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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 4 of 13 — ← Prev page 1 2 3 [4] 5 6 … 13 Next page →
| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2019-09-17 14:04 +0100 |
| Message-ID | <87impr9iq3.fsf@bsb.me.uk> |
| In reply to | #210892 |
Fred <Fred@nowhere.invalid> writes:
> On 16/09/2019 23:49, Ben Bacarisse wrote:
>> Fred <Fred@nowhere.invalid> writes:
<cut>
>>> The definition I gave appears on page 4 of volume I of Kuratowski's
>>> /Topology/, Academic Press, 1966. He write Limes for my lim, Lim inf
>>> for my liminf, and Lim sup for my limsup. I rest my case.
>>
>> Essentially the same definition (though given rather informally) can be
>> found early on in Hausdorff's "Mengenlehre" (1937). I relied on a 1957
>> English the translation where it's called "Set Theory".
>>
> Thank you.
>
> I've also found it in Kuratowski's /Introduction to set theory and
> topology/, a wholly elementary text. Kuratowski has 'Lim' for 'Limes'
> in his other book. An exercise (7 on p.61) asks the reader to show
> that, with A a set and
>
> ( 1 if x in A
> f_A(x) = (
> ( 0 if x in A^c (A's complement)
>
> F = Lim F_n iff f_F = lim f_{F_n}(x)
>
> where F_n is a sequence of sets and lim is the usual numerical limit.
And this is the point that Prof. Mueckenheim can't avoid but simply
denies. His textbook defines these usual limits but he wants to deny
that set sequence limits make sense. As a result, he has to deny that
usual point-wide limit as defined in his book gives the zero function
for the limit of the f_{E_n} (using Kuratowski's notation and with E_n
{x | x >= n}.
Of course that's nonsense for another reason unrelated to characteristic
functions: functions are sets, so function sequences are set sequences
and function sequence limits are set sequence limits. Even if he
rejects all others, his textbook has two examples of set sequence
limits, fully worked out!
--
Ben.
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| From | George Greene <greeneg@email.unc.edu> |
|---|---|
| Date | 2019-09-17 14:55 -0700 |
| Message-ID | <e180997f-7784-495f-8d8e-2d5bfd42b619@googlegroups.com> |
| In reply to | #210892 |
On Tuesday, September 17, 2019 at 3:53:16 AM UTC-4, Fred wrote: > where F_n is a sequence of sets and lim is the usual numerical limit. IF we are TALKING ABOUT the limit of a sequence of sets THEN lim CANNOT MEAN the usual numerical limit because MOST SETS HAVE THE PROPERTY that their elements *A*R*E* *N*O*T* numerical!!! Hausdorff can talk about it because he's doing point-set topology but IN GENERAL "limit of a sequence of sets" *I*S* *N*O*T* *E*V*E*N* *A* *THING*!!!!
[toc] | [prev] | [next] | [standalone]
| From | Exfalso Quodlibet <exflaso.quodlibet@gmail.com> |
|---|---|
| Date | 2019-09-17 16:17 -0700 |
| Message-ID | <e487a3a8-eb12-4e2e-ba5f-6cbb034e1321@googlegroups.com> |
| In reply to | #210965 |
On Tuesday, September 17, 2019 at 5:55:59 PM UTC-4, George Greene wrote: > IN GENERAL > "limit of a sequence of sets" > *I*S* *N*O*T* *E*V*E*N* > *A* *THING*!!!! https://en.wikipedia.org/wiki/Set-theoretic_limit EFQ
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| From | George Greene <greeneg@email.unc.edu> |
|---|---|
| Date | 2019-09-17 21:29 -0700 |
| Message-ID | <2862e538-15b8-4448-ba87-a36df2f3f014@googlegroups.com> |
| In reply to | #210967 |
On Tuesday, September 17, 2019 at 7:17:11 PM UTC-4, Exfalso Quodlibet wrote: > On Tuesday, September 17, 2019 at 5:55:59 PM UTC-4, George Greene wrote: > > IN GENERAL > > "limit of a sequence of sets" > > *I*S* *N*O*T* *E*V*E*N* > > *A* *THING*!!!! > > https://en.wikipedia.org/wiki/Set-theoretic_limit Oh, SHUT UP. THE VERY FIRST LINE of that treatment is "In mathematics, the limit of a sequence of sets A1, A2, ... (subsets of a common set X) " ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ WHICH IS ALREADY TOO LIMITING TO BE RELEVANT for almost any purpose. IN GENERAL a countably infinite sequence of sets do NOT need to all be subsets of some common set X unless the sequence is itself also a set (in which case they could be common subsets of its union). AS I was saying, that "common set X" is VERY USUALLY LIKELY to be a set OF POINTS OR NUMBERS, or at least of things with a natural METRIC, DISTANCE or ORDER between them (LIKE REAL NUMBERS). It is relevant, though, that this treatment is compatible with WM's "set of endsegments of N" example, since all of those are subsets of N and since lim sup and lim inf are both defined for that sequence and since they are both in fact the empty set. But this does sort of raise the question why you didn't reply to the original post with this wikipedia citation.
