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Groups > sci.math > #412942 > unrolled thread
| Started by | David Petry <davidlpetry@gmail.com> |
|---|---|
| First post | 2018-05-11 11:20 -0700 |
| Last post | 2018-05-17 12:29 -0700 |
| Articles | 20 on this page of 227 — 16 participants |
Back to article view | Back to sci.math
The fundamental theorem of set theory is problematic David Petry <davidlpetry@gmail.com> - 2018-05-11 11:20 -0700
Re: The fundamental theorem of set theory is problematic Alan Mackenzie <acm@muc.de> - 2018-05-11 19:33 +0000
Re: The fundamental theorem of set theory is problematic David Petry <davidlpetry@gmail.com> - 2018-05-11 13:16 -0700
Re: The fundamental theorem of set theory is problematic "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-05-11 16:48 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-11 18:31 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-11 18:46 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-11 18:49 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-12 00:59 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-12 02:20 -0700
Re: The fundamental theorem of set theory is problematic David Petry <davidlpetry@gmail.com> - 2018-05-12 12:43 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-12 14:09 -0700
Re: The fundamental theorem of set theory is problematic David Petry <davidlpetry@gmail.com> - 2018-05-12 12:40 -0700
Re: The fundamental theorem of set theory is problematic Alan Mackenzie <acm@muc.de> - 2018-05-12 20:48 +0000
Re: The fundamental theorem of set theory is problematic David Petry <davidlpetry@gmail.com> - 2018-05-12 14:50 -0700
Re: The fundamental theorem of set theory is problematic Alan Mackenzie <acm@muc.de> - 2018-05-13 01:00 +0000
Re: The fundamental theorem of set theory is problematic David Petry <davidlpetry@gmail.com> - 2018-05-13 15:10 -0700
Re: The fundamental theorem of set theory is problematic Alan Mackenzie <acm@muc.de> - 2018-05-14 17:33 +0000
Re: The fundamental theorem of set theory is problematic David Petry <davidlpetry@gmail.com> - 2018-05-14 21:09 -0700
Re: The fundamental theorem of set theory is problematic Alan Mackenzie <acm@muc.de> - 2018-05-15 19:19 +0000
Re: The fundamental theorem of set theory is problematic "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-05-15 13:50 -0700
Re: The fundamental theorem of set theory is problematic "Shobe, Martin" <martin.shobe@yahoo.com> - 2018-05-15 21:34 -0500
Re: The fundamental theorem of set theory is problematic FredJeffries <fredjeffries@gmail.com> - 2018-05-16 09:10 -0700
Re: The fundamental theorem of set theory is problematic FredJeffries <fredjeffries@gmail.com> - 2018-05-16 09:15 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-15 05:33 -0700
Re: The fundamental theorem of set theory is problematic Jim Burns <james.g.burns@att.net> - 2018-05-12 19:02 -0400
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-13 01:38 -0700
Re: The fundamental theorem of set theory is problematic David Petry <davidlpetry@gmail.com> - 2018-05-14 21:40 -0700
Re: The fundamental theorem of set theory is problematic Jim Burns <james.g.burns@att.net> - 2018-05-15 05:22 -0400
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-13 01:31 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-13 03:51 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-13 05:18 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-13 06:59 -0700
Re: The fundamental theorem of set theory is problematic Alan Mackenzie <acm@muc.de> - 2018-05-13 11:02 +0000
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-13 05:37 -0700
Re: The fundamental theorem of set theory is problematic Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2018-05-13 21:57 +0100
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-14 04:44 -0700
Re: The fundamental theorem of set theory is problematic Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2018-05-14 13:29 +0100
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-14 07:42 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-14 09:36 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-14 09:42 -0700
Re: The fundamental theorem of set theory is problematic Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2018-05-14 18:29 +0100
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-14 12:03 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-14 12:06 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-14 12:10 -0700
Re: The fundamental theorem of set theory is problematic "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-05-15 13:44 -0700
Re: The fundamental theorem of set theory is problematic Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2018-05-14 18:16 +0100
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-15 05:44 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-14 06:38 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-12 17:58 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-12 18:04 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-12 18:19 -0700
Re: The fundamental theorem of set theory is problematic FredJeffries <fredjeffries@gmail.com> - 2018-05-15 09:14 -0700
Re: The fundamental theorem of set theory is problematic David Petry <davidlpetry@gmail.com> - 2018-05-15 10:07 -0700
Re: The fundamental theorem of set theory is problematic FredJeffries <fredjeffries@gmail.com> - 2018-05-16 09:26 -0700
Re: The fundamental theorem of set theory is problematic David Petry <davidlpetry@gmail.com> - 2018-05-16 11:18 -0700
Re: The fundamental theorem of set theory is problematic FredJeffries <fredjeffries@gmail.com> - 2018-05-16 12:26 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-16 14:06 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-16 15:53 -0700
Re: The fundamental theorem of set theory is problematic Jim Burns <james.g.burns@att.net> - 2018-05-16 19:07 -0400
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-17 01:17 -0700
Re: The fundamental theorem of set theory is problematic Jim Burns <james.g.burns@att.net> - 2018-05-17 09:42 -0400
Re: The fundamental theorem of set theory is problematic David Petry <davidlpetry@gmail.com> - 2018-05-16 17:40 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-15 11:06 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-12 01:05 -0700
Re: The fundamental theorem of set theory is problematic David Petry <davidlpetry@gmail.com> - 2018-05-12 12:57 -0700
Re: The fundamental theorem of set theory is problematic Alan Mackenzie <acm@muc.de> - 2018-05-13 00:20 +0000
Re: The fundamental theorem of set theory is problematic "Shobe, Martin" <martin.shobe@yahoo.com> - 2018-05-11 20:03 -0500
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-12 00:59 -0700
Re: The fundamental theorem of set theory is problematic Julio Di Egidio <julio@diegidio.name> - 2018-05-12 01:59 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-12 02:55 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-12 03:04 -0700
Re: The fundamental theorem of set theory is problematic Julio Di Egidio <julio@diegidio.name> - 2018-05-12 03:06 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-12 03:09 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-12 03:13 -0700
Re: The fundamental theorem of set theory is problematic Julio Di Egidio <julio@diegidio.name> - 2018-05-12 03:20 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-12 03:28 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-12 03:24 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-12 01:14 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-13 09:07 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-13 11:08 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-13 12:05 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-13 12:53 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-15 11:02 -0700
Re: The fundamental theorem of set theory is problematic j4n bur53 <janburse@fastmail.fm> - 2018-05-15 21:13 +0200
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-16 11:30 -0700
Re: The fundamental theorem of set theory is problematic Zeit Geist <Tucsondrew@me.com> - 2018-05-16 12:23 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-16 14:00 -0700
