Groups | Search | Server Info | Keyboard shortcuts | Login | Register [http] [https] [nntp] [nntps]
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
Page 3 of 12 — ← Prev page 1 2 [3] 4 5 … 12 Next page →
| From | Alan Smaill <smaill@SPAMinf.ed.ac.uk> |
|---|---|
| Date | 2018-05-14 18:29 +0100 |
| Message-ID | <fwea7t2gn7z.fsf@foxtrot.inf.ed.ac.uk> |
| In reply to | #413083 |
bursejan@gmail.com writes: > 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 This result depends on the treatment of extensionality; it does not hold for example in Martin-Lof type theory. If anyone is interested: "100 years of Zermelo’s axiom of choice: what was the problem with it?" Per Martin-Löf https://pdfs.semanticscholar.org/68d6/790cdbfda26cf71311e13756d87f164838da.pdf Not sure abot access to that; it's a good summary of the developments and issues. -- Alan Smaill
[toc] | [prev] | [next] | [standalone]
| From | bursejan@gmail.com |
|---|---|
| Date | 2018-05-14 12:03 -0700 |
| Message-ID | <0b44a634-947d-44ba-ab36-080016b078e7@googlegroups.com> |
| In reply to | #413086 |
This one is also nice: "This stronger, extensional axiom of choice is not derivable in intensional type theory. In fact, the extensional axiom of choice implies the law of the excluded middle. The key difference between set theory and type theory is that in set theory one can form quotient sets for arbitrary equivalence relations. In intensional type theory, one cannot form quotient types." Russell O’Connor http://r6.ca/blog/20050604T143800Z.html Jesper Carlström http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.97.9535 No quotient types, no real numbers? At least the usual Cantor construction of real numbers might be complicated, and well I don't understand yet what is meant no quotient types... Am Montag, 14. Mai 2018 19:30:07 UTC+2 schrieb Alan Smaill: > bursejan@gmail.com writes: > > > 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 > > This result depends on the treatment of extensionality; > it does not hold for example in Martin-Lof type theory. > > If anyone is interested: > > "100 years of Zermelo’s axiom of choice: > what was the problem with it?" Per Martin-Löf > https://pdfs.semanticscholar.org/68d6/790cdbfda26cf71311e13756d87f164838da.pdf > > Not sure abot access to that; it's a good summary of the developments > and issues. > > -- > Alan Smaill
[toc] | [prev] | [next] | [standalone]
| From | bursejan@gmail.com |
|---|---|
| Date | 2018-05-14 12:06 -0700 |
| Message-ID | <9f1709c8-78aa-4d19-a1d6-fde9f87d1b3d@googlegroups.com> |
| In reply to | #413088 |
Levy, Basic Set Theory, has his own take on quotient types. He basically does a lot of set theory to only arrive at something of quotient types, just before he starts talking about real spaces. Its not heavily used by him, but he does it very meticulous, what if the equivalence relation is a class? Am Montag, 14. Mai 2018 21:03:29 UTC+2 schrieb burs...@gmail.com: > This one is also nice: > > "This stronger, extensional axiom of choice is > not derivable in intensional type theory. In > fact, the extensional axiom of choice implies > the law of the excluded middle. The key difference > between set theory and type theory is that in > set theory one can form quotient sets for > arbitrary equivalence relations. In intensional > type theory, one cannot form quotient types." > > Russell O’Connor > http://r6.ca/blog/20050604T143800Z.html > > Jesper Carlström > http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.97.9535 > > No quotient types, no real numbers? At least > the usual Cantor construction of real numbers > might be complicated, and well I don't understand > yet what is meant no quotient types... > > Am Montag, 14. Mai 2018 19:30:07 UTC+2 schrieb Alan Smaill: > > bursejan@gmail.com writes: > > > > > 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 > > > > This result depends on the treatment of extensionality; > > it does not hold for example in Martin-Lof type theory. > > > > If anyone is interested: > > > > "100 years of Zermelo’s axiom of choice: > > what was the problem with it?" Per Martin-Löf > > https://pdfs.semanticscholar.org/68d6/790cdbfda26cf71311e13756d87f164838da.pdf > > > > Not sure abot access to that; it's a good summary of the developments > > and issues. > > > > -- > > Alan Smaill
[toc] | [prev] | [next] | [standalone]
| From | bursejan@gmail.com |
|---|---|
| Date | 2018-05-14 12:10 -0700 |
| Message-ID | <2fe62513-9542-45c6-b728-9a0b49b56628@googlegroups.com> |
| In reply to | #413089 |
Dont remember, or maybe Levy did quotient types
before ordinal or cardinal numbers. Anyway, you
can ask wether this:
{ x | xRy }
or that is a set
{ y | yRx }
and depending, you can proceed with some theorems
or not you can. If its a set, you can use AC,
otherwise there is no AC for you.
