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Groups > sci.logic > #210293 > unrolled thread
| Started by | Ganzhinterseher <claus.vonnesser@gmail.com> |
|---|---|
| First post | 2019-09-08 09:49 -0700 |
| Last post | 2019-09-11 14:00 -0700 |
| Articles | 20 on this page of 260 — 11 participants |
Back to article view | Back to sci.logic
Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-08 09:49 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-08 11:23 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-09 13:17 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-10 20:31 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-11 13:40 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-09-11 23:12 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-12 06:45 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Fred <Fred@nowhere.invalid> - 2019-09-08 21:14 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-09 12:51 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-09 12:59 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-09 13:36 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-09 19:56 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-10 08:33 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Python <python@invalid> - 2019-09-10 17:54 +0200
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-10 13:31 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Python <python@invalid> - 2019-09-11 15:45 +0200
Re: Can the empty set be the limit of a sequence of non-empty sets? jvr <jrennenkampff@googlemail.com> - 2019-09-11 07:08 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-11 13:38 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? jvr <jrennenkampff@googlemail.com> - 2019-09-11 14:48 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-12 07:06 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? jvr <jrennenkampff@googlemail.com> - 2019-09-12 10:31 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Python <python@invalid> - 2019-09-12 12:00 +0200
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-11 13:29 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-10 08:59 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-10 13:39 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-10 13:56 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-10 13:44 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-10 13:57 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-11 13:16 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-11 13:52 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-11 14:01 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-11 14:35 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-12 06:54 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-12 07:46 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-12 08:39 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-12 08:54 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-12 13:19 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-09-12 13:21 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-12 14:27 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-13 04:13 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-14 12:13 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-15 03:00 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? jvr <jrennenkampff@googlemail.com> - 2019-09-15 03:38 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-15 05:42 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? jvr <jrennenkampff@googlemail.com> - 2019-09-15 06:02 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-15 13:31 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-15 07:54 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-15 07:27 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? jvr <jrennenkampff@googlemail.com> - 2019-09-15 09:37 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-15 07:26 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-16 10:33 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-15 07:52 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Fred <Fred@nowhere.invalid> - 2019-09-15 16:13 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Fred <Fred@nowhere.invalid> - 2019-09-15 21:29 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-15 16:55 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Fred <Fred@nowhere.invalid> - 2019-09-16 12:59 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Fred <Fred@nowhere.invalid> - 2019-09-16 13:12 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Fred <Fred@nowhere.invalid> - 2019-09-16 15:34 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-16 23:49 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Fred <Fred@nowhere.invalid> - 2019-09-17 08:53 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-17 14:04 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-17 14:55 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-17 16:17 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-17 21:29 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-17 21:39 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Fred <Fred@nowhere.invalid> - 2019-09-18 18:35 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Fred <Fred@nowhere.invalid> - 2019-09-18 18:32 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-18 03:46 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-17 21:39 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-18 12:46 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Fred <Fred@nowhere.invalid> - 2019-09-18 17:52 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Python <python@invalid> - 2019-09-18 18:58 +0200
Re: Can the empty set be the limit of a sequence of non-empty sets? Fred <Fred@nowhere.invalid> - 2019-09-18 18:47 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-17 14:53 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-18 03:37 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-17 21:34 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-18 12:37 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-18 21:01 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-19 22:11 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Fred <Fred@nowhere.invalid> - 2019-09-19 22:21 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? jvr <jrennenkampff@googlemail.com> - 2019-09-20 00:51 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Fred <Fred@nowhere.invalid> - 2019-09-18 18:28 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2019-09-16 15:38 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Fred <Fred@nowhere.invalid> - 2019-09-16 15:53 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? jvr <jrennenkampff@googlemail.com> - 2019-09-16 08:43 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Python <python@invalid> - 2019-09-16 17:51 +0200
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-16 13:29 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-16 16:02 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-17 13:23 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-15 17:04 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-16 10:44 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-16 13:09 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-17 09:28 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Python <python@invalid> - 2019-09-17 18:38 +0200
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-17 13:28 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-17 13:43 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-17 14:33 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-18 09:29 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-18 10:45 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-18 14:13 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-18 15:40 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-19 04:51 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-19 14:23 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-20 08:22 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-20 09:55 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-21 12:13 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-21 12:33 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-21 13:47 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-21 14:17 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-23 12:02 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-09-23 12:26 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-24 06:14 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-24 07:54 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-23 12:46 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-23 14:23 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-23 15:25 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Python <python@python.invalid> - 2019-09-24 00:37 +0200
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-24 07:12 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-24 06:17 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Python <python@python.invalid> - 2019-09-24 15:26 +0200