[toc] | [prev] | [next] | [standalone]
| From | Exfalso Quodlibet <exflaso.quodlibet@gmail.com> |
|---|---|
| Date | 2019-09-17 21:39 -0700 |
| Message-ID | <70b9f80b-35de-448d-b22f-073e9196f191@googlegroups.com> |
| In reply to | #210984 |
On Wednesday, September 18, 2019 at 12:29:10 AM UTC-4, George Greene wrote: > On Tuesday, September 17, 2019 at 7:17:11 PM UTC-4, Exfalso Quodlibet wrote: > > On Tuesday, September 17, 2019 at 5:55:59 PM UTC-4, George Greene wrote: > > > IN GENERAL > > > "limit of a sequence of sets" > > > *I*S* *N*O*T* *E*V*E*N* > > > *A* *THING*!!!! > > > > https://en.wikipedia.org/wiki/Set-theoretic_limit > > THE VERY FIRST LINE of that treatment is > "In mathematics, > the limit of a sequence of sets A1, A2, ... > (subsets of a common set X) " > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ It's not hard to come up with an X: just take the union of the sequence, and viola, the sets in the sequence are all subsets of a common set! > WHICH IS ALREADY TOO LIMITING TO BE RELEVANT for almost any purpose. But it's relevant for this thread! > It is relevant, though, that this treatment is compatible with > WM's "set of endsegments of N" example, See? > But this does sort of raise the question why you didn't reply to the original > post with this wikipedia citation. Because I found the citation after reading the post I replied to. EFQ
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| From | Fred <Fred@nowhere.invalid> |
|---|---|
| Date | 2019-09-18 18:35 +0100 |
| Message-ID | <qltpth$28u$1@news.albasani.net> |
| In reply to | #210987 |
On 18/09/2019 05:39, Exfalso Quodlibet wrote: > On Wednesday, September 18, 2019 at 12:29:10 AM UTC-4, George Greene wrote: >> On Tuesday, September 17, 2019 at 7:17:11 PM UTC-4, Exfalso Quodlibet wrote: >>> On Tuesday, September 17, 2019 at 5:55:59 PM UTC-4, George Greene wrote: >>>> IN GENERAL >>>> "limit of a sequence of sets" >>>> *I*S* *N*O*T* *E*V*E*N* >>>> *A* *THING*!!!! >>> >>> https://en.wikipedia.org/wiki/Set-theoretic_limit >> >> THE VERY FIRST LINE of that treatment is >> "In mathematics, >> the limit of a sequence of sets A1, A2, ... >> (subsets of a common set X) " >> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ > > It's not hard to come up with an X: just take the union of the sequence, and viola, the sets in the sequence are all subsets of a common set! > >> WHICH IS ALREADY TOO LIMITING TO BE RELEVANT for almost any purpose. > > But it's relevant for this thread! Don't worry. For every sequence of sets there is a set which they are all subsets of (viz the union of the sequence). The Wikipedia comment is redundant. > >> It is relevant, though, that this treatment is compatible with >> WM's "set of endsegments of N" example, > > See? > >> But this does sort of raise the question why you didn't reply to the original >> post with this wikipedia citation. > > Because I found the citation after reading the post I replied to. > > EFQ >
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| From | Fred <Fred@nowhere.invalid> |
|---|---|
| Date | 2019-09-18 18:32 +0100 |
| Message-ID | <qltpnm$4sf$1@news.albasani.net> |
| In reply to | #210967 |
On 18/09/2019 00:17, Exfalso Quodlibet wrote: > On Tuesday, September 17, 2019 at 5:55:59 PM UTC-4, George Greene wrote: >> IN GENERAL >> "limit of a sequence of sets" >> *I*S* *N*O*T* *E*V*E*N* >> *A* *THING*!!!! > > https://en.wikipedia.org/wiki/Set-theoretic_limit > > EFQ > Thank you! I spent some time rummaging among my set theory books and I found only two that defined lim A_n. It didn't occur to me to look at Pikiwedia.
