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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 | David Petry <davidlpetry@gmail.com> |
|---|---|
| Date | 2018-05-11 11:20 -0700 |
| Subject | The fundamental theorem of set theory is problematic |
| Message-ID | <92d7be26-4688-49f3-8875-b54aae5cd3ed@googlegroups.com> |
The essence of mathematics is proof. The idea behind proof is that if we believe we have proved some assertion, then we must also believe that everyone has no choice but to believe that the assertion is actually true. And consistent with the idea that the essence of mathematics is proof, is the idea that the domain of discourse for mathematics is the things that are provably well-defined. So now let's look at the fundamental theorem of set theory (Cantor's theorem). If we have a provably well-defined list of provably well-defined real numbers (so that every digit of every real number in the list can be computed), then the diagonal argument gives us a provably well-defined real number that is not in that list. So if we start with the premiss that the domain of discourse for mathematics is the things that are provably well-defined, then the diagonal argument most certainly does not prove to us that there must exist anything outside of that domain. At the same time, the domain of discourse cannot be larger than "countable", since proofs are objects in a language, and languages are countable. The conclusion being that there is no reason to think we are compelled to believe in the existence of uncountable sets, which contradicts what Cantor's theorem says, and furthermore, there is good reason to say that Cantor's theorem is false. Note that I'm not claiming that set theory is inconsistent as a game consisting of symbols and rules for manipulating strings of those symbols, as long as we recognize that the symbols have no meaning, and the strings of symbols have no connection to truth and reality. The argument I just presented for not accepting set theory as part of mathematics has been around right from the start of set theory. I believe that Julius Konig was the first to present that argument (or some variation of it). The fact that many first rate mathematicians do not believe that Cantor's formal proof proves that his theorem is actually true, should be seen as justification for believing that set theory is not part of mathematics: mathematics is all about things we have no choice but to believe are actually true. Mathematics has important connections to real life. Real life is not just a game. Set theory is nothing but a game. It is not mathematics.
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| From | Alan Mackenzie <acm@muc.de> |
|---|---|
| Date | 2018-05-11 19:33 +0000 |
| Message-ID | <pd4r6o$228$1@news.muc.de> |
| In reply to | #412942 |
David Petry <davidlpetry@gmail.com> wrote: > The essence of mathematics is proof. The idea behind proof is that if > we believe we have proved some assertion, then we must also believe > that everyone has no choice but to believe that the assertion is > actually true. And consistent with the idea that the essence of > mathematics is proof, is the idea that the domain of discourse for > mathematics is the things that are provably well-defined. This last is false. What maths discusses also includes things which haven't been proven, and indeed can't be proven. For exampe proof by contradiction discusses such things. > So now let's look at the fundamental theorem of set theory (Cantor's > theorem). > If we have a provably well-defined list of provably well-defined real > numbers (so that every digit of every real number in the list can be > computed), .... Is that your definition of "provably well-defined"? If so, I suggest it is not a useful definition. "Every real number between 0 and 1 which isn't rational" is provably well defined, by any reasonable interpretation of that phrase. > .... then the diagonal argument gives us a provably well-defined real > number that is not in that list. Yes, demonstrating that the list cannot have been complete in a well understood sense. > So if we start with the premiss that the domain of discourse for > mathematics is the things that are provably well-defined, then the > diagonal argument most certainly does not prove to us that there must > exist anything outside of that domain. That's not what the diagonal argument proves. This is that a supposed countable list of all real numbers cannot exist. Hence the real numbers are not countable. > At the same time, the domain of discourse cannot be larger than > "countable", since proofs are objects in a language, and languages are > countable. You're confusing two distinct meanings of "domain of discourse": (i) the domain of proofs; this is indeed countable; (ii) The entities these proofs deal with, which need not be countable. > .... The conclusion being that there is no reason to think we are > compelled to believe in the existence of uncountable sets, ..... If you accept the axiom of completion (which distinguishes the real numbers from countable subsets of them), you are compelled to believe in uncountable sets. > .... which contradicts what Cantor's theorem says, and furthermore, > there is good reason to say that Cantor's theorem is false. Cantor rigorously demonstrated the existence of non-countable sets, based only on the representation of numbers as arbitrary decimal expansions. If you wanted this theorem to be false, you'd have to somehow do away with decimal expansions