Re: Can the empty set be the limit of a sequence of non-empty sets? jvr <jrennenkampff@googlemail.com> - 2019-09-24 07:03 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-24 07:22 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? jvr <jrennenkampff@googlemail.com> - 2019-09-24 07:39 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-24 08:33 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-24 08:49 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? WM <mueckenh@rz.fh-augsburg.de> - 2019-09-24 09:05 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-24 08:09 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? WM <mueckenh@rz.fh-augsburg.de> - 2019-09-24 09:06 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-24 09:18 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-24 13:20 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-24 13:43 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-25 09:41 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-25 09:49 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-27 13:27 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-27 13:49 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-28 12:27 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-28 12:36 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-28 13:08 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-28 13:24 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-28 14:00 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-28 14:39 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-29 06:11 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-29 08:00 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-29 11:05 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-29 12:02 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-29 13:34 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? jvr <jrennenkampff@googlemail.com> - 2019-09-29 13:45 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-29 13:47 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-30 09:52 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-30 10:13 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-30 11:30 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-30 11:57 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-30 23:44 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-10-01 01:19 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-10-01 07:39 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-10-01 11:14 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-10-01 12:35 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-10-01 13:23 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-10-02 02:23 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-10-02 02:29 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-10-02 02:58 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-10-02 03:24 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-10-02 08:45 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-10-02 09:02 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-10-03 08:51 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-10-03 10:52 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-10-03 10:56 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? jvr <jrennenkampff@googlemail.com> - 2019-10-02 10:27 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-10-02 10:31 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-10-03 08:46 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-10-03 10:13 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-10-04 13:59 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? jvr <jrennenkampff@googlemail.com> - 2019-09-30 11:10 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-09-29 14:30 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-24 08:00 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-23 14:19 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-24 07:09 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? jvr <jrennenkampff@googlemail.com> - 2019-09-24 06:51 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-24 07:15 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? jvr <jrennenkampff@googlemail.com> - 2019-09-21 14:22 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Python <python@python.invalid> - 2019-09-22 03:07 +0200
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-25 20:14 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-27 13:27 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-18 20:51 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-19 04:50 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-13 12:44 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-13 14:28 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-13 12:52 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-13 14:28 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-12 15:36 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-13 04:38 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-14 12:17 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-15 05:53 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-09-15 06:24 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-15 13:23 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-09-15 13:29 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-09-15 13:30 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-09 13:08 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-09 13:44 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-14 18:31 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? jvr <jrennenkampff@googlemail.com> - 2019-09-15 02:52 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-15 05:29 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-15 16:49 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-15 13:30 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-09-15 13:33 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-16 10:40 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-09-16 10:46 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-09-16 10:48 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-16 13:06 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-17 09:21 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-17 13:23 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-17 13:47 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-15 17:09 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-16 10:47 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-16 13:10 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-17 09:23 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-17 13:25 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-17 13:34 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-17 14:31 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-18 09:26 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-16 02:39 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-16 10:51 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-09-16 12:08 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-17 01:34 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-16 13:27 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-17 09:39 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-15 02:55 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-08 22:38 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-09 13:09 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-09 13:21 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-10 03:27 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-10 08:14 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-11 02:25 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-11 13:16 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-12 02:15 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-12 07:23 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-12 17:03 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-12 13:27 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-13 00:40 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-13 04:49 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-09-13 06:00 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-13 06:24 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-09-13 06:37 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-09-13 06:39 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-13 06:45 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-09-13 09:40 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-13 14:28 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-09-13 14:51 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-14 05:14 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-14 02:15 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-14 05:49 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-14 15:29 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-14 08:38 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-15 02:11 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-11 15:11 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-12 02:23 +0100
Re: Can the empty set be the limit of a sequence of non-empty sets? jvr <jrennenkampff@googlemail.com> - 2019-09-12 02:08 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? George Greene <greeneg@email.unc.edu> - 2019-09-11 15:10 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Ganzhinterseher <claus.vonnesser@gmail.com> - 2019-09-12 07:29 -0700
Re: Can the empty set be the limit of a sequence of non-empty sets? Transfinite Numbers <bursejan@gmail.com> - 2019-09-11 14:00 -0700
Page 12 of 13 — ← Prev page 1 … 10 11 [12] 13 Next page →
| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2019-09-16 02:39 +0100 |
| Message-ID | <87d0g1c951.fsf@bsb.me.uk> |
| In reply to | #210774 |
Ganzhinterseher <claus.vonnesser@gmail.com> writes: (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des Unendlichen" at Hochschule Augsburg.) > Am Sonntag, 15. September 2019 17:50:01 UTC+2 schrieb Ben Bacarisse: >> Ganzhinterseher <claus.vonnesser@gmail.com> writes: > >> > ... the rationals will never be enumerated. >> >> In mathematics, a set either is or is not countable. It's not a >> process, and the word "never" is inappropriate. > > Counting is a procedure. The sequence of natnumbers can be interrupted > and checked at every natnumber. Denying this shows that you are trying > to cheat. Shame on you. Proofs establish facts about every element of a set (for example that it is in the image of some function) in a finite number of steps. Proving things in mathematics is not cheating. It's your pretending that a never ending process is required that is dishonest. >> The rationals are countable because there are bijections between Q and >> N, examples of which have been posted numerous times (though, for >> convenience, Q+ is usually used). > > They are only valid for finite initial segments because every > definable number belongs to a finite initial segment. All that is required is the fact of a bijection. Your waffle is of no consequence. It's unfortunate for you that B(n) = S^n(1) (see below) is provably bijective even in WMaths! >> For example, Q+ can be obviously enumerated because >> >> S(a/b) = b / (qb + b - r) (*) > > Every number that can be introduced belongs to a finite initial > segment. Never an actual infinity is reached, None the less, bijective functions between Q+ and N provably exist. The astute reader will notice that you try to avoid denying obvious facts. So you don't say "the B you gave is not bijective" or "the image if N under B is not Q+", you just add some waffle that makes it sound like you don't agree, but which don't impinge on the argument at all. -- Ben.