Re: The fundamental theorem of set theory is problematic Zeit Geist <Tucsondrew@me.com> - 2018-05-16 16:18 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-16 16:27 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-17 05:01 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-17 08:34 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-17 09:34 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-17 10:12 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-17 10:57 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-17 11:56 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-17 12:00 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-17 12:29 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-17 13:38 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-18 08:12 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-18 12:23 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-19 04:53 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-19 07:21 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-19 09:11 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-20 03:20 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-20 04:48 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-20 05:19 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-20 10:09 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-20 13:26 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-20 18:10 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-21 06:05 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-21 07:24 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-21 08:59 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-21 11:48 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-22 05:45 -0700
Re: The fundamental theorem of set theory is problematic Python <python@python.invalid> - 2018-05-22 14:51 +0200
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-22 06:31 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-22 13:26 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-22 14:22 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-23 06:06 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-23 07:19 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-23 11:56 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-23 12:08 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-23 13:22 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-23 18:11 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-24 02:47 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-24 06:00 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-24 07:43 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-24 08:10 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-24 09:37 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-24 09:45 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-25 09:05 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-25 11:09 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-25 15:26 -0700
Re: The fundamental theorem of set theory is problematic Zelos Malum <zelos.malum@gmail.com> - 2018-05-25 23:14 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-26 03:23 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-26 06:37 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-26 06:50 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-26 07:30 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-26 08:39 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-26 09:15 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-27 05:37 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-27 08:58 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-28 03:53 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-28 05:04 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-28 05:10 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-28 05:19 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-29 04:53 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-29 06:22 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-29 06:36 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-29 07:12 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-29 08:35 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-29 08:52 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-30 04:25 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-30 04:50 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-31 06:17 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-31 07:05 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-31 09:53 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-31 11:47 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-06-01 03:58 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-06-01 05:07 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-06-01 05:24 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-06-01 06:27 -0700
Re: The fundamental theorem of set theory is problematic "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-06-01 10:21 -0700
Re: The fundamental theorem of set theory is problematic "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-06-02 14:04 -0700
Re: The fundamental theorem of set theory is problematic "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-06-02 16:27 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-06-02 07:23 -0700
Re: The fundamental theorem of set theory is problematic Zelos Malum <zelos.malum@gmail.com> - 2018-06-04 00:11 -0700
Re: The fundamental theorem of set theory is problematic Zelos Malum <zelos.malum@gmail.com> - 2018-05-27 23:12 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-23 12:49 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-23 12:51 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-23 13:24 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-24 02:09 -0700
Re: The fundamental theorem of set theory is problematic Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2018-05-22 19:09 +0100
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-22 13:29 -0700
Re: The fundamental theorem of set theory is problematic Zelos Malum <zelos.malum@gmail.com> - 2018-05-23 22:32 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-24 03:57 -0700
Re: The fundamental theorem of set theory is problematic Zelos Malum <zelos.malum@gmail.com> - 2018-05-24 04:48 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-24 07:29 -0700
Re: The fundamental theorem of set theory is problematic Python <python@python.invalid> - 2018-05-24 16:39 +0200
Re: The fundamental theorem of set theory is problematic Python <python@python.invalid> - 2018-05-18 01:19 +0200
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-18 08:14 -0700
Re: The fundamental theorem of set theory is problematic Python <python@python.invalid> - 2018-05-18 17:31 +0200
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-18 10:09 -0700
Re: The fundamental theorem of set theory is problematic FredJeffries <fredjeffries@gmail.com> - 2018-05-18 10:53 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-18 13:54 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-18 13:55 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-18 14:09 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-18 14:19 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-18 14:23 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-18 14:37 -0700
Re: The fundamental theorem of set theory is problematic Zeit Geist <Tucsondrew@me.com> - 2018-05-18 14:33 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-19 04:58 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-19 05:28 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-19 05:34 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-19 05:36 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-19 05:46 -0700
Re: The fundamental theorem of set theory is problematic Zeit Geist <Tucsondrew@me.com> - 2018-05-19 09:00 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-20 03:21 -0700
Re: The fundamental theorem of set theory is problematic Python <python@python.invalid> - 2018-05-19 02:01 +0200
Re: The fundamental theorem of set theory is problematic Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2018-05-21 16:40 +0100
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-21 09:06 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-21 13:52 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-22 05:45 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-22 08:27 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-22 13:29 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-22 13:43 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-22 13:52 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-23 06:06 -0700