of numbers. But why would you want this? > Note that I'm not claiming that set theory is inconsistent as a game > consisting of symbols and rules for manipulating strings of those > symbols, as long as we recognize that the symbols have no meaning, and > the strings of symbols have no connection to truth and reality. You could say the same about any mathematical argument. As for truth and reality, that's more the domain of philosophers than mathematicians. > The argument I just presented for not accepting set theory as part of > mathematics has been around right from the start of set theory. I > believe that Julius Konig was the first to present that argument (or > some variation of it). The fact that many first rate mathematicians do > not believe that Cantor's formal proof .... Fact? That assertion is screaming out for a citation. As far as I am aware, Cantor's arguments are now universally accepted by mathematicians. > .... proves that his theorem is actually true, should be seen as > justification for believing that set theory is not part of mathematics: > mathematics is all about things we have no choice but to believe are > actually true. Fermat's Last Theorem has been part of maths for several hundred years, and up until ~20 years ago one had the choice to believe it or not. I think you have mis-characterised maths. There are many mathematical topics which one has the choice to believe or not (areas of research). At another level you can chose to "believe" (or not) the axiom of choice, for example. You'll get somewhat different systems of maths depending on your choice. > Mathematics has important connections to real life. Real life is not > just a game. Set theory is nothing but a game. It is not mathematics. Set theory is part of mathematics. Not all of maths has important connections with real life (whatever that might be). -- Alan Mackenzie (Nuremberg, Germany).
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| From | David Petry <davidlpetry@gmail.com> |
|---|---|
| Date | 2018-05-11 13:16 -0700 |
| Message-ID | <be32e459-cdae-422f-9e2b-05abebd2e0a8@googlegroups.com> |
| In reply to | #412944 |
> As far as I am > aware, Cantor's arguments are now universally accepted by mathematicians. "Set theory is based on polite lies, things we agree on even though we know they're not true" --William Thurston for example.
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| From | "Ross A. Finlayson" <ross.finlayson@gmail.com> |
|---|---|
| Date | 2018-05-11 16:48 -0700 |
| Message-ID | <61633987-0bd1-4a20-b12a-0e1a20688da3@googlegroups.com> |
| In reply to | #412946 |
Seems as easy to fabricate a usual "lesser" model of the reals as via EF and line continuity (sweep principle of a continuum) for a certain countable model of the linear continuum, then tack the moderb mathematic's abstract algebra's countable character of the real analysis onto (and after) that. Then the reals of the complete ordered field are uncountable but there's also a countable model of the reals. (And for some that's rather the more fundamental.)
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| From | bursejan@gmail.com |
|---|---|
| Date | 2018-05-11 18:31 -0700 |
| Message-ID | <37464760-151e-486d-a8a6-a0cfda1df2ff@googlegroups.com> |
| In reply to | #412946 |
Maybe set theory should start with the young. "Students in college mathematics courses are unresponsive. They are afraid to speculate and afraid to reach into themselves for ideas. When one visits classrooms at different grade levels, one sees a dramatic decline in liveliness and spontaneity with age. One gets the impression that the natural interest and curiosity of young children in mathematics has been weeded out." --William Thurston Am Freitag, 11. Mai 2018 22:16:23 UTC+2 schrieb David Petry: > > As far as I am > > aware, Cantor's arguments are now universally accepted by mathematicians. > > "Set theory is based on polite lies, things we agree on even though we know they're not true" --William Thurston > > for example.
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| From | bursejan@gmail.com |
|---|---|
| Date | 2018-05-11 18:46 -0700 |
| Message-ID | <44cf53ec-c53c-4516-92eb-efc7acf973b8@googlegroups.com> |
| In reply to | #412953 |
On the other hand, what does "do" mean here: "G ̈odel’s incompleteness theorem implies that there can be no formal system that is consistent, yet powerful enough to serve as a basis for all of the mathematics that we do." --William Thurston Had William Thurston the idea that mathematics can be communicated above formal systems and transcending what formal system can "do", by their own "doing". How is this "done"? Am Samstag, 12. Mai 2018 03:31:34 UTC+2 schrieb burs...@gmail.com: > Maybe set theory should start with the young. > > "Students in college mathematics courses > are unresponsive. They are afraid to > speculate and afraid to reach into > themselves for ideas. When one visits > classrooms at different grade levels, > one sees a dramatic decline in liveliness > and spontaneity with age. One gets the > impression that the natural interest and > curiosity of young children in mathematics > has been weeded out." > --William Thurston > > Am Freitag, 11. Mai 2018 22:16:23 UTC+2 schrieb David Petry: > > > As far as I am > > > aware, Cantor's arguments are now universally accepted by mathematicians. > > > > "Set theory is based on polite lies, things we agree on even though we know they're not true" --William Thurston > > > > for example.
