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Groups > sci.math > #639398 > unrolled thread
| Started by | WM <wolfgang.mueckenheim@tha.de> |
|---|---|
| First post | 2025-07-30 19:29 +0200 |
| Last post | 2025-08-09 07:35 -0700 |
| Articles | 20 on this page of 268 — 10 participants |
Back to article view | Back to sci.math
Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-07-30 19:29 +0200
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-07-30 19:09 +0000
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-07-30 14:03 -0700
Re: Conquer the Binary Tree FromTheRafters <FTR@nomail.afraid.org> - 2025-07-30 17:17 -0400
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-07-31 16:04 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-07-31 17:34 +0200
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-07-31 15:53 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-07-31 18:56 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-01 18:23 +0200
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-08-01 19:44 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-02 12:40 +0200
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-08-02 11:15 +0000
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-08-02 11:33 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-02 14:54 +0200
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-08-02 12:59 +0000
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-08-02 13:03 +0000
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-02 12:51 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-02 22:46 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-02 20:20 -0700
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-02 20:39 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-03 12:55 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-03 13:04 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-04 12:37 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-04 12:22 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-04 21:29 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-04 12:41 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-04 21:44 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-04 12:52 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-04 22:34 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-05 15:44 -0700
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-03 13:06 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-04 12:39 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-04 14:29 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-04 16:45 +0200
Re: Conquer the Binary Tree joes <noreply@example.org> - 2025-08-04 20:19 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-04 22:37 +0200
Re: Conquer the Binary Tree joes <noreply@example.org> - 2025-08-05 08:22 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-05 12:10 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-04 20:36 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-04 22:45 +0200
Re: Conquer the Binary Tree joes <noreply@example.org> - 2025-08-05 08:28 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-05 12:16 +0200
Re: Conquer the Binary Tree joes <noreply@example.org> - 2025-08-05 21:41 +0000
Re: Conquer the Binary Tree FromTheRafters <FTR@nomail.afraid.org> - 2025-08-05 18:41 -0400
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-05 16:23 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-06 19:16 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-05 15:39 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-05 15:48 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-05 14:13 +0000
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-05 16:30 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-05 17:09 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-05 17:37 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-05 19:56 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-05 19:21 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-05 22:01 +0200
Re: Conquer the Binary Tree joes <noreply@example.org> - 2025-08-05 21:33 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-06 12:10 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-06 19:11 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-06 16:17 -0700
Re: Conquer the Binary Tree FromTheRafters <FTR@nomail.afraid.org> - 2025-08-06 19:25 -0400
Re: Conquer the Binary Tree FromTheRafters <FTR@nomail.afraid.org> - 2025-08-06 19:29 -0400
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-06 16:42 -0700
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-06 12:43 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-06 16:56 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-06 16:59 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-06 19:34 +0200
Re: Conquer the Binary Tree joes <noreply@example.org> - 2025-08-06 20:12 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-07 19:06 +0200
Re: Conquer the Binary Tree joes <noreply@example.org> - 2025-08-07 21:20 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-08 20:39 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-06 16:23 -0700
Re: Conquer the Binary Tree Ben Bacarisse <ben@bsb.me.uk> - 2025-08-10 23:31 +0100
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-11 01:38 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-11 02:02 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-11 16:25 +0200
Re: Conquer the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2025-08-10 21:37 -0700
Re: Conquer the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2025-08-10 21:50 -0700
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-11 12:28 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-11 16:05 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-11 15:56 +0200
Re: Conquer the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2025-08-12 19:40 -0700
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-11 17:02 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-11 22:24 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-11 22:57 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-12 00:27 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-12 15:20 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-12 15:10 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-12 13:16 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-12 16:24 +0200
Re: Conquer the Binary Tree joes <noreply@example.org> - 2025-08-12 15:27 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-12 17:59 +0200
Re: Conquer the Binary Tree joes <noreply@example.org> - 2025-08-12 15:30 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-06 19:03 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-06 17:46 +0000
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-08-06 18:00 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-06 22:35 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-06 16:38 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-07 17:37 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-07 18:20 +0000
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-07 12:48 -0700
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-07 22:03 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-07 22:43 +0200
Re: Conquer the Binary Tree joes <noreply@example.org> - 2025-08-07 21:01 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-08 14:23 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-08 13:00 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-08 22:41 +0200
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-08-08 20:47 +0000
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-08 22:53 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-09 15:15 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-09 15:10 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-08 20:53 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-09 15:26 +0200
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-08-09 13:37 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-09 18:32 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-09 11:10 -0700
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-08 14:32 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-09 15:13 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-09 11:08 -0700
Re: Conquer the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2025-08-09 11:13 -0700
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-15 22:21 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-07 22:32 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-07 13:47 -0700
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-08 00:50 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-07 16:53 -0700
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-08 00:25 -0700
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-09 11:11 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-10 15:08 +0200
Re: Conquer the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2025-08-10 07:51 -0700
Re: Conquer the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2025-08-10 08:08 -0700
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-10 12:56 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-11 15:44 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-11 12:37 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-11 16:16 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-12 14:04 +0000
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-12 16:40 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-12 16:41 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-12 15:28 +0000
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-12 17:56 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-12 18:15 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-12 18:17 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-12 21:58 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-12 22:45 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-13 16:53 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-12 18:11 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-12 17:15 +0000
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-12 22:19 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-12 22:35 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-13 16:58 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-13 17:01 +0200