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| From | Ganzhinterseher <claus.vonnesser@gmail.com> |
|---|---|
| Date | 2019-09-16 10:51 -0700 |
| Message-ID | <8b00b31c-54fb-4359-8066-889e04a9185b@googlegroups.com> |
| In reply to | #210781 |
Am Montag, 16. September 2019 03:39:09 UTC+2 schrieb Ben Bacarisse: > Ganzhinterseher <claus.vonnesser@gmail.com> writes: > > Every number that can be introduced belongs to a finite initial > > segment. Never an actual infinity is reached, > > None the less, bijective functions between Q+ and N provably exist. They exist up to every finite initial segment. That is never an infinite set. > The > astute reader will notice that you try to avoid denying obvious facts. Your claim is obviously wrong. Every definable pair of a bijection with |N belongs to a tiny initial segment of |N and is followed by almost all elements of |N. Try to find a counter-example. Regards, WM
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| From | Transfinite Numbers <bursejan@gmail.com> |
|---|---|
| Date | 2019-09-16 12:08 -0700 |
| Message-ID | <80104cb0-086f-4b56-a640-5cdd760beee5@googlegroups.com> |
| In reply to | #210827 |
Well thats a property of countable infinite, that we have the following: |E(k)| = |omega| Whats your prolololoblem? Something to do with Augsburg Crank institute? Maybe some Volkswagen Omlette? On Monday, September 16, 2019 at 7:51:52 PM UTC+2, Ganzhinterseher wrote: > Am Montag, 16. September 2019 03:39:09 UTC+2 schrieb Ben Bacarisse: > > Ganzhinterseher <claus.vonnesser@gmail.com> writes: > > > > > Every number that can be introduced belongs to a finite initial > > > segment. Never an actual infinity is reached, > > > > None the less, bijective functions between Q+ and N provably exist. > > They exist up to every finite initial segment. That is never an infinite set. > > > The > > astute reader will notice that you try to avoid denying obvious facts. > > Your claim is obviously wrong. Every definable pair of a bijection with |N belongs to a tiny initial segment of |N and is followed by almost all elements of |N. Try to find a counter-example. > > Regards, WM
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2019-09-17 01:34 +0100 |
| Message-ID | <87muf3bw11.fsf@bsb.me.uk> |
| In reply to | #210827 |
Ganzhinterseher <claus.vonnesser@gmail.com> writes: (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des Unendlichen" at Hochschule Augsburg.) > Am Montag, 16. September 2019 03:39:09 UTC+2 schrieb Ben Bacarisse: >> Ganzhinterseher <claus.vonnesser@gmail.com> writes: > >> > Every number that can be introduced belongs to a finite initial >> > segment. Never an actual infinity is reached, >> >> None the less, bijective functions between Q+ and N provably exist. > > They exist up to every finite initial segment. That is never an > infinite set. > >> The >> astute reader will notice that you try to avoid denying obvious facts. > > Your claim is obviously wrong. Ah! Getting somewhere. So B(n) = S^n(1) where S(a/b) = b / (qb + b - r) (a/b is a rational written in it's lowest terms, and q and r are the quotient and remainder of a divided by b.) is (according to you) not a bijection between N and Q+. To be a bijection from N to Q+ you require it to be (1) a function (2) surjektiv (oder Abbildung auf Q+), wenn jedes y ∈ Q+ ein Bild ist (3) injektiv (oder eindeutig), wenn aus B(x1) = B(x2) folgt x1 = x2 Which of (1), (2) or (3) is not provable for B? (Technically, we only really care about (1) and (2) since a surjective function will do for countability.) -- Ben.
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| From | George Greene <greeneg@email.unc.edu> |
|---|---|
| Date | 2019-09-16 13:27 -0700 |
| Message-ID | <07c117b4-df12-4285-bf50-c6baa44f0a10@googlegroups.com> |
| In reply to | #210781 |
On Sunday, September 15, 2019 at 9:39:09 PM UTC-4, Ben Bacarisse wrote: > Proofs establish facts about every element of a set (for example that it > is in the image of some function) in a finite number of steps. Of course, but they can do this in two DIFFERENT WAYS depending on whether the set is finite or infinite. If the set is finite then you will still have a finite proof even if you prove a separate theorem for every element of the set -- their conjunction is also finite and therefore defined, AND IS the theorem ABOUT THE WHOLE set (that its EVERY element has some property). WHEN THE SET IS INFINITE, THE ABOVE will NECESSARILY NOT work. You HAVE to do something different. In practice, believe it or not, proving that Every element of an Infinite set has some property IS EASIER THAN proving n instances for elements of a size-n set, because in the infinite case, YOU ONLY HAVE TO PROVE THAT *ONE* element has the property -- but it has to be A GENERIC element -- you have to conduct the proof while designating the element BY A NAME that is NOT MENTINOED IN THE AXIOMS -- you therefore have to conduct the proof WITH NO INFORMATION ABOUT the element that you are proving has the property -- EXCEPT that it is an element of the set. This is OBVIOUSLY SOUND because you could simply repeat the proof WITH that variable-name INSTANTIATED TO ANY element of the set and the proof would look typogrpahically identical to(and be correct for all the same reasons as) the proof you just presented with the variable-name. So WM's contention that the proof for the infinite case is somehow "harder" is just laughable. His claim that you need an infinite process is even worse. Once you have proved the property for the generic case, all infinity of the specific cases follow IN PARALLEL, NOT serially -- not that it would matter, since math doesn't acknowledge ANY lower limit on TIME required to complete an individual case.