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| From | bursejan@gmail.com |
|---|---|
| Date | 2018-05-11 18:49 -0700 |
| Message-ID | <4fafb0ef-fde0-4632-b8c1-57312c353471@googlegroups.com> |
| In reply to | #412954 |
Either William Thurston was a complete moron or he had a secret that he didnt tell anybody. Am Samstag, 12. Mai 2018 03:46:35 UTC+2 schrieb burs...@gmail.com: > On the other hand, what does "do" mean here: > > "G ̈odel’s incompleteness theorem implies that > there can be no formal system that is consistent, > yet powerful enough to serve as a basis for all > of the mathematics that we do." > --William Thurston > > Had William Thurston the idea that mathematics > can be communicated above formal systems and > transcending what formal system can "do", by > their own "doing". How is this "done"? > > Am Samstag, 12. Mai 2018 03:31:34 UTC+2 schrieb burs...@gmail.com: > > Maybe set theory should start with the young. > > > > "Students in college mathematics courses > > are unresponsive. They are afraid to > > speculate and afraid to reach into > > themselves for ideas. When one visits > > classrooms at different grade levels, > > one sees a dramatic decline in liveliness > > and spontaneity with age. One gets the > > impression that the natural interest and > > curiosity of young children in mathematics > > has been weeded out." > > --William Thurston > > > > Am Freitag, 11. Mai 2018 22:16:23 UTC+2 schrieb David Petry: > > > > As far as I am > > > > aware, Cantor's arguments are now universally accepted by mathematicians. > > > > > > "Set theory is based on polite lies, things we agree on even though we know they're not true" --William Thurston > > > > > > for example.
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| From | WM <wolfgang.mueckenheim@hs-augsburg.de> |
|---|---|
| Date | 2018-05-12 00:59 -0700 |
| Message-ID | <f8a5c513-5083-458b-9f0a-e65ad70c308b@googlegroups.com> |
| In reply to | #412955 |
Am Samstag, 12. Mai 2018 03:49:20 UTC+2 schrieb burs...@gmail.com: > Either William Thurston was a complete moron or Burse. Regards, WM
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| From | bursejan@gmail.com |
|---|---|
| Date | 2018-05-12 02:20 -0700 |
| Message-ID | <fbe3e659-6f79-45f2-8bae-e51292cd15e8@googlegroups.com> |
| In reply to | #412961 |
Here we see how old, WM, cannot grip irrational numbers, maybe set theory should start with the young, when old, its too late. Yes, usual proof is that |R|>|N|, there is a paper by Cantor with comment by Zermelo, very famous paper, that shows that. Maybe the notion potential infinite makes an exception here, but this notion is not found in ZFC. So if R and N are sets, then there is no bijection f : N -> R. Am Samstag, 12. Mai 2018 09:59:43 UTC+2 schrieb WM: > Am Samstag, 12. Mai 2018 03:49:20 UTC+2 schrieb burs...@gmail.com: > > Either William Thurston was a complete moron > > or Burse. > > Regards, WM
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| From | David Petry <davidlpetry@gmail.com> |
|---|---|
| Date | 2018-05-12 12:43 -0700 |
| Message-ID | <f86ce98c-1d40-4d44-a37f-d101cc5c7b5f@googlegroups.com> |
| In reply to | #412955 |
> Either William Thurston was a complete moron > or he had a secret that he didnt tell anybody. Or maybe he tried hard to tell everybody the "secret", but people won't listen.