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2019-09-18 03:46 +0100 |
| Message-ID | <87v9tq8go6.fsf@bsb.me.uk> |
| In reply to | #210965 |
George Greene <greeneg@email.unc.edu> writes: > On Tuesday, September 17, 2019 at 3:53:16 AM UTC-4, Fred wrote: >> where F_n is a sequence of sets and lim is the usual numerical limit. > > IF we are TALKING ABOUT the limit of a sequence of sets > THEN lim CANNOT MEAN the usual numerical limit because MOST SETS HAVE THE PROPERTY > that their elements *A*R*E* *N*O*T* numerical!!! I think you missed what was being said. The lim is usual numerical limit of a sequence of functions -- the characteristic functions of the F_n. > Hausdorff can talk about it because he's doing point-set topology but > IN GENERAL > "limit of a sequence of sets" > *I*S* *N*O*T* *E*V*E*N* > *A* *THING*!!!! -- Ben.
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| From | George Greene <greeneg@email.unc.edu> |
|---|---|
| Date | 2019-09-17 21:39 -0700 |
| Message-ID | <041e6a9f-2262-472e-a085-632788f015e7@googlegroups.com> |
| In reply to | #210969 |
On Tuesday, September 17, 2019 at 10:46:51 PM UTC-4, Ben Bacarisse wrote: > I think you missed what was being said. The lim is usual numerical > limit of a sequence of functions -- the characteristic functions of the > F_n. I didn't miss THAT. THAT is irrelevant. That is explained AS ISOMORPHIC to the OTHER description with unions of intersections for lim inf and intersections of unions for lim sup. In any case, it DOES MATTER that in the wikipedia treatment, all the examples given ARE numerical (i.e. the bounding set X is a set OF NUMBERS).
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2019-09-18 12:46 +0100 |
| Message-ID | <87muf17rol.fsf@bsb.me.uk> |
| In reply to | #210988 |
George Greene <greeneg@email.unc.edu> writes: > On Tuesday, September 17, 2019 at 10:46:51 PM UTC-4, Ben Bacarisse wrote: >> I think you missed what was being said. The lim is usual numerical >> limit of a sequence of functions -- the characteristic functions of the >> F_n. > > I didn't miss THAT. THAT is irrelevant. So why do you think that requires the elements be numerical? And why is the usual function limit irrelevant? It's an alternative definition given in both the books cited and applies to very general sequences of sets. > That is explained AS ISOMORPHIC to the OTHER description with unions > of intersections for lim inf and intersections of unions for lim sup. > > In any case, it DOES MATTER that in the wikipedia treatment, all the > examples given ARE numerical (i.e. the bounding set X is a set OF > NUMBERS). Why does that matter? It's not a key property for a limit to exist, simply a convenient source of examples. Hausdorff's first example is entirely general. He gives the sequence M, N, M, N, ... for any two sets and shows that the upper limit is M ∪ N and the lower limit is M ∩ N so the sequence converges only when M = N. -- Ben.
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| From | Fred <Fred@nowhere.invalid> |
|---|---|
| Date | 2019-09-18 17:52 +0100 |
| Message-ID | <qltndb$a42$1@news.albasani.net> |
| In reply to | #210965 |
On 17/09/2019 22:55, George Greene wrote: > On Tuesday, September 17, 2019 at 3:53:16 AM UTC-4, Fred wrote: >> where F_n is a sequence of sets and lim is the usual numerical limit. > > IF we are TALKING ABOUT the limit of a sequence of sets > THEN lim CANNOT MEAN the usual numerical limit because MOST SETS HAVE THE PROPERTY > that their elements *A*R*E* *N*O*T* numerical!!! > Hausdorff can talk about it because he's doing point-set topology but > IN GENERAL > "limit of a sequence of sets" > *I*S* *N*O*T* *E*V*E*N* > *A* *THING*!!!! limsup and liminf (as defined in an earlier post) both exist for any sequence of sets. They have nothing to do with numbers or topology. If they are equal then the limit of the sequence is that common value. If Kuratowski and Hausdorff agree on that, then I am content with it.