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| From | Ganzhinterseher <claus.vonnesser@gmail.com> |
|---|---|
| Date | 2019-09-17 09:39 -0700 |
| Message-ID | <eaf9eaaa-ced3-4e6c-bd13-037a5516e24b@googlegroups.com> |
| In reply to | #210842 |
Am Montag, 16. September 2019 22:27:22 UTC+2 schrieb George Greene:
> In practice, believe it or not, proving that Every
> element of an Infinite set has some property IS EASIER THAN proving n
> instances for elements of a size-n set, because in the infinite case, YOU
> ONLY HAVE TO PROVE THAT *ONE* element has the property -- but it has to be
> A GENERIC element --
and the readers are in general too polite to mention that every element is not generic but belongs to a tiny finite segment. For instance every endsegmnet appearing in
∀n ∈ ℕ: E(1) ∩ E(2) ∩ ... ∩ E(n) =/= { }
is useless and invalid to contribute to
E(1) ∩ E(2) ∩ E(3) ∩ ... = { } .
But the readers are too polite to ask what causes the emptiness of the result.
Regards, WM
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| From | Ganzhinterseher <claus.vonnesser@gmail.com> |
|---|---|
| Date | 2019-09-15 02:55 -0700 |
| Message-ID | <11000a23-2f28-4827-a554-53798d489f68@googlegroups.com> |
| In reply to | #210725 |
Am Sonntag, 15. September 2019 03:31:20 UTC+2 schrieb George Greene: > On Monday, September 9, 2019 at 4:44:11 PM UTC-4, Ganzhinterseher wrote: > > > > See > > > > 2.16 Set-theoretical limits of sequences of sets > > > > https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p. 55 > > > > > > why would anybody take THAT seriously as a source? > > > > Every learned person could check and veryify it. You cannot. > > *WHO* cannot *what*? You cannot verify that p. 55 defines the limit of sequences of sets. > You're damn right that every learned person can verify > that you cannot define the limit of a sequence of sets, and that p.55 cannot define it either. Obviously you do not belong to the set of learned persons. Regards, WM
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2019-09-08 22:38 +0100 |
| Message-ID | <878sqywjrq.fsf@bsb.me.uk> |
| In reply to | #210293 |
Ganzhinterseher <claus.vonnesser@gmail.com> writes:
(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
Unendlichen" at Hochschule Augsburg.)
> But if a sequence of sets has exclusively non-empty elements, then the
> limit cannot be empty
What is the point-wise limit of the sequence of real functions
e_n(x) = 1 if x >= n, 0 otherwise
? I ask (as you know) because point-wise convergence of indicator
functions is one way to define the limit you are puzzled by.
You could use this definition of point-wise convergence if you like:
Let (f_n) be a sequence of functions and B ⊆ D. The sequence of
functions is said to be convergent in the usual sense or point-wise
convergent on B, if a function f: B -> R exists, such that for every x ∈
B we have f(x) = lim_{n->oo} f_n(x).
The f_n are functions from D -> R with D ⊆ R.
--
Ben.
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| From | George Greene <greeneg@email.unc.edu> |
|---|---|
| Date | 2019-09-09 13:09 -0700 |
| Message-ID | <7fcb2ec7-bbba-48e8-a480-220b4dbde31c@googlegroups.com> |
| In reply to | #210308 |
On Sunday, September 8, 2019 at 5:38:51 PM UTC-4, Ben Bacarisse wrote: > Ganzhinterseher <claus.vonnesser@gmail.com> writes: > (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des > Unendlichen" at Hochschule Augsburg.) > > > But if a sequence of sets has exclusively non-empty elements, then the > > limit cannot be empty > > What is the point-wise limit of the sequence of real functions This is a stupid question. He ISN'T TALKING about real functions. He's talking ABOUT SETS. The elements of sets in ZF are OTHER SETS, NOT "points".
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| From | Ganzhinterseher <claus.vonnesser@gmail.com> |
|---|---|
| Date | 2019-09-09 13:21 -0700 |
| Message-ID | <e94c23c3-47ea-449b-8181-52ca0c32460f@googlegroups.com> |
| In reply to | #210308 |
Am Sonntag, 8. September 2019 23:38:51 UTC+2 schrieb Ben Bacarisse: > Ganzhinterseher <claus.vonnesser@gmail.com> writes: > (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des > Unendlichen" at Hochschule Augsburg.) > > > But if a sequence of sets has exclusively non-empty elements, then the > > limit cannot be empty > > What is the point-wise limit of the sequence of real functions > > e_n(x) = 1 if x >= n, 0 otherwise > > ? I ask (as you know) because point-wise convergence of indicator > functions is one way to define the limit you are puzzled by. Let's apply it to this problem: How many natnumbers can we remove from 1, 2, 3, ... without emptying this set? Regards, WM
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2019-09-10 03:27 +0100 |
| Message-ID | <87y2ywlwb7.fsf@bsb.me.uk> |
| In reply to | #210346 |
Ganzhinterseher <claus.vonnesser@gmail.com> writes:
> Am Sonntag, 8. September 2019 23:38:51 UTC+2 schrieb Ben Bacarisse:
>> Ganzhinterseher <claus.vonnesser@gmail.com> writes:
>> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
>> Unendlichen" at Hochschule Augsburg.)
>>
>> > But if a sequence of sets has exclusively non-empty elements, then the
>> > limit cannot be empty
>>
>> What is the point-wise limit of the sequence of real functions
>>
>> e_n(x) = 1 if x >= n, 0 otherwise
>>
>> ? I ask (as you know) because point-wise convergence of indicator
>> functions is one way to define the limit you are puzzled by.
