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Groups > sci.logic > #210463 > unrolled thread
| Started by | peteolcott <Here@Home> |
|---|---|
| First post | 2019-09-11 12:05 -0500 |
| Last post | 2019-09-11 23:15 -0500 |
| Articles | 20 on this page of 322 — 13 participants |
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Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) peteolcott <Here@Home> - 2019-09-11 12:05 -0500
Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) Fred <Fred@nowhere.invalid> - 2019-09-11 18:37 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-11 18:32 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Fred <Fred@nowhere.invalid> - 2019-09-12 01:15 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-11 19:30 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) temakoutera62@gmail.com - 2019-09-11 17:32 -0700
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Fred <Fred@nowhere.invalid> - 2019-09-12 01:38 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Fred <Fred@nowhere.invalid> - 2019-09-12 01:49 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) George Greene <greeneg@email.unc.edu> - 2019-09-14 18:22 -0700
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Fred <Fred@nowhere.invalid> - 2019-09-15 02:32 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-14 21:19 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-15 03:29 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-14 21:41 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-15 04:19 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-14 22:46 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-15 13:02 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-16 10:26 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-17 00:21 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-16 18:24 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-17 02:10 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-16 20:41 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-17 03:34 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-16 22:43 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-16 23:12 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Fred <Fred@nowhere.invalid> - 2019-09-17 09:27 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-17 14:26 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-17 11:53 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-18 03:55 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-17 22:19 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 21:23 -0600
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-17 22:31 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 22:04 -0600
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-17 23:21 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 22:40 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 00:27 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 23:55 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 00:59 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 00:12 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 10:15 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 09:35 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 11:00 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 10:14 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 11:26 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 10:42 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 13:10 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 12:36 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 13:49 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 12:59 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 14:12 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) Fred <Fred@nowhere.invalid> - 2019-09-18 20:25 +0100
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 13:30 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 14:45 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 14:14 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 15:29 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 14:38 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 15:56 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 20:38 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 22:03 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 21:33 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) Fred <Fred@nowhere.invalid> - 2019-09-19 16:35 +0100
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-19 10:35 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) Fred <Fred@nowhere.invalid> - 2019-09-19 16:46 +0100
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-19 11:42 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) Fred <Fred@nowhere.invalid> - 2019-09-19 17:58 +0100
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-19 12:43 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) Fred <Fred@nowhere.invalid> - 2019-09-19 19:34 +0100
Re: Provably unprovable eliminates incompleteness (NO GAPS) Fred <Fred@nowhere.invalid> - 2019-09-18 19:42 +0100
Re: Provably unprovable eliminates incompleteness (NO GAPS) Fred <Fred@nowhere.invalid> - 2019-09-18 18:16 +0100
Re: Provably unprovable eliminates incompleteness (NO GAPS) Fred <Fred@nowhere.invalid> - 2019-09-18 18:12 +0100
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-19 10:24 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) Fred <Fred@nowhere.invalid> - 2019-09-19 16:39 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Fred <Fred@nowhere.invalid> - 2019-09-18 18:05 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-17 20:41 -0700
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-18 05:15 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-17 23:32 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 23:21 -0600
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-18 18:42 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-18 12:58 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Fred <Fred@nowhere.invalid> - 2019-09-18 19:05 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-19 01:42 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-18 20:46 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Fred <Fred@nowhere.invalid> - 2019-09-19 02:52 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-19 22:24 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-19 17:13 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-20 01:05 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-19 19:49 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-20 02:34 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-19 20:54 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-20 03:32 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Fred <Fred@nowhere.invalid> - 2019-09-20 14:16 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Fred <Fred@nowhere.invalid> - 2019-09-20 17:49 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-20 07:16 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-20 21:37 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-20 15:53 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-21 00:10 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-21 09:45 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Fred <Fred@nowhere.invalid> - 2019-09-21 17:20 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Jim Burns <james.g.burns@att.net> - 2019-09-21 14:39 -0400
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-22 13:32 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-23 19:50 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-24 02:27 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-24 15:22 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-21 15:22 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Fred <Fred@nowhere.invalid> - 2019-09-18 18:02 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Fred <Fred@nowhere.invalid> - 2019-09-17 09:10 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Fred <Fred@nowhere.invalid> - 2019-09-17 09:06 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-14 22:54 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Fred <Fred@nowhere.invalid> - 2019-09-15 12:06 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-12 01:42 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-11 20:03 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-12 02:50 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-11 20:59 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-12 04:07 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-11 22:16 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-12 04:20 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-11 22:22 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-12 16:33 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-12 11:14 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-13 02:00 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Deceptive Strawman by Ben peteolcott <Here@Home> - 2019-09-12 21:39 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Deceptive Strawman by Ben Fred <Fred@nowhere.invalid> - 2019-09-13 16:38 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Deceptive Strawman by Ben peteolcott <Here@Home> - 2019-09-13 11:48 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Deceptive Strawman by Ben Fred <Fred@nowhere.invalid> - 2019-09-13 17:54 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-14 00:51 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-14 17:23 +0100
Re: Cantor was wrong about cardinality --- APOLOGY peteolcott <Here@Home> - 2019-09-14 13:09 -0500
Re: Cantor was wrong about cardinality --- APOLOGY Fred <Fred@nowhere.invalid> - 2019-09-14 19:28 +0100
Re: Cantor was wrong about cardinality --- APOLOGY Fred <Fred@nowhere.invalid> - 2019-09-14 19:33 +0100
Re: Cantor was wrong about cardinality --- APOLOGY peteolcott <Here@Home> - 2019-09-14 14:27 -0500
Re: Cantor was wrong about cardinality --- APOLOGY Fred <Fred@nowhere.invalid> - 2019-09-14 20:30 +0100
Re: Cantor was wrong about cardinality --- APOLOGY peteolcott <Here@Home> - 2019-09-14 14:41 -0500
Re: Cantor was wrong about cardinality --- APOLOGY Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-14 13:02 -0700
Re: Cantor was wrong about cardinality --- APOLOGY André G. Isaak <agisaak@gm.invalid> - 2019-09-14 14:08 -0600
Re: Cantor was wrong about cardinality --- APOLOGY Fred <Fred@nowhere.invalid> - 2019-09-14 21:18 +0100
Re: Cantor was wrong about cardinality --- APOLOGY André G. Isaak <agisaak@gm.invalid> - 2019-09-14 14:25 -0600
Re: Cantor was wrong about cardinality --- APOLOGY peteolcott <Here@Home> - 2019-09-14 16:03 -0500
Re: Cantor was wrong about cardinality --- APOLOGY Fred <Fred@nowhere.invalid> - 2019-09-14 22:19 +0100
Re: Cantor was wrong about cardinality --- APOLOGY peteolcott <Here@Home> - 2019-09-14 15:45 -0500
Re: Cantor was wrong about cardinality --- APOLOGY Fred <Fred@nowhere.invalid> - 2019-09-14 21:55 +0100
Re: Cantor was wrong about cardinality --- APOLOGY peteolcott <Here@Home> - 2019-09-14 16:06 -0500
Re: Cantor was wrong about cardinality --- APOLOGY Fred <Fred@nowhere.invalid> - 2019-09-14 22:20 +0100
Re: Cantor was wrong about cardinality --- APOLOGY Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-15 00:54 +0100
Re: Cantor was wrong about cardinality --- APOLOGY Exfalso Quodlibet <exflaso.quodlibet@gmail.com> - 2019-09-14 14:07 -0700
Re: Cantor was wrong about cardinality --- APOLOGY Fred <Fred@nowhere.invalid> - 2019-09-14 21:14 +0100
Re: Cantor was wrong about cardinality --- APOLOGY peteolcott <Here@Home> - 2019-09-14 16:01 -0500
Re: Cantor was wrong about cardinality --- APOLOGY Fred <Fred@nowhere.invalid> - 2019-09-14 22:17 +0100
Re: Cantor was wrong about cardinality --- APOLOGY Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-15 00:47 +0100
Re: Cantor was wrong about cardinality --- APOLOGY peteolcott <Here@Home> - 2019-09-14 19:24 -0500
Re: Cantor was wrong about cardinality --- APOLOGY Fred <Fred@nowhere.invalid> - 2019-09-15 02:20 +0100
Re: Cantor was wrong about cardinality --- APOLOGY Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-15 02:43 +0100
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-14 21:33 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-15 04:09 +0100
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-14 22:40 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? André G. Isaak <agisaak@gm.invalid> - 2019-09-14 22:40 -0600
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? Fred <Fred@nowhere.invalid> - 2019-09-15 12:05 +0100
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-15 12:33 +0100
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-16 10:17 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? André G. Isaak <agisaak@gm.invalid> - 2019-09-16 09:26 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 10:36 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Fred <Fred@nowhere.invalid> - 2019-09-16 16:50 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Fred <Fred@nowhere.invalid> - 2019-09-16 16:55 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 10:06 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 11:22 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 10:38 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 11:59 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 11:10 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 13:27 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 12:34 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 14:19 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Fred <Fred@nowhere.invalid> - 2019-09-16 20:28 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 13:35 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 14:51 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Fred <Fred@nowhere.invalid> - 2019-09-16 21:02 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 14:15 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 16:18 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 16:00 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Keith Thompson <kst-u@mib.org> - 2019-09-16 15:29 -0700
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 18:14 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Jeff Barnett <jbb@notatt.com> - 2019-09-16 23:34 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Fred <Fred@nowhere.invalid> - 2019-09-17 08:16 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 14:25 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 16:21 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 16:01 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Fred <Fred@nowhere.invalid> - 2019-09-17 08:21 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Keith Thompson <kst-u@mib.org> - 2019-09-16 15:46 -0700
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 18:07 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Me <invalid@invalid.de> - 2019-09-17 01:09 +0200
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2019-09-17 02:33 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 20:53 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 21:55 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Keith Thompson <kst-u@mib.org> - 2019-09-16 19:21 -0700
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 21:40 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Keith Thompson <kst-u@mib.org> - 2019-09-16 21:38 -0700
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (some concurrence). peteolcott <Here@Home> - 2019-09-17 00:24 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (some concurrence). peteolcott <Here@Home> - 2019-09-17 11:39 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (some concurrence). Fred <Fred@nowhere.invalid> - 2019-09-17 17:55 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (some concurrence). Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2019-09-17 18:38 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (some concurrence). peteolcott <Here@Home> - 2019-09-17 13:17 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (some concurrence). Keith Thompson <kst-u@mib.org> - 2019-09-17 11:13 -0700