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| From | bursejan@gmail.com |
|---|---|
| Date | 2018-05-12 14:09 -0700 |
| Message-ID | <f1dc5f78-603c-47dd-90e2-77bdb2f12a44@googlegroups.com> |
| In reply to | #412996 |
You mean like Witgensteins understanding of Gödel? "Since the publication of Wittgenstein's Nachlass in 2000, a series of papers in philosophy have sought to evaluate whether the original criticism of Wittgenstein's remarks was justified. Floyd and Putnam (2000) argue that Wittgenstein had a more complete understanding of the incompleteness theorem than was previously assumed." https://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems#Wittgenstein Ha Ha, " ... more complete ... of the incompletness ..." sounds very funny. Maybe we can simply conclude William Thurston didn't understand the theorem, maybe not exactly in the same fashion like Wittgenstein: "It is clear from the passages you cite that Wittgenstein did not understand [the first incompleteness theorem] (or pretended not to understand it). He interpreted it as a kind of logical paradox, while in fact is just the opposite, namely a mathematical theorem within an absolutely uncontroversial part of mathematics (finitary number theory or combinatorics)." (Wang 1996:179) I don't know. I depends what William Thurston exactly meant by "do"? AP brain farto went berserk with a crazy post about God, IS and DOES a few hours ago. Ha Ha laughing my ass off. https://groups.google.com/forum/#!topic/sci.math/sUR7ftnGeFU Maybe his kitchen sink exploded because of the many conic sections that were oval, but not ellipse. Am Samstag, 12. Mai 2018 21:43:51 UTC+2 schrieb David Petry: > > Either William Thurston was a complete moron > > or he had a secret that he didnt tell anybody. > > Or maybe he tried hard to tell everybody the "secret", but people won't listen.
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| From | David Petry <davidlpetry@gmail.com> |
|---|---|
| Date | 2018-05-12 12:40 -0700 |
| Message-ID | <2e2bf3e4-92b6-45a9-9606-8d58a7e39c48@googlegroups.com> |
| In reply to | #412953 |
> Maybe set theory should start with the young. > > "Students in college mathematics courses > are unresponsive. They are afraid to > speculate and afraid to reach into > themselves for ideas. When one visits > classrooms at different grade levels, > one sees a dramatic decline in liveliness > and spontaneity with age. One gets the > impression that the natural interest and > curiosity of young children in mathematics > has been weeded out." > --William Thurston That's a very interesting quote from Thurston. Thank you for posting it. I'd like to encourage people to read that two or three times. "Students in college mathematics courses ... are afraid to ... reach into themselves for ideas." One of the ideas that I have been promoting over the years is that mathematics comes from introspection. It comes from seeking to understand how we reason about the real world. This should include seeking to understand how we distinguish truth from fiction, and how we distinguish reality from fantasy. And then mathematics could be defined as a formalization of that undertanding of ourselves. And then, for example, it would be seen that the idea of falsifiability, which scientists use to distinguish science from pseudoscience, can and should be formalized and integrated into the foundations of mathematics, where it could be used to distinguish between mathematics and pseudo-mathematics. And set theory would be seen at pseudo-mathematics. If we start teaching set theory to the very young, it will only make the problems that Thurston delineates much worse.
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| From | Alan Mackenzie <acm@muc.de> |
|---|---|
| Date | 2018-05-12 20:48 +0000 |
| Message-ID | <pd7jur$nh4$1@news.muc.de> |
| In reply to | #412995 |
David Petry <davidlpetry@gmail.com> wrote: >> Maybe set theory should start with the young. >> >> "Students in college mathematics courses >> are unresponsive. They are afraid to >> speculate and afraid to reach into >> themselves for ideas. When one visits >> classrooms at different grade levels, >> one sees a dramatic decline in liveliness >> and spontaneity with age. One gets the >> impression that the natural interest and >> curiosity of young children in mathematics >> has been weeded out." >> --William Thurston > That's a very interesting quote from Thurston. Thank you for posting > it. I'd like to encourage people to read that two or three times. > "Students in college mathematics courses ... are afraid to ... reach > into themselves for ideas." > One of the ideas that I have been promoting over the years is that > mathematics comes from introspection. It comes from seeking to > understand how we reason about the real world. This should include > seeking to understand how we distinguish truth from fiction, and how we > distinguish reality from fantasy. And then mathematics could be > defined as a formalization of that undertanding of ourselves. And > then, for example, it would be seen that the idea of falsifiability, > which scientists use to distinguish science from pseudoscience, can and > should be formalized and integrated into the foundations of > mathematics, where it could be used to distinguish between mathematics > and pseudo-mathematics. And set theory would be seen as > pseudo-mathematics. This last sentence is garbage. Dangerous garbage. It's like arguing to impressionable youngsters that the Earth is flat. Your opening post to this thread strongly suggests you do not understand Cantor's proof - that you do not understand proof by contradiction at all. If so, then anything you opine about maths is suspect. Set theory is established mathematics, just as a round Earth is established geography, evolution is established biology, and electromagnetism is established physics. To pretend otherwise is to act as a crank, and to spread ignorance. > 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. -- Alan Mackenzie (Nuremberg, Germany).