Re: The fundamental theorem of set theory is problematic Zeit Geist <Tucsondrew@me.com> - 2018-05-22 17:30 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-23 06:06 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-15 17:14 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-16 11:26 -0700
Re: The fundamental theorem of set theory is problematic wpihughes@gmail.com - 2018-05-16 16:35 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-17 05:03 -0700
Re: The fundamental theorem of set theory is problematic Zelos Malum <zelos.malum@gmail.com> - 2018-05-17 05:33 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-17 09:31 -0700
Re: The fundamental theorem of set theory is problematic Sergio <invalid@invalid.com> - 2018-05-13 12:40 -0500
Re: The fundamental theorem of set theory is problematic Zelos Malum <zelos.malum@gmail.com> - 2018-05-13 22:36 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-14 04:49 -0700
Re: The fundamental theorem of set theory is problematic Zelos Malum <zelos.malum@gmail.com> - 2018-05-17 01:34 -0700
Re: The fundamental theorem of set theory is problematic WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-17 05:08 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-17 10:03 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-17 10:06 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-17 10:19 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-17 10:26 -0700
Re: The fundamental theorem of set theory is problematic bursejan@gmail.com - 2018-05-17 10:30 -0700
Re: The fundamental theorem of set theory is problematic FredJeffries <fredjeffries@gmail.com> - 2018-05-17 12:29 -0700
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| From | "Shobe, Martin" <martin.shobe@yahoo.com> |
|---|---|
| Date | 2018-05-15 21:34 -0500 |
| Message-ID | <pdg5cf$4h0$1@dont-email.me> |
| In reply to | #413132 |
On 5/15/2018 2:19 PM, Alan Mackenzie wrote: > David Petry <davidlpetry@gmail.com> wrote: > >> Alan Mackenzie wrote: > >>> I've skimmed the article in the English wikipedia on Constructivism. >>> I don't find it, as yet, an attractive philosophy. > > >> The basic idea behind constructivism is that the logic and reasoning >> used by the constructivists is much closer to the reasoning used in >> science and technology than the logic used in classical pure >> mathematics. > > OK. That doesn't make me like it, though. I don't think it's accurate either. Martin Shobe
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| From | FredJeffries <fredjeffries@gmail.com> |
|---|---|
| Date | 2018-05-16 09:10 -0700 |
| Message-ID | <c434f53d-7b04-4fa4-9297-7c08acc0f8e9@googlegroups.com> |
| In reply to | #413156 |
On Tuesday, May 15, 2018 at 7:35:02 PM UTC-7, Shobe, Martin wrote: > On 5/15/2018 2:19 PM, Alan Mackenzie wrote: > > David Petry <davidlpetry@gmail.com> wrote: > > > >> Alan Mackenzie wrote: > > > >>> I've skimmed the article in the English wikipedia on Constructivism. > >>> I don't find it, as yet, an attractive philosophy. > > > > > >> The basic idea behind constructivism is that the logic and reasoning > >> used by the constructivists is much closer to the reasoning used in > >> science and technology than the logic used in classical pure > >> mathematics. > > > > OK. That doesn't make me like it, though. > > I don't think it's accurate either. Interested readers may consult profitably the video of Andrej Bauer's talk "Five stages of accepting constructive mathematics" at the Institute for Advanced Study and subsequent article in the Bulletin of the American Mathematical Society: https://www.youtube.com/watch?v=21qPOReu4FI http://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf http://math.andrej.com/2016/10/10/five-stages-of-accepting-constructive-mathematics/
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| From | FredJeffries <fredjeffries@gmail.com> |
|---|---|
| Date | 2018-05-16 09:15 -0700 |
| Message-ID | <cb98c384-1985-46bd-bcf2-07fa28622ade@googlegroups.com> |
| In reply to | #413156 |
On Tuesday, May 15, 2018 at 7:35:02 PM UTC-7, Shobe, Martin wrote: > On 5/15/2018 2:19 PM, Alan Mackenzie wrote: > > David Petry <davidlpetry@gmail.com> wrote: > > > >> Alan Mackenzie wrote: > > > >>> I've skimmed the article in the English wikipedia on Constructivism. > >>> I don't find it, as yet, an attractive philosophy. > > > > > >> The basic idea behind constructivism is that the logic and reasoning > >> used by the constructivists is much closer to the reasoning used in > >> science and technology than the logic used in classical pure > >> mathematics. > > > > OK. That doesn't make me like it, though. > > I don't think it's accurate either. For one thing, it assumes that there is one unique technique of "reasoning used in science and technology"
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| From | WM <wolfgang.mueckenheim@hs-augsburg.de> |
|---|---|
| Date | 2018-05-15 05:33 -0700 |
| Message-ID | <ee7310df-a87a-47d7-8124-42c5bf513286@googlegroups.com> |
| In reply to | #413087 |
Am Montag, 14. Mai 2018 19:33:11 UTC+2 schrieb Alan Mackenzie: > At one point you were talking about manipulating the individual elements > of a list that does not, and cannot exist. You introduce this impossible > list by saying "if we have a provably well-defined list of provable > well-defined real numbers ....". You begin by assuming the converse of > what Cantor proves. You seem to be ingnorant of the fact that in mathematics at Cantor's time everything had to be well defined and that also Cantor accepted only well-defined real numbers? Know then that besmirching of mathematics only happened after it had turned out that clean mathematics does not allow for uncountable sets involving undefinable objects: "Infinite definitions" (that do not happen in finite time) are non-things (Undinge). If König's theorem was true, according to which all "finitely definable" real numbers form a set of cardinality aleph_0, this would imply that the whole continuum was countable, and that is certainly false. [G. Cantor, letter to D. Hilbert (8 Aug 1906)] Regards, WM
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| From | Jim Burns <james.g.burns@att.net> |
|---|---|
| Date | 2018-05-12 19:02 -0400 |
| Message-ID | <c63702b8-7bd1-b6bd-63e0-96665574d0cf@att.net> |
| In reply to | #412999 |
On 5/12/2018 4:48 PM, Alan Mackenzie wrote: > David Petry <davidlpetry@gmail.com> wrote: >> If we start teaching set theory to the very young, it will only >> make the problems that Thurston delineates much worse. > > I agree, but not for the reasons you would put forward. Set theory > as taught to the very young consists of putting intersecting circles > on a blackboard with small numbers of objects in each, and > announcing the definitions of union and intersection. This is > pointless to the children, and is dull, dull, dull. > > Set theory only becomes interesting with infinite sets. Young > children lack the intellectual development to cope with the infinite. > Set theory should only be taught to older age groups, when they are > ready for it. I am completely inexperienced in this area. But I would like to see first order predicate logic taught at as young an age as could be managed, for the sake of its beauty, though it will also have many important applications to some of the students at some eventual later point. I also would like these students to learn foreign languages and art and music and dance and history and science -- for much the same reason. There is wide, wonderful world out there, and I want them to get at least a taste of things that they might want more of later. It seems to me that the toughest concept in FOPL is "all". Wouldn't children be familiar with the natural language version of "all" at a fairly young age? I don't know, I'm asking. I wonder if a course in formal logic (perhaps with computer assistance for gritty details) is something that is out there, somewhere. And, once we have FOPL, coping _formally_ with the infinite becomes possible, it seems to me. Actually _understanding_ the infinite might need to wait until the 12th of Never. I doubt that I understand it myself. It's easy for me to overlook this momentarily, though, since I've gotten so familiar with the formalism.