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (some concurrence). peteolcott <Here@Home> - 2019-09-17 13:29 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Fred <Fred@nowhere.invalid> - 2019-09-17 08:36 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (Concurrence) peteolcott <Here@Home> - 2019-09-17 11:20 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (Concurrence) Fred <Fred@nowhere.invalid> - 2019-09-17 17:31 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (Concurrence) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 10:56 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (Concurrence) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 11:29 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2019-09-17 18:32 +0100
Re: Cantor was wrong about cardinality --- (Correction) peteolcott <Here@Home> - 2019-09-17 12:49 -0500
Re: Cantor was wrong about cardinality --- (Correction) Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2019-09-17 19:59 +0100
Re: Cantor was wrong about cardinality --- (Correction) peteolcott <Here@Home> - 2019-09-17 14:09 -0500
Re: Cantor was wrong about cardinality --- (Correction) Fred <Fred@nowhere.invalid> - 2019-09-17 20:33 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Keith Thompson <kst-u@mib.org> - 2019-09-17 11:42 -0700
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-17 14:01 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Fred <Fred@nowhere.invalid> - 2019-09-17 20:29 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-17 17:51 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Fred <Fred@nowhere.invalid> - 2019-09-18 17:55 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-17 15:58 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-18 04:36 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Fred <Fred@nowhere.invalid> - 2019-09-17 08:27 +0100
Re: Cantor was wrong about cardinality --- (Mike Terry Concurrence ] peteolcott <Here@Home> - 2019-09-19 11:46 -0500
Re: Cantor was wrong about cardinality --- (Mike Terry Concurrence ] Fred <Fred@nowhere.invalid> - 2019-09-19 18:00 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Fred <Fred@nowhere.invalid> - 2019-09-17 08:23 +0100
Re: Cantor was wrong about cardinality --- [ Keith Thompson Concurrence ] peteolcott <Here@Home> - 2019-09-19 11:52 -0500
Re: Cantor was wrong about cardinality --- [ Keith Thompson Concurrence ] peteolcott <Here@Home> - 2019-09-19 11:55 -0500
Re: Cantor was wrong about cardinality --- [ Keith Thompson Concurrence ] Fred <Fred@nowhere.invalid> - 2019-09-19 18:09 +0100
Re: Cantor was wrong about cardinality --- [ Keith Thompson Concurrence ] Fred <Fred@nowhere.invalid> - 2019-09-19 18:22 +0100
Re: Cantor was wrong about cardinality --- [ Keith Thompson Concurrence ] Keith Thompson <kst-u@mib.org> - 2019-09-19 14:44 -0700
Re: Cantor was wrong about cardinality --- [ Keith Thompson Concurrence ] peteolcott <Here@Home> - 2019-09-19 17:31 -0500
Re: Cantor was wrong about cardinality --- [ Keith Thompson Concurrence ] Fred <Fred@nowhere.invalid> - 2019-09-20 14:11 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Fred <Fred@nowhere.invalid> - 2019-09-16 19:47 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Fred <Fred@nowhere.invalid> - 2019-09-16 18:16 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 11:02 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Fred <Fred@nowhere.invalid> - 2019-09-16 18:08 +0100
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-17 01:49 +0100
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-16 20:01 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-17 03:25 +0100
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-16 22:39 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-17 14:26 +0100
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-17 11:55 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? Python <python@invalid> - 2019-09-17 18:59 +0200
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-18 04:01 +0100
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-17 22:23 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-18 05:19 +0100
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? Keith Thompson <kst-u@mib.org> - 2019-09-18 01:39 -0700
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-18 09:59 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? André G. Isaak <agisaak@gm.invalid> - 2019-09-18 09:21 -0600
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-18 10:41 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? Keith Thompson <kst-u@mib.org> - 2019-09-18 13:55 -0700
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-18 16:17 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? Fred <Fred@nowhere.invalid> - 2019-09-18 23:09 +0100
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? Fred <Fred@nowhere.invalid> - 2019-09-18 23:02 +0100
Re: Cantor was wrong about cardinality --- [ CONCURRENCE ] peteolcott <Here@Home> - 2019-09-18 11:10 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? Fred <Fred@nowhere.invalid> - 2019-09-18 18:39 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Fred <Fred@nowhere.invalid> - 2019-09-12 12:15 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-12 10:37 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Fred <Fred@nowhere.invalid> - 2019-09-12 16:56 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 00:27 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Fred <Fred@nowhere.invalid> - 2019-09-13 16:43 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 11:51 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Fred <Fred@nowhere.invalid> - 2019-09-13 18:12 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-12 09:05 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-12 10:49 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-12 10:02 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 00:18 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-12 23:24 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 10:01 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-13 09:13 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-13 09:56 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 11:40 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-13 11:03 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 12:48 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-13 12:25 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-13 22:35 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-14 16:42 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Fred <Fred@nowhere.invalid> - 2019-09-14 22:54 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 11:28 -0500
Re: Cantor was wrong about cardinality --- COMPLETE peteolcott <Here@Home> - 2019-09-14 11:34 -0500
Re: Cantor was wrong about cardinality --- COMPLETE Fred <Fred@nowhere.invalid> - 2019-09-14 18:05 +0100
Re: Cantor was wrong about cardinality --- COMPLETE (Andy Goofed) peteolcott <Here@Home> - 2019-09-14 18:00 -0500
Re: Cantor was wrong about cardinality --- COMPLETE (Andy Goofed) Fred <Fred@nowhere.invalid> - 2019-09-15 02:04 +0100
Re: Cantor was wrong about cardinality --- COMPLETE (Andy really did Goof) peteolcott <Here@Home> - 2019-09-16 09:42 -0500
Re: Cantor was wrong about cardinality --- COMPLETE (Andy really did Goof) André G. Isaak <agisaak@gm.invalid> - 2019-09-16 09:13 -0600
Re: Cantor was wrong about cardinality --- COMPLETE (Andy really did Goof) Fred <Fred@nowhere.invalid> - 2019-09-16 16:15 +0100
Re: Cantor was wrong about cardinality --- COMPLETE Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-15 01:26 +0100
Re: Cantor was wrong about cardinality --- COMPLETE peteolcott <Here@Home> - 2019-09-14 19:32 -0500
Re: Cantor was wrong about cardinality --- COMPLETE peteolcott <Here@Home> - 2019-09-14 19:36 -0500
Re: Cantor was wrong about cardinality --- COMPLETE Fred <Fred@nowhere.invalid> - 2019-09-15 02:26 +0100
Re: Cantor was wrong about cardinality --- COMPLETE Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-15 02:54 +0100
Re: Cantor was wrong about cardinality --- COMPLETE peteolcott <Here@Home> - 2019-09-14 21:38 -0500
Re: Cantor was wrong about cardinality --- COMPLETE Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-15 04:14 +0100
Re: Cantor was wrong about cardinality --- COMPLETE peteolcott <Here@Home> - 2019-09-14 22:45 -0500
Re: Cantor was wrong about cardinality --- COMPLETE Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-15 12:53 +0100
Re: Cantor was wrong about cardinality --- COMPLETE peteolcott <Here@Home> - 2019-09-16 10:21 -0500
Re: Cantor was wrong about cardinality --- COMPLETE Fred <Fred@nowhere.invalid> - 2019-09-16 16:27 +0100
Re: Cantor was wrong about cardinality --- COMPLETE Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-17 00:35 +0100
Re: Cantor was wrong about cardinality --- COMPLETE Fred <Fred@nowhere.invalid> - 2019-09-15 11:53 +0100
Re: Cantor was wrong about cardinality --- COMPLETE Fred <Fred@nowhere.invalid> - 2019-09-15 02:22 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Fred <Fred@nowhere.invalid> - 2019-09-12 17:04 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 00:10 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-12 23:16 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 10:09 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-13 09:41 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 11:36 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-13 11:14 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 13:03 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Fred <Fred@nowhere.invalid> - 2019-09-13 19:08 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-13 12:34 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-14 16:38 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Fred <Fred@nowhere.invalid> - 2019-09-14 22:49 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-14 22:00 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Fred <Fred@nowhere.invalid> - 2019-09-13 16:45 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Fred <Fred@nowhere.invalid> - 2019-09-13 16:57 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 11:55 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-14 02:30 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-14 00:53 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) BEN LIED !!! peteolcott <Here@Home> - 2019-09-14 00:55 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) BEN LIED !!! peteolcott <Here@Home> - 2019-09-17 11:33 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-14 17:14 +0100
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) George Greene <greeneg@email.unc.edu> - 2019-09-14 18:24 -0700
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-14 21:25 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Fred <Fred@nowhere.invalid> - 2019-09-15 11:50 +0100
Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) George Greene <greeneg@email.unc.edu> - 2019-09-11 20:55 -0700
Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) peteolcott <Here@Home> - 2019-09-11 23:15 -0500
Page 2 of 17 — ← Prev page 1 [2] 3 4 … 17 Next page →
| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-16 20:41 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) |
| Message-ID | <trqdnSVNLtM7pB3AnZ2dnUU7-WHNnZ2d@giganews.com> |
| In reply to | #210863 |
On 9/16/2019 8:10 PM, Ben Bacarisse wrote: > peteolcott <Here@Home> writes: > >> On 9/16/2019 6:21 PM, Ben Bacarisse wrote: >>> peteolcott <Here@Home> writes: >>> <cut> >>>> https://www.encyclopediaofmath.org/index.php/Interval_and_segment >>>> An interval (open interval) is a set of points on a line lying between >>>> two fixed points A and B, where A and B themselves are considered not >>>> to belong to the interval. >>>> >>>> This clearly specifies that there is a point immediately after A and >>>> before B. >>> >>> No it doesn't. I don't think there is any way forward here because you >>> are simply reading things into that text that are not there. Normally >>> I'd suggest you go and read about the development of the reals, but >>> that's a non-starter for you. >> >> When we remove the beginning 0.5 from [0.0, 2.0] we derive [0.5, 2.0] >> with a first point of 0.5. > > You could be clearer, but you are not wrong: > > [0, 2] \ [0, 0.5) = [0.5, 2] with first point 0.5 > > but > > [0, 2] \ [0, 0.5] = (0.5, 2] with no first point. > You keep getting confused by this. [0, 2] and we remove the very first point (0, 2] then necessarily all of the rest of the points remain. Just because we don't have a real number to express the first point does not prove that there is no first point as everyone assumes. all of the rest of the points remain and they remain in their same order. >> When we remove the beginning 0.1 from [0.0, 2.0] we derive [0.1, 2.0] >> with a first point of 0.1. >> >> When we remove the beginning 0.0001 from [0.0, 2.0] we derive [0.0001, 2.0] >> with a first point of 0.0001. >> >> So when we remove a single point from the beginning of [0.0, 1.0] >> we end up with (0.0, 1.0] with beginning point that cannot be named. > > No, you /can/ name it: min (0, 1]. It still does not exist. > >> WE DON'T END UP WITHOUT A BEGINNING POINT. > > How often will you say that? I'll keep saying that you are wrong until > you learn what an open set is. > > Note that I am talking about real intervals because that is the context > of the definition you cite above. PO-points and PO-intervals are not > interesting to me. Can you at least agree that for the real interval > (0, 1] there is no first number? > > Despite prior experience, you might possible answer this question > because you seem to accept that, for reals at east, if x = min (0, 1] > then x/2 must be be in the real interval (0, 1] and yet less than min > (0, 1]. I say you accept this, because I've seen you claim that your > "points" are not subject to this argument. > -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2019-09-17 03:34 +0100 |
| Subject | Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) |
| Message-ID | <87sgovabwc.fsf@bsb.me.uk> |
| In reply to | #210866 |
peteolcott <Here@Home> writes: > On 9/16/2019 8:10 PM, Ben Bacarisse wrote: >> peteolcott <Here@Home> writes: >> >>> On 9/16/2019 6:21 PM, Ben Bacarisse wrote: >>>> peteolcott <Here@Home> writes: >>>> <cut> >>>>> https://www.encyclopediaofmath.org/index.php/Interval_and_segment >>>>> An interval (open interval) is a set of points on a line lying between >>>>> two fixed points A and B, where A and B themselves are considered not >>>>> to belong to the interval. >>>>> >>>>> This clearly specifies that there is a point immediately after A and >>>>> before B. >>>> >>>> No it doesn't. I don't think there is any way forward here because you >>>> are simply reading things into that text that are not there. Normally >>>> I'd suggest you go and read about the development of the reals, but >>>> that's a non-starter for you. >>> >>> When we remove the beginning 0.5 from [0.0, 2.0] we derive [0.5, 2.0] >>> with a first point of 0.5. >> >> You could be clearer, but you are not wrong: >> >> [0, 2] \ [0, 0.5) = [0.5, 2] with first point 0.5 >> >> but >> >> [0, 2] \ [0, 0.5] = (0.5, 2] with no first point. > > You keep getting confused by this. > [0, 2] and we remove the very first point (0, 2] > then necessarily all of the rest of the points remain. This is true. I did not say otherwise. > Just because we don't have a real number to express the > first point does not prove that there is no first point > as everyone assumes. No one assumes it. It's proven. > all of the rest of the points remain and they remain in > their same order. They do indeed. I don't see what you think I am confused by. You are simply assuming the set of remaining points has a least member in that "same order". It does not, for the reasons that have been pointed out many times. The following is important, but you don't read what people write, do you? >> Note that I am talking about real intervals because that is the context >> of the definition you cite above. PO-points and PO-intervals are not >> interesting to me. Can you at least agree that for the real interval >> (0, 1] there is no first number? I very much hope (against experience) that you will answer this question. >> Despite prior experience, you might possibly answer this question >> because you seem to accept that, for reals at east, if x = min (0, 1] >> then x/2 must be be in the real interval (0, 1] and yet less than min >> (0, 1]. I say you accept this, because I've seen you claim that your >> "points" are not subject to this argument. If your points are no real numbers, then can only keep talking about reals and real intervals until you defined PO-points and PO-intervals. -- Ben.