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| From | Python <python@invalid> |
|---|---|
| Date | 2019-09-18 18:58 +0200 |
| Message-ID | <5d826234$0$20334$426a74cc@news.free.fr> |
| In reply to | #211015 |
Fred wrote: > On 17/09/2019 22:55, George Greene wrote: >> On Tuesday, September 17, 2019 at 3:53:16 AM UTC-4, Fred wrote: >>> where F_n is a sequence of sets and lim is the usual numerical limit. >> >> IF we are TALKING ABOUT the limit of a sequence of sets >> THEN lim CANNOT MEAN the usual numerical limit because MOST SETS HAVE >> THE PROPERTY >> that their elements *A*R*E* *N*O*T* numerical!!! >> Hausdorff can talk about it because he's doing point-set topology but >> IN GENERAL >> "limit of a sequence of sets" >> *I*S* *N*O*T* *E*V*E*N* >> *A* *THING*!!!! > > limsup and liminf (as defined in an earlier post) both exist for any > sequence of sets. They have nothing to do with numbers or topology. If > they are equal then the limit of the sequence is that common value. It is not that clear that Set Limits have nothing to do with "usual" limit for numbers-like kind of object. As in modern math almost everything is defined as some kind of set, I asked the question, last year - in a similar thread initiated by some stupid rant from Crank Wolfgang Mueckenheim, from Hochschule Augsburg - if there were some study about Set Limits and "topological-metric" limits being the same. As a matter of fact there is, as it has been noticed then, for instance, for Dedekind Cuts it can be shown that Set Theorical Limits AND topological-metric limits for sequences of real numbers coincides exactly. My guess is that it may be done for a lot more of similar structures, given that you pick up the adequate set style representation.
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| From | Fred <Fred@nowhere.invalid> |
|---|---|
| Date | 2019-09-18 18:47 +0100 |
| Message-ID | <qltqjj$43t$1@news.albasani.net> |
| In reply to | #211017 |
On 18/09/2019 17:58, Python wrote: > Fred wrote: >> On 17/09/2019 22:55, George Greene wrote: >>> On Tuesday, September 17, 2019 at 3:53:16 AM UTC-4, Fred wrote: >>>> where F_n is a sequence of sets and lim is the usual numerical limit. >>> >>> IF we are TALKING ABOUT the limit of a sequence of sets >>> THEN lim CANNOT MEAN the usual numerical limit because MOST SETS HAVE >>> THE PROPERTY >>> that their elements *A*R*E* *N*O*T* numerical!!! >>> Hausdorff can talk about it because he's doing point-set topology but >>> IN GENERAL >>> "limit of a sequence of sets" >>> *I*S* *N*O*T* *E*V*E*N* >>> *A* *THING*!!!! >> >> limsup and liminf (as defined in an earlier post) both exist for any >> sequence of sets. They have nothing to do with numbers or topology. >> If they are equal then the limit of the sequence is that common value. > > It is not that clear that Set Limits have nothing to do with "usual" > limit for numbers-like kind of object. As in modern math almost I take your point. My remark was over-stating my point which was that George seemed to think that the definition of lim A_n *required* topological notions, or that the A_n be sets of numbers, but neither is so. > everything is defined as some kind of set, I asked the question, last > year - in a similar thread initiated by some stupid rant from > Crank Wolfgang Mueckenheim, from Hochschule Augsburg - if there were > some study about Set Limits and "topological-metric" limits being > the same. As a matter of fact there is, as it has been noticed then, > for instance, for Dedekind Cuts it can be shown that Set Theorical > Limits AND topological-metric limits for sequences of real numbers > coincides exactly. > > My guess is that it may be done for a lot more of similar structures, > given that you pick up the adequate set style representation. > >
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| From | George Greene <greeneg@email.unc.edu> |
|---|---|
| Date | 2019-09-17 14:53 -0700 |
| Message-ID | <a629f472-de09-49e4-8f87-6e63aeedb827@googlegroups.com> |
| In reply to | #210851 |
On Monday, September 16, 2019 at 6:49:53 PM UTC-4, Ben Bacarisse wrote: > Essentially the same definition (though given rather informally) can be > found early on in Hausdorff's "Mengenlehre" (1937). I relied on a 1957 > English the translation where it's called "Set Theory". I wasn't TALKING to YOU, and I don't GIVE A SHIT ABOUT "essentially"! IF THERE WERE A STANDARD definition, you would NOT HAVE TO GO BACK to 1937 or 1957! Hausdorff is about topology and ALL THE USUAL definitions of "limit" are about limits of sets OF POINTS or limits of sequences OF POINTS! *N*O*BODY much TALKS about limits OF SETS EXCEPT in the context of LIMIT ORDINALS, which are called that because they need to be the sets OF INDICES of-sequences-that-have-limits -- even THEY are NOT generally talked of as HAVING limits themSELVES!!