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-08-13 15:06 +0000
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-14 00:24 +0200
Re: Conquer the Binary Tree Hugh Kalambetov <ahuebbl@htlhkm.ru> - 2025-08-14 09:34 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-14 13:00 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-14 12:33 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-14 15:01 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-14 13:40 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-14 16:08 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-14 14:18 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-15 14:59 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-15 13:59 +0000
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-15 17:21 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-15 17:29 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-15 21:30 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-15 13:23 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-15 18:03 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-15 16:45 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-15 18:52 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-16 00:10 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-15 22:25 -0700
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-16 13:57 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-16 13:55 +0200
Re: Conquer the Binary Tree joes <noreply@example.org> - 2025-08-16 14:26 +0000
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-08-15 15:00 +0000
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-15 17:27 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-13 16:48 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-13 16:13 +0000
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-08-13 16:23 +0000
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-13 19:10 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-14 14:38 +0200
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-08-14 12:42 +0000
Re: Conquer the Binary Tree joes <noreply@example.org> - 2025-08-14 15:33 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-15 15:04 +0200
Re: Conquer the Binary Tree joes <noreply@example.org> - 2025-08-15 13:38 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-15 19:01 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-15 10:04 -0700
Re: Conquer the Binary Tree joes <noreply@example.org> - 2025-08-15 17:36 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-16 13:46 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-16 12:11 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-16 14:29 +0200
Re: Conquer the Binary Tree joes <noreply@example.org> - 2025-08-16 14:27 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-16 18:25 +0200
Re: Conquer the Binary Tree joes <noreply@example.org> - 2025-08-17 06:06 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-17 12:44 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-16 16:54 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-16 17:01 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-16 18:29 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-17 12:40 +0200
Re: Conquer the Binary Tree joes <noreply@example.org> - 2025-08-16 14:24 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-14 14:24 +0200
Re: Conquer the Binary Tree joes <noreply@example.org> - 2025-08-14 15:42 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-15 15:37 +0200
Re: Conquer the Binary Tree joes <noreply@example.org> - 2025-08-13 17:37 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-14 14:51 +0200
Re: Conquer the Binary Tree FromTheRafters <FTR@nomail.afraid.org> - 2025-08-14 11:02 -0400
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-14 11:23 -0700
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-14 00:43 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-14 00:57 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-14 14:56 +0200
Re: Conquer the Binary Tree Alan Mackenzie <acm@muc.de> - 2025-08-14 13:09 +0000
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-08-14 13:12 +0000
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-14 17:54 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-14 17:57 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-14 18:06 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-14 15:55 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-14 17:43 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-14 17:56 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-14 18:03 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-15 01:10 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-15 17:55 +0200
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-08-15 16:03 +0000
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-15 18:44 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-15 18:58 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-15 23:45 +0200
Re: Conquer the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2025-08-15 20:52 -0700
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-15 09:57 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-16 14:08 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-16 13:48 -0700
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-12 17:04 -0700
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-11 16:44 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-11 17:16 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-11 17:18 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-11 17:28 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-11 17:42 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-12 01:57 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-12 01:57 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-11 18:38 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-11 22:42 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-12 15:16 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-11 22:47 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-12 02:00 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-12 02:01 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-07 00:11 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-06 16:35 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-06 22:19 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-06 16:38 -0700
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-06 16:28 -0700
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-06 16:19 -0700
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-08-05 15:47 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-06 19:19 +0200
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-08-06 17:31 +0000
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-07 00:58 +0200
Re: Conquer the Binary Tree Moebius <invalid@example.invalid> - 2025-08-07 00:58 +0200
Ben Bacarisse's "debunking" attempt WM <wolfgang.mueckenheim@tha.de> - 2025-08-18 16:07 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-02 15:03 +0200
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-08-02 13:17 +0000
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-08-02 13:24 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-02 19:39 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-08-02 20:12 +0200
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-07-31 15:28 +0200
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-07-31 13:35 +0000
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-07-31 16:49 +0200
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-07-31 14:53 +0000
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-07-31 11:52 -0700
Re: Conquer the Binary Tree WM <wolfgang.mueckenheim@tha.de> - 2025-07-31 22:55 +0200
Re: Conquer the Binary Tree "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-07-31 13:58 -0700
Re: Conquer the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2025-08-09 07:11 -0700
Re: Conquer the Binary Tree Python <jp@python.invalid> - 2025-08-09 14:15 +0000
Re: Conquer the Binary Tree Ross Finlayson <ross.a.finlayson@gmail.com> - 2025-08-09 07:35 -0700
Page 4 of 14 — ← Prev page 1 2 3 [4] 5 6 … 14 Next page →
| From | FromTheRafters <FTR@nomail.afraid.org> |
|---|---|
| Date | 2025-08-06 19:29 -0400 |
| Message-ID | <1070ogh$3m5a7$1@dont-email.me> |
| In reply to | #639509 |
Chris M. Thomasson formulated on Wednesday : > On 8/6/2025 10:11 AM, WM wrote: >> On 05.08.2025 23:33, joes wrote: >>> Am Tue, 05 Aug 2025 22:01:23 +0200 schrieb WM: >> >>>> When dealing with Cantor's mappings between infinite sets, it is argued >>>> usually that these mappings require a "limit" to be completed or that >>>> they cannot be completed. Such arguing has to be rejected flatly. >>> Weren't you the one who complained that the process never finished? >> >> Either the enumeration of the rationals is never finished, or, if it is >> claimed to be finished, dark numbers are needed. > > Do you think complete and/or all means finite? He seems to mean that finite means no dark numbers are needed, therefore infinite means they are needed. :D He is insane.
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| From | "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> |
|---|---|
| Date | 2025-08-06 16:42 -0700 |
| Message-ID | <1070p9f$3lg2e$6@dont-email.me> |
| In reply to | #639514 |
On 8/6/2025 4:29 PM, FromTheRafters wrote: > Chris M. Thomasson formulated on Wednesday : >> On 8/6/2025 10:11 AM, WM wrote: >>> On 05.08.2025 23:33, joes wrote: >>>> Am Tue, 05 Aug 2025 22:01:23 +0200 schrieb WM: >>> >>>>> When dealing with Cantor's mappings between infinite sets, it is >>>>> argued >>>>> usually that these mappings require a "limit" to be completed or that >>>>> they cannot be completed. Such arguing has to be rejected flatly. >>>> Weren't you the one who complained that the process never finished? >>> >>> Either the enumeration of the rationals is never finished, or, if it >>> is claimed to be finished, dark numbers are needed. >> >> Do you think complete and/or all means finite? > > He seems to mean that finite means no dark numbers are needed, therefore > infinite means they are needed. :D > > He is insane. Shit happens! Well now... Perhaps this song was written about him (WM)? Not sure quite yet... Perhaps? https://youtu.be/eQNI1KfGXBA?list=RDeQNI1KfGXBA wow! perhaps indeed... ;^) ?