[toc] | [prev] | [next] | [standalone]
| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-16 22:43 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) |
| Message-ID | <j66dncFr9pPeyx3AnZ2dnUU7-XvNnZ2d@giganews.com> |
| In reply to | #210870 |
On 9/16/2019 9:34 PM, Ben Bacarisse wrote: > peteolcott <Here@Home> writes: > >> On 9/16/2019 8:10 PM, Ben Bacarisse wrote: >>> peteolcott <Here@Home> writes: >>> >>>> On 9/16/2019 6:21 PM, Ben Bacarisse wrote: >>>>> peteolcott <Here@Home> writes: >>>>> <cut> >>>>>> https://www.encyclopediaofmath.org/index.php/Interval_and_segment >>>>>> An interval (open interval) is a set of points on a line lying between >>>>>> two fixed points A and B, where A and B themselves are considered not >>>>>> to belong to the interval. >>>>>> >>>>>> This clearly specifies that there is a point immediately after A and >>>>>> before B. >>>>> >>>>> No it doesn't. I don't think there is any way forward here because you >>>>> are simply reading things into that text that are not there. Normally >>>>> I'd suggest you go and read about the development of the reals, but >>>>> that's a non-starter for you. >>>> >>>> When we remove the beginning 0.5 from [0.0, 2.0] we derive [0.5, 2.0] >>>> with a first point of 0.5. >>> >>> You could be clearer, but you are not wrong: >>> >>> [0, 2] \ [0, 0.5) = [0.5, 2] with first point 0.5 >>> >>> but >>> >>> [0, 2] \ [0, 0.5] = (0.5, 2] with no first point. >> >> You keep getting confused by this. >> [0, 2] and we remove the very first point (0, 2] >> then necessarily all of the rest of the points remain. > > This is true. I did not say otherwise. > >> Just because we don't have a real number to express the >> first point does not prove that there is no first point >> as everyone assumes. > > No one assumes it. It's proven. If you look at it on the basis of real numbers it is proven. If you look at it on the basis of infinitesimal numbers it is not proven. > >> all of the rest of the points remain and they remain in >> their same order. > > They do indeed. I don't see what you think I am confused by. You are > simply assuming the set of remaining points has a least member in that > "same order". It does not, for the reasons that have been pointed out > many times. > > The following is important, but you don't read what people write, do > you? > >>> Note that I am talking about real intervals because that is the context >>> of the definition you cite above. PO-points and PO-intervals are not >>> interesting to me. Can you at least agree that for the real interval >>> (0, 1] there is no first number? > > I very much hope (against experience) that you will answer this question. > >>> Despite prior experience, you might possibly answer this question >>> because you seem to accept that, for reals at east, if x = min (0, 1] >>> then x/2 must be be in the real interval (0, 1] and yet less than min >>> (0, 1]. I say you accept this, because I've seen you claim that your >>> "points" are not subject to this argument. > > If your points are no real numbers, then can only keep talking about > reals and real intervals until you defined PO-points and PO-intervals. > -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2019-09-16 23:12 -0600 |
| Subject | Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) |
| Message-ID | <qlpq0g$r2t$1@dont-email.me> |
| In reply to | #210874 |
On 2019-09-16 9:43 p.m., peteolcott wrote: > On 9/16/2019 9:34 PM, Ben Bacarisse wrote: >> peteolcott <Here@Home> writes: >> >>> Just because we don't have a real number to express the >>> first point does not prove that there is no first point >>> as everyone assumes. >> >> No one assumes it. It's proven. > > If you look at it on the basis of real numbers it is proven. > If you look at it on the basis of infinitesimal numbers it is not proven. We can't prove anything on the basis of infinitesimals until you actually explain how infinitesimals work. That means not just coming up with your notation R[I] where R is the real part and I the infinitesimal part, but also explaining how arithmetic works over these and how to evaluate relations like <, > etc. I've already pointed out that the definition of multiplication you offered doesn't work, but you've never come back to that point and offered a coherent account of how your system works. So here's a simple question: What is the actual *point* of having the infinitesimal portion in your numbers? You claim that your infinitesimal is indivisible such that there does not exist x such that 0.0[0] < x < 0.0[1]. But, for the real numbers, for all y and z such that y < z there is *always* some x such that y < z < z. So if the 'real' portion of your R[I] is actually real, we should be able to have some number a[0] which is less than 0.0[1]. Given this, any number expressible as R[I] should also be expressible as a simple real without making use of the infinitesimal portion. What makes your system any more useful than the following 'system': In my intermediatesimal system, every number consists of a rational part Q, and hundredths portion [H] which specifies an offsets in hundredths. So, for example, ⅜[5] specifies the rational number ⅝ with an offset of 5 one hundredths. Of what use would such a system be given that any number expressible in this system could be expressible by the rational portion alone? André -- To email remove 'invalid' & replace 'gm' with well known Google mail service.
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| From | Fred <Fred@nowhere.invalid> |
|---|---|
| Date | 2019-09-17 09:27 +0100 |
| Subject | Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) |
| Message-ID | <qlq5cr$sai$1@news.albasani.net> |
| In reply to | #210874 |
On 17/09/2019 04:43, peteolcott wrote: > On 9/16/2019 9:34 PM, Ben Bacarisse wrote: >> peteolcott <Here@Home> writes: >> >>> On 9/16/2019 8:10 PM, Ben Bacarisse wrote: >>>> peteolcott <Here@Home> writes: >>>> >>>>> On 9/16/2019 6:21 PM, Ben Bacarisse wrote: >>>>>> peteolcott <Here@Home> writes: >>>>>> <cut> >>>>>>> https://www.encyclopediaofmath.org/index.php/Interval_and_segment >>>>>>> An interval (open interval) is a set of points on a line lying >>>>>>> between >>>>>>> two fixed points A and B, where A and B themselves are considered >>>>>>> not >>>>>>> to belong to the interval. >>>>>>> >>>>>>> This clearly specifies that there is a point immediately after A and >>>>>>> before B. >>>>>> >>>>>> No it doesn't. I don't think there is any way forward here >>>>>> because you >>>>>> are simply reading things into that text that are not there. >>>>>> Normally >>>>>> I'd suggest you go and read about the development of the reals, but >>>>>> that's a non-starter for you. >>>>> >>>>> When we remove the beginning 0.5 from [0.0, 2.0] we derive [0.5, 2.0] >>>>> with a first point of 0.5. >>>> >>>> You could be clearer, but you are not wrong: >>>> >>>> [0, 2] \ [0, 0.5) = [0.5, 2] with first point 0.5 >>>> >>>> but >>>> >>>> [0, 2] \ [0, 0.5] = (0.5, 2] with no first point. >>> >>> You keep getting confused by this. >>> [0, 2] and we remove the very first point (0, 2] >>> then necessarily all of the rest of the points remain. >> >> This is true. I did not say otherwise. >> >>> Just because we don't have a real number to express the >>> first point does not prove that there is no first point >>> as everyone assumes. >> >> No one assumes it. It's proven. > > If you look at it on the basis of real numbers it is proven. > If you look at it on the basis of infinitesimal numbers it is not proven. You are in no position to make either claim. You seem not to know what the real numbers are. If you look in an analysis text you will find one (or maybe both) of two approaches - (a) axioms for real numbers, (b) a definition of real numbers in terms of rational numbers. If it's just (a) then it may be just assumed that there is something that satisfies those axioms. If it's both, then what's defined in (b) satisfies what's stated in (a). As for infinitesimals, there are a number of theories of them. I think (I'm far from sure) that the best known of them are Abraham Robinson's non-standard analysis and Edward Nelson's internal set theory. They are both over your head. I wonder if Keisler's /Elementary calculus, an infinitesimal approach/ is useful. It can be found here - http://www.math.wisc.edu/~keisler/calc.html. I know nothing about it other than that it exists.
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2019-09-17 14:26 +0100 |
| Subject | Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) |
| Message-ID | <87a7b39hqr.fsf@bsb.me.uk> |
| In reply to | #210874 |
peteolcott <Here@Home> writes: > On 9/16/2019 9:34 PM, Ben Bacarisse wrote: >> peteolcott <Here@Home> writes: >> >>> On 9/16/2019 8:10 PM, Ben Bacarisse wrote: >>>> peteolcott <Here@Home> writes: >>>> >>>>> On 9/16/2019 6:21 PM, Ben Bacarisse wrote: >>>>>> peteolcott <Here@Home> writes: >>>>>> <cut> >>>>>>> https://www.encyclopediaofmath.org/index.php/Interval_and_segment >>>>>>> An interval (open interval) is a set of points on a line lying between >>>>>>> two fixed points A and B, where A and B themselves are considered not >>>>>>> to belong to the interval. >>>>>>> >>>>>>> This clearly specifies that there is a point immediately after A and >>>>>>> before B. >>>>>> >>>>>> No it doesn't. I don't think there is any way forward here because you >>>>>> are simply reading things into that text that are not there. Normally >>>>>> I'd suggest you go and read about the development of the reals, but >>>>>> that's a non-starter for you. >>>>> >>>>> When we remove the beginning 0.5 from [0.0, 2.0] we derive [0.5, 2.0] >>>>> with a first point of 0.5. >>>> >>>> You could be clearer, but you are not wrong: >>>> >>>> [0, 2] \ [0, 0.5) = [0.5, 2] with first point 0.5 >>>> >>>> but >>>> >>>> [0, 2] \ [0, 0.5] = (0.5, 2] with no first point. >>> >>> You keep getting confused by this. >>> [0, 2] and we remove the very first point (0, 2] >>> then necessarily all of the rest of the points remain. >> >> This is true. I did not say otherwise. >> >>> Just because we don't have a real number to express the >>> first point does not prove that there is no first point >>> as everyone assumes. >> >> No one assumes it. It's proven. > > If you look at it on the basis of real numbers it is proven. I am glad you accept that. It is proven for any set that like those in the article you cited so you will forgive me (an others) for think you were talking about such sets. > If you look at it on the basis of infinitesimal numbers it is not > proven. It is also proven for all infinitesimals defined by actual mathematicians, but you can be forgiven for not know this because of the value you place on ignorance. It will also be true of your infinitesimals unless they are useless. You claim they are countable so they do not serve as a model for points on a line, but ayway, you can't define them (and trying to get you to will be like getting blood out of a stone) so exactly why they are useless will remain a mystery. -- Ben.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-17 11:53 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) |
| Message-ID | <mpudnd2-lMQNkhzAnZ2dnUU7-eXNnZ2d@giganews.com> |
| In reply to | #210907 |
On 9/17/2019 8:26 AM, Ben Bacarisse wrote: > peteolcott <Here@Home> writes: > >> On 9/16/2019 9:34 PM, Ben Bacarisse wrote: >>> peteolcott <Here@Home> writes: >>> >>>> On 9/16/2019 8:10 PM, Ben Bacarisse wrote: >>>>> peteolcott <Here@Home> writes: >>>>> >>>>>> On 9/16/2019 6:21 PM, Ben Bacarisse wrote: >>>>>>> peteolcott <Here@Home> writes: >>>>>>> <cut> >>>>>>>> https://www.encyclopediaofmath.org/index.php/Interval_and_segment >>>>>>>> An interval (open interval) is a set of points on a line lying between >>>>>>>> two fixed points A and B, where A and B themselves are considered not >>>>>>>> to belong to the interval. >>>>>>>> >>>>>>>> This clearly specifies that there is a point immediately after A and >>>>>>>> before B. >>>>>>> >>>>>>> No it doesn't. I don't think there is any way forward here because you >>>>>>> are simply reading things into that text that are not there. Normally >>>>>>> I'd suggest you go and read about the development of the reals, but >>>>>>> that's a non-starter for you. >>>>>> >>>>>> When we remove the beginning 0.5 from [0.0, 2.0] we derive [0.5, 2.0] >>>>>> with a first point of 0.5. >>>>> >>>>> You could be clearer, but you are not wrong: >>>>> >>>>> [0, 2] \ [0, 0.5) = [0.5, 2] with first point 0.5 >>>>> >>>>> but >>>>> >>>>> [0, 2] \ [0, 0.5] = (0.5, 2] with no first point. >>>> >>>> You keep getting confused by this. >>>> [0, 2] and we remove the very first point (0, 2] >>>> then necessarily all of the rest of the points remain. >>> >>> This is true. I did not say otherwise. >>> >>>> Just because we don't have a real number to express the >>>> first point does not prove that there is no first point >>>> as everyone assumes. >>> >>> No one assumes it. It's proven. >> >> If you look at it on the basis of real numbers it is proven. > > I am glad you accept that. It is proven for any set that like those in > the article you cited so you will forgive me (an others) for think you > were talking about such sets. > >> If you look at it on the basis of