>
> Let's apply it to this problem: How many natnumbers can we remove from
> 1, 2, 3, ... without emptying this set?
Non-answer number 1. (I expect you not to answer this question at least
a dozen times, so I may as well start counting.)
As for your question: as many as you like. It's a silly question
because whether N \ E = {} (for E ⊆ N) depends one which numbers are in
E as much as one how many numbers are in E.
Now, will you answer my question?
--
Ben.
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| From | Ganzhinterseher <claus.vonnesser@gmail.com> |
|---|---|
| Date | 2019-09-10 08:14 -0700 |
| Message-ID | <5bf8f6f2-82dd-438a-89f0-0d708a7e8966@googlegroups.com> |
| In reply to | #210360 |
Am Dienstag, 10. September 2019 04:28:02 UTC+2 schrieb Ben Bacarisse:
> Ganzhinterseher <claus.vonnesser@gmail.com> writes:
> > Let's apply it to this problem: How many natnumbers can we remove from
> > 1, 2, 3, ... without emptying this set?
>
> Non-answer number 1.
It is the answer. All removable numbers must first be defined. The limit function is 0 for every definable number. But most numbers are not definable.
> As for your question: as many as you like. It's a silly question
> because whether N \ E = {} (for E ⊆ N) depends one which numbers are in
> E as much as one how many numbers are in E.
Of course I meant definable numbers, i.e., numbers where a decimal repesentation can be given. The set of these numbers is infinite, because for every defined number a larger one can be defined. But not all can be defined, because every defined number has infinitely many successors.
>
> Now, will you answer my question?
I did. The limit function is zero for every definable number. The set of definable numbers is infinite. But it is a proper subset of |N which is the smallest infinite set according to ZFC.
Regards, WM
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2019-09-11 02:25 +0100 |
| Message-ID | <87impzlj3j.fsf@bsb.me.uk> |
| In reply to | #210377 |
Ganzhinterseher <claus.vonnesser@gmail.com> writes:
(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
Unendlichen" at Hochschule Augsburg.)
> Am Dienstag, 10. September 2019 04:28:02 UTC+2 schrieb Ben Bacarisse:
>> Ganzhinterseher <claus.vonnesser@gmail.com> writes:
>
>> > Let's apply it to this problem: How many natnumbers can we remove from
>> > 1, 2, 3, ... without emptying this set?
>>
>> Non-answer number 1.
>
> It is the answer. All removable numbers must first be defined. The
> limit function is 0 for every definable number. But most numbers are
> not definable.
No it was not an answer. The point-wise limit has nothing to do with
"removable" numbers, and you failed to even mention "definable" numbers
in your claimed answer.
Here's what is actually meant by the point-wise limit:
Let (f_n) be a sequence of functions and B ⊆ D. The sequence of
functions is said to be convergent in the usual sense or point-wise
convergent on B, if a function f: B -> R exists, such that for every x
∈ B we have f(x) = lim_{n->oo} f_n(x). (The f_n are functions from D
-> R with D ⊆ R.)
See? No reference to removing anything and no reference to definable
numbers. The function e(x) = 0 meets this definition for the limit of
the e_n. A reminder:
e_n(x) = 1 if x >= n, 0 otherwise
If you have some other definition of the point-wise limit function, what
is it? It would have to exclude e(x) = 0 being the limit of the
sequence (e_n).
But definability is just a ruse. You want to talk about anything other
than the plain fact that e(x) = 0 meets the above definition, and
waffling about words like "definable" is your favourite pass time now
that you are retired.
>> As for your question: as many as you like. It's a silly question
>> because whether N \ E = {} (for E ⊆ N) depends one which numbers are in
>> E as much as one how many numbers are in E.
>
> Of course I meant definable numbers,
You should have said so then, but the answer remains the same: "How many
definable natnumbers can we remove from 1, 2, 3, ... without emptying
this set?" Answer: as many as you like. It remains a silly question
for the reason given.
--
Ben.
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| From | Ganzhinterseher <claus.vonnesser@gmail.com> |
|---|---|
| Date | 2019-09-11 13:16 -0700 |
| Message-ID | <c053632c-71fb-40d8-a989-83a95784e59e@googlegroups.com> |
| In reply to | #210427 |
Am Mittwoch, 11. September 2019 03:25:38 UTC+2 schrieb Ben Bacarisse:
> Ganzhinterseher <claus.vonnesser@gmail.com> writes:
> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> Unendlichen" at Hochschule Augsburg.)
>
> > Am Dienstag, 10. September 2019 04:28:02 UTC+2 schrieb Ben Bacarisse:
> >> Ganzhinterseher <claus.vonnesser@gmail.com> writes:
> >
> >> > Let's apply it to this problem: How many natnumbers can we remove from
> >> > 1, 2, 3, ... without emptying this set?
> >>
> >> Non-answer number 1.
> >
> > It is the answer. All removable numbers must first be defined. The
> > limit function is 0 for every definable number. But most numbers are
> > not definable.
>
> No it was not an answer. The point-wise limit has nothing to do with
> "removable" numbers, and you failed to even mention "definable" numbers
> in your claimed answer.
Definable is simple: A decimal representation can be given.
>
> Here's what is actually meant by the point-wise limit:
>
> Let (f_n) be a sequence of functions and B ⊆ D. The sequence of
> functions is said to be convergent in the usual sense or point-wise
> convergent on B, if a function f: B -> R exists, such that for every x
> ∈ B we have f(x) = lim_{n->oo} f_n(x). (The f_n are functions from D
> -> R with D ⊆ R.)