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| From | David Petry <davidlpetry@gmail.com> |
|---|---|
| Date | 2018-05-12 14:50 -0700 |
| Message-ID | <ac275d5b-5a14-47d9-9f0f-b029e83712b4@googlegroups.com> |
| In reply to | #412999 |
Alan Mackenzie wrote: > Set theory is established mathematics, just as a round Earth is > established geography, ... So now we see your true colors -- you're an antidisestablishmentarianist! You're saying that whatever has been "established", cannot possibly in a million years be disestablished! I've been waiting so long for a chance to use that word! All kidding aside, I am suggesting that reasoning about the foundations of mathematics requires non-monotonic reasoning. I'm suggesting that we start with the idea that mathematics is a framework for reasoning about the real world, and then as we gain a better understanding of what "real world" even means, and a better understanding of how we reason about the real world, we must be willing to clear out the "deadwood" from mathematics when it has no relevance for reasoning about the real world, even if that deadwood is believed to be "established".
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| From | Alan Mackenzie <acm@muc.de> |
|---|---|
| Date | 2018-05-13 01:00 +0000 |
| Message-ID | <pd82mv$656$1@news.muc.de> |
| In reply to | #413002 |
David Petry <davidlpetry@gmail.com> wrote: > Alan Mackenzie wrote: >> Set theory is established mathematics, just as a round Earth is >> established geography, ... > So now we see your true colors -- you're an > antidisestablishmentarianist! You're saying that whatever has been > "established", cannot possibly in a million years be disestablished! > I've been waiting so long for a chance to use that word! Not at all. Newtonian mechanics, a well established theory, was displaced by Einstein's relativity. Einstein understood Newtonian mechanics very well, and enhanced it to make it fit new observations. I am all in favour of the progress and spread of knowledge. What I'm not in favour of is the spreading of falsehoods by ignorant cranks. You are trying to smear set theory largely because you don't understand it. Let me repeat, because you ignored it in my last post: there is no sign in the post with which you opened this thread that you understand Cantor's proof, in particular, or proof by contradiction in general. This is so basic to mathematics, that lack of this understanding renders your opinions about maths inconsequential. > All kidding aside, I am suggesting that reasoning about the foundations > of mathematics requires non-monotonic reasoning. I'm suggesting that > we start with the idea that mathematics is a framework for reasoning > about the real world, and then as we gain a better understanding of > what "real world" even means, and a better understanding of how we > reason about the real world, we must be willing to clear out the > "deadwood" from mathematics when it has no relevance for reasoning > about the real world, even if that deadwood is believed to be > "established". I am suggesting you lack the background to make any such pronouncements. Aside from that, much of maths has little or no connection with the "real world" and that is just fine - take John Wiles's proof of Fermat's last theorem as an example - yet it is a triumph of human endeavour, fully comparable with the creation of great music or visual art. Anyway, "deadwood" does get cleared from maths, by people moving on from it to other things. How many people still study Euclidean geometry or quaternions, for example? But the study of modular forms, once old fashioned, got a kick from the said John Wiles's and colleagues' investigations into FLT. -- Alan Mackenzie (Nuremberg, Germany).
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| From | David Petry <davidlpetry@gmail.com> |
|---|---|
| Date | 2018-05-13 15:10 -0700 |
| Message-ID | <59359449-8767-4235-90ae-184881394e91@googlegroups.com> |
| In reply to | #413009 |
Alan Mackenzie wrote: > there is no > sign in the post with which you opened this thread that you understand > Cantor's proof, in particular, or proof by contradiction in general. Since you've been rather rude to me, I'm going to offer you some advice. Learn something about constructivism. Try to understand why the constructivists reject proof by contradiction. The Wikipedia is a good place to start.