infinitesimal numbers it is not >> proven. > > It is also proven for all infinitesimals defined by actual > mathematicians, but you can be forgiven for not know this because of the > value you place on ignorance. > > It will also be true of your infinitesimals unless they are useless. > You claim they are countable so they do not serve as a model for points > on a line, but ayway, you can't define them (and trying to get you to > will be like getting blood out of a stone) so exactly why they are > useless will remain a mystery. > Keith Thompson understands that they could refute differing sized cardinality of infinite sets. On 9/16/2019 11:38 PM, Keith Thompson wrote: > Arbitrarily inventing a new notation, let _(0,1]_ be a set of, um, > something or other. It includes all the real numbers in (0,1], > and it may also include some other things, each of which is greater > than 0, less than or equal to 1, and somewhere on the "number line" > between 0 and 1. Some of the non-real elements might be points, > or infinitesimals. I'll assert that for any two members of *this* > set, exactly one of a<b, a=b, a>b is true. > > I can imagine that *this* set has a least element. For example > we can define "iota" as an infinitesimal, something that is not a > real number but that has a location on the number line, directly > adjacent to the real number 0. It is greater than 0 and less than > any positive real number. > > For the same argument I used to demonstrate that x is not the least > member of the interval *not* to apply to iota, one of the following > must be true: > > - iota/2 <= 0 > - iota/2 >= iota > - iota/2 is not defined > > Though we can define something like this, I'm having difficulty > thinking of a way to define it that is consistent *and useful*. > We need the non-real members of the interval to behave radically > differently under common arithmetic operations. Not that there's > anything wrong with that, but it seems to require inventing a set > that's even denser than the reals, but is nevertheless in some > vague sense discrete rather than continuous. As far as I can tell > (and it's very likely I've missed something), most proposed systems > of infinitesimals do not have this property. > -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2019-09-18 03:55 +0100 |
| Subject | Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) |
| Message-ID | <87pnjy8g96.fsf@bsb.me.uk> |
| In reply to | #210923 |
peteolcott <Here@Home> writes: > On 9/17/2019 8:26 AM, Ben Bacarisse wrote: >> peteolcott <Here@Home> writes: >> >>> On 9/16/2019 9:34 PM, Ben Bacarisse wrote: >>>> peteolcott <Here@Home> writes: >>>> >>>>> On 9/16/2019 8:10 PM, Ben Bacarisse wrote: >>>>>> peteolcott <Here@Home> writes: >>>>>> >>>>>>> On 9/16/2019 6:21 PM, Ben Bacarisse wrote: >>>>>>>> peteolcott <Here@Home> writes: >>>>>>>> <cut> >>>>>>>>> https://www.encyclopediaofmath.org/index.php/Interval_and_segment >>>>>>>>> An interval (open interval) is a set of points on a line lying between >>>>>>>>> two fixed points A and B, where A and B themselves are considered not >>>>>>>>> to belong to the interval. >>>>>>>>> >>>>>>>>> This clearly specifies that there is a point immediately after A and >>>>>>>>> before B. >>>>>>>> >>>>>>>> No it doesn't. I don't think there is any way forward here because you >>>>>>>> are simply reading things into that text that are not there. Normally >>>>>>>> I'd suggest you go and read about the development of the reals, but >>>>>>>> that's a non-starter for you. >>>>>>> >>>>>>> When we remove the beginning 0.5 from [0.0, 2.0] we derive [0.5, 2.0] >>>>>>> with a first point of 0.5. >>>>>> >>>>>> You could be clearer, but you are not wrong: >>>>>> >>>>>> [0, 2] \ [0, 0.5) = [0.5, 2] with first point 0.5 >>>>>> >>>>>> but >>>>>> >>>>>> [0, 2] \ [0, 0.5] = (0.5, 2] with no first point. >>>>> >>>>> You keep getting confused by this. >>>>> [0, 2] and we remove the very first point (0, 2] >>>>> then necessarily all of the rest of the points remain. >>>> >>>> This is true. I did not say otherwise. >>>> >>>>> Just because we don't have a real number to express the >>>>> first point does not prove that there is no first point >>>>> as everyone assumes. >>>> >>>> No one assumes it. It's proven. >>> >>> If you look at it on the basis of real numbers it is proven. >> >> I am glad you accept that. It is proven for any set that like those in >> the article you cited so you will forgive me (an others) for think you >> were talking about such sets. >> >>> If you look at it on the basis of infinitesimal numbers it is not >>> proven. >> >> It is also proven for all infinitesimals defined by actual >> mathematicians, but you can be forgiven for not know this because of the >> value you place on ignorance. >> >> It will also be true of your infinitesimals unless they are useless. >> You claim they are countable so they do not serve as a model for points >> on a line, but ayway, you can't define them (and trying to get you to >> will be like getting blood out of a stone) so exactly why they are >> useless will remain a mystery. > > Keith Thompson understands that they could refute differing sized > cardinality of infinite sets. No he doesn't. The trouble is, you don't understand what he's saying. In fact he agrees with me though he puts it more cautiously. I'll leave it here for those that /can/ follow: > On 9/16/2019 11:38 PM, Keith Thompson wrote: >> Arbitrarily inventing a new notation, let _(0,1]_ be a set of, um, >> something or other. It includes all the real numbers in (0,1], >> and it may also include some other things, each of which is greater >> than 0, less than or equal to 1, and somewhere on the "number line" >> between 0 and 1. Some of the non-real elements might be points, >> or infinitesimals. I'll assert that for any two members of *this* >> set, exactly one of a<b, a=b, a>b is true. >> >> I can imagine that *this* set has a least element. For example >> we can define "iota" as an infinitesimal, something that is not a >> real number but that has a location on the number line, directly >> adjacent to the real number 0. It is greater than 0 and less than >> any positive real number. >> >> For the same argument I used to demonstrate that x is not the least >> member of the interval *not* to apply to iota, one of the following >> must be true: >> >> - iota/2 <= 0 >> - iota/2 >= iota >> - iota/2 is not defined >> >> Though we can define something like this, I'm having difficulty >> thinking of a way to define it that is consistent *and useful*. >> We need the non-real members of the interval to behave radically >> differently under common arithmetic operations. Not that there's >> anything wrong with that, but it seems to require inventing a set >> that's even denser than the reals, but is nevertheless in some >> vague sense discrete rather than continuous. As far as I can tell >> (and it's very likely I've missed something), most proposed systems >> of infinitesimals do not have this property. -- Ben.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-17 22:19 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) |
| Message-ID | <q_Gdne4MsYTKPxzAnZ2dnUU7-fPNnZ2d@giganews.com> |
| In reply to | #210970 |
On 9/17/2019 9:55 PM, Ben Bacarisse wrote:
> peteolcott <Here@Home> writes:
>
>> On 9/17/2019 8:26 AM, Ben Bacarisse wrote:
>>> peteolcott <Here@Home> writes:
>>>
>>>> On 9/16/2019 9:34 PM, Ben Bacarisse wrote:
>>>>> peteolcott <Here@Home> writes:
>>>>>
>>>>>> On 9/16/2019 8:10 PM, Ben Bacarisse wrote:
>>>>>>> peteolcott <Here@Home> writes:
>>>>>>>
>>>>>>>> On 9/16/2019 6:21 PM, Ben Bacarisse wrote:
>>>>>>>>> peteolcott <Here@Home> writes:
>>>>>>>>> <cut>
>>>>>>>>>> https://www.encyclopediaofmath.org/index.php/Interval_and_segment
>>>>>>>>>> An interval (open interval) is a set of points on a line lying between
>>>>>>>>>> two fixed points A and B, where A and B themselves are considered not
>>>>>>>>>> to belong to the interval.
>>>>>>>>>>
>>>>>>>>>> This clearly specifies that there is a point immediately after A and
>>>>>>>>>> before B.
>>>>>>>>>
>>>>>>>>> No it doesn't. I don't think there is any way forward here because you
>>>>>>>>> are simply reading things into that text that are not there. Normally
>>>>>>>>> I'd suggest you go and read about the development of the reals, but
>>>>>>>>> that's a non-starter for you.
>>>>>>>>
>>>>>>>> When we remove the beginning 0.5 from [0.0, 2.0] we derive [0.5, 2.0]
>>>>>>>> with a first point of 0.5.
>>>>>>>
>>>>>>> You could be clearer, but you are not wrong:
>>>>>>>
>>>>>>> [0, 2] \ [0, 0.5) = [0.5, 2] with first point 0.5
>>>>>>>
>>>>>>> but
>>>>>>>
>>>>>>> [0, 2] \ [0, 0.5] = (0.5, 2] with no first point.
>>>>>>
>>>>>> You keep getting confused by this.
>>>>>> [0, 2] and we remove the very first point (0, 2]
>>>>>> then necessarily all of the rest of the points remain.
>>>>>
>>>>> This is true. I did not say otherwise.
>>>>>
>>>>>> Just because we don't have a real number to express the
>>>>>> first point does not prove that there is no first point
>>>>>> as everyone assumes.
>>>>>
>>>>> No one assumes it. It's proven.
>>>>
>>>> If you look at it on the basis of real numbers it is proven.
>>>
>>> I am glad you accept that. It is proven for any set that like those in
>>> the article you cited so you will forgive me (an others) for think you
>>> were talking about such sets.
>>>
>>>> If you look at it on the basis of infinitesimal numbers it is not
>>>> proven.
>>>
>>> It is also proven for all infinitesimals defined by actual
>>> mathematicians, but you can be forgiven for not know this because of the
>>> value you place on ignorance.
>>>
>>> It will also be true of your infinitesimals unless they are useless.
>>> You claim they are countable so they do not serve as a model for points
>>> on a line, but ayway, you can't define them (and trying to get you to
>>> will be like getting blood out of a stone) so exactly why they are
>>> useless will remain a mystery.
>>
>> Keith Thompson understands that they could refute differing sized
>> cardinality of infinite sets.
>
> No he doesn't. The trouble is, you don't understand what he's saying.
> In fact he agrees with me though he puts it more cautiously. I'll leave
> it here for those that /can/ follow:
>
I SUMMED IT UP MORE CONCISELY HERE:
THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0
On 9/16/2019 11:38 PM, Keith Thompson wrote:
> we can define "iota" as an infinitesimal, something that is not a
> real number but that has a location on the number line, directly
> adjacent to the real number 0. It is greater than 0 and less than
> any positive real number.
WE CAN SUBTRACT A SINGLE POINT FROM AN INTERVAL DERIVING ANOTHER
INTERVAL THAT DOES NOT HAVE THIS POINT
On 9/16/2019 8:33 PM, Mike Terry wrote:
> OK, so (A,B] = [A,B] - {A}, and that can be counted as an "operation
in which we get (A,B] from [A,B].
Mike Terry agreed with this verbatim: [A, B] - {A} = (A, B]
Infinitesimal Numbers would encode (A, B] as [A[1], B]
We merely extrapolate the very next elaboration
[A[1], B] - {A[1]} = (A[1], B] encoded as [A[2], B]
This shows how Infinitesimal Numbers directly reference immediately
adjacent points on a number line.
0.0[0] = 0.0
0.0[1] = the next point after 0.0
0.0[2] = 2 points after 0.0
0.0[3] = 3 points after 0.0
--
Copyright 2019 Pete Olcott All rights reserved
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein
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| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2019-09-17 21:23 -0600 |
| Subject | Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) |
| Message-ID | <qls7vu$1l9$1@dont-email.me> |
| In reply to | #210972 |
On 2019-09-17 9:19 p.m., peteolcott wrote:
> On 9/17/2019 9:55 PM, Ben Bacarisse wrote:
>> peteolcott <Here@Home> writes:
>>
<snip>
>>> Keith Thompson understands that they could refute differing sized
>>> cardinality of infinite sets.
>>
>> No he doesn't. The trouble is, you don't understand what he's saying.
>> In fact he agrees with me though he puts it more cautiously. I'll leave
>> it here for those that /can/ follow:
>>
>
> I SUMMED IT UP MORE CONCISELY HERE:
>
> THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0
> On 9/16/2019 11:38 PM, Keith Thompson wrote:
> > we can define "iota" as an infinitesimal, something that is not a
> > real number but that has a location on the number line, directly
> > adjacent to the real number 0. It is greater than 0 and less than
> > any positive real number.
>
> WE CAN SUBTRACT A SINGLE POINT FROM AN INTERVAL DERIVING ANOTHER
> INTERVAL THAT DOES NOT HAVE THIS POINT
> On 9/16/2019 8:33 PM, Mike Terry wrote:
> > OK, so (A,B] = [A,B] - {A}, and that can be counted as an "operation
> in which we get (A,B] from [A,B].
>
> Mike Terry agreed with this verbatim: [A, B] - {A} = (A, B]
> Infinitesimal Numbers would encode (A, B] as [A[1], B]
>
> We merely extrapolate the very next elaboration
> [A[1], B] - {A[1]} = (A[1], B] encoded as [A[2], B]
>
> This shows how Infinitesimal Numbers directly reference immediately
> adjacent points on a number line.
>
> 0.0[0] = 0.0
> 0.0[1] = the next point after 0.0
> 0.0[2] = 2 points after 0.0
> 0.0[3] = 3 points after 0.0
I'd asked earlier but got no response, so I'll ask again.