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| From | Alan Mackenzie <acm@muc.de> |
|---|---|
| Date | 2025-08-06 12:43 +0000 |
| Message-ID | <106vim1$6tc$1@news.muc.de> |
| In reply to | #639476 |
WM <wolfgang.mueckenheim@tha.de> wrote: > On 05.08.2025 21:21, Alan Mackenzie wrote: >> WM <wolfgang.mueckenheim@tha.de> wrote: >>> On 05.08.2025 16:13, Alan Mackenzie wrote: >>>> WM <wolfgang.mueckenheim@tha.de> wrote: >>>> I already told you in a post yesterday. The O's "move" steadily away >>>> from the origin. In the limit they have "moved all the way to infinity", >>>> every last one of them. >>> The O's are exchanged with X's, never deleted, never leaving the finite >>> domain. All exchanges happen at finite places. >> Of course. But you don't understand the concept "limit as the steps >> tend to infinity". > I understand .... You don't understand at all. > .... that never an O can be deleted. In no case! In the current scenario, O's don't get deleted. They just move away to an unbounded distance. >> For every step the X's indeed never leave the finite >> domain, but in the limit they have vanished. > You mean the O's. Yes. Sorry. > So you wish to apply magic, I prefer mathematics. Your preferences are beyond your abilities. What you call "magic" is established mathematics, as developed by minds far superior to either of ours over the last few centuries. That you fail to understand this "magic" should be your problem alone. >> You don't understand that, and you're not trying to understand it. > I am refuting to apply magic. You are refusing to apply established mathematics. >> Every mathematics undergraduate understands it, but you don't. > Unfortunately they have been spoilt by stupid teachers. That statement confirms you as a crank, but we knew that anyway. >>>> Please forgive me trying to explain it in terms you might understand. >>> You should try to understand that all happens at finite places. No O >>> will ever reach an infinite index. >> As already said, you don't understand "limit ... tends to infinity". > Note that Cantor does not accept a limit. As a pre-eminent mathematician, Cantor understood full well what limits were and how to use them. You don't. >>> How can you dare to propose such illogical nonsense! >> It's basic mathematics. > Cantor, rejecting the limit idea > When dealing with Cantor's mappings between infinite sets, it is argued > usually that these mappings require a "limit" to be completed or that > they cannot be completed. Don't know about "usually", I never heard any such silly arguments till I encountered this newsgroup. > Such arguing has to be rejected flatly. Such arguing is non-sensical, based on misunderstandings of basic maths. > For this reason some of Cantor's statements are quoted below. None of the following cites (which I would normally snip as they are off topic) comes close to "rejecting the limit idea". That you cite them as such shows how poor your understanding of the topic is. > "If we think the numbers p/q in such an order [...] then every number > p/q comes at an absolutely fixed position of a simple infinite sequence" > [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und > philosophischen Inhalts", Springer, Berlin (1932) p. 126] > "The infinite sequence thus defined has the peculiar property to contain > the positive rational numbers completely, and each of them only once at > a determined place." [G. Cantor, letter to R. Lipschitz (19 Nov 1883)] > "thus we get the epitome (ω) of all real algebraic numbers [...] and > with respect to this order we can talk about the th algebraic number > where not a single one of this epitome () has been forgotten." [E. > Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und > philosophischen Inhalts", Springer, Berlin (1932) p. 116] > "such that every element of the set stands at a definite position of > this sequence" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen > mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 152] > The clarity of these expressions is noteworthy: all and every, > completely, at an absolutely fixed position, th number, where not a > single one has been forgotten. > Regards, WM -- Alan Mackenzie (Nuremberg, Germany).
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| From | WM <wolfgang.mueckenheim@tha.de> |
|---|---|
| Date | 2025-08-06 16:56 +0200 |
| Message-ID | <95dd0ab5-6c24-41bf-b677-31aefa969cbc@tha.de> |
| In reply to | #639489 |
On 06.08.2025 14:43, Alan Mackenzie wrote: > WM <wolfgang.mueckenheim@tha.de> wrote: > In the current scenario, O's don't get deleted. They just move away to > an unbounded distance. Whih however is always finite. So the O's remain in the matrix. >> So you wish to apply magic, I prefer mathematics. > > Your preferences are beyond your abilities. What you call "magic" is > established mathematics, as developed by minds far superior to either of > ours over the last few centuries. No, that has not been the subject of analysis. It is only Cantor's invention. And he has not used infnite numbers to enumerate. > That you fail to understand this > "magic" should be your problem alone. You claim that exchange with X's at finite places can remove O's into the infinite. This is simply nonsense. Regards, WM
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| From | Alan Mackenzie <acm@muc.de> |
|---|---|
| Date | 2025-08-06 16:59 +0000 |
| Message-ID | <10701l6$19ml$1@news.muc.de> |
| In reply to | #639491 |
WM <wolfgang.mueckenheim@tha.de> wrote: > On 06.08.2025 14:43, Alan Mackenzie wrote: >> WM <wolfgang.mueckenheim@tha.de> wrote: >> In the current scenario, O's don't get deleted. They just move away to >> an unbounded distance. > Which however is always finite. .... Yes. > .... So the O's remain in the matrix. Yes. For any number of steps. But NOT in the limit. Using an analogous, but simpler example, consider the sequence of real numbers in decimal: 1.1, 1.01, 1.001, 1.0001, ...... Every element of that sequence has two non-zero digits. The limit of the sequence (I hope you can agree to this) is 1. This limit has only one non-zero digit. At no element of the sequence does the second 1 get "deleted". That 1 "remains in the number". But in the limit, it has gone. This is essentially the same thing which is happening to your X's and O's. >>> So you wish to apply magic, I prefer mathematics. >> Your preferences are beyond your abilities. What you call "magic" is >> established mathematics, as developed by minds far superior to either of >> ours over the last few centuries. > No, that has not been the subject of analysis. It is only Cantor's > invention. And he has not used infinite numbers to enumerate. To what do your "that" and your "it" refer? >> That you fail to understand this >> "magic" should be your problem alone. > You claim that exchange with X's at finite places can remove O's into > the infinite. This is simply nonsense. It may be nonsense, but it was a plausible argument to try to get you to understand the notion of limits. Maybe the sequence I depict above might do a better job. > Regards, WM -- Alan Mackenzie (Nuremberg, Germany).
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| From | WM <wolfgang.mueckenheim@tha.de> |
|---|---|
| Date | 2025-08-06 19:34 +0200 |
| Message-ID | <10703n4$3gk0n$3@dont-email.me> |
| In reply to | #639494 |
On 06.08.2025 18:59, Alan Mackenzie wrote: > WM <wolfgang.mueckenheim@tha.de> wrote: >> On 06.08.2025 14:43, Alan Mackenzie wrote: >>> WM <wolfgang.mueckenheim@tha.de> wrote: > >>> In the current scenario, O's don't get deleted. They just move away to >>> an unbounded distance. > >> Which however is always finite. .... > > Yes. > >> .... So the O's remain in the matrix. > > Yes. For any number of steps. But NOT in the limit. How and where do they leave? But don't try to stultify students. If all n fail to enumerate the infinitely many fractions equipped with O's but you claim that afterwards all fractions are indexed (by what), then every intelligent student recognizes that you don't tell the truth. > > Using an analogous, but simpler example, consider the sequence of real > numbers in decimal: > > 1.1, 1.01, 1.001, 1.0001, ...... > > Every element of that sequence has two non-zero digits. And the limit is never reached by any f(n). That's the same as with Cantor's enumeration. Note that indexing is only possible by natural numbers, not by ω. > > The limit of the sequence (I hope you can agree to this) is 1. This > limit has only one non-zero digit. > > At no element of the sequence does the second 1 get "deleted". That 1 > "remains in the number". But in the limit, it has gone. > > This is essentially the same thing which is happening to your X's and > O's. Not by a long way. There are infinitely many O's. They cannot vanish immediately. >> No, that has not been the subject of analysis. It is only Cantor's >> invention. And he has not used infinite numbers to enumerate. > > To what do your "that" and your "it" refer? The completion of an enumeration. > >>> That you fail to understand this >>> "magic" should be your problem alone. > >> You claim that exchange with X's at finite places can remove O's into >> the infinite. This is simply nonsense. > > It may be nonsense, but it was a plausible argument to try to get you to > understand the notion of limits. Maybe that this confusion of limits has supported the Cantor-superstition. > Maybe the sequence I depict above > might do a better job. This is a nice example. It shows that you confuse the enumeration of all terms of the sequence (which in fact is done by the position of the second 1) and the limit which has nothing to do with this enumeration. Regards, WM