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| From | WM <wolfgang.mueckenheim@hs-augsburg.de> |
|---|---|
| Date | 2018-05-13 01:38 -0700 |
| Message-ID | <f662a6da-2ff7-4b7b-b54f-05972da6fb04@googlegroups.com> |
| In reply to | #413005 |
Am Sonntag, 13. Mai 2018 01:02:47 UTC+2 schrieb Jim Burns: > On 5/12/2018 4:48 PM, Alan Mackenzie wrote: > > David Petry <davidlpetry@gmail.com> wrote: > > >> If we start teaching set theory to the very young, it will only > >> make the problems that Thurston delineates much worse. > > > > I agree, but not for the reasons you would put forward. Set theory > > as taught to the very young consists of putting intersecting circles > > on a blackboard with small numbers of objects in each, and > > announcing the definitions of union and intersection. This is > > pointless to the children, and is dull, dull, dull. > > > > Set theory only becomes interesting with infinite sets. Young > > children lack the intellectual development to cope with the infinite. > > Set theory should only be taught to older age groups, when they are > > ready for it. > > I am completely inexperienced in this area. But I would like to > see first order predicate logic taught at as young an age as > could be managed, for the sake of its beauty, though it will also > have many important applications to some of the students at some > eventual later point. For instance this one: What means "all natural numbers"? Does it mean only those natural numbers which have the characteristic property of every natural number, i.e., to be followed by infinitely many natural numbers? Then you don't have all of them because always infinitely many are remaining. Or does it mean all natural numbers with no exception? Then you include some which are not followed by infinitely many and hence are not natural numbers because they are lacking the characteristic property of every natural number. That means you get more than all natural numbers. Logic? Regards, WM
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| From | David Petry <davidlpetry@gmail.com> |
|---|---|
| Date | 2018-05-14 21:40 -0700 |
| Message-ID | <ae99943a-723f-478f-8e4e-937ea6dd800c@googlegroups.com> |
| In reply to | #413005 |
Jim Burns wrote: > But I would like to > see first order predicate logic taught at as young an age as > could be managed, for That might leave students with the impression that what they are taught in school is nothing but boring irrelevant games. That's already a big problem which could be made worse by teaching them first order predicate logic.
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| From | Jim Burns <james.g.burns@att.net> |
|---|---|
| Date | 2018-05-15 05:22 -0400 |
| Message-ID | <fbcb8433-9963-c490-14ad-a71620e08d40@att.net> |
| In reply to | #413102 |
On 5/15/2018 12:40 AM, David Petry wrote: > Jim Burns wrote: >> I am completely inexperienced in this area. But I would like to >> see first order predicate logic taught at as young an age as >> could be managed, for the sake of its beauty, though it will also >> have many important applications to some of the students at some >> eventual later point. >> >> I also would like these students to learn foreign languages and >> art and music and dance and history and science -- for much the >> same reason. There is wide, wonderful world out there, and I >> want them to get at least a taste of things that they might want >> more of later. > > That might leave students with the impression that what they are > taught in school is nothing but boring irrelevant games. > That's already a big problem which could be made worse by > teaching them first order predicate logic. De gustibus non est disputandum. I apparently have a higher opinion of young people than you do. Boring? Irrelevant? I'm pretty sure music is boring and irrelevant when you don't like it. And you're not going to like it, some of the ways it's "taught". The problem, if there is a problem, is in the basic set-up, not in the subject material, it seems to me. Have you seen some of the games they play? https://en.wikipedia.org/wiki/Magic:_The_Gathering I don't think complexity is a barrier, especially with computer-assisted syntax and rule-checking. And FOPL is pretty damn close to a real-life Magic Spell language.
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| From | WM <wolfgang.mueckenheim@hs-augsburg.de> |
|---|---|
| Date | 2018-05-13 01:31 -0700 |
| Message-ID | <37430d59-3adc-4c4a-832e-6d64256c7564@googlegroups.com> |
| In reply to | #412999 |
Am Samstag, 12. Mai 2018 22:48:33 UTC+2 schrieb Alan Mackenzie:
> Your opening post to
> this thread strongly suggests you do not understand Cantor's proof
That is the usual snooty behaviour of students who obviously had problems when Cantor's proof was presented to them. Is it real possible to not understand this simple idea? You must have made very bad experience with it when you believe so. By the way, there is no mathematics required to understand Cantor, according to his own testimony. At least in five of his preserved letters (and that is merely a small fraction of his complete correspondence) Cantor emphasizes that very little mathematical knowledge is required to understand his theory:
"To understand the basic idea of the teaching of the transfinite no scholarly education in newer mathematics is required; this could even be a hindrance because in the so-called infinitesimal analysis the potential infinite has pushed to the fore and lead to the opinion, even of the heroes, as if they with their 'differentials' and 'integrals' mastered the heights of knowledge and skill. Strictly speaking, the potential infinite is always unthinkable without the foundational A. I. (which only most of those gentlemen will not or can not account for). So, if you expect to get an 'expertly' competent judgement in the current question from those circles you may find your expectations disappointed. The only forum here is the Empress Reason who does not acknowledge any reputation of privileged, scholarly, academical guilds. She persists and rules – we humans come and go." [G. Cantor, draft of a letter to A. Schmid (18 Apr 1887)]
"Should you have time to read my papers, you might find that very little previous mathematical knowledge is required for the understanding." [G. Cantor, letter to C.F. Heman (28 Jul 1887)]
"With respect to the question of the actual infinite in creatis I repeat first of all, what I wrote you one year ago: A scholarly preparation in mathematics is not at all necessary for the understanding of my relevant papers but a careful study of the latter is sufficient. Everyone, and in particular the trained philosopher, is able to scrutinize the teaching of the transfinite and to convince himself from its correctness and truth." [G. Cantor, letter to C.F. Heman (2 Jun 1888)]
"The comprehension of the elements of the teaching of the transfinite does not require any scholarly preparation in the newer mathematics. That could rather be disadvantageous than useful for this purpose because the modern mathematicians, in their majority, have got into a victory flush [...] which lets them degenerate into material one-sidedness and makes them blind for any objective-metaphysical recognition and therefore also for the foundations of their own science." [G. Cantor, letter to I. Jeiler (20 May 1888, Whitsun)]
"I mention that great previous knowledge of mathematics is not required to understand my teachings, but only extensive philosophical knowledge, as it is learned best and most beautifully at your institution." [G. Cantor, letter to A. Baumgartner (27 Dec 1893)]
And finally, Cantor gives a general advice: "As a philosopher you do well, in my opinion, to be very sceptical against mathematical authorities in all mathematical-philosophical questions, in memory of Pascal's true saying: 'Il est rare, que les géomètres soient fins, et que les fins soient géomètres.'" {{It is rare that the mathematicians are sharp-witted and that the sharp-witted are mathematicians.}} [G. Cantor, letter to A. Schmid (8 May 1887)]
> Set theory is established mathematics, just as a round Earth is
> established geography,
Set theory is nonsensical matheology. The finished infinite is absurd. The violation of logic is hard to swallow for health minds.