>
> See? No reference to removing anything and no reference to definable
> numbers.
"Every x" means every definable x.
>
> "How many
> definable natnumbers can we remove from 1, 2, 3, ... without emptying
> this set?" Answer: as many as you like.
Then define all elements individually such that none remains.
You must see it by yourself that this is an impossible task.
Regards, WM
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2019-09-12 02:15 +0100 |
| Message-ID | <87y2yujow1.fsf@bsb.me.uk> |
| In reply to | #210472 |
Ganzhinterseher <claus.vonnesser@gmail.com> writes:
> Am Mittwoch, 11. September 2019 03:25:38 UTC+2 schrieb Ben Bacarisse:
>> Ganzhinterseher <claus.vonnesser@gmail.com> writes:
>> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
>> Unendlichen" at Hochschule Augsburg.)
>>
>> > Am Dienstag, 10. September 2019 04:28:02 UTC+2 schrieb Ben Bacarisse:
>> >> Ganzhinterseher <claus.vonnesser@gmail.com> writes:
>> >
>> >> > Let's apply it to this problem: How many natnumbers can we remove from
>> >> > 1, 2, 3, ... without emptying this set?
>> >>
>> >> Non-answer number 1.
>> >
>> > It is the answer. All removable numbers must first be defined. The
>> > limit function is 0 for every definable number. But most numbers are
>> > not definable.
>>
>> No it was not an answer. The point-wise limit has nothing to do with
>> "removable" numbers, and you failed to even mention "definable" numbers
>> in your claimed answer.
>
> Definable is simple: A decimal representation can be given.
It's a red herring because e(x) = 0 meets the definition of the
point-wise limit of the (e_n).
>> Here's what is actually meant by the point-wise limit:
>>
>> Let (f_n) be a sequence of functions and B ⊆ D. The sequence of
>> functions is said to be convergent in the usual sense or point-wise
>> convergent on B, if a function f: B -> R exists, such that for every x
>> ∈ B we have f(x) = lim_{n->oo} f_n(x). (The f_n are functions from D
>> -> R with D ⊆ R.)
>>
>> See? No reference to removing anything and no reference to definable
>> numbers.
>
> "Every x" means every definable x.
Still a red herring. e(x) = 0 meets this definition of the limit of the
(e_n). Even if we include the red herring, you must agree that for
every definable x, lim_{n->oo} e_n(x) = 0 = e(x).
Are you claiming some other function or functions are also limits of
that sequence?
(The e_n are e_n(x) = 1 if x >= n, 0 otherwise.)
>> "How many
>> definable natnumbers can we remove from 1, 2, 3, ... without emptying
>> this set?" Answer: as many as you like.
>
> Then define all elements individually such that none remains.
That's not what you asked. I answered the question you asked despite it
being a silly one with an obvious answer.
> You must see it by yourself that this is an impossible task.
Do you think there are natural numbers for which a decimal
representation can not be given?
--
Ben.
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| From | Ganzhinterseher <claus.vonnesser@gmail.com> |
|---|---|
| Date | 2019-09-12 07:23 -0700 |
| Message-ID | <7e3802de-0a9e-472c-9d04-5a8bb0096f1b@googlegroups.com> |
| In reply to | #210506 |
Am Donnerstag, 12. September 2019 03:15:44 UTC+2 schrieb Ben Bacarisse:
> Ganzhinterseher <claus.vonnesser@gmail.com> writes:
> >> > It is the answer. All removable numbers must first be defined. The
> >> > limit function is 0 for every definable number. But most numbers are
> >> > not definable.
> >>
> >> No it was not an answer. The point-wise limit has nothing to do with
> >> "removable" numbers, and you failed to even mention "definable" numbers
> >> in your claimed answer.
> >
> > Definable is simple: A decimal representation can be given.
>
> It's a red herring because e(x) = 0 meets the definition of the
> point-wise limit of the (e_n).
But n does not meet the definition.
Upon every definable natnumber, there are infinitely many undefinable following. Thsi is in the simplest ways proved by endsegments:
For every definable k ∈ ℕ: {E(1),E(2),...,E(n),...}\{E(2),..., E(k)}
take the intersection of the result: ∩{E(1), E(k+1),E(k+2), E(k+3),...} = { }
>
> >> Here's what is actually meant by the point-wise limit:
> >>
> >> Let (f_n) be a sequence of functions and B ⊆ D. The sequence of
> >> functions is said to be convergent in the usual sense or point-wise
> >> convergent on B, if a function f: B -> R exists, such that for every x
> >> ∈ B we have f(x) = lim_{n->oo} f_n(x). (The f_n are functions from D
> >> -> R with D ⊆ R.)
> >>
> >> See? No reference to removing anything and no reference to definable
> >> numbers.
> >
> > "Every x" means every definable x.
>
> Still a red herring. e(x) = 0 meets this definition of the limit of the
> (e_n).
But not every x. See above.
Even if we include the red herring, you must agree that for
> every definable x, lim_{n->oo} e_n(x) = 0 = e(x).
>
> Are you claiming some other function or functions are also limits of
> that sequence?
No, for every definable x and n we get e_n(x) = 0 = e(x).
>
> Do you think there are natural numbers for which a decimal
> representation can not be given?