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| From | joes <noreply@example.org> |
|---|---|
| Date | 2025-08-06 20:12 +0000 |
| Message-ID | <1070d05$3h67l$1@dont-email.me> |
| In reply to | #639500 |
Am Wed, 06 Aug 2025 19:34:27 +0200 schrieb WM: > On 06.08.2025 18:59, Alan Mackenzie wrote: >> WM <wolfgang.mueckenheim@tha.de> wrote: >>> On 06.08.2025 14:43, Alan Mackenzie wrote: >>>> WM <wolfgang.mueckenheim@tha.de> wrote: >> >>>> In the current scenario, O's don't get deleted. They just move away >>>> to an unbounded distance. >>> Which however is always finite. So the O's remain in the matrix. >> Yes. For any number of steps. But NOT in the limit. > How and where do they leave? > But don't try to stultify students. If all n fail to enumerate the > infinitely many fractions equipped with O's but you claim that > afterwards all fractions are indexed (by what), then every intelligent > student recognizes that you don't tell the truth. There is no index "where" they leave, it happens in the limit process, the total of all the steps. (You also tried to shift the quantifiers again.) >> Using an analogous, but simpler example, consider the sequence of real >> numbers in decimal: >> 1.1, 1.01, 1.001, 1.0001, ...... >> Every element of that sequence has two non-zero digits. > And the limit is never reached by any f(n). That's the same as with > Cantor's enumeration. Note that indexing is only possible by natural > numbers, not by ω. Of course. The limit is not a term. >> The limit of the sequence (I hope you can agree to this) is 1. This >> limit has only one non-zero digit. >> At no element of the sequence does the second 1 get "deleted". That 1 >> "remains in the number". But in the limit, it has gone. >> This is essentially the same thing which is happening to your X's and >> O's. > Not by a long way. There are infinitely many O's. They cannot vanish > immediately. They don't. Do you agree that the limit is 1? >>> No, that has not been the subject of analysis. It is only Cantor's >>> invention. And he has not used infinite numbers to enumerate. >> To what do your "that" and your "it" refer? > The completion of an enumeration. Cantor invented completion? >> Maybe the sequence I depict above might do a better job. > This is a nice example. It shows that you confuse the enumeration of all > terms of the sequence (which in fact is done by the position of the > second 1) and the limit which has nothing to do with this enumeration. So which term enumerates all positions? The limit is a handy compression of a sequence. -- Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math: It is not guaranteed that n+1 exists for every n.
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| From | WM <wolfgang.mueckenheim@tha.de> |
|---|---|
| Date | 2025-08-07 19:06 +0200 |
| Message-ID | <1072mfh$4d5m$1@dont-email.me> |
| In reply to | #639503 |
On 06.08.2025 22:12, joes wrote: > Am Wed, 06 Aug 2025 19:34:27 +0200 schrieb WM: >> On 06.08.2025 18:59, Alan Mackenzie wrote: >>> WM <wolfgang.mueckenheim@tha.de> wrote: >>>> On 06.08.2025 14:43, Alan Mackenzie wrote: >>>>> WM <wolfgang.mueckenheim@tha.de> wrote: >>> >>>>> In the current scenario, O's don't get deleted. They just move away >>>>> to an unbounded distance. >>>> Which however is always finite. So the O's remain in the matrix. >>> Yes. For any number of steps. But NOT in the limit. >> How and where do they leave? >> But don't try to stultify students. If all n fail to enumerate the >> infinitely many fractions equipped with O's but you claim that >> afterwards all fractions are indexed (by what), then every intelligent >> student recognizes that you don't tell the truth. > There is no index "where" they leave, it happens in the limit process, > the total of all the steps. There is no limit process. There is only the process of enumerating. It is defined by exchange of O and X >>> Using an analogous, but simpler example, consider the sequence of real >>> numbers in decimal: >>> 1.1, 1.01, 1.001, 1.0001, ...... >>> Every element of that sequence has two non-zero digits. >> And the limit is never reached by any f(n). That's the same as with >> Cantor's enumeration. Note that indexing is only possible by natural >> numbers, not by ω. > Of course. The limit is not a term. But indexing is done in the terms only. > >>> The limit of the sequence (I hope you can agree to this) is 1. This >>> limit has only one non-zero digit. >>> At no element of the sequence does the second 1 get "deleted". That 1 >>> "remains in the number". But in the limit, it has gone. >>> This is essentially the same thing which is happening to your X's and >>> O's. >> Not by a long way. There are infinitely many O's. They cannot vanish >> immediately. > They don't. > Do you agree that the limit is 1? Of course. But it has no bearing on the indexing. > >>>> No, that has not been the subject of analysis. It is only Cantor's >>>> invention. And he has not used infinite numbers to enumerate. >>> To what do your "that" and your "it" refer? >> The completion of an enumeration. > Cantor invented completion? So he said. "Die so definirte unendliche Reihe hat nun das merkwürdige an sich, sämmtliche positiven rationalen Zahlen und jede von ihnen nur einmal an einer bestimmten Stelle zu enthalten." Note: sämmtliche. > >>> Maybe the sequence I depict above might do a better job. >> This is a nice example. It shows that you confuse the enumeration of all >> terms of the sequence (which in fact is done by the position of the >> second 1) and the limit which has nothing to do with this enumeration. > So which term enumerates all positions? > The limit is a handy compression of a sequence. Alas it has nothing to do with counting of the terms. Regards, WM >
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| From | joes <noreply@example.org> |
|---|---|
| Date | 2025-08-07 21:20 +0000 |
| Message-ID | <10735am$3s7pj$5@dont-email.me> |
| In reply to | #639523 |
Am Thu, 07 Aug 2025 19:06:58 +0200 schrieb WM: > On 06.08.2025 22:12, joes wrote: >> Am Wed, 06 Aug 2025 19:34:27 +0200 schrieb WM: >>> On 06.08.2025 18:59, Alan Mackenzie wrote: >>>> WM <wolfgang.mueckenheim@tha.de> wrote: >>>>> Which however is always finite. So the O's remain in the matrix. >>>> Yes. For any number of steps. But NOT in the limit. >>> How and where do they leave? If all n fail to enumerate the >>> infinitely many fractions equipped with O's but you claim that >>> afterwards all fractions are indexed (by what), then every intelligent >>> student recognizes that you don't tell the truth. >> There is no index "where" they leave, it happens in the limit process, >> the total of all the steps. > There is no limit process. There is only the process of enumerating. It > is defined by exchange of O and X Yes yes, the limit is the result of that. No single step finishes it. >>>> Using an analogous, but simpler example, consider the sequence of >>>> real numbers in decimal: 1.1, 1.01, 1.001, 1.0001, ...... >>>> Every element of that sequence has two non-zero digits. >>> And the limit is never reached by any f(n). That's the same as with >>> Cantor's enumeration. Note that indexing is only possible by natural >>> numbers, not by ω. >> Of course. The limit is not a term. > But indexing is done in the terms only. What's your point? >>>> The limit of the sequence (I hope you can agree to this) is 1. This >>>> limit has only one non-zero digit. >>>> At no element of the sequence does the second 1 get "deleted". That >>>> 1 "remains in the number". But in the limit, it has gone. >>>> This is essentially the same thing which is happening to your X's and >>>> O's. >>> Not by a long way. There are infinitely many O's. They cannot vanish >>> immediately. >> They don't. Do you agree that the limit is 1? > Of course. But it has no bearing on the indexing. How can the 1 disappear? >>>>> No, that has not been the subject of analysis. It is only Cantor's >>>>> invention. And he has not used infinite numbers to enumerate. >>>> To what do your "that" and your "it" refer? >>> The completion of an enumeration. >> Cantor invented completion? > So he said. "Die so definirte unendliche Reihe hat nun das merkwürdige > an sich, sämtliche positiven rationalen Zahlen und jede von ihnen nur > einmal an einer bestimmten Stelle zu enthalten." Note: sämtliche. That refers to the sequence given by his formula being a bijection. >>> This is a nice example. It shows that you confuse the enumeration of >>> all terms of the sequence (which in fact is done by the position of >>> the second 1) and the limit which has nothing to do with this >>> enumeration. >> So which term enumerates all positions? >> The limit is a handy compression of a sequence. > Alas it has nothing to do with counting of the terms. Answer the question. -- Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math: It is not guaranteed that n+1 exists for every n.