> Set theory only becomes interesting with infinite sets. Young children
> lack the intellectual development to cope with the infinite.
There is no intellect required, but only its absence, when the finished infinite is taught.
What means all natural numbers? Does it mean to take only those natural numbers which have the characteristic property of every natural number, i.e., to be followed by infinitely many natural numbers? Then you don't get all of them because always infinitely many are remaining. Or do you take all natural numbers with no exception? Then you include some which are not followed by infinitely many and hence are not natural numbers because they are lacking the characteristic property of every natural number. That means you get more than all natural numbers.
Regards, WM
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| From | bursejan@gmail.com |
|---|---|
| Date | 2018-05-13 03:51 -0700 |
| Message-ID | <c609fce4-926a-4937-9065-dffcf277279b@googlegroups.com> |
| In reply to | #413020 |
Well you find serious discussions on the internet that notions such us "injective" and "surjective" where doubtful inventions by the Nicolas Bourbaki group, possibly as wrong as Cantors theorem. And mathematics was better before these new inventions, that only lead to pseudo mathematics. The derogative is Bourbaki's style mathematics. https://de.wikipedia.org/wiki/Nicolas_Bourbaki#Nachwirkung But we find grown ups still today, that don't understand that there is a bijection f : N -> Q, or what is the difference between saying something is a function or something is surjective. So the answer is Cantors theorem is difficult, anything that uses abstraction and that should be exact is difficult. Irony has it that Cantors Theorem is also valid in the finite, not only in the infinite. So you can actually show |A| < |P(A)| for a finite A very easily. And you can use a different proof to show it for all finite A. But from this it follows not automatically that it also holds for infinite A. So yes, there are many challenges lurking, although it makes use of very little of ZFC. Its challenge lies in logic. Am Sonntag, 13. Mai 2018 10:31:40 UTC+2 schrieb WM: > Am Samstag, 12. Mai 2018 22:48:33 UTC+2 schrieb Alan Mackenzie: > > > Your opening post to > > this thread strongly suggests you do not understand Cantor's proof > > That is the usual snooty behaviour of students who obviously had problems when Cantor's proof was presented to them.
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| From | WM <wolfgang.mueckenheim@hs-augsburg.de> |
|---|---|
| Date | 2018-05-13 05:18 -0700 |
| Message-ID | <ee96d38a-813e-4929-b9c4-f1914312fdc9@googlegroups.com> |
| In reply to | #413023 |
Am Sonntag, 13. Mai 2018 12:51:27 UTC+2 schrieb burs...@gmail.com: > Irony has it that Cantors Theorem is also valid in > the finite, That is not surprising. The theorem rquires a finished list. Finite lists are finished, infinity lists are not. Only mathematicians who are unable to understand that the infinite is infinite can stick to Cantor's "theorem". Regards, WM
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| From | bursejan@gmail.com |
|---|---|
| Date | 2018-05-13 06:59 -0700 |
| Message-ID | <1f7d5bf0-c19b-4c65-86b8-2708aa587460@googlegroups.com> |
| In reply to | #413026 |
What does this mean "the infinite is infinite can stick to Cantor's "theorem"." Something from Volkswagen Omlette. Does it make any sense what you wrote? "... the infinite is infinite can stick ..." You changed profession. Your now in the bublemgum industry, you want to stick something to a theorem. How is this done? Very interesting Augsburg Crank institute nonsense as usual. Am Sonntag, 13. Mai 2018 14:18:47 UTC+2 schrieb WM: > Am Sonntag, 13. Mai 2018 12:51:27 UTC+2 schrieb burs...@gmail.com: > > > > Irony has it that Cantors Theorem is also valid in > > the finite, > > That is not surprising. The theorem rquires a finished list. Finite lists are finished, infinity lists are not. Only mathematicians who are unable to understand that the infinite is infinite can stick to Cantor's "theorem". > > Regards, WM
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| From | Alan Mackenzie <acm@muc.de> |
|---|---|
| Date | 2018-05-13 11:02 +0000 |
| Message-ID | <pd960t$u45$1@news.muc.de> |
| In reply to | #413020 |
WM <wolfgang.mueckenheim@hs-augsburg.de> wrote:
> Am Samstag, 12. Mai 2018 22:48:33 UTC+2 schrieb Alan Mackenzie:
>> Your opening post to
>> this thread strongly suggests you do not understand Cantor's proof
> That is the usual snooty behaviour of students who obviously had
> problems when Cantor's proof was presented to them. Is it real possible
> to not understand this simple idea?
I had no difficulty at any stage in understanding Cantor's proof. I
wonder about you and David Petry. Is it really possible for an
intelligent human being not to understand it? The two of you may well be
feigning misunderstanding, with the object of spreading falsehood and
doubt.
> You must have made very bad experience with it when you believe so. By
> the way, there is no mathematics required to understand Cantor,
> according to his own testimony. At least in five of his preserved
> letters (and that is merely a small fraction of his complete
> correspondence) Cantor emphasizes that very little mathematical
> knowledge is required to understand his theory:
Cantor was an academic, thus also a politician. Much of the political
stuff what he wrote was likely insincere, just as it would be today
(unfortunately).
But your modus operandi in this conversation is to quote mathematicians
from 120, 130 years ago, who had genuine doubts and misgivings about set
theory. At the time set theory was fresh, new, and controversial. These
doubts have long been settled, and it is not clever of you to keep
bringing up all this old stuff as though there were still some
controversy.