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2019-09-18 03:37 +0100 |
| Message-ID | <871rwe9vnv.fsf@bsb.me.uk> |
| In reply to | #210964 |
George Greene <greeneg@email.unc.edu> writes: > On Monday, September 16, 2019 at 6:49:53 PM UTC-4, Ben Bacarisse wrote: >> Essentially the same definition (though given rather informally) can be >> found early on in Hausdorff's "Mengenlehre" (1937). I relied on a 1957 >> English the translation where it's called "Set Theory". > > I wasn't TALKING to YOU, Nor I to you, but here we are! > and I don't GIVE A SHIT ABOUT > "essentially"! IF THERE WERE A STANDARD definition, you would NOT HAVE TO > GO BACK to 1937 or 1957! Hausdorff is about topology and ALL THE USUAL > definitions of "limit" are about limits of sets OF POINTS or limits of sequences OF POINTS! > *N*O*BODY > much TALKS about limits OF SETS > EXCEPT > in the context of > LIMIT ORDINALS, I mentioned that book because it's an example of someone who /does/ talk about limits of sequences of sets. He does so using lim sup and lim inf, and then using characteristic functions. > which are called that because they need to be the sets > OF INDICES > of-sequences-that-have-limits -- even THEY are NOT generally > talked of as HAVING limits themSELVES!! -- Ben.
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| From | George Greene <greeneg@email.unc.edu> |
|---|---|
| Date | 2019-09-17 21:34 -0700 |
| Message-ID | <356154d3-d826-4774-841b-2ce4043fab2c@googlegroups.com> |
| In reply to | #210968 |
On Tuesday, September 17, 2019 at 10:37:44 PM UTC-4, Ben Bacarisse wrote:
> I mentioned that book because it's an example of someone who /does/ talk
> about limits of sequences of sets.
Well, sure, but the point is, if that's what you had to do, then that looks
OLD AND RARE.
> He does so using lim sup and lim inf,
> and then using characteristic functions.
Well, that, on the other hand, is the same thing the current wikipedia treatment says. The wikipedia treatment, however, has the disadvantage that in the VERY FIRST line, it says
>>> limit of a sequence of sets A1, A2,
>>> ... (subsets of a common set X)
^^^^^^^^^^^^^^^^^^^^^^^
And the requirement that every set in the sequence be a subset of the same set
really is almost restrictive enough to make the concept generally non-existent,
unless the sequence is itself representable as 1 set and the bounding set X
could then just be its union. And AGAIN, THAT is a rare case -- it specifically
invites X to be a set OF NUMBERS as opposed to of sets. Or at least of things with an order/metric/distance between them.
The only thing that can be said in favor of this definition in this context
is that it directly refutes WM (the bounding set is N), so why wasn't it the first reply??
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2019-09-18 12:37 +0100 |
| Message-ID | <87sgot7s3g.fsf@bsb.me.uk> |
| In reply to | #210986 |
George Greene <greeneg@email.unc.edu> writes:
> On Tuesday, September 17, 2019 at 10:37:44 PM UTC-4, Ben Bacarisse wrote:
>> I mentioned that book because it's an example of someone who /does/ talk
>> about limits of sequences of sets.
>
> Well, sure, but the point is, if that's what you had to do, then that looks
> OLD AND RARE.
Old matters because it's not a new and trendy notion.
>> He does so using lim sup and lim inf,
>> and then using characteristic functions.