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| From | WM <wolfgang.mueckenheim@tha.de> |
|---|---|
| Date | 2025-08-08 20:39 +0200 |
| Message-ID | <1075g8j$s7gc$1@dont-email.me> |
| In reply to | #639532 |
On 07.08.2025 23:20, joes wrote: > Am Thu, 07 Aug 2025 19:06:58 +0200 schrieb WM: >> On 06.08.2025 22:12, joes wrote: >>> Am Wed, 06 Aug 2025 19:34:27 +0200 schrieb WM: >>>> On 06.08.2025 18:59, Alan Mackenzie wrote: >>>>> WM <wolfgang.mueckenheim@tha.de> wrote: > >>>>>> Which however is always finite. So the O's remain in the matrix. >>>>> Yes. For any number of steps. But NOT in the limit. >>>> How and where do they leave? If all n fail to enumerate the >>>> infinitely many fractions equipped with O's but you claim that >>>> afterwards all fractions are indexed (by what), then every intelligent >>>> student recognizes that you don't tell the truth. >>> There is no index "where" they leave, it happens in the limit process, >>> the total of all the steps. >> There is no limit process. There is only the process of enumerating. It >> is defined by exchange of O and X > Yes yes, the limit is the result of that. No single step finishes it. The enumeration is done by all single finite steps. No further limit can contribute. > >>>>> Using an analogous, but simpler example, consider the sequence of >>>>> real numbers in decimal: 1.1, 1.01, 1.001, 1.0001, ...... >>>>> Every element of that sequence has two non-zero digits. >>>> And the limit is never reached by any f(n). That's the same as with >>>> Cantor's enumeration. Note that indexing is only possible by natural >>>> numbers, not by ω. >>> Of course. The limit is not a term. >> But indexing is done in the terms only. > What's your point? There is no limit. > >>>>> The limit of the sequence (I hope you can agree to this) is 1. This >>>>> limit has only one non-zero digit. >>>>> At no element of the sequence does the second 1 get "deleted". That >>>>> 1 "remains in the number". But in the limit, it has gone. >>>>> This is essentially the same thing which is happening to your X's and >>>>> O's. >>>> Not by a long way. There are infinitely many O's. They cannot vanish >>>> immediately. >>> They don't. Do you agree that the limit is 1? >> Of course. But it has no bearing on the indexing. > How can the 1 disappear? It cannot disappear within the numerated terms. > >>>>>> No, that has not been the subject of analysis. It is only Cantor's >>>>>> invention. And he has not used infinite numbers to enumerate. >>>>> To what do your "that" and your "it" refer? >>>> The completion of an enumeration. >>> Cantor invented completion? >> So he said. "Die so definirte unendliche Reihe hat nun das merkwürdige >> an sich, sämtliche positiven rationalen Zahlen und jede von ihnen nur >> einmal an einer bestimmten Stelle zu enthalten." Note: sämtliche. > That refers to the sequence given by his formula being a bijection. It shows completion. > >>>> This is a nice example. It shows that you confuse the enumeration of >>>> all terms of the sequence (which in fact is done by the position of >>>> the second 1) and the limit which has nothing to do with this >>>> enumeration. >>> So which term enumerates all positions? >>> The limit is a handy compression of a sequence. >> Alas it has nothing to do with counting of the terms. > Answer the question. All terms together enumerate as many positions as natural indices can do. Regards, WM >
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| From | "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> |
|---|---|
| Date | 2025-08-06 16:23 -0700 |
| Message-ID | <1070o64$3lg2e$3@dont-email.me> |
| In reply to | #639494 |
On 8/6/2025 9:59 AM, Alan Mackenzie wrote: > WM <wolfgang.mueckenheim@tha.de> wrote: >> On 06.08.2025 14:43, Alan Mackenzie wrote: >>> WM <wolfgang.mueckenheim@tha.de> wrote: > >>> In the current scenario, O's don't get deleted. They just move away to >>> an unbounded distance. > >> Which however is always finite. .... > > Yes. > >> .... So the O's remain in the matrix. > > Yes. For any number of steps. But NOT in the limit. > > Using an analogous, but simpler example, consider the sequence of real > numbers in decimal: > > 1.1, 1.01, 1.001, 1.0001, ...... > > Every element of that sequence has two non-zero digits. It tends to 1. At no point in the step-by-step sequence is it ever equal to 1. > > The limit of the sequence (I hope you can agree to this) is 1. This > limit has only one non-zero digit. Right. > > At no element of the sequence does the second 1 get "deleted". That 1 > "remains in the number". But in the limit, it has gone. > > This is essentially the same thing which is happening to your X's and > O's. > >>>> So you wish to apply magic, I prefer mathematics. > >>> Your preferences are beyond your abilities. What you call "magic" is >>> established mathematics, as developed by minds far superior to either of >>> ours over the last few centuries. > >> No, that has not been the subject of analysis. It is only Cantor's >> invention. And he has not used infinite numbers to enumerate. > > To what do your "that" and your "it" refer? > >>> That you fail to understand this >>> "magic" should be your problem alone. > >> You claim that exchange with X's at finite places can remove O's into >> the infinite. This is simply nonsense. > > It may be nonsense, but it was a plausible argument to try to get you to > understand the notion of limits. Maybe the sequence I depict above > might do a better job. > >> Regards, WM >
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| From | Ben Bacarisse <ben@bsb.me.uk> |
|---|---|
| Date | 2025-08-10 23:31 +0100 |
| Message-ID | <87cy92u7vc.fsf@bsb.me.uk> |
| In reply to | #639494 |
Alan Mackenzie <acm@muc.de> writes:
> WM <wolfgang.mueckenheim@tha.de> wrote:
>> On 06.08.2025 14:43, Alan Mackenzie wrote:
>>> WM <wolfgang.mueckenheim@tha.de> wrote:
>
>>> In the current scenario, O's don't get deleted. They just move away to
>>> an unbounded distance.