[ .... ]
>> Set theory is established mathematics, just as a round Earth is
>> established geography,
> Set theory is nonsensical matheology. The finished infinite is absurd.
> The violation of logic is hard to swallow for health minds.
Whatever "matheology" and "finished infinite" mean, things you seem
unable to state.
>> Set theory only becomes interesting with infinite sets. Young children
>> lack the intellectual development to cope with the infinite.
> There is no intellect required, but only its absence, when the finished
> infinite is taught.
> What means all natural numbers?
It means {0, 1, 2, 3, 4, 5, 6, 7, ....}. No doubt you're going to get
all delicate about the "....". Well don't: there's an axiom of set
theory which states that if (i) 0 is a member of a set; and (ii) whenever
n is a member of a set, so is its successor; then the set contains all
the natural numbers.
> Does it mean to take only those natural numbers which have the
> characteristic property of every natural number, i.e., to be followed
> by infinitely many natural numbers? Then you don't get all of them
> because always infinitely many are remaining. Or do you take all
> natural numbers with no exception? Then you include some which are not
> followed by infinitely many and hence are not natural numbers because
> they are lacking the characteristic property of every natural number.
> That means you get more than all natural numbers.
That's pure garbage, sorry, but it's not worth replying to. Either
you're trying to stir up confusion in the mathematically less
sophisticated, or you're genuinely confused yourself.
> Regards, WM
--
Alan Mackenzie (Nuremberg, Germany).
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| From | WM <wolfgang.mueckenheim@hs-augsburg.de> |
|---|---|
| Date | 2018-05-13 05:37 -0700 |
| Message-ID | <6cf08551-2341-4e81-bc47-68d70651d79a@googlegroups.com> |
| In reply to | #413024 |
Am Sonntag, 13. Mai 2018 13:03:00 UTC+2 schrieb Alan Mackenzie:
> WM <wolfgang.mueckenheim@hs-augsburg.de> wrote:
> I had no difficulty at any stage in understanding Cantor's proof. I
> wonder about you and David Petry. Is it really possible for an
> intelligent human being not to understand it?
It is obvious that Cantor requires the finished infinite, as he frequently confessed. Only those who cannot comprehend the infinite but think that it is somehow finished, can "understand" Cantor's "proof".
Only a small elite is able to recognize the mistake:
Cantor's second number class does not exist. [L.E.J. Brouwer]
The ordinary diagonal Verfahren I believe to involve a patent confusion of the program and object aspects of the decimal fraction, which must be apparent to any who imagines himself actually carrying out the operations demanded in the proof. In fact, I find it difficult to understand how such a situation should have been capable of persisting in mathematics. [P.W. Bridgman, Nobel laureate]
I never met a mathematician who to a higher degree than Hermite has been a realist in the sense of Plato, and yet I can claim that I never met a more decided opponent of the Cantorian ideas. This is the more a seeming contradiction, as he himself stated frankly: I am an opponent of Cantor because I am a realist. [H. Poincaré]
> > You must have made very bad experience with it when you believe so. By
> > the way, there is no mathematics required to understand Cantor,
> > according to his own testimony. At least in five of his preserved
> > letters (and that is merely a small fraction of his complete
> > correspondence) Cantor emphasizes that very little mathematical
> > knowledge is required to understand his theory:
>
> Cantor was an academic, thus also a politician. Much of the political
> stuff what he wrote was likely insincere, just as it would be today
> (unfortunately).
But fact is that his diagonal proof is easier than most of analysis.
>
> But your modus operandi in this conversation is to quote mathematicians
> from 120, 130 years ago, who had genuine doubts and misgivings about set
> theory. At the time set theory was fresh, new, and controversial. These
> doubts have long been settled, and it is not clever of you to keep
> bringing up all this old stuff as though there were still some
> controversy.
Look into https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, chapter V. There are many modern mathematicians and other intelligent people who have grasped the truth. For instance
Herren Geheimrat Hilbert und Prof. Dr. Cantor, I'd like to be Excused from your "Paradise": It is a Paradise of Fools, and besides feels more like Hell. [Doron Zeilberger]
If you have an elaborate theory of 'hierarchies upon hierarchies of infinite sets', in which you cannot even in principle decide whether there is anything between the first and second 'infinity' on your list, then it's time to admit that you are no longer doing mathematics. [Norman Wildberger]
>
> Whatever "matheology" and "finished infinite" mean, things you seem
> unable to state.
Things you are unwilling to learn. I stated them in detail here: https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf
>
> > What means all natural numbers?
>
> It means {0, 1, 2, 3, 4, 5, 6, 7, ....}. No doubt you're going to get
> all delicate about the "....". Well don't: there's an axiom of set
> theory which states that if (i) 0 is a member of a set; and (ii) whenever
> n is a member of a set, so is its successor; then the set contains all
> the natural numbers.
That is the self-contradictory statement.
>
> > Does it mean to take only those natural numbers which have the
> > characteristic property of every natural number, i.e., to be followed
> > by infinitely many natural numbers? Then you don't get all of them
> > because always infinitely many are remaining. Or do you take all
> > natural numbers with no exception? Then you include some which are not
> > followed by infinitely many and hence are not natural numbers because
> > they are lacking the characteristic property of every natural number.
> > That means you get more than all natural numbers.
>
> That's pure garbage, sorry, but it's not worth replying to.
Not worth or not possible? Is every natural number followed by infinitely many?