Do you accept that there is no *greatest* integer?
André
--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-17 22:31 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) |
| Message-ID | <UP6dnfNIrbNsORzAnZ2dnUU7-XfNnZ2d@giganews.com> |
| In reply to | #210974 |
On 9/17/2019 10:23 PM, André G. Isaak wrote:
> On 2019-09-17 9:19 p.m., peteolcott wrote:
>> On 9/17/2019 9:55 PM, Ben Bacarisse wrote:
>>> peteolcott <Here@Home> writes:
>>>
>
> <snip>
>
>>>> Keith Thompson understands that they could refute differing sized
>>>> cardinality of infinite sets.
>>>
>>> No he doesn't. The trouble is, you don't understand what he's saying.
>>> In fact he agrees with me though he puts it more cautiously. I'll leave
>>> it here for those that /can/ follow:
>>>
>>
>> I SUMMED IT UP MORE CONCISELY HERE:
>>
>> THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0
>> On 9/16/2019 11:38 PM, Keith Thompson wrote:
>> > we can define "iota" as an infinitesimal, something that is not a
>> > real number but that has a location on the number line, directly
>> > adjacent to the real number 0. It is greater than 0 and less than
>> > any positive real number.
>>
>> WE CAN SUBTRACT A SINGLE POINT FROM AN INTERVAL DERIVING ANOTHER
>> INTERVAL THAT DOES NOT HAVE THIS POINT
>> On 9/16/2019 8:33 PM, Mike Terry wrote:
>> > OK, so (A,B] = [A,B] - {A}, and that can be counted as an "operation
>> in which we get (A,B] from [A,B].
>>
>> Mike Terry agreed with this verbatim: [A, B] - {A} = (A, B]
>> Infinitesimal Numbers would encode (A, B] as [A[1], B]
>>
>> We merely extrapolate the very next elaboration
>> [A[1], B] - {A[1]} = (A[1], B] encoded as [A[2], B]
>>
>> This shows how Infinitesimal Numbers directly reference immediately
>> adjacent points on a number line.
>>
>> 0.0[0] = 0.0
>> 0.0[1] = the next point after 0.0
>> 0.0[2] = 2 points after 0.0
>> 0.0[3] = 3 points after 0.0
>
> I'd asked earlier but got no response, so I'll ask again.
>
> Do you accept that there is no *greatest* integer?
>
> André
>
Infinity is a process rather than a value, thus there is no greatest
integer and all infinite sets are likewise this exact same limitlessness.
Integer N = 0;
HERE:
N = N + 1;
goto HERE;
--
Copyright 2019 Pete Olcott All rights reserved
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein
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| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2019-09-17 22:04 -0600 |
| Subject | Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) |
| Message-ID | <qlsac4$bgp$1@dont-email.me> |
| In reply to | #210975 |
On 2019-09-17 9:31 p.m., peteolcott wrote:
> On 9/17/2019 10:23 PM, André G. Isaak wrote:
>> On 2019-09-17 9:19 p.m., peteolcott wrote:
>>> On 9/17/2019 9:55 PM, Ben Bacarisse wrote:
>>>> peteolcott <Here@Home> writes:
>>>>
>>
>> <snip>
>>
>>>>> Keith Thompson understands that they could refute differing
>>>>> sized cardinality of infinite sets.
>>>>
>>>> No he doesn't. The trouble is, you don't understand what he's
>>>> saying. In fact he agrees with me though he puts it more
>>>> cautiously. I'll leave it here for those that /can/ follow:
>>>>
>>>
>>> I SUMMED IT UP MORE CONCISELY HERE:
>>>
>>> THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0 On
>>> 9/16/2019 11:38 PM, Keith Thompson wrote:
>>>> we can define "iota" as an infinitesimal, something that is not
>>>> a real number but that has a location on the number line,
>>>> directly adjacent to the real number 0. It is greater than 0
>>>> and less than any positive real number.
>>>
>>> WE CAN SUBTRACT A SINGLE POINT FROM AN INTERVAL DERIVING ANOTHER
>>> INTERVAL THAT DOES NOT HAVE THIS POINT On 9/16/2019 8:33 PM, Mike
>>> Terry wrote:
>>>> OK, so (A,B] = [A,B] - {A}, and that can be counted as an
>>>> "operation
>>> in which we get (A,B] from [A,B].
>>>
>>> Mike Terry agreed with this verbatim: [A, B] - {A} = (A, B]
>>> Infinitesimal Numbers would encode (A, B] as [A[1], B]
>>>
>>> We merely extrapolate the very next elaboration [A[1], B] -
>>> {A[1]} = (A[1], B] encoded as [A[2], B]
>>>
>>> This shows how Infinitesimal Numbers directly reference
>>> immediately adjacent points on a number line.
>>>
>>> 0.0[0] = 0.0 0.0[1] = the next point after 0.0 0.0[2] = 2 points
>>> after 0.0 0.0[3] = 3 points after 0.0
>>
>> I'd asked earlier but got no response, so I'll ask again.
>>
>> Do you accept that there is no *greatest* integer?
>>
>> André
>>
>
> Infinity is a process rather than a value, thus there is no greatest
> integer and all infinite sets are likewise this exact same
> limitlessness.
OK. So if you agree that the set of positive integers {1, 2, 3, 4...}
has no greatest element, then you should also agree that this subset of
the positive rationals (in descending order) {1/1, 1/2, 1/3, 1/4...} has
no least element.
But if the rationals (which are a subset of the reals) have no least
element, then how do you reconcile this with your view that there exists
some smallest non-zero number which is immediately adjacent to zero?
André
--
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-17 23:21 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) |
| Message-ID | <Sb2dne2sKfMmLRzAnZ2dnUU7-a_NnZ2d@giganews.com> |
| In reply to | #210979 |
On 9/17/2019 11:04 PM, André G. Isaak wrote:
> On 2019-09-17 9:31 p.m., peteolcott wrote:
>> On 9/17/2019 10:23 PM, André G. Isaak wrote:
>>> On 2019-09-17 9:19 p.m., peteolcott wrote:
>>>> On 9/17/2019 9:55 PM, Ben Bacarisse wrote:
>>>>> peteolcott <Here@Home> writes:
>>>>>
>>>
>>> <snip>
>>>
>>>>>> Keith Thompson understands that they could refute differing
>>>>>> sized cardinality of infinite sets.
>>>>>
>>>>> No he doesn't. The trouble is, you don't understand what he's
>>>>> saying. In fact he agrees with me though he puts it more
>>>>> cautiously. I'll leave it here for those that /can/ follow:
>>>>>
>>>>
>>>> I SUMMED IT UP MORE CONCISELY HERE:
>>>>
>>>> THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0 On
>>>> 9/16/2019 11:38 PM, Keith Thompson wrote:
>>>>> we can define "iota" as an infinitesimal, something that is not
>>>>> a real number but that has a location on the number line,
>>>>> directly adjacent to the real number 0. It is greater than 0
>>>>> and less than any positive real number.
>>>>
>>>> WE CAN SUBTRACT A SINGLE POINT FROM AN INTERVAL DERIVING ANOTHER INTERVAL THAT DOES NOT HAVE THIS POINT On 9/16/2019 8:33 PM, Mike
>>>> Terry wrote:
>>>>> OK, so (A,B] = [A,B] - {A}, and that can be counted as an
>>>>> "operation
>>>> in which we get (A,B] from [A,B].
>>>>
>>>> Mike Terry agreed with this verbatim: [A, B] - {A} = (A, B] Infinitesimal Numbers would encode (A, B] as [A[1], B]
>>>>
>>>> We merely extrapolate the very next elaboration [A[1], B] -
>>>> {A[1]} = (A[1], B] encoded as [A[2], B]
>>>>
>>>> This shows how Infinitesimal Numbers directly reference
>>>> immediately adjacent points on a number line.
>>>>
>>>> 0.0[0] = 0.0 0.0[1] = the next point after 0.0 0.0[2] = 2 points
>>>> after 0.0 0.0[3] = 3 points after 0.0
>>>
>>> I'd asked earlier but got no response, so I'll ask again.
>>>
>>> Do you accept that there is no *greatest* integer?
>>>
>>> André
>>>
>>
>> Infinity is a process rather than a value, thus there is no greatest integer and all infinite sets are likewise this exact same
>> limitlessness.
>
> OK. So if you agree that the set of positive integers {1, 2, 3, 4...} has no greatest element, then you should also agree that this subset of the positive rationals (in descending order) {1/1, 1/2, 1/3, 1/4...} has no least element.
>
> But if the rationals (which are a subset of the reals) have no least element, then how do you reconcile this with your view that there exists some smallest non-zero number which is immediately adjacent to zero?
>
> André
>
>
Infinitesimal Numbers account for all of the points on the number line.
Real numbers do not account for all of the points on a number line.
--
Copyright 2019 Pete Olcott All rights reserved
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein
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| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2019-09-17 22:40 -0600 |
| Subject | Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) |
| Message-ID | <qlsch0$jle$1@dont-email.me> |
| In reply to | #210982 |
On 2019-09-17 10:21 p.m., peteolcott wrote:
> On 9/17/2019 11:04 PM, André G. Isaak wrote:
>> On 2019-09-17 9:31 p.m., peteolcott wrote:
>>> On 9/17/2019 10:23 PM, André G. Isaak wrote:
>>>> On 2019-09-17 9:19 p.m., peteolcott wrote:
>>>>> On 9/17/2019 9:55 PM, Ben Bacarisse wrote:
>>>>>> peteolcott <Here@Home> writes:
>>>>>>
>>>>
>>>> <snip>
>>>>
>>>>>>> Keith Thompson understands that they could refute differing
>>>>>>> sized cardinality of infinite sets.
>>>>>>
>>>>>> No he doesn't. The trouble is, you don't understand what he's
>>>>>> saying. In fact he agrees with me though he puts it more
>>>>>> cautiously. I'll leave it here for those that /can/ follow:
>>>>>>
>>>>>
>>>>> I SUMMED IT UP MORE CONCISELY HERE:
>>>>>
>>>>> THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0 On
>>>>> 9/16/2019 11:38 PM, Keith Thompson wrote:
>>>>>> we can define "iota" as an infinitesimal, something that is not
>>>>>> a real number but that has a location on the number line,
>>>>>> directly adjacent to the real number 0. It is greater than 0
>>>>>> and less than any positive real number.
>>>>>
>>>>> WE CAN SUBTRACT A SINGLE POINT FROM AN INTERVAL DERIVING ANOTHER
>>>>> INTERVAL THAT DOES NOT HAVE THIS POINT On 9/16/2019 8:33 PM, Mike
>>>>> Terry wrote:
>>>>>> OK, so (A,B] = [A,B] - {A}, and that can be counted as an
>>>>>> "operation
>>>>> in which we get (A,B] from [A,B].
>>>>>
>>>>> Mike Terry agreed with this verbatim: [A, B] - {A} = (A, B]
>>>>> Infinitesimal Numbers would encode (A, B] as [A[1], B]
>>>>>
>>>>> We merely extrapolate the very next elaboration [A[1], B] -
>>>>> {A[1]} = (A[1], B] encoded as [A[2], B]
>>>>>
>>>>> This shows how Infinitesimal Numbers directly reference
>>>>> immediately adjacent points on a number line.
>>>>>
>>>>> 0.0[0] = 0.0 0.0[1] = the next point after 0.0 0.0[2] = 2 points
>>>>> after 0.0 0.0[3] = 3 points after 0.0
>>>>
>>>> I'd asked earlier but got no response, so I'll ask again.
>>>>
>>>> Do you accept that there is no *greatest* integer?
>>>>
>>>> André
>>>>
>>>
>>> Infinity is a process rather than a value, thus there is no greatest
>>> integer and all infinite sets are likewise this exact same
>>> limitlessness.
>>
>> OK. So if you agree that the set of positive integers {1, 2, 3, 4...}
>> has no greatest element, then you should also agree that this subset
>> of the positive rationals (in descending order) {1/1, 1/2, 1/3,
>> 1/4...} has no least element.
>>
>> But if the rationals (which are a subset of the reals) have no least
>> element, then how do you reconcile this with your view that there
>> exists some smallest non-zero number which is immediately adjacent to
>> zero?
>>
>> André
>>
>>
>
> Infinitesimal Numbers account for all of the points on the number line.
> Real numbers do not account for all of the points on a number line.
But this is simply an assertion on your part. If you want to convince
anyone of this you'd need to define your number line in a sufficiently
precise way that you can actually show that there are things on it which
aren't representable by the reals but which are by your infinitesimals.
Can you provide an example of a number which can be shown to not be
included in the reals but which is in your infinitesimals?