>
>> Which however is always finite. ....
>
> Yes.
>
>> .... So the O's remain in the matrix.
>
> Yes. For any number of steps. But NOT in the limit.
>
> Using an analogous, but simpler example, consider the sequence of real
> numbers in decimal:
One of the things I used to think was odd was the complexity a WM's
examples. But then I decided this was deliberate.
> 1.1, 1.01, 1.001, 1.0001, ......
>
> Every element of that sequence has two non-zero digits.
>
> The limit of the sequence (I hope you can agree to this) is 1. This
> limit has only one non-zero digit.
>
> At no element of the sequence does the second 1 get "deleted". That 1
> "remains in the number". But in the limit, it has gone.
>
> This is essentially the same thing which is happening to your X's and
> O's.
Years ago I used a very specific simpler example, using 0 and 1 rather
than X an 0 and a one-dimensional "grid". One can use (the Cantor index
of) fractions or, even simpler, start with an alternating sequence and,
step by step, just swap the first 1 with the first following 0:
s_0 = 0, 1, 0, 1, 0, 1, 0, 1, 0, ...
s_1 = 0, 0, 1, 1, 0, 1, 0, 1, 0, ...
s_2 = 0, 0, 0, 1, 1, 1, 0, 1, 0, ...
s_3 = 0, 0, 0, 0, 1, 1, 1, 1, 0, ...
In the limit, this sequence is all zeros. "Where did all the 1s go?" he
might ask his students.
One day he might get a student who (a) points out that such sequences
are just functions from N to {0,1}. (b) The sequence of functions s_n
has a well-defined limit. (c) MW's own textbook defines this limit and
shows how to calculate it!
[Also, he used to vehemently deny that any non-constant set sequences
have limits. But his textbook defines functions as sets (sets of pairs)
and defines limits for certain sequences of such sets.]
--
Ben.
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| From | Moebius <invalid@example.invalid> |
|---|---|
| Date | 2025-08-11 01:38 +0200 |
| Message-ID | <107bagu$278bi$1@dont-email.me> |
| In reply to | #639583 |
Am 11.08.2025 um 00:31 schrieb Ben Bacarisse: > Years ago I used a very specific simpler example, using 0 and 1 rather > than X an 0 and a one-dimensional "grid". One can use (the Cantor index > of) fractions or, even simpler, start with an alternating sequence and, > step by step, just swap the first 1 with the first following 0: > > s_0 = 0, 1, 0, 1, 0, 1, 0, 1, 0, ... > s_1 = 0, 0, 1, 1, 0, 1, 0, 1, 0, ... > s_2 = 0, 0, 0, 1, 1, 1, 0, 1, 0, ... > s_3 = 0, 0, 0, 0, 1, 1, 1, 1, 0, ... > > In the limit, this sequence is all zeros. "Where did all the 1s go?" he > might ask his students. Recently, I posted a similar example in de.sci.mathematic: Wir betrachten eine Folge von Folgen und deren (punktweisen) Grenzwert. Die Folge sei (f_0, f_1, f_2, f_3, ...) mit f_0 = (0, 1, 0, 1, 0, 1, 0, 1, ...) f_1 = (1, 0, 0, 1, 0, 1, 0, 1, ...) f_2 = (1, 1, 0, 0, 0, 1, 0, 1, ...) f_3 = (1, 1, 1, 0, 0, 0, 0, 1, ...) usw. Es ist dann lim f_n = (1, 1, 1, 1, ...) = (a_n)_(n e IN) mit a_n = 1 für alle n e IN. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Of course, no meaningful answer from crank Wolfgang Mückenheim. . . . .
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| From | Moebius <invalid@example.invalid> |
|---|---|
| Date | 2025-08-11 02:02 +0200 |
| Message-ID | <107bbv8$278bi$5@dont-email.me> |
| In reply to | #639584 |
Am 11.08.2025 um 01:38 schrieb Moebius: > Am 11.08.2025 um 00:31 schrieb Ben Bacarisse: > > > Years ago I used a very specific simpler example, using 0 and 1 rather > > than X an 0 and a one-dimensional "grid". One can use (the Cantor index > > of) fractions or, even simpler, start with an alternating sequence and, > > step by step, just swap the first 1 with the first following 0: > > > > s_0 = 0, 1, 0, 1, 0, 1, 0, 1, 0, ... > > s_1 = 0, 0, 1, 1, 0, 1, 0, 1, 0, ... > > s_2 = 0, 0, 0, 1, 1, 1, 0, 1, 0, ... > > s_3 = 0, 0, 0, 0, 1, 1, 1, 1, 0, ... > > > > In the limit, this sequence is all zeros. "Where did all the 1s go?" he > > might ask his students. > > Recently, I posted a similar example in de.sci.mathematic: > > Wir betrachten eine Folge von Folgen und deren (punktweisen) Grenzwert. > > Die Folge sei (f_0, f_1, f_2, f_3, ...) mit > > f_0 = (0, 1, 0, 1, 0, 1, 0, 1, ...) > f_1 = (1, 0, 0, 1, 0, 1, 0, 1, ...) > f_2 = (1, 1, 0, 0, 0, 1, 0, 1, ...) > f_3 = (1, 1, 1, 0, 0, 0, 0, 1, ...) > usw. > > Es ist dann lim f_n = (1, 1, 1, 1, ...) = (a_n)_(n e IN) mit a_n = 1 für > alle n e IN. > > ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ > > Of course, no meaningful answer from crank Wolfgang Mückenheim. He tends to ignore such (relevant) replies (objections). > . > . > . > . >
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| From | WM <wolfgang.mueckenheim@tha.de> |
|---|---|
| Date | 2025-08-11 16:25 +0200 |
| Message-ID | <107cuga$2iuu8$4@dont-email.me> |
| In reply to | #639585 |
On 11.08.2025 02:02, Moebius wrote: >> Recently, I posted a similar example in de.sci.mathematic: >> >> Of course, no meaningful answer I am satisfied that you must tell the untruth because you have no arguments. Of course I told you that Cantor uses natiural nu8mbers for indexing, not limits. Regards, WM
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| From | Ross Finlayson <ross.a.finlayson@gmail.com> |
|---|---|
| Date | 2025-08-10 21:37 -0700 |
| Message-ID | <VIucnROp9oDl7gT1nZ2dnZfqnPGdnZ2d@giganews.com> |
| In reply to | #639584 |