Regards, WM
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| From | Alan Smaill <smaill@SPAMinf.ed.ac.uk> |
|---|---|
| Date | 2018-05-13 21:57 +0100 |
| Message-ID | <fwek1s7i89v.fsf@foxtrot.inf.ed.ac.uk> |
| In reply to | #413027 |
WM <wolfgang.mueckenheim@hs-augsburg.de> writes: > Am Sonntag, 13. Mai 2018 13:03:00 UTC+2 schrieb Alan Mackenzie: >> WM <wolfgang.mueckenheim@hs-augsburg.de> wrote: > > >> I had no difficulty at any stage in understanding Cantor's proof. I >> wonder about you and David Petry. Is it really possible for an >> intelligent human being not to understand it? > > It is obvious that Cantor requires the finished infinite, as he > frequently confessed. Only those who cannot comprehend the infinite > but think that it is somehow finished, can "understand" Cantor's > "proof". > > Only a small elite is able to recognize the mistake: > > Cantor's second number class does not exist. [L.E.J. Brouwer] Oh look, WM suggests that Brouwer is on his side when it comes to completed infinity. In fact Brouwer wrote: "all mathematical sets of units which are entitled to that name can be developed out of the basal intuition, and this can only be done by combining a finite number of times the two operations 'to create a finite ordinal number' and 'to create the infinite ordinal number omega'; " (Bulletin of AMS, 1913) Please do not confuse the acceptance of omega as a completed totality with the acceptance of larger transfinite cardinals. > > Regards, WM -- Alan Smaill
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| From | WM <wolfgang.mueckenheim@hs-augsburg.de> |
|---|---|
| Date | 2018-05-14 04:44 -0700 |
| Message-ID | <43297190-0618-41b8-b909-164ee9438a47@googlegroups.com> |
| In reply to | #413039 |
Am Sonntag, 13. Mai 2018 23:00:06 UTC+2 schrieb Alan Smaill: > WM <wolfgang.mueckenheim@hs-augsburg.de> writes: > > > Am Sonntag, 13. Mai 2018 13:03:00 UTC+2 schrieb Alan Mackenzie: > >> WM <wolfgang.mueckenheim@hs-augsburg.de> wrote: > > > > > >> I had no difficulty at any stage in understanding Cantor's proof. I > >> wonder about you and David Petry. Is it really possible for an > >> intelligent human being not to understand it? > > > > It is obvious that Cantor requires the finished infinite, as he > > frequently confessed. Only those who cannot comprehend the infinite > > but think that it is somehow finished, can "understand" Cantor's > > "proof". > > > > Only a small elite is able to recognize the mistake: > > > > Cantor's second number class does not exist. [L.E.J. Brouwer] > > Oh look, WM suggests that Brouwer is on his side when it comes to > completed infinity. > > In fact Brouwer wrote: "all mathematical sets of units which are > entitled to that name can be developed out of the basal intuition, and > this can only be done by combining a finite number of times the two > operations 'to create a finite ordinal number' and 'to create the > infinite ordinal number omega'; " (Bulletin of AMS, 1913) > > Please do not confuse the acceptance of omega as a completed totality > with the acceptance of larger transfinite cardinals. Brouwer denied uncountable sets. That means he denied that Cantor's "proof" is valid. Further Brouwer accepted omega only when constructed from equal elements like periodic decimals: We can create in mathematics nothing but finite sequences, and further, on the ground of the clearly conceived "and so on", the order type omega, but only consisting of equal elements, so that we can never imagine the arbitrary infinite binary fractions as finished. [L.E.J. Brouwer: "Over de grondslagen der wiskunde", Thesis, Univ. Amsterdam (1907) p. 142f] Regards, WM
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| From | Alan Smaill <smaill@SPAMinf.ed.ac.uk> |
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| Date | 2018-05-14 13:29 +0100 |
| Message-ID | <fwer2meh13z.fsf@foxtrot.inf.ed.ac.uk> |
| In reply to | #413069 |
WM <wolfgang.mueckenheim@hs-augsburg.de> writes: > Am Sonntag, 13. Mai 2018 23:00:06 UTC+2 schrieb Alan Smaill: >> WM <wolfgang.mueckenheim@hs-augsburg.de> writes: >> >> > Am Sonntag, 13. Mai 2018 13:03:00 UTC+2 schrieb Alan Mackenzie: >> >> WM <wolfgang.mueckenheim@hs-augsburg.de> wrote: >> > >> > >> >> I had no difficulty at any stage in understanding Cantor's proof. I >> >> wonder about you and David Petry. Is it really possible for an >> >> intelligent human being not to understand it? >> > >> > It is obvious that Cantor requires the finished infinite, as he >> > frequently confessed. Only those who cannot comprehend the infinite >> > but think that it is somehow finished, can "understand" Cantor's >> > "proof". >> > >> > Only a small elite is able to recognize the mistake: >> > >> > Cantor's second number class does not exist. [L.E.J. Brouwer] >> >> Oh look, WM suggests that Brouwer is on his side when it comes to >> completed infinity. >> >> In fact Brouwer wrote: "all mathematical sets of units which are >> entitled to that name can be developed out of the basal intuition, and >> this can only be done by combining a finite number of times the two >> operations 'to create a finite ordinal number' and 'to create the >> infinite ordinal number omega'; " (Bulletin of AMS, 1913) >> >> Please do not confuse the acceptance of omega as a completed totality >> with the acceptance of larger transfinite cardinals. > > Brouwer denied uncountable sets. That means he denied that Cantor's > "proof" is valid. Brouwer denied that Cantor's argument shows the existence of uncountable sets. He did not claim, as you do, that classical mathematics is inconsistent. However, as the quote shows, he was perfectly happy to accept the existence as a unit (collection) the ordinal omega. That is a different question entirely. > Further Brouwer accepted omega only when constructed from equal > elements like periodic decimals: So, you agree that he accepted the existence of omega as an intelligible mathematical unit. -- Alan Smaill
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| From | WM <wolfgang.mueckenheim@hs-augsburg.de> |
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| Date | 2018-05-14 07:42 -0700 |
| Message-ID | <0a036bed-b1e8-488c-bcad-ba633e6cb70c@googlegroups.com> |
| In reply to | #413071 |
Am Montag, 14. Mai 2018 14:30:06 UTC+2 schrieb Alan Smaill: > WM <wolfgang.mueckenheim@hs-augsburg.de> writes: > > Brouwer denied uncountable sets. That means he denied that Cantor's > > "proof" is valid. > > Brouwer denied that Cantor's argument shows the existence of > uncountable sets. He denied that real numbers can be defined by infinite digit sequences. Examples of (according to Brouwer) meaningless word play are the second number class and the higher power sets. > He did not claim, as you do, that classical > mathematics is inconsistent. The assumption that real numbers can be defined by infinite digit sequences is inconsistent. > > However, as the quote shows, he was perfectly happy to accept > the existence as a unit (collection) the ordinal omega. That is merely another name for potential infinity, since no higher infinities are possible. > > Further Brouwer accepted omega only when constructed from equal > > elements like periodic decimals: > > So, you agree that he accepted the existence of omega as > an intelligible mathematical unit. Infinity can be named by what you like: omega, aleph_0, infinity. REgards, WM