>
> Well, that, on the other hand, is the same thing the current wikipedia
> treatment says. The wikipedia treatment, however, has the
> disadvantage that in the VERY FIRST line, it says
>>>> limit of a sequence of sets A1, A2,
>>>> ... (subsets of a common set X)
> ^^^^^^^^^^^^^^^^^^^^^^^
> And the requirement that every set in the sequence be a subset of the same set
> really is almost restrictive enough to make the concept generally non-existent,
> unless the sequence is itself representable as 1 set and the bounding set X
> could then just be its union. And AGAIN, THAT is a rare case -- it specifically
> invites X to be a set OF NUMBERS as opposed to of sets. Or at least
> of things with an order/metric/distance between them.
The "metric" between elements of the individual sets is equality. You
seem to be knowledgeable person (and I am not knowledgeable in this
area) so I wonder which one of us has the wrong end of the stick.
> The only thing that can be said in favor of this definition in this
> context is that it directly refutes WM (the bounding set is N),
Again, I'm not sure what you mean here by the bounding set being N. His
original post was very general and said "if a sequence of sets has
exclusively non-empty elements, then the limit cannot be empty"
following on from other threads (which he just walks away from when the
inconsistencies become overwhelming) discussing lim { k∈N | k >= n }.
Shortly afterwards he gave another example -- the limit of a set of open
intervals of R: lim (0, 1/k).
> so why wasn't it the first reply??
One of the very first replies asked him to define the notion of set
sequence limit and he gave the "usual" one ("scare quotes" because how
usual it is is the subject of this sub-thread) and his own.
--
Ben.
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| From | George Greene <greeneg@email.unc.edu> |
|---|---|
| Date | 2019-09-18 21:01 -0700 |
| Message-ID | <12eb49b8-1885-4e1f-a03a-2594ea626889@googlegroups.com> |
| In reply to | #210997 |
On Wednesday, September 18, 2019 at 7:37:42 AM UTC-4, Ben Bacarisse wrote:
> George Greene <greeneg@email.unc.edu> writes:
>
> > On Tuesday, September 17, 2019 at 10:37:44 PM UTC-4, Ben Bacarisse wrote:
> >> I mentioned that book because it's an example of someone who /does/ talk
> >> about limits of sequences of sets.
> >
> > Well, sure, but the point is, if that's what you had to do, then that looks
> > OLD AND RARE.
>
> Old matters because it's not a new and trendy notion.
It does NOT matter, not in the way YOU think it does.
It matters in the way *I* think it does because IT IS NOT COMMONLY CURRENTLY
USED FOR ANYTHING BY ANYBODY, except maybe topology, BECAUSE IT IS *NOT*ANY*
*GOOD* for anything!! That is HOW IT GOT TO BE *OLD*,
as OPPOSED to CURRENT.
About the definitions of lim sup and lim inf in this treatment,
Did you or did you not notice that one of them is
a union of intersections, while the other is an intersection of unions?
GUESS WHAT: SINCE BOTH INTERSECTION AND UNION ARE COMMUTATIVE,
the ORDER of the SEQUENCE
*DOES*
*NOT*
*M*A*T*T*E*R*!!
What is fundamentally going on has NOTHING WHATSOEVER TO DO
with SEQUENCE in that definition!!!
> > And the requirement that every set in the
> > sequence be a subset of the same set
> > really is almost restrictive enough to make
> > the concept generally non-existent,
> > unless the sequence is itself representable as 1 set
> > and the bounding set X
> > could then just be its union. And AGAIN, THAT is a rare
> > case -- it specifically
> > invites X to be a set OF NUMBERS as opposed to of sets. Or at least
> > of things with an order/metric/distance between them.
>
> The "metric" between elements of the individual sets is equality.
That's ALL it is and THAT AIN'T SHIT.
Good grief. Have YOU ever seen a vector space where every two things
that were not identical were the same maximal distance away from each
other? COME ON!! THIS version AIN'T SHIT. This version IS DEGENERATE.
It is also, I repeat, NOT EVEN SEQUENTIAL.
> > The only thing that can be said in favor of this definition in this
> > context is that it directly refutes WM (the bounding set is N),
>
> Again, I'm not sure what you mean here by the bounding set being N.
Okay, it is FULLY within my power TO CLARIFY that for you.
THIS DEFINITION SAYS that when defining lim for a sequence of sets,
ALL THE SETS IN THE SEQUENCE MUST BE ELEMENTS OF THE POWERSET OF
*ONE*AND*THE*SAME* set, X. Otherwise the definition DOES NOT EVEN APPLY
(I repeat, it's worthless outside a few specialized areas like topology).
