3/1, 3/2, 3/3, 3/4, ... >>>>> 4/1, 4/2, 4/3, 4/4, ... >>>>> ... >>>>> >>>>> can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 >>>>> + m which attaches the index k to the fraction m/n in Cantor's >>>>> sequence >>>>> >>>>> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, >>>>> 4/2, 5/1, 1/6, 2/5, 3/4, ... . >>>> >>>> Obviously, what Cantor did is doable. >>> >>> But what he believed to have done is not. >> >> Cantor's theological opinons are not relevant to mathematics. >> >>>>> Its terms can be represented by matrices. When we attach all >>>>> indeXes k = 1, 2, 3, ..., for clarity represented by X, to the >>>>> integer fractions m/1 and indicate missing indexes by hOles O, then >>>>> we get the matrix M(0) as starting position: >>>>> >>>>> XOOO... XXOO... XXOO... XXXO... >>>>> XOOO... OOOO... XOOO... XOOO... >>>>> XOOO... XOOO... OOOO... OOOO... >>>>> XOOO... XOOO... XOOO... OOOO... >>>>> ... ... ... ... >>>>> M(0) M(2) M(3) M(4) ... >>>> >>>> The description of the process is not clear. What does "missing index" >>>> mean? >>> >>> "Missing index" is said of a fraction that has no index.> >>>>> M(1) is the same as M(0) because index 1 remains at 1/1. >>>> >>>> Before explaining why M(1) is what it is you should tell us what it is. >>> >>> I did above. >> >> No, you did not. You only told what M(0) is. >> >>> But again: All indices 1, 2, 3, ... are used to index all unit >>> fractions in the first column. No.one can claim that there are too >>> few indices in the matrix. Then these indices are distributed >>> according to Cantor's description. Most critics of this proof claim >>> that the limit has other properties as the sequence.(https:// >>> www.reddit.com/r/ AspectsOfTheInfinite/comments/1tc6v1l/ >>> proof_of_the_existence_of_dark_numbers/) but there is no limit in >>> Cantor's theory but only the complete sequence. >> >> Indeed, no concept of limit is needed for Cantor's proof. All that >> matters is that every positive rational number is covered by at least >> one positive integer and that positive integers are a subset of positive >> rationals. >> >> It is possible to start with a bijection between a finite set of >> positive fractions and a finite set of positive integers and then >> to note that when the sizes of those sets are increased beyond any >> limit so that there always is a bijuction between then tnen every >> fraction and every integer is included, sooner or later. But that >> is not the way Cantor did it. Anyway, the conclusion is the same. >> > > Basically wrt unsigned integers, its (Cantor pairing) a lossless > bijection between a single X to a unique pair (A, B), and back from (A, > B) to X... After pairing positibe rationals with positive integers it is trivial to pair negative rationals with negative integers and the zero rational with the zero integer. -- Mikko
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| From | "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> |
|---|---|
| Date | 2026-06-08 13:04 -0700 |
| Message-ID | <110777o$3g593$1@dont-email.me> |
| In reply to | #346499 |
On 6/7/2026 1:37 AM, Mikko wrote: > On 06/06/2026 23:35, Chris M. Thomasson wrote: >> On 6/6/2026 2:19 AM, Mikko wrote: >>> On 05/06/2026 17:37, wm wrote: >>>> Am 05.06.2026 um 10:44 schrieb Mikko: >>>>> On 04/06/2026 23:52, WM wrote: >>>> >>>>>> Simplest by showing the nonsense of Cantor's bijections. >>>>>> >>>>>> According to Cantor all positive fractions >>>>>> >>>>>> 1/1, 1/2, 1/3, 1/4, ... >>>>>> 2/1, 2/2, 2/3, 2/4, ... >>>>>> 3/1, 3/2, 3/3, 3/4, ... >>>>>> 4/1, 4/2, 4/3, 4/4, ... >>>>>> ... >>>>>> >>>>>> can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 >>>>>> + m which attaches the index k to the fraction m/n in Cantor's >>>>>> sequence >>>>>> >>>>>> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, >>>>>> 4/2, 5/1, 1/6, 2/5, 3/4, ... . >>>>> >>>>> Obviously, what Cantor did is doable. >>>> >>>> But what he believed to have done is not. >>> >>> Cantor's theological opinons are not relevant to mathematics. >>> >>>>>> Its terms can be represented by matrices. When we attach all >>>>>> indeXes k = 1, 2, 3, ..., for clarity represented by X, to the >>>>>> integer fractions m/1 and indicate missing indexes by hOles O, >>>>>> then we get the matrix M(0) as starting position: >>>>>> >>>>>> XOOO... XXOO... XXOO... XXXO... >>>>>> XOOO... OOOO... XOOO... XOOO... >>>>>> XOOO... XOOO... OOOO... OOOO... >>>>>> XOOO... XOOO... XOOO... OOOO... >>>>>> ... ... ... ... >>>>>> M(0) M(2) M(3) M(4) ... >>>>> >>>>> The description of the process is not clear. What does "missing index" >>>>> mean? >>>> >>>> "Missing index" is said of a fraction that has no index.> >>>>>> M(1) is the same as M(0) because index 1 remains at 1/1. >>>>> >>>>> Before explaining why M(1) is what it is you should tell us what it >>>>> is. >>>> >>>> I did above. >>> >>> No, you did not. You only told what M(0) is. >>> >>>> But again: All indices 1, 2, 3, ... are used to index all unit >>>> fractions in the first column. No.one can claim that there are too >>>> few indices in the matrix. Then these indices are distributed >>>> according to Cantor's description. Most critics of this proof claim >>>> that the limit has other properties as the sequence.(https:// >>>> www.reddit.com/r/ AspectsOfTheInfinite/comments/1tc6v1l/ >>>> proof_of_the_existence_of_dark_numbers/) but there is no limit in >>>> Cantor's theory but only the complete sequence. >>> >>> Indeed, no concept of limit is needed for Cantor's proof. All that >>> matters is that every positive rational number is covered by at least >>> one positive integer and that positive integers are a subset of positive >>> rationals. >>> >>> It is possible to start with a bijection between a finite set of >>> positive fractions and a finite set of positive integers and then >>> to note that when the sizes of those sets are increased beyond any >>> limit so that there always is a bijuction between then tnen every >>> fraction and every integer is included, sooner or later. But that >>> is not the way Cantor did it. Anyway, the conclusion is the same. >>> >> >> Basically wrt unsigned integers, its (Cantor pairing) a lossless >> bijection between a single X to a unique pair (A, B), and back from >> (A, B) to X... > > After pairing positibe rationals with positive integers it is trivial > to pair negative rationals with negative integers and the zero rational > with the zero integer. > Fwiw, I don't think Cantor Pairing is fun to work with in the integers. ;^) Only the unsigned integers... Well how to pair X with say (-A, +B), what X goes with (+A, -B)? X is a single integer here.
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| From | Moebius <moebius@example.invalid> |
|---|---|
| Date | 2026-06-08 22:21 +0200 |
| Message-ID | <110787j$3gept$1@dont-email.me> |
| In reply to | #346565 |
Am 08.06.2026 um 22:04 schrieb Chris M. Thomasson:
> Fwiw, I don't think Cantor Pairing is fun to work with in the integers.
Huh?!
0 <-> 0
1 <-> -1
2 <-> 1
3 <-> -2
4 <-> 2
5 <-> -3
6 <-> 3
:
So if pairing works with the elements in IN it also works with elements
in Z.
Not that hard, I'd say.
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