Note: by provide an example, I don't mean simply naming a point such as
0.0[1], but explaining its properties sufficiently that it is clear what
it means and that it does not correspond to some real number].
André
--
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-18 00:27 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <f72dnawZ6-62XRzAnZ2dnUU7-T3NnZ2d@giganews.com> |
| In reply to | #210989 |
On 9/17/2019 11:40 PM, André G. Isaak wrote:
> On 2019-09-17 10:21 p.m., peteolcott wrote:
>> On 9/17/2019 11:04 PM, André G. Isaak wrote:
>>> On 2019-09-17 9:31 p.m., peteolcott wrote:
>>>> On 9/17/2019 10:23 PM, André G. Isaak wrote:
>>>>> On 2019-09-17 9:19 p.m., peteolcott wrote:
>>>>>> On 9/17/2019 9:55 PM, Ben Bacarisse wrote:
>>>>>>> peteolcott <Here@Home> writes:
>>>>>>>
>>>>>
>>>>> <snip>
>>>>>
>>>>>>>> Keith Thompson understands that they could refute differing
>>>>>>>> sized cardinality of infinite sets.
>>>>>>>
>>>>>>> No he doesn't. The trouble is, you don't understand what he's
>>>>>>> saying. In fact he agrees with me though he puts it more
>>>>>>> cautiously. I'll leave it here for those that /can/ follow:
>>>>>>>
>>>>>>
>>>>>> I SUMMED IT UP MORE CONCISELY HERE:
>>>>>>
>>>>>> THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0 On
>>>>>> 9/16/2019 11:38 PM, Keith Thompson wrote:
>>>>>>> we can define "iota" as an infinitesimal, something that is not
>>>>>>> a real number but that has a location on the number line,
>>>>>>> directly adjacent to the real number 0. It is greater than 0
>>>>>>> and less than any positive real number.
>>>>>>
>>>>>> WE CAN SUBTRACT A SINGLE POINT FROM AN INTERVAL DERIVING ANOTHER INTERVAL THAT DOES NOT HAVE THIS POINT On 9/16/2019 8:33 PM, Mike
>>>>>> Terry wrote:
>>>>>>> OK, so (A,B] = [A,B] - {A}, and that can be counted as an
>>>>>>> "operation
>>>>>> in which we get (A,B] from [A,B].
>>>>>>
>>>>>> Mike Terry agreed with this verbatim: [A, B] - {A} = (A, B] Infinitesimal Numbers would encode (A, B] as [A[1], B]
>>>>>>
>>>>>> We merely extrapolate the very next elaboration [A[1], B] -
>>>>>> {A[1]} = (A[1], B] encoded as [A[2], B]
>>>>>>
>>>>>> This shows how Infinitesimal Numbers directly reference
>>>>>> immediately adjacent points on a number line.
>>>>>>
>>>>>> 0.0[0] = 0.0 0.0[1] = the next point after 0.0 0.0[2] = 2 points
>>>>>> after 0.0 0.0[3] = 3 points after 0.0
>>>>>
>>>>> I'd asked earlier but got no response, so I'll ask again.
>>>>>
>>>>> Do you accept that there is no *greatest* integer?
>>>>>
>>>>> André
>>>>>
>>>>
>>>> Infinity is a process rather than a value, thus there is no greatest integer and all infinite sets are likewise this exact same
>>>> limitlessness.
>>>
>>> OK. So if you agree that the set of positive integers {1, 2, 3, 4...} has no greatest element, then you should also agree that this subset of the positive rationals (in descending order) {1/1, 1/2, 1/3, 1/4...} has no least element.
>>>
>>> But if the rationals (which are a subset of the reals) have no least element, then how do you reconcile this with your view that there exists some smallest non-zero number which is immediately adjacent to zero?
>>>
>>> André
>>>
>>>
>>
>> Infinitesimal Numbers account for all of the points on the number line.
>> Real numbers do not account for all of the points on a number line.
>
> But this is simply an assertion on your part. If you want to convince anyone of this you'd need to define your number line in a sufficiently precise way that you can actually show that there are things on it which aren't representable by the reals
The point immediately adjacent to 0.0 cannot be represented by Reals.
Everyone always assumes that it does not exist yet they make this
assumption entirely on the basis that it cannot be expressed as a real.
If there is no point immediately adjacent to 0.0 this entails that an
infinitesimal gap must exist. If an infinitesimal gap does not exist
then there must be a point immediately adjacent to 0.0.
> but
> which are by your infinitesimals. Can you provide an example of a number which can be shown to not be included in the reals but which is in your infinitesimals?
>
> Note: by provide an example, I don't mean simply naming a point such as 0.0[1], but explaining its properties sufficiently that it is clear what it means and that it does not correspond to some real number].
>
> André
>
--
Copyright 2019 Pete Olcott All rights reserved
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein
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| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2019-09-17 23:55 -0600 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <qlsgta$68o$1@dont-email.me> |
| In reply to | #210991 |
On 2019-09-17 11:27 p.m., peteolcott wrote:
> On 9/17/2019 11:40 PM, André G. Isaak wrote:
>> On 2019-09-17 10:21 p.m., peteolcott wrote:
>>> On 9/17/2019 11:04 PM, André G. Isaak wrote:
>>>> On 2019-09-17 9:31 p.m., peteolcott wrote:
>>>>> On 9/17/2019 10:23 PM, André G. Isaak wrote:
>>>>>> On 2019-09-17 9:19 p.m., peteolcott wrote:
>>>>>>> On 9/17/2019 9:55 PM, Ben Bacarisse wrote:
>>>>>>>> peteolcott <Here@Home> writes:
>>>>>>>>
>>>>>>
>>>>>> <snip>
>>>>>>
>>>>>>>>> Keith Thompson understands that they could refute differing
>>>>>>>>> sized cardinality of infinite sets.
>>>>>>>>
>>>>>>>> No he doesn't. The trouble is, you don't understand what he's
>>>>>>>> saying. In fact he agrees with me though he puts it more
>>>>>>>> cautiously. I'll leave it here for those that /can/ follow:
>>>>>>>>
>>>>>>>
>>>>>>> I SUMMED IT UP MORE CONCISELY HERE:
>>>>>>>
>>>>>>> THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0 On
>>>>>>> 9/16/2019 11:38 PM, Keith Thompson wrote:
>>>>>>>> we can define "iota" as an infinitesimal, something that is not
>>>>>>>> a real number but that has a location on the number line,
>>>>>>>> directly adjacent to the real number 0. It is greater than 0
>>>>>>>> and less than any positive real number.
>>>>>>>
>>>>>>> WE CAN SUBTRACT A SINGLE POINT FROM AN INTERVAL DERIVING ANOTHER
>>>>>>> INTERVAL THAT DOES NOT HAVE THIS POINT On 9/16/2019 8:33 PM, Mike
>>>>>>> Terry wrote:
>>>>>>>> OK, so (A,B] = [A,B] - {A}, and that can be counted as an
>>>>>>>> "operation
>>>>>>> in which we get (A,B] from [A,B].
>>>>>>>
>>>>>>> Mike Terry agreed with this verbatim: [A, B] - {A} = (A, B]
>>>>>>> Infinitesimal Numbers would encode (A, B] as [A[1], B]
>>>>>>>
>>>>>>> We merely extrapolate the very next elaboration [A[1], B] -
>>>>>>> {A[1]} = (A[1], B] encoded as [A[2], B]
>>>>>>>
>>>>>>> This shows how Infinitesimal Numbers directly reference
>>>>>>> immediately adjacent points on a number line.
>>>>>>>
>>>>>>> 0.0[0] = 0.0 0.0[1] = the next point after 0.0 0.0[2] = 2 points
>>>>>>> after 0.0 0.0[3] = 3 points after 0.0
>>>>>>
>>>>>> I'd asked earlier but got no response, so I'll ask again.
>>>>>>
>>>>>> Do you accept that there is no *greatest* integer?
>>>>>>
>>>>>> André
>>>>>>
>>>>>
>>>>> Infinity is a process rather than a value, thus there is no
>>>>> greatest integer and all infinite sets are likewise this exact same
>>>>> limitlessness.
>>>>
>>>> OK. So if you agree that the set of positive integers {1, 2, 3,
>>>> 4...} has no greatest element, then you should also agree that this
>>>> subset of the positive rationals (in descending order) {1/1, 1/2,
>>>> 1/3, 1/4...} has no least element.
>>>>
>>>> But if the rationals (which are a subset of the reals) have no least
>>>> element, then how do you reconcile this with your view that there
>>>> exists some smallest non-zero number which is immediately adjacent
>>>> to zero?
>>>>
>>>> André
>>>>
>>>>
>>>
>>> Infinitesimal Numbers account for all of the points on the number line.
>>> Real numbers do not account for all of the points on a number line.
>>
>> But this is simply an assertion on your part. If you want to convince
>> anyone of this you'd need to define your number line in a sufficiently
>> precise way that you can actually show that there are things on it
>> which aren't representable by the reals
>
> The point immediately adjacent to 0.0 cannot be represented by Reals.
> Everyone always assumes that it does not exist yet they make this
> assumption entirely on the basis that it cannot be expressed as a real.
>
> If there is no point immediately adjacent to 0.0 this entails that an
> infinitesimal gap must exist.If an infinitesimal gap does not exist
> then there must be a point immediately adjacent to 0.0.
Why does this entail a gap? That seems to be based on the assumption
that numbers are discrete, and while that is true of the integers, it is
not true of the reals.
To the extent that I understand your proposal, you seem to be assuming
that both reals and infinitesimals are represented on the same number
line, and that the reals are a subset of the infinitesimals.
You also assume there is some number, call it I (which I take
abbreviates 0.0[1]) which is an infinitesimal such that I > 0 but there
is no number between zero and I.
But since the reals are a subset of your numbers, and for the reals it
can be proven that for any real numbers x and y such that x < y, there
will always be a z such that x < z < y, how exactly do you preclude the
existence of some real number which lies in between zero and your I?
If you claim no such real exists, then you are effectively claiming that
there is a least positive real number, but that is easily shown to be false.
If you allow for such a real to exist, then this contradicts your claim
that I is immediately adjacent to zero.
André
--
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-18 00:59 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <Z_-dnX0lPJNKWhzAnZ2dnUU7-cWdnZ2d@giganews.com> |
| In reply to | #210992 |
On 9/18/2019 12:55 AM, André G. Isaak wrote:
> On 2019-09-17 11:27 p.m., peteolcott wrote:
>> On 9/17/2019 11:40 PM, André G. Isaak wrote:
>>> On 2019-09-17 10:21 p.m., peteolcott wrote:
>>>> On 9/17/2019 11:04 PM, André G. Isaak wrote:
>>>>> On 2019-09-17 9:31 p.m., peteolcott wrote:
>>>>>> On 9/17/2019 10:23 PM, André G. Isaak wrote:
>>>>>>> On 2019-09-17 9:19 p.m., peteolcott wrote:
>>>>>>>> On 9/17/2019 9:55 PM, Ben Bacarisse wrote:
>>>>>>>>> peteolcott <Here@Home> writes:
>>>>>>>>>
>>>>>>>
>>>>>>> <snip>
>>>>>>>
>>>>>>>>>> Keith Thompson understands that they could refute differing
>>>>>>>>>> sized cardinality of infinite sets.
>>>>>>>>>
>>>>>>>>> No he doesn't. The trouble is, you don't understand what he's
>>>>>>>>> saying. In fact he agrees with me though he puts it more
>>>>>>>>> cautiously. I'll leave it here for those that /can/ follow:
>>>>>>>>>
>>>>>>>>
>>>>>>>> I SUMMED IT UP MORE CONCISELY HERE:
>>>>>>>>
>>>>>>>> THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0 On
>>>>>>>> 9/16/2019 11:38 PM, Keith Thompson wrote:
>>>>>>>>> we can define "iota" as an infinitesimal, something that is not
>>>>>>>>> a real number but that has a location on the number line,
>>>>>>>>> directly adjacent to the real number 0. It is greater than 0
>>>>>>>>> and less than any positive real number.
>>>>>>>>
>>>>>>>> WE CAN SUBTRACT A SINGLE POINT FROM AN INTERVAL DERIVING ANOTHER INTERVAL THAT DOES NOT HAVE THIS POINT On 9/16/2019 8:33 PM, Mike
>>>>>>>> Terry wrote:
>>>>>>>>> OK, so (A,B] = [A,B] - {A}, and that can be counted as an
>>>>>>>>> "operation
>>>>>>>> in which we get (A,B] from [A,B].