Am Montag, 14. Mai 2018 21:06:37 UTC+2 schrieb burs...@gmail.com:
> Levy, Basic Set Theory, has his own take on
> quotient types. He basically does a lot of
> set theory to only arrive at something of
>
> quotient types, just before he starts talking
> about real spaces. Its not heavily used by him,
> but he does it very meticulous,
>
> what if the equivalence relation is a class?
>
> Am Montag, 14. Mai 2018 21:03:29 UTC+2 schrieb burs...@gmail.com:
> > This one is also nice:
> >
> > "This stronger, extensional axiom of choice is
> > not derivable in intensional type theory. In
> > fact, the extensional axiom of choice implies
> > the law of the excluded middle. The key difference
> > between set theory and type theory is that in
> > set theory one can form quotient sets for
> > arbitrary equivalence relations. In intensional
> > type theory, one cannot form quotient types."
> >
> > Russell O’Connor
> > http://r6.ca/blog/20050604T143800Z.html
> >
> > Jesper Carlström
> > http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.97.9535
> >
> > No quotient types, no real numbers? At least
> > the usual Cantor construction of real numbers
> > might be complicated, and well I don't understand
> > yet what is meant no quotient types...
> >
> > Am Montag, 14. Mai 2018 19:30:07 UTC+2 schrieb Alan Smaill:
> > > bursejan@gmail.com writes:
> > >
> > > > 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
> > >
> > > This result depends on the treatment of extensionality;
> > > it does not hold for example in Martin-Lof type theory.
> > >
> > > If anyone is interested:
> > >
> > > "100 years of Zermelo’s axiom of choice:
> > > what was the problem with it?" Per Martin-Löf
> > > https://pdfs.semanticscholar.org/68d6/790cdbfda26cf71311e13756d87f164838da.pdf
> > >
> > > Not sure abot access to that; it's a good summary of the developments
> > > and issues.
> > >
> > > --
> > > Alan Smaill
[toc] | [prev] | [next] | [standalone]
| From | "Ross A. Finlayson" <ross.finlayson@gmail.com> |
|---|---|
| Date | 2018-05-15 13:44 -0700 |
| Message-ID | <da49a1d8-8eab-4495-afc6-cd8239e30446@googlegroups.com> |
| In reply to | #413083 |
On Monday, May 14, 2018 at 9:36:27 AM UTC-7, j4n bur53 wrote: > 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 Maybe plainly there's quite all the "higher infinities" as simply via standard, regular, well-founded cardinal exponentiation, then those trail off to Kunen which is the result of an extended education and survey in the modern standard, regular, well-founded, where Kunen then says again "you ignored ORD is its own order type, contains itself, is extra-ordinary and non- well-founded as a set, and your projection of the finite combinatorics upon the regular, ordinary omega finds a neat incomplete closure", where clearly "the" foundation of mathematics has it all.