If there are more than a potentially infinite set of natnumbers, then there must be such natnumbers. See my proof above. Or simply try to define a natural number k for which
∩{E(1), E(k+1),E(k+2), E(k+3),...} =/= { }
Regards, WM
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2019-09-12 17:03 +0100 |
| Message-ID | <87zhj9ijrs.fsf@bsb.me.uk> |
| In reply to | #210537 |
Ganzhinterseher <claus.vonnesser@gmail.com> writes:
(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
Unendlichen" at Hochschule Augsburg.)
> Am Donnerstag, 12. September 2019 03:15:44 UTC+2 schrieb Ben Bacarisse:
>> Ganzhinterseher <claus.vonnesser@gmail.com> writes:
This is Me:
>> >> Here's what is actually meant by the point-wise limit:
>> >>
>> >> Let (f_n) be a sequence of functions and B ⊆ D. The sequence of
>> >> functions is said to be convergent in the usual sense or point-wise
>> >> convergent on B, if a function f: B -> R exists, such that for every x
>> >> ∈ B we have f(x) = lim_{n->oo} f_n(x). (The f_n are functions from D
>> >> -> R with D ⊆ R.)
>> >>
>> >> See? No reference to removing anything and no reference to definable
>> >> numbers.
>> >
>> > "Every x" means every definable x.
>>
>> Still a red herring. e(x) = 0 meets this definition of the limit of the
>> (e_n).
>
> But not every x. See above.
(A reminder: e_n(x) = 1 if x >= n, 0 otherwise. You seem to have
forgotten this given what you write later on.)
In this case we are interested in the convergence on B = N. I take it
you agree that there is a function e: N -> R given by e(x) = 0? It is
simply the set N x {0}. You have previous stated that for every x ∈ N
lim_{n->oo} e_n(x) = 0.
Are you retracting that? That would undermine almost everything in your
book about limits of sequences.
>> Even if we include the red herring, you must agree that for
>> every definable x, lim_{n->oo} e_n(x) = 0 = e(x).
>>
>> Are you claiming some other function or functions are also limits of
>> that sequence?
>
> No, for every definable x and n we get e_n(x) = 0 = e(x).
Either you have miss-understood something very basic here, or you have
not written what you mean.
e_n(x) = 1 if x >= n, 0 otherwise
so for the eminently definable x = 1 and n = 1
e_1(1) = 1 =/= 0.
<cut>
>> Do you think there are natural numbers for which a decimal
>> representation can not be given?
>
> If there are more than a potentially infinite set of natnumbers, then
> there must be such natnumbers.
How can there be more? Is there a natural number that is not in the
potentially infinite set N? Maybe you have a secret meaning for "more".
--
Ben.
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| From | Ganzhinterseher <claus.vonnesser@gmail.com> |
|---|---|
| Date | 2019-09-12 13:27 -0700 |
| Message-ID | <6efefb29-1d42-4c17-86bd-37099e97b354@googlegroups.com> |
| In reply to | #210554 |
Am Donnerstag, 12. September 2019 18:03:53 UTC+2 schrieb Ben Bacarisse:
> Ganzhinterseher <claus.vonnesser@gmail.com> writes:
> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> Unendlichen" at Hochschule Augsburg.)
>
> > Am Donnerstag, 12. September 2019 03:15:44 UTC+2 schrieb Ben Bacarisse:
> >> Ganzhinterseher <claus.vonnesser@gmail.com> writes:
> This is Me:
> >> >> Here's what is actually meant by the point-wise limit:
> >> >>
> >> >> Let (f_n) be a sequence of functions and B ⊆ D. The sequence of
> >> >> functions is said to be convergent in the usual sense or point-wise
> >> >> convergent on B, if a function f: B -> R exists, such that for every x
> >> >> ∈ B we have f(x) = lim_{n->oo} f_n(x). (The f_n are functions from D
> >> >> -> R with D ⊆ R.)
> >> >>
> >> >> See? No reference to removing anything and no reference to definable
> >> >> numbers.
> >> >
> >> > "Every x" means every definable x.
> >>
> >> Still a red herring. e(x) = 0 meets this definition of the limit of the
> >> (e_n).
> >
> > But not every x. See above.
>
> (A reminder: e_n(x) = 1 if x >= n, 0 otherwise. You seem to have
> forgotten this given what you write later on.)
No.
>
> In this case we are interested in the convergence on B = N. I take it
> you agree that there is a function e: N -> R given by e(x) = 0? It is
> simply the set N x {0}. You have previous stated that for every x ∈ N
>
> lim_{n->oo} e_n(x) = 0.
That is true for potential infinity (my book) and for every definable n (there are infinitely many in set theory), followed by infinitely many undefinable natnumbers.
>
> Are you retracting that? That would undermine almost everything in your
> book about limits of sequences.
That is true for potential infinity (my book) and for every definable n (there are infinitely many in set theory), followed by infinitely many undefinable natnumbers.
>
> >> Even if we include the red herring, you must agree that for
> >> every definable x, lim_{n->oo} e_n(x) = 0 = e(x).
> >>
> >> Are you claiming some other function or functions are also limits of
> >> that sequence?
> >
> > No, for every definable x and n we get e_n(x) = 0 = e(x).
>
> Either you have miss-understood something very basic here, or you have
> not written what you mean.
I meant: The limit function is 0 for every definable x
> >> Do you think there are natural numbers for which a decimal
> >> representation can not be given?
> >
> > If there are more than a potentially infinite set of natnumbers, then
> > there must be such natnumbers.
>
> How can there be more? Is there a natural number that is not in the
> potentially infinite set N?
In set theory there must be more. There is no potential infinity.