>>>>>>>>
>>>>>>>> Mike Terry agreed with this verbatim: [A, B] - {A} = (A, B] Infinitesimal Numbers would encode (A, B] as [A[1], B]
>>>>>>>>
>>>>>>>> We merely extrapolate the very next elaboration [A[1], B] -
>>>>>>>> {A[1]} = (A[1], B] encoded as [A[2], B]
>>>>>>>>
>>>>>>>> This shows how Infinitesimal Numbers directly reference
>>>>>>>> immediately adjacent points on a number line.
>>>>>>>>
>>>>>>>> 0.0[0] = 0.0 0.0[1] = the next point after 0.0 0.0[2] = 2 points
>>>>>>>> after 0.0 0.0[3] = 3 points after 0.0
>>>>>>>
>>>>>>> I'd asked earlier but got no response, so I'll ask again.
>>>>>>>
>>>>>>> Do you accept that there is no *greatest* integer?
>>>>>>>
>>>>>>> André
>>>>>>>
>>>>>>
>>>>>> Infinity is a process rather than a value, thus there is no greatest integer and all infinite sets are likewise this exact same
>>>>>> limitlessness.
>>>>>
>>>>> OK. So if you agree that the set of positive integers {1, 2, 3, 4...} has no greatest element, then you should also agree that this subset of the positive rationals (in descending order) {1/1, 1/2, 1/3, 1/4...} has no least element.
>>>>>
>>>>> But if the rationals (which are a subset of the reals) have no least element, then how do you reconcile this with your view that there exists some smallest non-zero number which is immediately adjacent to zero?
>>>>>
>>>>> André
>>>>>
>>>>>
>>>>
>>>> Infinitesimal Numbers account for all of the points on the number line.
>>>> Real numbers do not account for all of the points on a number line.
>>>
>>> But this is simply an assertion on your part. If you want to convince anyone of this you'd need to define your number line in a sufficiently precise way that you can actually show that there are things on it which aren't representable by the reals
>>
>> The point immediately adjacent to 0.0 cannot be represented by Reals.
>> Everyone always assumes that it does not exist yet they make this
>> assumption entirely on the basis that it cannot be expressed as a real.
>>
>> If there is no point immediately adjacent to 0.0 this entails that an
>> infinitesimal gap must exist.If an infinitesimal gap does not exist
>> then there must be a point immediately adjacent to 0.0.
>
> Why does this entail a gap? That seems to be based on the assumption that numbers are discrete, and while that is true of the integers, it is not true of the reals.
>
If it is impossible to name the point immediately adjacent to 0.0
and there are no gaps in the number line then naming is insufficient.
> To the extent that I understand your proposal, you seem to be assuming that both reals and infinitesimals are represented on the same number line, and that the reals are a subset of the infinitesimals.
>
> You also assume there is some number, call it I (which I take abbreviates 0.0[1]) which is an infinitesimal such that I > 0 but there is no number between zero and I.
>
> But since the reals are a subset of your numbers, and for the reals it can be proven that for any real numbers x and y such that x < y, there will always be a z such that x < z < y, how exactly do you preclude the existence of some real number which lies
> in between zero and your I?
>
> If you claim no such real exists, then you are effectively claiming that there is a least positive real number, but that is easily shown to be false.
>
> If you allow for such a real to exist, then this contradicts your claim that I is immediately adjacent to zero.
>
>
> André
>
>
--
Copyright 2019 Pete Olcott All rights reserved
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein
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| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2019-09-18 00:12 -0600 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <qlshsf$atm$1@dont-email.me> |
| In reply to | #210993 |
On 2019-09-17 11:59 p.m., peteolcott wrote:
> On 9/18/2019 12:55 AM, André G. Isaak wrote:
>> On 2019-09-17 11:27 p.m., peteolcott wrote:
>>> On 9/17/2019 11:40 PM, André G. Isaak wrote:
>>>> On 2019-09-17 10:21 p.m., peteolcott wrote:
>>>>> On 9/17/2019 11:04 PM, André G. Isaak wrote:
>>>>>> On 2019-09-17 9:31 p.m., peteolcott wrote:
>>>>>>> On 9/17/2019 10:23 PM, André G. Isaak wrote:
>>>>>>>> On 2019-09-17 9:19 p.m., peteolcott wrote:
>>>>>>>>> On 9/17/2019 9:55 PM, Ben Bacarisse wrote:
>>>>>>>>>> peteolcott <Here@Home> writes:
>>>>>>>>>>
>>>>>>>>
>>>>>>>> <snip>
>>>>>>>>
>>>>>>>>>>> Keith Thompson understands that they could refute differing
>>>>>>>>>>> sized cardinality of infinite sets.
>>>>>>>>>>
>>>>>>>>>> No he doesn't. The trouble is, you don't understand what he's
>>>>>>>>>> saying. In fact he agrees with me though he puts it more
>>>>>>>>>> cautiously. I'll leave it here for those that /can/ follow:
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> I SUMMED IT UP MORE CONCISELY HERE:
>>>>>>>>>
>>>>>>>>> THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0 On
>>>>>>>>> 9/16/2019 11:38 PM, Keith Thompson wrote:
>>>>>>>>>> we can define "iota" as an infinitesimal, something that is not
>>>>>>>>>> a real number but that has a location on the number line,
>>>>>>>>>> directly adjacent to the real number 0. It is greater than 0
>>>>>>>>>> and less than any positive real number.
>>>>>>>>>
>>>>>>>>> WE CAN SUBTRACT A SINGLE POINT FROM AN INTERVAL DERIVING
>>>>>>>>> ANOTHER INTERVAL THAT DOES NOT HAVE THIS POINT On 9/16/2019
>>>>>>>>> 8:33 PM, Mike
>>>>>>>>> Terry wrote:
>>>>>>>>>> OK, so (A,B] = [A,B] - {A}, and that can be counted as an
>>>>>>>>>> "operation
>>>>>>>>> in which we get (A,B] from [A,B].
>>>>>>>>>
>>>>>>>>> Mike Terry agreed with this verbatim: [A, B] - {A} = (A, B]
>>>>>>>>> Infinitesimal Numbers would encode (A, B] as [A[1], B]
>>>>>>>>>
>>>>>>>>> We merely extrapolate the very next elaboration [A[1], B] -
>>>>>>>>> {A[1]} = (A[1], B] encoded as [A[2], B]
>>>>>>>>>
>>>>>>>>> This shows how Infinitesimal Numbers directly reference
>>>>>>>>> immediately adjacent points on a number line.
>>>>>>>>>
>>>>>>>>> 0.0[0] = 0.0 0.0[1] = the next point after 0.0 0.0[2] = 2 points
>>>>>>>>> after 0.0 0.0[3] = 3 points after 0.0
>>>>>>>>
>>>>>>>> I'd asked earlier but got no response, so I'll ask again.
>>>>>>>>
>>>>>>>> Do you accept that there is no *greatest* integer?
>>>>>>>>
>>>>>>>> André
>>>>>>>>
>>>>>>>
>>>>>>> Infinity is a process rather than a value, thus there is no
>>>>>>> greatest integer and all infinite sets are likewise this exact same
>>>>>>> limitlessness.
>>>>>>
>>>>>> OK. So if you agree that the set of positive integers {1, 2, 3,
>>>>>> 4...} has no greatest element, then you should also agree that
>>>>>> this subset of the positive rationals (in descending order) {1/1,
>>>>>> 1/2, 1/3, 1/4...} has no least element.
>>>>>>
>>>>>> But if the rationals (which are a subset of the reals) have no
>>>>>> least element, then how do you reconcile this with your view that
>>>>>> there exists some smallest non-zero number which is immediately
>>>>>> adjacent to zero?
>>>>>>
>>>>>> André
>>>>>>
>>>>>>
>>>>>
>>>>> Infinitesimal Numbers account for all of the points on the number
>>>>> line.
>>>>> Real numbers do not account for all of the points on a number line.
>>>>
>>>> But this is simply an assertion on your part. If you want to
>>>> convince anyone of this you'd need to define your number line in a
>>>> sufficiently precise way that you can actually show that there are
>>>> things on it which aren't representable by the reals
>>>
>>> The point immediately adjacent to 0.0 cannot be represented by Reals.
>>> Everyone always assumes that it does not exist yet they make this
>>> assumption entirely on the basis that it cannot be expressed as a real.
>>>
>>> If there is no point immediately adjacent to 0.0 this entails that an
>>> infinitesimal gap must exist.If an infinitesimal gap does not exist
>>> then there must be a point immediately adjacent to 0.0.
>>
>> Why does this entail a gap? That seems to be based on the assumption
>> that numbers are discrete, and while that is true of the integers, it
>> is not true of the reals.
>>
>
> If it is impossible to name the point immediately adjacent to 0.0
> and there are no gaps in the number line then naming is insufficient.
Or it means no such point exists. In a discrete system, the notion of
adjacency makes sense. In a continuous system, it does not. The reals
are continuous.
>> To the extent that I understand your proposal, you seem to be assuming
>> that both reals and infinitesimals are represented on the same number
>> line, and that the reals are a subset of the infinitesimals.
>>
>> You also assume there is some number, call it I (which I take
>> abbreviates 0.0[1]) which is an infinitesimal such that I > 0 but
>> there is no number between zero and I.
>>
>> But since the reals are a subset of your numbers, and for the reals it
>> can be proven that for any real numbers x and y such that x < y, there
>> will always be a z such that x < z < y, how exactly do you preclude
>> the existence of some real number which lies in between zero and your I?
>>
>> If you claim no such real exists, then you are effectively claiming
>> that there is a least positive real number, but that is easily shown
>> to be false.
>>
>> If you allow for such a real to exist, then this contradicts your
>> claim that I is immediately adjacent to zero.
You entirely skipped my more salient points above. If you want to claim
your definition of infinitesimals is actually consistent, you need to
address the above.
André
--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.
[toc] | [prev] | [next] | [standalone]
| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-18 10:15 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <lO6dnW6aP9lg1B_AnZ2dnUU7-KfNnZ2d@giganews.com> |
| In reply to | #210994 |
On 9/18/2019 1:12 AM, André G. Isaak wrote:
> On 2019-09-17 11:59 p.m., peteolcott wrote:
>> On 9/18/2019 12:55 AM, André G. Isaak wrote:
>>> On 2019-09-17 11:27 p.m., peteolcott wrote:
>>>> On 9/17/2019 11:40 PM, André G. Isaak wrote:
>>>>> On 2019-09-17 10:21 p.m., peteolcott wrote:
>>>>>> On 9/17/2019 11:04 PM, André G. Isaak wrote:
>>>>>>> On 2019-09-17 9:31 p.m., peteolcott wrote:
>>>>>>>> On 9/17/2019 10:23 PM, André G. Isaak wrote:
>>>>>>>>> On 2019-09-17 9:19 p.m., peteolcott wrote:
>>>>>>>>>> On 9/17/2019 9:55 PM, Ben Bacarisse wrote:
>>>>>>>>>>> peteolcott <Here@Home> writes:
>>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> <snip>
>>>>>>>>>
>>>>>>>>>>>> Keith Thompson understands that they could refute differing
>>>>>>>>>>>> sized cardinality of infinite sets.
>>>>>>>>>>>
>>>>>>>>>>> No he doesn't. The trouble is, you don't understand what he's
>>>>>>>>>>> saying. In fact he agrees with me though he puts it more
>>>>>>>>>>> cautiously. I'll leave it here for those that /can/ follow:
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> I SUMMED IT UP MORE CONCISELY HERE:
>>>>>>>>>>
>>>>>>>>>> THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0 On
>>>>>>>>>> 9/16/2019 11:38 PM, Keith Thompson wrote:
>>>>>>>>>>> we can define "iota" as an infinitesimal, something that is not
>>>>>>>>>>> a real number but that has a location on the number line,
>>>>>>>>>>> directly adjacent to the real number 0. It is greater than 0
>>>>>>>>>>> and less than any positive real number.
>>>>>>>>>>
>>>>>>>>>> WE CAN SUBTRACT A SINGLE POINT FROM AN INTERVAL DERIVING ANOTHER INTERVAL THAT DOES NOT HAVE THIS POINT On 9/16/2019 8:33 PM, Mike
>>>>>>>>>> Terry wrote:
>>>>>>>>>>> OK, so (A,B] = [A,B] - {A}, and that can be counted as an
>>>>>>>>>>> "operation
>>>>>>>>>> in which we get (A,B] from [A,B].