On 08/10/2025 04:38 PM, Moebius wrote: > Am 11.08.2025 um 00:31 schrieb Ben Bacarisse: > > > Years ago I used a very specific simpler example, using 0 and 1 rather > > than X an 0 and a one-dimensional "grid". One can use (the Cantor index > > of) fractions or, even simpler, start with an alternating sequence and, > > step by step, just swap the first 1 with the first following 0: > > > > s_0 = 0, 1, 0, 1, 0, 1, 0, 1, 0, ... > > s_1 = 0, 0, 1, 1, 0, 1, 0, 1, 0, ... > > s_2 = 0, 0, 0, 1, 1, 1, 0, 1, 0, ... > > s_3 = 0, 0, 0, 0, 1, 1, 1, 1, 0, ... > > > > In the limit, this sequence is all zeros. "Where did all the 1s go?" he > > might ask his students. > > Recently, I posted a similar example in de.sci.mathematic: > > Wir betrachten eine Folge von Folgen und deren (punktweisen) Grenzwert. > > Die Folge sei (f_0, f_1, f_2, f_3, ...) mit > > f_0 = (0, 1, 0, 1, 0, 1, 0, 1, ...) > f_1 = (1, 0, 0, 1, 0, 1, 0, 1, ...) > f_2 = (1, 1, 0, 0, 0, 1, 0, 1, ...) > f_3 = (1, 1, 1, 0, 0, 0, 0, 1, ...) > usw. > > Es ist dann lim f_n = (1, 1, 1, 1, ...) = (a_n)_(n e IN) mit a_n = 1 für > alle n e IN. > > ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ > > Of course, no meaningful answer from crank Wolfgang Mückenheim. > > . > . > . > . > Maybe think of it as being a Hilbert's Hotel franchisee. So, you have not one, but three Hilbert's Hotels, and they're full. So, the bellboy, has that each Hilbert Hotel is only one long corridor, so, the bellboy can not reach any room unless first passing all the preceding rooms, where each has a natural number. Then, it's to make it more like a balls-and-vase problem where you're not allowed to break the rules by claiming some capriciously arbitrary construction exists, instead that here these sort of things have to be done in an order. So, imagine Hotels 2 and 3 don't have any towels, while Hotel 1 does, so, due their clamoring complaints, you send the bellboy to take towels from Hotel 1 and back-and-forth provide towels to Hotels 2 and 3. Yet, the bellboy's lazy, and will only serve the first room respectively with or without a towel, depending on whether he is without or with a towel. So, you can provide towels to Hotels 2 and 3, yet now Hotel 1 has none. Then, in this case the guy only had two hotels in his franchise to begin with, and when you come up with a third hotel and this "Cantor Pairing", there's nothing he can do about it, because he would have to start all over with a brand new hotel with infinitely many towels, and another bellboy. Or, you know, you could start right away, yet maybe one of the reasons the bellboy is so lazy is because he's constantly doing busywork with no recognition. So, in mathematics, given that sort of contrivance, it's pretty simple that you can't wish it away. I.e., you're starting all over.
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| From | Ross Finlayson <ross.a.finlayson@gmail.com> |
|---|---|
| Date | 2025-08-10 21:50 -0700 |
| Message-ID | <0JWcneD7j8kO6wT1nZ2dnZfqn_adnZ2d@giganews.com> |
| In reply to | #639587 |
On 08/10/2025 09:37 PM, Ross Finlayson wrote: > On 08/10/2025 04:38 PM, Moebius wrote: >> Am 11.08.2025 um 00:31 schrieb Ben Bacarisse: >> >> > Years ago I used a very specific simpler example, using 0 and 1 rather >> > than X an 0 and a one-dimensional "grid". One can use (the Cantor >> index >> > of) fractions or, even simpler, start with an alternating sequence >> and, >> > step by step, just swap the first 1 with the first following 0: >> > >> > s_0 = 0, 1, 0, 1, 0, 1, 0, 1, 0, ... >> > s_1 = 0, 0, 1, 1, 0, 1, 0, 1, 0, ... >> > s_2 = 0, 0, 0, 1, 1, 1, 0, 1, 0, ... >> > s_3 = 0, 0, 0, 0, 1, 1, 1, 1, 0, ... >> > >> > In the limit, this sequence is all zeros. "Where did all the 1s >> go?" he >> > might ask his students. >> >> Recently, I posted a similar example in de.sci.mathematic: >> >> Wir betrachten eine Folge von Folgen und deren (punktweisen) Grenzwert. >> >> Die Folge sei (f_0, f_1, f_2, f_3, ...) mit >> >> f_0 = (0, 1, 0, 1, 0, 1, 0, 1, ...) >> f_1 = (1, 0, 0, 1, 0, 1, 0, 1, ...) >> f_2 = (1, 1, 0, 0, 0, 1, 0, 1, ...) >> f_3 = (1, 1, 1, 0, 0, 0, 0, 1, ...) >> usw. >> >> Es ist dann lim f_n = (1, 1, 1, 1, ...) = (a_n)_(n e IN) mit a_n = 1 für >> alle n e IN. >> >> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ >> >> Of course, no meaningful answer from crank Wolfgang Mückenheim. >> >> . >> . >> . >> . >> > > Maybe think of it as being a Hilbert's Hotel franchisee. So, you > have not one, but three Hilbert's Hotels, and they're full. So, > the bellboy, has that each Hilbert Hotel is only one long corridor, > so, the bellboy can not reach any room unless first passing all > the preceding rooms, where each has a natural number. > > Then, it's to make it more like a balls-and-vase problem where > you're not allowed to break the rules by claiming some capriciously > arbitrary construction exists, instead that here these sort of > things have to be done in an order. > > So, imagine Hotels 2 and 3 don't have any towels, while Hotel 1 > does, so, due their clamoring complaints, you send the bellboy > to take towels from Hotel 1 and back-and-forth provide towels > to Hotels 2 and 3. Yet, the bellboy's lazy, and will only serve > the first room respectively with or without a towel, depending > on whether he is without or with a towel. > > So, you can provide towels to Hotels 2 and 3, yet now Hotel 1 has none. > > Then, in this case the guy only had two hotels in his franchise to > begin with, and when you come up with a third hotel and this "Cantor > Pairing", there's nothing he can do about it, because he would have > to start all over with a brand new hotel with infinitely many towels, > and another bellboy. > > Or, you know, you could start right away, yet maybe one of the > reasons the bellboy is so lazy is because he's constantly doing > busywork with no recognition. > > > So, in mathematics, given that sort of contrivance, it's pretty simple > that you can't wish it away. > > I.e., you're starting all over. > > I suppose you might say "well as a mathematician, I'll simply define the bellboy not lazy, or, imagine another full hotel in my empire", and then you have a different problem, a lazy mathematician. At least one of which is absent a towel, .... There's a usual thought setting often about Dirichlet principle in infinite induction, "lions: eat, or sleep". In a world of lions, lions: if woken, immediately eat the nearest lion then go back to sleep. So, is it a world of dozing lions, or empty? Maybe just one, a sort of king of nothing? This is about usual concepts of supertasks, then, and about that the integers, themselves, do not have a standard model, and, there are models where they do, and models where they don't. (Complete.) Related rates, counting arguments, and combinatorics, in finite means, then as for the exhaustion, limits, and completions, in the infinitary analysis, have that either way it's directly demonstrable that points can't make a line, and lines can't make points: then for that both of those are, well let's say "incomplete".