In WM's case, we are talking about a sequence OF ENDSEGMENTS OF N,
so every set in the sequence is a subset of N, so THIS DEFINITION DOES
apply. N is "the bounding set for" his sequence Every element of the sequence
is a set that is bounded by (is a subset of, does not have an element outside
the bounds of) N.
His
> original post was very general and said "if a sequence of sets has
> exclusively non-empty elements, then the limit cannot be empty"
> following on from other threads (which he just walks away from when the
> inconsistencies become overwhelming) discussing lim { k∈N | k >= n }.
Yes, but he later started talking about the sequence of endsegments of N
as a classic example of a sequence of non-empty set that "could not have" an
empty limit, EVEN THOUGH IT DOES -- even though lim sup and lim inf of
that sequence according to the definition you are recommending are both
the empty set.
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2019-09-19 22:11 +0100 |
| Message-ID | <87muf056ue.fsf@bsb.me.uk> |
| In reply to | #211066 |
George Greene <greeneg@email.unc.edu> writes: > On Wednesday, September 18, 2019 at 7:37:42 AM UTC-4, Ben Bacarisse wrote: >> George Greene <greeneg@email.unc.edu> writes: >> >> > On Tuesday, September 17, 2019 at 10:37:44 PM UTC-4, Ben Bacarisse wrote: >> >> I mentioned that book because it's an example of someone who /does/ talk >> >> about limits of sequences of sets. >> > >> > Well, sure, but the point is, if that's what you had to do, then that looks >> > OLD AND RARE. >> >> Old matters because it's not a new and trendy notion. > > It does NOT matter, not in the way YOU think it does. > It matters in the way *I* think it does because IT IS NOT COMMONLY CURRENTLY > USED FOR ANYTHING BY ANYBODY, except maybe topology, BECAUSE IT IS *NOT*ANY* > *GOOD* for anything!! That is HOW IT GOT TO BE *OLD*, > as OPPOSED to CURRENT. OK. I don't know the field so I don't know what's current. I was taught this stuff decades ago. > About the definitions of lim sup and lim inf in this treatment, > Did you or did you not notice that one of them is > a union of intersections, while the other is an intersection of unions? > GUESS WHAT: SINCE BOTH INTERSECTION AND UNION ARE COMMUTATIVE, > the ORDER of the SEQUENCE > *DOES* > *NOT* > *M*A*T*T*E*R*!! > What is fundamentally going on has NOTHING WHATSOEVER TO DO > with SEQUENCE in that definition!!! Yes of course I do. I talk of sequences because WM does, so that he can't go off on yet another a tangent. And set sequences have cropped up in another guise -- as sequences of functions where the order matters, a least superficially. WM's textbook gives two definitions for limits of function sequences (the two usual ones) so however he likes to paint it, he endorses /some/ meaning for the term "limit" when applied to sequences of sets (two, infact). That contradicts his claim in the head post. >> > And the requirement that every set in the >> > sequence be a subset of the same set >> > really is almost restrictive enough to make >> > the concept generally non-existent, >> > unless the sequence is itself representable as 1 set >> > and the bounding set X >> > could then just be its union. And AGAIN, THAT is a rare >> > case -- it specifically >> > invites X to be a set OF NUMBERS as opposed to of sets. Or at least >> > of things with an order/metric/distance between them. >> >> The "metric" between elements of the individual sets is equality. > > That's ALL it is and THAT AIN'T SHIT. > Good grief. Have YOU ever seen a vector space where every two things > that were not identical were the same maximal distance away from each > other? COME ON!! THIS version AIN'T SHIT. This version IS DEGENERATE. > It is also, I repeat, NOT EVEN SEQUENTIAL. > > >> > The only thing that can be said in favor of this definition in this >> > context is that it directly refutes WM (the bounding set is N), >> >> Again, I'm not sure what you mean here by the bounding set being N. > > Okay, it is FULLY within my power TO CLARIFY that for you. > THIS DEFINITION SAYS that when defining lim for a sequence of sets, > ALL THE SETS IN THE SEQUENCE MUST BE ELEMENTS OF THE POWERSET OF > *ONE*AND*THE*SAME* set, X. Otherwise the definition DOES NOT EVEN APPLY > (I repeat, it's worthless outside a few specialized areas like > topology). OK. I've not heard the term used like that before. Can you say more about what you meant further up: "unless the sequence is itself representable as 1 set and the bounding set X"? I did not follow that at all, and now that I know you meant the LUB of the collection of sets, what is the "1 set"? I get the feeling there is something important here. -- Ben.