[toc] | [prev] | [next] | [standalone]
| From | Alan Smaill <smaill@SPAMinf.ed.ac.uk> |
|---|---|
| Date | 2018-05-14 18:16 +0100 |
| Message-ID | <fwek1s6gnss.fsf@foxtrot.inf.ed.ac.uk> |
| In reply to | #413078 |
WM <wolfgang.mueckenheim@hs-augsburg.de> writes: > 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. My take is the more accurate. > Examples of (according to Brouwer) meaningless word play are the > second number class and the higher power sets. Yes, of course. *But* you started out by claiming that the Original Sin was the acceptance of completed infinites, and claimed that Brouwer supported you on this. He clearly did not. >> 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. That is *your* claim, not Brouwer's. Or do you find Brouwer saying this? >> 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. No, read what was said. If you think that acceptance of any completed infinity entails acceptance of higher infinities, then you need to do more than simply repeat the claim. Try reading Feferman, for example. Or Hermann Weyl. Try thinking for yourself. >> > 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. "A basal mathematical intuition": this is not about the *name*, that you think so important and magical, but about the nature of mathematical entities themselves. > > REgards, WM -- Alan Smaill
[toc] | [prev] | [next] | [standalone]
| From | WM <wolfgang.mueckenheim@hs-augsburg.de> |
|---|---|
| Date | 2018-05-15 05:44 -0700 |
| Message-ID | <5e2e71e7-2ba3-4af1-a3be-e62af8440cb1@googlegroups.com> |
| In reply to | #413085 |
Am Montag, 14. Mai 2018 19:20:07 UTC+2 schrieb Alan Smaill: > WM <wolfgang.mueckenheim@hs-augsburg.de> writes: > > > 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. > > My take is the more accurate. "we can never imagine the arbitrary infinite binary fractions as finished", i.e., as defining something. > > > Examples of (according to Brouwer) meaningless word play are the > > second number class and the higher power sets. > > Yes, of course. > > *But* you started out by claiming that the Original Sin was the > acceptance of completed infinites, and claimed that Brouwer supported > you on this. He clearly did not. He used omega, but not as completed. > > >> 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. > > That is *your* claim, not Brouwer's. Or do you find Brouwer saying this? "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." i.e., defining something. > > >> 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. > > No, read what was said. I did. > > If you think that acceptance of any completed infinity entails > acceptance of higher infinities, I do not. But acceptance of any completed infinity entails already contradictions. > > > Infinity can be named by what you like: omega, aleph_0, infinity. > > "A basal mathematical intuition": this is not about the *name*, that you > think so important and magical, but about the nature of mathematical > entities themselves. Completed infinity is not only a contradictio in adjecto but also in mathematics: 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
[toc] | [prev] | [next] | [standalone]
| From | bursejan@gmail.com |
|---|---|
| Date | 2018-05-14 06:38 -0700 |
| Message-ID | <637a8586-ee12-41cf-9067-bd69a0e99b8b@googlegroups.com> |
| In reply to | #413069 |
What does he mean by "finished". If the infinite binary fractions
are infinite sequences binary representation b0.b1 b2 b3 ... where
bj in {0,1} for j>0, then Brower is right. This infinite sequence
is not a sequence which has an end element, so all elements are
equal. This is also the case in ZFC, when we model such as sequence
as f : omega -> A, and not as f : omega+1 -> A.
So Brower basically says he doesn't want to work with McDuck that
lives over eternity, there is only McDucks earnings and spending
f : omage -> A, but there is no Augsburg Crank institute
nonsense f : omega+1 -> A in connection with McDuck.
Am Montag, 14. Mai 2018 13:44:58 UTC+2 schrieb WM:
> 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
[toc] | [prev] | [next] | [standalone]
| From | bursejan@gmail.com |
|---|---|
| Date | 2018-05-12 17:58 -0700 |
| Message-ID | <862aa97d-7d1f-4d05-b3c6-ed67919476f4@googlegroups.com> |
| In reply to | #412995 |
Thanks for obsfuscating that I picked the quote. But the quote by William Thurston is even not needed. Those who teach themselves can confirm it, and those who do not teach temselves, might see it elsewhere, your own children or others people children. From lively to almost dead in a few years, thats the result of a terrifying school, where its all about marks. But what David Pretry writes is of course nonsense. There is no such thing as formalization being a leeway to pseudo mathematics. Such a claim I call pseudo philosophy of science. Formalization and falsifiability do not exclude each other. You are free to accept axioms or not. Thats completely orthogonal. So basically David Petry belongs to the same kind of cranks such as WM, AP, former John Gabriel, etc.. etc.. who have absolutely no clue how axiomatics works. Am Samstag, 12. Mai 2018 21:40:49 UTC+2 schrieb David Petry: > > 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.