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| From | Alan Mackenzie <acm@muc.de> |
|---|---|
| Date | 2025-08-11 12:28 +0000 |
| Message-ID | <107cnl2$2u5l$1@news.muc.de> |
| In reply to | #639583 |
Hello, Ben.
Ben Bacarisse <ben@bsb.me.uk> wrote:
> Alan Mackenzie <acm@muc.de> writes:
>> WM <wolfgang.mueckenheim@tha.de> wrote:
>>> On 06.08.2025 14:43, Alan Mackenzie wrote:
>>>> WM <wolfgang.mueckenheim@tha.de> wrote:
>>>> In the current scenario, O's don't get deleted. They just move away to
>>>> an unbounded distance.
>>> Which however is always finite. ....
>> Yes.
>>> .... So the O's remain in the matrix.
>> Yes. For any number of steps. But NOT in the limit.
>> Using an analogous, but simpler example, consider the sequence of real
>> numbers in decimal:
> One of the things I used to think was odd was the complexity a WM's
> examples. But then I decided this was deliberate.
Possibly. On the other hand, it takes understanding to reduce
complicated things to their essentials.
>> 1.1, 1.01, 1.001, 1.0001, ......
>> Every element of that sequence has two non-zero digits.
>> The limit of the sequence (I hope you can agree to this) is 1. This
>> limit has only one non-zero digit.
>> At no element of the sequence does the second 1 get "deleted". That 1
>> "remains in the number". But in the limit, it has gone.
>> This is essentially the same thing which is happening to your X's and
>> O's.
> Years ago I used a very specific simpler example, using 0 and 1 rather
> than X an 0 and a one-dimensional "grid". One can use (the Cantor index
> of) fractions or, even simpler, start with an alternating sequence and,
> step by step, just swap the first 1 with the first following 0:
> s_0 = 0, 1, 0, 1, 0, 1, 0, 1, 0, ...
> s_1 = 0, 0, 1, 1, 0, 1, 0, 1, 0, ...
> s_2 = 0, 0, 0, 1, 1, 1, 0, 1, 0, ...
> s_3 = 0, 0, 0, 0, 1, 1, 1, 1, 0, ...
That's a neat example!
> In the limit, this sequence is all zeros. "Where did all the 1s go?" he
> might ask his students.
> One day he might get a student who (a) points out that such sequences
> are just functions from N to {0,1}. (b) The sequence of functions s_n
> has a well-defined limit. (c) WM's own textbook defines this limit and
> shows how to calculate it!
> [Also, he used to vehemently deny that any non-constant set sequences
> have limits. But his textbook defines functions as sets (sets of pairs)
> and defines limits for certain sequences of such sets.]
WM is lacking basic abstract understanding. He doesn't understand what a
limit actually is. It is too much to expect consistency from him.
> --
> Ben.
--
Alan Mackenzie (Nuremberg, Germany).
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| From | WM <wolfgang.mueckenheim@tha.de> |
|---|---|
| Date | 2025-08-11 16:05 +0200 |
| Message-ID | <107ctad$2iuu8$2@dont-email.me> |
| In reply to | #639589 |
On 11.08.2025 14:28, Alan Mackenzie wrote: > Ben Bacarisse <ben@bsb.me.uk> wrote: > >> One of the things I used to think was odd was the complexity a WM's >> examples. But then I decided this was deliberate. > > Possibly. On the other hand, it takes understanding to reduce > complicated things to their essentials. And to see these essentials. Every term of the following sequence is enumerated by the position of the second 1. The limit is not enumerated and in the limit nothing is enumerated. > >>> 1.1, 1.01, 1.001, 1.0001, ...... >> s_0 = 0, 1, 0, 1, 0, 1, 0, 1, 0, ... >> s_1 = 0, 0, 1, 1, 0, 1, 0, 1, 0, ... >> s_2 = 0, 0, 0, 1, 1, 1, 0, 1, 0, ... >> s_3 = 0, 0, 0, 0, 1, 1, 1, 1, 0, ... > > That's a neat example! Alas the limit has noting to do with Cantor. Had he declared that his enumeration become comple in the limit, no-one would know his name today. > WM is lacking basic abstract understanding. He doesn't understand what a > limit actually is. Liar. You can learn it from my text books. But Cantor does not use it. Only confusion like yours has helped to keep this nonsense alive. "If we think the numbers p/q in such an order [...] then every number p/q comes at an absolutely fixed position of a simple infinite sequence" No limit involved. "The infinite sequence thus defined has the peculiar property to contain the positive rational numbers completely, and each of them only once at a determined place." No limit involved. Regards, WM
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| From | WM <wolfgang.mueckenheim@tha.de> |
|---|---|
| Date | 2025-08-11 15:56 +0200 |
| Message-ID | <107csrb$2iuu8$1@dont-email.me> |
| In reply to | #639583 |
On 11.08.2025 00:31, Ben Bacarisse wrote:
> Years ago I used a very specific simpler example, using 0 and 1 rather
> than X an 0 and a one-dimensional "grid". One can use (the Cantor index
> of) fractions or, even simpler, start with an alternating sequence and,
> step by step, just swap the first 1 with the first following 0:
>
> s_0 = 0, 1, 0, 1, 0, 1, 0, 1, 0, ...
> s_1 = 0, 0, 1, 1, 0, 1, 0, 1, 0, ...
> s_2 = 0, 0, 0, 1, 1, 1, 0, 1, 0, ...
> s_3 = 0, 0, 0, 0, 1, 1, 1, 1, 0, ...
>
> In the limit, this sequence is all zeros. "Where did all the 1s go?" he
> might ask his students.
Cantor's enumeration has nothing to do with the analytic limit. Only the
terms of a sequence can index.
>
> One day he might get a student who (a) points out that such sequences
> are just functions from N to {0,1}. (b) The sequence of functions s_n
> has a well-defined limit. (c) MW's own textbook defines this limit and
> shows how to calculate it!
But it has no bearing on indexing. If Cantor had claimed that in the
limit all fractions were indexed, nobody would have paid attention.
>
> [Also, he used to vehemently deny that any non-constant set sequences
> have limits.
I use limits where they are appropriate. For instance the number of
indices in the first column of an n*n-matrix is n. Its share is n/n^2.
Here the limit tells us about the share of enumerated fractions in the
infinite matrix: lim(n-->oo) n/n^2 = 0.
Result: Every mathematician accepting the analytical limit must deny
Cantor's claims.
Regards, WM
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