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| From | Fred <Fred@nowhere.invalid> |
|---|---|
| Date | 2019-09-19 22:21 +0100 |
| Message-ID | <qm0rgd$rum$1@news.albasani.net> |
| In reply to | #211066 |
On 19/09/2019 05:01, George Greene wrote:
> On Wednesday, September 18, 2019 at 7:37:42 AM UTC-4, Ben Bacarisse wrote:
>> George Greene <greeneg@email.unc.edu> writes:
>>
>>> On Tuesday, September 17, 2019 at 10:37:44 PM UTC-4, Ben Bacarisse wrote:
>>>> I mentioned that book because it's an example of someone who /does/ talk
>>>> about limits of sequences of sets.
>>>
>>> Well, sure, but the point is, if that's what you had to do, then that looks
>>> OLD AND RARE.
>>
>> Old matters because it's not a new and trendy notion.
>
> It does NOT matter, not in the way YOU think it does.
> It matters in the way *I* think it does because IT IS NOT COMMONLY CURRENTLY
> USED FOR ANYTHING BY ANYBODY, except maybe topology, BECAUSE IT IS *NOT*ANY*
> *GOOD* for anything!! That is HOW IT GOT TO BE *OLD*,
> as OPPOSED to CURRENT.
> About the definitions of lim sup and lim inf in this treatment,
> Did you or did you not notice that one of them is
> a union of intersections, while the other is an intersection of unions?
> GUESS WHAT: SINCE BOTH INTERSECTION AND UNION ARE COMMUTATIVE,
> the ORDER of the SEQUENCE
> *DOES*
> *NOT*
> *M*A*T*T*E*R*!!
> What is fundamentally going on has NOTHING WHATSOEVER TO DO
> with SEQUENCE in that definition!!!
>
>>> And the requirement that every set in the
>>> sequence be a subset of the same set
>>> really is almost restrictive enough to make
>>> the concept generally non-existent,
>>> unless the sequence is itself representable as 1 set
>>> and the bounding set X
>>> could then just be its union. And AGAIN, THAT is a rare
>>> case -- it specifically
>>> invites X to be a set OF NUMBERS as opposed to of sets. Or at least
>>> of things with an order/metric/distance between them.
>>
>> The "metric" between elements of the individual sets is equality.
>
> That's ALL it is and THAT AIN'T SHIT.
> Good grief. Have YOU ever seen a vector space where every two things
> that were not identical were the same maximal distance away from each
> other? COME ON!! THIS version AIN'T SHIT. This version IS DEGENERATE.
> It is also, I repeat, NOT EVEN SEQUENTIAL.
>
>
>>> The only thing that can be said in favor of this definition in this
>>> context is that it directly refutes WM (the bounding set is N),
>>
>> Again, I'm not sure what you mean here by the bounding set being N.
>
> Okay, it is FULLY within my power TO CLARIFY that for you.
> THIS DEFINITION SAYS that when defining lim for a sequence of sets,
> ALL THE SETS IN THE SEQUENCE MUST BE ELEMENTS OF THE POWERSET OF
> *ONE*AND*THE*SAME* set, X. Otherwise the definition DOES NOT EVEN APPLY
The definition applies whenever lim inf = lim sup.
> (I repeat, it's worthless outside a few specialized areas like topology).
>
> In WM's case, we are talking about a sequence OF ENDSEGMENTS OF N,
> so every set in the sequence is a subset of N, so THIS DEFINITION DOES
> apply. N is "the bounding set for" his sequence Every element of the sequence
> is a set that is bounded by (is a subset of, does not have an element outside
> the bounds of) N.
>
> His
>> original post was very general and said "if a sequence of sets has
>> exclusively non-empty elements, then the limit cannot be empty"
>> following on from other threads (which he just walks away from when the
>> inconsistencies become overwhelming) discussing lim { k∈N | k >= n }.
>
> Yes, but he later started talking about the sequence of endsegments of N
> as a classic example of a sequence of non-empty set that "could not have" an
> empty limit, EVEN THOUGH IT DOES -- even though lim sup and lim inf of
> that sequence according to the definition you are recommending are both
> the empty set.
>
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