[toc] | [prev] | [next] | [standalone]
| From | bursejan@gmail.com |
|---|---|
| Date | 2018-05-12 18:04 -0700 |
| Message-ID | <372d4a1e-f88c-4c73-a1f8-111f82ef4b30@googlegroups.com> |
| In reply to | #413008 |
Also such terms as "provably well-defined" do not exist. Thats just a nonsense invention. Do you want to proof well definedness? What should this mean. Thats John Gabriel like hogwash, who was ranting all the time about some real numbers that are not well defined. Ha Ha, laughinh my ass off. Am Sonntag, 13. Mai 2018 02:58:29 UTC+2 schrieb burs...@gmail.com: > Thanks for obsfuscating that I picked the quote. > But the quote by William Thurston is even not > needed. Those who teach themselves can confirm > it, and those who do not teach temselves, > > might see it elsewhere, your own children > or others people children. From lively to > almost dead in a few years, thats the result > of a terrifying school, where its all about marks. > > But what David Pretry writes is of course nonsense. > There is no such thing as formalization being a > leeway to pseudo mathematics. Such a claim I call > pseudo philosophy of science. Formalization and > > falsifiability do not exclude each other. You are > free to accept axioms or not. Thats completely orthogonal. > So basically David Petry belongs to the same kind > of cranks such as WM, AP, former John Gabriel, etc.. > > etc.. who have absolutely no clue how axiomatics works. > > Am Samstag, 12. Mai 2018 21:40:49 UTC+2 schrieb David Petry: > > > 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.
[toc] | [prev] | [next] | [standalone]
| From | bursejan@gmail.com |
|---|---|
| Date | 2018-05-12 18:19 -0700 |
| Message-ID | <fe3e307e-2c40-48ef-bdfa-6b2e688cd524@googlegroups.com> |
| In reply to | #413010 |
For example in logic there are WFF. Only when you have WFF you can even start talking formally about a domain. But this is not really a proof step, thats more kind of sanity to enter the formal world and to some extend domain independent, what the logical constants concerns. But of course with WFF you can express all kind of stuff, for example my unicorns are all red as an axiom. unicorn(x) => red(x) And then from the above theory, if jack is a unicorn we should be able to derive that jack is red. Right? Such things could be a lot of fun for young age, Like for example Einstein's riddle or the Zebra puzzle. It is often claimed that only 2% of the population can solve the puzzle. But I guess current demand would be that https://en.wikipedia.org/wiki/Zebra_Puzzle everybody can solve this puzzle, otherwise robots will do it for us. Ha Ha. Maybe the new requirement should be that everybody can program a robot to solve it. Or maybe the future is like that: - We have to teach our children how they can teach their robots - Which includes of course fluent use of all kind of stuff like ZFC, etc.. See also: Driverless cars are hitting the road, powered by artificial intelligence. Robots can climb stairs, open doors, win Jeopardy, analyze stocks, work in factories, find parking spaces, advise oncologists. In the past, automation was considered a threat to low-skilled labor. Now, many high-skilled functions, including interpreting medical images, doing legal research, and analyzing data, are within the skill sets of machines. How can higher education prepare students for their professional lives when professions themselves are disappearing? In Robot-Proof, Northeastern University president Joseph Aoun proposes a way to educate the next generation of college students to invent, to create, and to discover—to fill needs in society that even the most sophisticated artificial intelligence agent cannot. https://mitpress.mit.edu/books/robot-proof Am Sonntag, 13. Mai 2018 03:04:26 UTC+2 schrieb burs...@gmail.com: > Also such terms as "provably well-defined" do > not exist. Thats just a nonsense invention. > Do you want to proof well definedness? What > > should this mean. Thats John Gabriel like > hogwash, who was ranting all the time about > some real numbers that are not well defined. > > Ha Ha, laughinh my ass off. > > Am Sonntag, 13. Mai 2018 02:58:29 UTC+2 schrieb burs...@gmail.com: > > Thanks for obsfuscating that I picked the quote. > > But the quote by William Thurston is even not > > needed. Those who teach themselves can confirm > > it, and those who do not teach temselves, > > > > might see it elsewhere, your own children > > or others people children. From lively to > > almost dead in a few years, thats the result > > of a terrifying school, where its all about marks. > > > > But what David Pretry writes is of course nonsense. > > There is no such thing as formalization being a > > leeway to pseudo mathematics. Such a claim I call > > pseudo philosophy of science. Formalization and > > > > falsifiability do not exclude each other. You are > > free to accept axioms or not. Thats completely orthogonal. > > So basically David Petry belongs to the same kind > > of cranks such as WM, AP, former John Gabriel, etc.. > > > > etc.. who have absolutely no clue how axiomatics works. > > > > Am Samstag, 12. Mai 2018 21:40:49 UTC+2 schrieb David Petry: > > > > 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.