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| From | bursejan@gmail.com |
|---|---|
| Date | 2018-05-14 09:36 -0700 |
| Message-ID | <c78a92af-4743-4559-8415-0574bfca8496@googlegroups.com> |
| In reply to | #413078 |
Well by Cantors Theorem obviously higher infinities are possible, for example |omega| < |P(omega)|. This is something from ZFC. But Browser was up to something else, he tried to develop even logic, that is more restricted than FOL and would subsequently also yield different set theories. But even in an intuitionistic setting, when you have for example choice, you get classic logic again. This might not have been known to Brouwer: "Diaconescu's theorem the full axiom of choice is sufficient to derive the law of the excluded middle" https://en.wikipedia.org/wiki/Diaconescu's_theorem But even if in Brouwers realms no higher infinities would exist, what does this prove about ZFC? Nothing. In PA there are also no higher infinities. See also: 3.4 The continuum "In intuitionism, the continuum is both an extension and a restriction of its classical counterpart. In its full form, both notions are incomparable since the intuitionistic real numbers possess properties that the classical real numbers do not have. https://plato.stanford.edu/entries/intuitionism/#TheCon law of trichotomy ∀x(x<y∨x=y∨x>y) is not true on the intuitionistic continuum." https://plato.stanford.edu/entries/intuitionism/#TheCon So in my opinion Brouwer might be a better theory for computability, but I doubt that is the better theory for real spaces. Maybe both is needed side by side ZFC and Brouwer realm. Am Montag, 14. Mai 2018 16:42:28 UTC+2 schrieb WM: > Am Montag, 14. Mai 2018 14:30:06 UTC+2 schrieb Alan Smaill: > > WM <wolfgang.mueckenheim@hs-augsburg.de> writes: > > > > > Brouwer denied uncountable sets. That means he denied that Cantor's > > > "proof" is valid. > > > > Brouwer denied that Cantor's argument shows the existence of > > uncountable sets. > > He denied that real numbers can be defined by infinite digit sequences. > > Examples of (according to Brouwer) meaningless word play are the second number class and the higher power sets. > > > He did not claim, as you do, that classical > > mathematics is inconsistent. > > The assumption that real numbers can be defined by infinite digit sequences is inconsistent. > > > > However, as the quote shows, he was perfectly happy to accept > > the existence as a unit (collection) the ordinal omega. > > That is merely another name for potential infinity, since no higher infinities are possible. > > > > > Further Brouwer accepted omega only when constructed from equal > > > elements like periodic decimals: > > > > So, you agree that he accepted the existence of omega as > > an intelligible mathematical unit. > > Infinity can be named by what you like: omega, aleph_0, infinity. > > REgards, WM
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| From | bursejan@gmail.com |
|---|---|
| Date | 2018-05-14 09:42 -0700 |
| Message-ID | <9321600a-7788-4951-94a5-8f2b536225fc@googlegroups.com> |
| In reply to | #413083 |
BTW, you can do a lot of computability theory inside ZFC, like recursion theory and complexity theory. ZFC contains also non-analysis. Its not really necessary to revert to intuitionistic logic or alternative set theories for theories about computability. But it gives of course insights into computability attacking it from logic or set theory side... Am Montag, 14. Mai 2018 18:36:27 UTC+2 schrieb burs...@gmail.com: > Well by Cantors Theorem obviously higher infinities > are possible, for example |omega| < |P(omega)|. This > is something from ZFC. > > But Browser was up to something else, he tried to > develop even logic, that is more restricted than > FOL and would subsequently also yield different set > > theories. But even in an intuitionistic setting, > when you have for example choice, you get classic > logic again. This might not have > > been known to Brouwer: > > "Diaconescu's theorem > the full axiom of choice is sufficient to derive > the law of the excluded middle" > https://en.wikipedia.org/wiki/Diaconescu's_theorem > > But even if in Brouwers realms no higher infinities > would exist, what does this prove about ZFC? Nothing. > In PA there are also no higher infinities. > > See also: > > 3.4 The continuum > "In intuitionism, the continuum is both an extension > and a restriction of its classical counterpart. In > its full form, both notions are incomparable since > the intuitionistic real numbers possess properties > that the classical real numbers do not have. > https://plato.stanford.edu/entries/intuitionism/#TheCon > > law of trichotomy ∀x(x<y∨x=y∨x>y) is not true on the > intuitionistic continuum." > > https://plato.stanford.edu/entries/intuitionism/#TheCon > > So in my opinion Brouwer might be a better theory > for computability, but I doubt that is the better > theory for real spaces. Maybe both is needed side by > side ZFC and Brouwer realm. > > Am Montag, 14. Mai 2018 16:42:28 UTC+2 schrieb WM: > > Am Montag, 14. Mai 2018 14:30:06 UTC+2 schrieb Alan Smaill: > > > WM <wolfgang.mueckenheim@hs-augsburg.de> writes: > > > > > > > > Brouwer denied uncountable sets. That means he denied that Cantor's > > > > "proof" is valid. > > > > > > Brouwer denied that Cantor's argument shows the existence of > > > uncountable sets. > > > > He denied that real numbers can be defined by infinite digit sequences. > > > > Examples of (according to Brouwer) meaningless word play are the second number class and the higher power sets. > > > > > He did not claim, as you do, that classical > > > mathematics is inconsistent. > > > > The assumption that real numbers can be defined by infinite digit sequences is inconsistent. > > > > > > However, as the quote shows, he was perfectly happy to accept > > > the existence as a unit (collection) the ordinal omega. > > > > That is merely another name for potential infinity, since no higher infinities are possible. > > > > > > > > Further Brouwer accepted omega only when constructed from equal > > > > elements like periodic decimals: > > > > > > So, you agree that he accepted the existence of omega as > > > an intelligible mathematical unit. > > > > Infinity can be named by what you like: omega, aleph_0, infinity. > > > > REgards, WM
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