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| From | Alan Mackenzie <acm@muc.de> |
|---|---|
| Date | 2018-05-14 17:33 +0000 |
| Message-ID | <pdch8h$t87$1@news.muc.de> |
| In reply to | #413041 |
David Petry <davidlpetry@gmail.com> wrote: > Alan Mackenzie wrote: >> there is no >> sign in the post with which you opened this thread that you understand >> Cantor's proof, in particular, or proof by contradiction in general. > Since you've been rather rude to me, ..... Toughen up, this is Usenet. Your opening post in this thread made outrageous suggestions (just, as an example, that a part of standard maths isn't maths at all), and what you call rudeness is what you intended to provoke. 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. This doesn't suggest understanding, or a desire for clarity. Please settle the matter you quoted me writing. Do you understand Cantor's proof? Do you understand proof by contradiction? I will accept your (straightforward) answers. > .... I'm going to offer you some advice. > Learn something about constructivism. Try to understand why the > constructivists reject proof by contradiction. The Wikipedia is a good > place to start. I've skimmed the article in the English wikipedia on Constructivism. I don't find it, as yet, an attractive philosophy. It may surprise you to learn that I don't find the theory of transfinite numbers attractive either, but at least I'm not about to try to demote rigorous mathematical results to the realm of "non-mathematics". It strikes me, as a novice to the topic, that constructive mathematics is going to have difficulty proving theorems of the form "There does not exist ...." or "There exist an infinite number of ....", but no doubt that would become clearer from more intense study. Also, it seems constructive mathematics is going to be less rich, and less full than standard mathematics. To give up proof by contradiction would be to leave easily proven theorems provable only with great difficulty, or possibly not at all. All for what? -- Alan Mackenzie (Nuremberg, Germany).
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| From | David Petry <davidlpetry@gmail.com> |
|---|---|
| Date | 2018-05-14 21:09 -0700 |
| Message-ID | <8ed48cac-6165-478d-83e9-c3df88d1e5bb@googlegroups.com> |
| In reply to | #413087 |
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. In particular, the reasoning used in science and technology would never lead to the conclusion that there exists a super-infinite universe lying beyond the potentially infinite universe that science and technology deals with.
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| From | Alan Mackenzie <acm@muc.de> |
|---|---|
| Date | 2018-05-15 19:19 +0000 |
| Message-ID | <pdfbs0$2oat$1@news.muc.de> |
| In reply to | #413101 |
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. > In particular, the reasoning used in science and technology would never > lead to the conclusion that there exists a super-infinite universe > lying beyond the potentially infinite universe that science and > technology deals with. That's a self-referential statement that appears more meaningful than it is. It says "science ... would never ... beyond the potentially infinite univers dealt with by science". In fact it says nothing. The Super-infinite universe is the subject of earnest scientific research: it goes by the name "Multiverse". Who knows, maybe the topology of this postulated multiverse is modelled by the familiar transfinite numbers somehow. -- Alan Mackenzie (Nuremberg, Germany).
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| From | "Ross A. Finlayson" <ross.finlayson@gmail.com> |
|---|---|
| Date | 2018-05-15 13:50 -0700 |
| Message-ID | <821cdd90-e7fe-4402-bb43-0a36d2c3eeaa@googlegroups.com> |
| In reply to | #413132 |
On Tuesday, May 15, 2018 at 12:19:34 PM UTC-7, 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. > > > In particular, the reasoning used in science and technology would never > > lead to the conclusion that there exists a super-infinite universe > > lying beyond the potentially infinite universe that science and > > technology deals with. > > That's a self-referential statement that appears more meaningful than it > is. It says "science ... would never ... beyond the potentially infinite > univers dealt with by science". In fact it says nothing. > > The Super-infinite universe is the subject of earnest scientific > research: it goes by the name "Multiverse". Who knows, maybe the > topology of this postulated multiverse is modelled by the familiar > transfinite numbers somehow. > > -- > Alan Mackenzie (Nuremberg, Germany). Many these days are putting down the Multiple Worlds Interpretation because they'd all be in one again and that's just the one (eg that the Pauli versus Schroedinger or rather Heisenberg find that the objects do have their position and momentum, that measurement is in effect as the measurement or observer effect, rather mechanically or as courtesy energy and configuration of experiment). That the universe contains itself or for example that the universe of physical objects (eg, particles, waves, fields, or voids) as mathematical objects sees that the universe is infinite and full of those has also that it's an example of "Cantor's paradox" or that it's its own powerset and contains itself. So, the "trans-finite" is in a sense only as familiar as the polynomial or power series, and, there's quite all the rest of the "infinite" that the trans-finite is only a tiny part. Let's keep in mind that physics as science today and theory has conservation and symmetry as utmost principle.
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