[toc] | [prev] | [next] | [standalone]
| From | FredJeffries <fredjeffries@gmail.com> |
|---|---|
| Date | 2018-05-15 09:14 -0700 |
| Message-ID | <932f8518-d1d2-47e0-9b26-ece1b052a0a6@googlegroups.com> |
| In reply to | #412946 |
On Friday, May 11, 2018 at 1:16:23 PM UTC-7, David Petry wrote: > > 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. For anyone interested, this line is from a Scientific American article by science popularizer John Horgan (Horgan is quoting Thurston) "The Death of Proof" from October 1993. Thurston's reaction to the article was published in the letters to the editor of the January, 1994 edition (where he makes no comment on that particular quote). Thurston published his thoughts in "On Proof and Progress in Mathematics" in the Bulletin of the American Mathematics Society in April, 1994 (see quote below) https://arxiv.org/pdf/math/9404236v1.pdf Horgan reminisced about his article in August, 2012 on the occasion of Thurston's death: https://blogs.scientificamerican.com/cross-check/how-william-thurston-rip-helped-bring-about-the-death-of-proof/ The following may shed light on the point Thurston was trying to make to Horgan (which, by the way, was NOT the main point (or even an important point) of either article): <quote> On the most fundamental level, the foundations of mathematics are much shakier than the mathematics that we do. Most mathematicians adhere to foundational principles that are known to be polite fictions. For example, it is a theorem that there does not exist any way to ever actually construct or even define a well-ordering of the real numbers. There is considerable evidence (but no proof) that we can get away with these polite fictions without being caught out, but that doesn’t make them right. Set theorists construct many alternate and mutually contradictory "mathematical universes" such that if one is consistent, the others are too. This leaves very little confidence that one or the other is the right choice or the natural choice. Godel'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. </quote> --William Thurston in "On Proof and Progress in Mathematics"
[toc] | [prev] | [next] | [standalone]
| From | David Petry <davidlpetry@gmail.com> |
|---|---|
| Date | 2018-05-15 10:07 -0700 |
| Message-ID | <a4cb6007-4a79-4c08-8a90-6055ee44ac9e@googlegroups.com> |
| In reply to | #413117 |
Fred Jeffries gave us this quote from Thurston: "Set theorists construct many alternate and mutually contradictory "mathematical universes" such that if one is consistent, the others are too. This leaves very little confidence that one or the other is the right choice or the natural choice" Something I have been advocating is that we should seek a "minimal" foundation for mathematics. That is, it should include everything that we absolutely have to believe is true, and nothing more. That would have the effect that none of the "alternate and mutually contradictory 'mathematical universes'" that the set theorists construct should be seen as the foundation of mathematics. "everything should be as simple as possible, but no simpler" -- often attributed to A. Einstein
[toc] | [prev] | [next] | [standalone]
| From | FredJeffries <fredjeffries@gmail.com> |
|---|---|
| Date | 2018-05-16 09:26 -0700 |
| Message-ID | <b78b1de6-4487-4f0a-ae26-e56b1f99c20d@googlegroups.com> |
| In reply to | #413120 |
On Tuesday, May 15, 2018 at 10:07:59 AM UTC-7, David Petry wrote: > Something I have been advocating is that we should seek a "minimal" foundation for mathematics. That is, it should include everything that we absolutely have to believe is true, and nothing more. More accurately, it should include everything that David Petry believes to be true, whether he (absolutely) has to or not (and whatever he takes "believe" to mean).