Regards, WM
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2019-09-13 00:40 +0100 |
| Message-ID | <87pnk5hymu.fsf@bsb.me.uk> |
| In reply to | #210570 |
Ganzhinterseher <claus.vonnesser@gmail.com> writes:
(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
Unendlichen" at Hochschule Augsburg.)
> Am Donnerstag, 12. September 2019 18:03:53 UTC+2 schrieb Ben Bacarisse:
>> Ganzhinterseher <claus.vonnesser@gmail.com> writes:
>>
>> > Am Donnerstag, 12. September 2019 03:15:44 UTC+2 schrieb Ben Bacarisse:
>> >> Ganzhinterseher <claus.vonnesser@gmail.com> writes:
>> This is Me:
>> >> >> Here's what is actually meant by the point-wise limit:
>> >> >>
>> >> >> Let (f_n) be a sequence of functions and B ⊆ D. The sequence of
>> >> >> functions is said to be convergent in the usual sense or point-wise
>> >> >> convergent on B, if a function f: B -> R exists, such that for every x
>> >> >> ∈ B we have f(x) = lim_{n->oo} f_n(x). (The f_n are functions from D
>> >> >> -> R with D ⊆ R.)
Context: the e_n are a family of functions e_n(x) = 1 if x >=n, 0
otherwise.
>> In this case we are interested in the convergence on B = N. I take it
>> you agree that there is a function e: N -> R given by e(x) = 0? It is
>> simply the set N x {0}. You have previous stated that for every x ∈ N
>>
>> lim_{n->oo} e_n(x) = 0.
>
> That is true for potential infinity (my book) and for every definable
> n (there are infinitely many in set theory), followed by infinitely
> many undefinable natnumbers.
Oh! So in WMaths the limit function is e(x) = 0. I thought you denied
this.
And just to be clear, you think that in "set theory" (i.e. normal
mathematics) there are some x ∈ N for which lim_{n->oo} e_n(x) is not 0.
I won't ask you to prove this -- it's just useful for people to see your
beliefs clearly stated.
Presumably there is nothing special about the functions I picked. Can
you confirm that you think that in "set theory" there are no point-wise
limit functions because the "for every x" in the definition can never be
met?
<cut>
>> >> Do you think there are natural numbers for which a decimal
>> >> representation can not be given?
>> >
>> > If there are more than a potentially infinite set of natnumbers, then
>> > there must be such natnumbers.
>>
>> How can there be more? Is there a natural number that is not in the
>> potentially infinite set N?
>
> In set theory there must be more. There is no potential infinity.
As you say, there is no potential infinity so what on Earth did you
think you were asking about set theory? The question only makes sense
in WMaths.
Mathematics (what you call set theory) proves statements like
for every x ∈ N, lim_{n->oo} e_n(x) = 0.
There are no qualifications about "purple x", "honourable x" or
"definable x". The proof just applies to every x ∈ N.
--
Ben.
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| From | Ganzhinterseher <claus.vonnesser@gmail.com> |
|---|---|
| Date | 2019-09-13 04:49 -0700 |
| Message-ID | <be1ba1ee-3e30-478b-a328-40e5005a6e95@googlegroups.com> |
| In reply to | #210580 |
Am Freitag, 13. September 2019 01:40:30 UTC+2 schrieb Ben Bacarisse:
> Ganzhinterseher <claus.vonnesser@gmail.com> writes:
> >> lim_{n->oo} e_n(x) = 0.
> >
> > That is true for potential infinity (my book) and for every definable
> > n (there are infinitely many in set theory), followed by infinitely
> > many undefinable natnumbers.
>
> Oh! So in WMaths the limit function is e(x) = 0. I thought you denied
> this.
Up to every definable n this is correct. In mathematics there are only definable n. In set theory there is actual infinity, i.e., a set larger than all FISONs. Since all definable natnumbers are in FISONs, the actually infinite set must contain more than definable numbers.
>
> And just to be clear, you think that in "set theory" (i.e. normal
> mathematics) there are some x ∈ N for which lim_{n->oo} e_n(x) is not 0.
> I won't ask you to prove this -- it's just useful for people to see your
> beliefs clearly stated.
It is easy: Since all definable natnumbers are in FISONs, the actually infinite set (larger than all FISONs) must contain more than definable numbers.
>
> Presumably there is nothing special about the functions I picked. Can
> you confirm that you think that in "set theory" there are no point-wise
> limit functions because the "for every x" in the definition can never be
> met?
All limit functions are correctly described in potential infinity. The set of definable numbers in set theory is always finite without a last element, i.e., the same a the set of all natural numbers in potential infinity.
>
> >> How can there be more? Is there a natural number that is not in the
> >> potentially infinite set N?
> >
> > In set theory there must be more. There is no potential infinity.
>
> As you say, there is no potential infinity so what on Earth did you
> think you were asking about set theory?
Since all definable natnumbers are in FISONs, the actually infinite set (larger than all FISONs) must contain more than definable numbers.
>
> Mathematics (what you call set theory) proves statements like
>
> for every x ∈ N, lim_{n->oo} e_n(x) = 0.
It also proves ∀k ∈ ℕ ∩{E(k+1),E(k+2), E(k+3),...} = { } (*)
Therefore for all natnumbers there must be infinitely many successors. Even the intersection of these successor sets must be infinite. Otherwise (*) would not be true for all natnumbers k.
>
> There are no qualifications about "purple x", "honourable x" or
> "definable x". The proof just applies to every x ∈ N.
As you see here (*), it does not. It is only believes by people who are not looking through.
Regards, WM
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