[toc] | [prev] | [next] | [standalone]
| From | David Petry <davidlpetry@gmail.com> |
|---|---|
| Date | 2018-05-16 11:18 -0700 |
| Message-ID | <cd21abf3-8d1e-4fd3-be1a-2f0c831ef02f@googlegroups.com> |
| In reply to | #413182 |
FredJeffries wrote: > On Tuesday, May 15, 2018 at 10:07:59 AM UTC-7, David Petry wrote: >> Something I have been advocating is that we should seek a "minimal" foundation for mathematics. That is, it should include everything that we absolutely have to believe is true, and nothing more. > More accurately, it should include everything that David Petry believes to be true, whether he (absolutely) has to or not (and whatever he takes "believe" to mean). No, that's not more accurate. I was a little bit vague in an attempt to get people to think about the question of what we absolutely have to believe to be true and why we have to believe those things. The answer I've been suggesting is that the things that are essential for reasoning about nature and about rational thinking, are the things we absolutely have to believe to be true. And as the Hilbert points out in a quote we were discussing in another thread, by that standard, infinity has no role to play in the things we absolutely have to believe to be true.
[toc] | [prev] | [next] | [standalone]
| From | FredJeffries <fredjeffries@gmail.com> |
|---|---|
| Date | 2018-05-16 12:26 -0700 |
| Message-ID | <85476fdd-fed7-4f02-8bfe-9fe6fb0448ae@googlegroups.com> |
| In reply to | #413185 |
On Wednesday, May 16, 2018 at 11:18:42 AM UTC-7, David Petry wrote: > FredJeffries wrote: > > > On Tuesday, May 15, 2018 at 10:07:59 AM UTC-7, David Petry wrote: > > >> Something I have been advocating is that we should seek a "minimal" foundation for mathematics. That is, it should include everything that we absolutely have to believe is true, and nothing more. > > > More accurately, it should include everything that David Petry believes to be true, whether he (absolutely) has to or not (and whatever he takes "believe" to mean). > > > No, that's not more accurate. > > I was a little bit vague in an attempt to get people to think about the question of what we absolutely have to believe to be true and why we have to believe those things. > > The answer I've been suggesting is that the things that are essential for reasoning about nature and about rational thinking, are the things we absolutely have to believe to be true. And as the Hilbert points out in a quote we were discussing in another thread, by that standard, infinity has no role to play in the things we absolutely have to believe to be true. But Cantor showed that transfinite ordinals ARE necessary when reasoning convergence of trigonometric series. And Turing and others have shown that transfinite ordinals are useful, if not absolutely necessary, in reasoning about debugging, and about strengths of proofs, and about formal axiomatic systems as well as other areas.
[toc] | [prev] | [next] | [standalone]
| From | WM <wolfgang.mueckenheim@hs-augsburg.de> |
|---|---|
| Date | 2018-05-16 14:06 -0700 |
| Message-ID | <a857d94c-741c-4d6c-a681-46207665385e@googlegroups.com> |
| In reply to | #413196 |
Am Mittwoch, 16. Mai 2018 21:26:32 UTC+2 schrieb FredJeffries: > But Cantor showed that transfinite ordinals ARE necessary when reasoning convergence of trigonometric series. No. > > And Turing and others have shown that transfinite ordinals are useful, only in order to confuse newbies. > if not absolutely necessary, in reasoning about debugging, and about strengths of proofs, and about formal axiomatic systems as well as other areas. Sometimes this heap of junk is even extended by the claim that transfinite induction is useful for programmers: "Transfinite induction covers all possible ways in which one could show that a program terminates," [Andrej Bauer] I'd like to see a programmer who checks his program by transfinite induction! Regards, WM
[toc] | [prev] | [next] | [standalone]
| From | bursejan@gmail.com |
|---|---|
| Date | 2018-05-16 15:53 -0700 |
| Message-ID | <41d21d3d-a3b5-4ddd-ae7f-b17b17b06c75@googlegroups.com> |
| In reply to | #413200 |
Well transfinite induction, induction that is not based on omega, but on another ordinal but still countable ordinal is very common in program verification. See also: Proof Rules for Fair Termination It was observed that transfinite induction on ordinals higher than omega was necessary in general. Theory of Computation von Dexter C. Kozen Am Mittwoch, 16. Mai 2018 23:07:02 UTC+2 schrieb WM: > Am Mittwoch, 16. Mai 2018 21:26:32 UTC+2 schrieb FredJeffries: > > > But Cantor showed that transfinite ordinals ARE necessary when reasoning convergence of trigonometric series. > > No. > > > > And Turing and others have shown that transfinite ordinals are useful, > > only in order to confuse newbies. > > > if not absolutely necessary, in reasoning about debugging, and about strengths of proofs, and about formal axiomatic systems as well as other areas. > > Sometimes this heap of junk is even extended by the claim that transfinite induction is useful for programmers: > > "Transfinite induction covers all possible ways in which one could show that a program terminates," [Andrej Bauer] > > I'd like to see a programmer who checks his program by transfinite induction! > > Regards, WM
[toc] | [prev] | [next] | [standalone]
| From | Jim Burns <james.g.burns@att.net> |
|---|---|
| Date | 2018-05-16 19:07 -0400 |
| Message-ID | <a06248ae-50ac-cc20-0c74-dc682d8c9da8@att.net> |
| In reply to | #413201 |
On 5/16/2018 6:53 PM, bursejan@gmail.com wrote: > Well transfinite induction, induction that is > not based on omega, but on another ordinal but > still countable ordinal is very common in > program verification. I don't understand your qualification as _countable_ of the ordinals for which transfinite induction can be used. Countability is irrelevant. _Transfinite_ induction depends upon ordinals being well-ordered. If there is a _non-empty_ collection of ordinals, then there is a first ordinal in that collection. (well-order) If there is no _first_ counter-example to some predicate P(x) (as the condition for transfinite induction would imply), then there is no counter-example to P(x) at all. (transfinite induction) > See also: > Proof Rules for Fair Termination > It was observed that transfinite induction > on ordinals higher than omega was necessary > in general. > Theory of Computation > von Dexter C. Kozen
[toc] | [prev] | [next] | [standalone]
| From | bursejan@gmail.com |
|---|---|
| Date | 2018-05-17 01:17 -0700 |
| Message-ID | <e156598d-c018-4af0-96bc-572df452cf31@googlegroups.com> |
| In reply to | #413202 |
"countable ordinal" is a standard term. A countable ordinal, is an ordinal that has countable cardinality. There are also uncountable ordinals. Here is an example of an uncountable ordinal: "In mathematics, the first uncountable ordinal, traditionally denoted by ω1 or sometimes by Ω, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals." https://en.wikipedia.org/wiki/First_uncountable_ordinal I am not sure whether also uncountable ordinals appear in program verification. Maybe they also appear. In this book, there is a ordinal example with fairness and some binary tree, whether an uncountable ordinal involved, have to check: Proof Rules for Fair Termination It was observed that transfinite induction on ordinals higher than omega was necessary in general. Theory of Computation von Dexter C. Kozen For example the hydra game, see also: http://math.andrej.com/2008/02/02/the-hydra-game/ Uses only countable ordinals. And then even ε0 is still countable: "The original epsilon numbers were introduced by Georg Cantor in the context of ordinal arithmetic; ... The ordinal ε0 is still countable, as is any epsilon number whose index is countable (there exist uncountable ordinals, and uncountable epsilon numbers whose index is an uncountable ordinal)." https://en.wikipedia.org/wiki/Epsilon_numbers_%28mathematics%29 Am Donnerstag, 17. Mai 2018 01:07:58 UTC+2 schrieb Jim Burns: > On 5/16/2018 6:53 PM, bursejan@gmail.com wrote: > > > Well transfinite induction, induction that is > > not based on omega, but on another ordinal but > > still countable ordinal is very common in > > program verification. > > I don't understand your qualification as _countable_ of the ordinals > for which transfinite induction can be used. > > Countability is irrelevant. _Transfinite_ induction depends > upon ordinals being well-ordered. > > If there is a _non-empty_ collection of ordinals, then there is a > first ordinal in that collection. (well-order) > > If there is no _first_ counter-example to some predicate P(x) > (as the condition for transfinite induction would imply), > then there is no counter-example to P(x) at all. > (transfinite induction) > > > See also: > > Proof Rules for Fair Termination > > It was observed that transfinite induction > > on ordinals higher than omega was necessary > > in general. > > Theory of Computation > > von Dexter C. Kozen
[toc] | [prev] | [next] | [standalone]
Page 3 of 12 — ← Prev page 1 2 [3] 4 5 … 12 Next page →
Back to top | Article view | sci.math
csiph-web