>>>>>>>>>>
>>>>>>>>>> Mike Terry agreed with this verbatim: [A, B] - {A} = (A, B] Infinitesimal Numbers would encode (A, B] as [A[1], B]
>>>>>>>>>>
>>>>>>>>>> We merely extrapolate the very next elaboration [A[1], B] -
>>>>>>>>>> {A[1]} = (A[1], B] encoded as [A[2], B]
>>>>>>>>>>
>>>>>>>>>> This shows how Infinitesimal Numbers directly reference
>>>>>>>>>> immediately adjacent points on a number line.
>>>>>>>>>>
>>>>>>>>>> 0.0[0] = 0.0 0.0[1] = the next point after 0.0 0.0[2] = 2 points
>>>>>>>>>> after 0.0 0.0[3] = 3 points after 0.0
>>>>>>>>>
>>>>>>>>> I'd asked earlier but got no response, so I'll ask again.
>>>>>>>>>
>>>>>>>>> Do you accept that there is no *greatest* integer?
>>>>>>>>>
>>>>>>>>> André
>>>>>>>>>
>>>>>>>>
>>>>>>>> Infinity is a process rather than a value, thus there is no greatest integer and all infinite sets are likewise this exact same
>>>>>>>> limitlessness.
>>>>>>>
>>>>>>> OK. So if you agree that the set of positive integers {1, 2, 3, 4...} has no greatest element, then you should also agree that this subset of the positive rationals (in descending order) {1/1, 1/2, 1/3, 1/4...} has no least element.
>>>>>>>
>>>>>>> But if the rationals (which are a subset of the reals) have no least element, then how do you reconcile this with your view that there exists some smallest non-zero number which is immediately adjacent to zero?
>>>>>>>
>>>>>>> André
>>>>>>>
>>>>>>>
>>>>>>
>>>>>> Infinitesimal Numbers account for all of the points on the number line.
>>>>>> Real numbers do not account for all of the points on a number line.
>>>>>
>>>>> But this is simply an assertion on your part. If you want to convince anyone of this you'd need to define your number line in a sufficiently precise way that you can actually show that there are things on it which aren't representable by the reals
>>>>
>>>> The point immediately adjacent to 0.0 cannot be represented by Reals.
>>>> Everyone always assumes that it does not exist yet they make this
>>>> assumption entirely on the basis that it cannot be expressed as a real.
>>>>
>>>> If there is no point immediately adjacent to 0.0 this entails that an
>>>> infinitesimal gap must exist.If an infinitesimal gap does not exist
>>>> then there must be a point immediately adjacent to 0.0.
>>>
>>> Why does this entail a gap? That seems to be based on the assumption that numbers are discrete, and while that is true of the integers, it is not true of the reals.
>>>
>>
>> If it is impossible to name the point immediately adjacent to 0.0
>> and there are no gaps in the number line then naming is insufficient.
>
> Or it means no such point exists. In a discrete system, the notion of adjacency makes sense. In a continuous system, it does not. The reals are continuous.
If we have a definite boundary at one named point: 0.0, then the
mere lack of a name does not entail that we have a lack of a
definite boundary at other points.
If we use the idea of a continuous system that has only ishy
squishy boundaries then the same would apply to all points
not merely the unnamed points.
The point at 0.0 would blend in to the point before it and the
point after it thus ceasing to be at exactly 0.0.
>
>>> To the extent that I understand your proposal, you seem to be assuming that both reals and infinitesimals are represented on the same number line, and that the reals are a subset of the infinitesimals.
>>>
>>> You also assume there is some number, call it I (which I take abbreviates 0.0[1]) which is an infinitesimal such that I > 0 but there is no number between zero and I.
>>>
>>> But since the reals are a subset of your numbers, and for the reals it can be proven that for any real numbers x and y such that x < y, there will always be a z such that x < z < y, how exactly do you preclude the existence of some real number which
>>> lies in between zero and your I?
Infinitesimal numbers are defined as the entire set of immediately adjacent points on the number line. A single point on the number line is indivisible.
>>>
>>> If you claim no such real exists, then you are effectively claiming that there is a least positive real number, but that is easily shown to be false.
>>>
This is the least positive number: 0.0[1] one point larger than zero on the number line.
>>> If you allow for such a real to exist, then this contradicts your claim that I is immediately adjacent to zero.
>
> You entirely skipped my more salient points above. If you want to claim your definition of infinitesimals is actually consistent, you need to address the above.
>
> André
>
--
Copyright 2019 Pete Olcott All rights reserved
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein
[toc] | [prev] | [next] | [standalone]
| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2019-09-18 09:35 -0600 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <qltiss$4tu$1@dont-email.me> |
| In reply to | #211000 |
On 2019-09-18 9:15 a.m., peteolcott wrote:
> On 9/18/2019 1:12 AM, André G. Isaak wrote:
>> On 2019-09-17 11:59 p.m., peteolcott wrote:
>>> On 9/18/2019 12:55 AM, André G. Isaak wrote:
>>>> On 2019-09-17 11:27 p.m., peteolcott wrote:
>>>>> On 9/17/2019 11:40 PM, André G. Isaak wrote:
>>>>>> On 2019-09-17 10:21 p.m., peteolcott wrote:
>>>>>>> On 9/17/2019 11:04 PM, André G. Isaak wrote:
>>>>>>>> On 2019-09-17 9:31 p.m., peteolcott wrote:
>>>>>>>>> On 9/17/2019 10:23 PM, André G. Isaak wrote:
>>>>>>>>>> On 2019-09-17 9:19 p.m., peteolcott wrote:
>>>>>>>>>>> On 9/17/2019 9:55 PM, Ben Bacarisse wrote:
>>>>>>>>>>>> peteolcott <Here@Home> writes:
>>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> <snip>
>>>>>>>>>>
>>>>>>>>>>>>> Keith Thompson understands that they could refute differing
>>>>>>>>>>>>> sized cardinality of infinite sets.
>>>>>>>>>>>>
>>>>>>>>>>>> No he doesn't. The trouble is, you don't understand what he's
>>>>>>>>>>>> saying. In fact he agrees with me though he puts it more
>>>>>>>>>>>> cautiously. I'll leave it here for those that /can/ follow:
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> I SUMMED IT UP MORE CONCISELY HERE:
>>>>>>>>>>>
>>>>>>>>>>> THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0 On
>>>>>>>>>>> 9/16/2019 11:38 PM, Keith Thompson wrote:
>>>>>>>>>>>> we can define "iota" as an infinitesimal, something that is not
>>>>>>>>>>>> a real number but that has a location on the number line,
>>>>>>>>>>>> directly adjacent to the real number 0. It is greater than 0
>>>>>>>>>>>> and less than any positive real number.
>>>>>>>>>>>
>>>>>>>>>>> WE CAN SUBTRACT A SINGLE POINT FROM AN INTERVAL DERIVING
>>>>>>>>>>> ANOTHER INTERVAL THAT DOES NOT HAVE THIS POINT On 9/16/2019
>>>>>>>>>>> 8:33 PM, Mike
>>>>>>>>>>> Terry wrote:
>>>>>>>>>>>> OK, so (A,B] = [A,B] - {A}, and that can be counted as an
>>>>>>>>>>>> "operation
>>>>>>>>>>> in which we get (A,B] from [A,B].
>>>>>>>>>>>
>>>>>>>>>>> Mike Terry agreed with this verbatim: [A, B] - {A} = (A, B]
>>>>>>>>>>> Infinitesimal Numbers would encode (A, B] as [A[1], B]
>>>>>>>>>>>
>>>>>>>>>>> We merely extrapolate the very next elaboration [A[1], B] -
>>>>>>>>>>> {A[1]} = (A[1], B] encoded as [A[2], B]
>>>>>>>>>>>
>>>>>>>>>>> This shows how Infinitesimal Numbers directly reference
>>>>>>>>>>> immediately adjacent points on a number line.
>>>>>>>>>>>
>>>>>>>>>>> 0.0[0] = 0.0 0.0[1] = the next point after 0.0 0.0[2] = 2 points
>>>>>>>>>>> after 0.0 0.0[3] = 3 points after 0.0
>>>>>>>>>>
>>>>>>>>>> I'd asked earlier but got no response, so I'll ask again.
>>>>>>>>>>
>>>>>>>>>> Do you accept that there is no *greatest* integer?
>>>>>>>>>>
>>>>>>>>>> André
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> Infinity is a process rather than a value, thus there is no
>>>>>>>>> greatest integer and all infinite sets are likewise this exact
>>>>>>>>> same
>>>>>>>>> limitlessness.
>>>>>>>>
>>>>>>>> OK. So if you agree that the set of positive integers {1, 2, 3,
>>>>>>>> 4...} has no greatest element, then you should also agree that
>>>>>>>> this subset of the positive rationals (in descending order)
>>>>>>>> {1/1, 1/2, 1/3, 1/4...} has no least element.
>>>>>>>>
>>>>>>>> But if the rationals (which are a subset of the reals) have no
>>>>>>>> least element, then how do you reconcile this with your view
>>>>>>>> that there exists some smallest non-zero number which is
>>>>>>>> immediately adjacent to zero?
>>>>>>>>
>>>>>>>> André
>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>> Infinitesimal Numbers account for all of the points on the number
>>>>>>> line.
>>>>>>> Real numbers do not account for all of the points on a number line.
>>>>>>
>>>>>> But this is simply an assertion on your part. If you want to
>>>>>> convince anyone of this you'd need to define your number line in a
>>>>>> sufficiently precise way that you can actually show that there are
>>>>>> things on it which aren't representable by the reals
>>>>>
>>>>> The point immediately adjacent to 0.0 cannot be represented by Reals.
>>>>> Everyone always assumes that it does not exist yet they make this
>>>>> assumption entirely on the basis that it cannot be expressed as a
>>>>> real.
>>>>>
>>>>> If there is no point immediately adjacent to 0.0 this entails that an
>>>>> infinitesimal gap must exist.If an infinitesimal gap does not exist
>>>>> then there must be a point immediately adjacent to 0.0.
>>>>
>>>> Why does this entail a gap? That seems to be based on the assumption
>>>> that numbers are discrete, and while that is true of the integers,
>>>> it is not true of the reals.
>>>>
>>>
>>> If it is impossible to name the point immediately adjacent to 0.0
>>> and there are no gaps in the number line then naming is insufficient.
>>
>> Or it means no such point exists. In a discrete system, the notion of
>> adjacency makes sense. In a continuous system, it does not. The reals
>> are continuous.
>
> If we have a definite boundary at one named point: 0.0, then the
> mere lack of a name does not entail that we have a lack of a
> definite boundary at other points.
Intervals have boundaries. Points don't have boundaries. Points are
*dimensionless* entities. I don't think you fully appreciate what
dimensionless means. You may be familiar with the definition, but you
seem to be unable to conceptualize such entities.
> If we use the idea of a continuous system that has only ishy
> squishy boundaries then the same would apply to all points
> not merely the unnamed points.
>
> The point at 0.0 would blend in to the point before it and the
> point after it thus ceasing to be at exactly 0.0.
A continuous system isn't one in which points 'blend' into one another.
>>
>>>> To the extent that I understand your proposal, you seem to be
>>>> assuming that both reals and infinitesimals are represented on the
>>>> same number line, and that the reals are a subset of the
>>>> infinitesimals.
>>>>
>>>> You also assume there is some number, call it I (which I take
>>>> abbreviates 0.0[1]) which is an infinitesimal such that I > 0 but
>>>> there is no number between zero and I.
>>>>
>>>> But since the reals are a subset of your numbers, and for the reals
>>>> it can be proven that for any real numbers x and y such that x < y,
>>>> there will always be a z such that x < z < y, how exactly do you
>>>> preclude the existence of some real number which lies in between
>>>> zero and your I?
>
> Infinitesimal numbers are defined as the entire set of immediately
> adjacent points on the number line. A single point on the number line is
> indivisible.
Again, 'adjacent' is a relation that only exists for discrete systems.
And if your infinitesimals are discrete then they won't model the real
numbers.
BTW of course points are indivisible. When you divide a number, you
aren't dividing a point; you're dividing the line between two points.
>>>>
>>>> If you claim no such real exists, then you are effectively claiming
>>>> that there is a least positive real number, but that is easily shown
>>>> to be false.
>>>>
>
> This is the least positive number: 0.0[1] one point larger than zero on
> the number line.
You didn't actually address my objection, though. If you claim 0.0[1] is
the least positive number then either (a) there is some positive real
number which is smaller than all others (which is provably false) or (b)
there is some real number which is larger than zero and smaller than
0.0[1] (which contradicts your claim).
André
>>>> If you allow for such a real to exist, then this contradicts your
>>>> claim that I is immediately adjacent to zero.
>>
>> You entirely skipped my more salient points above. If you want to
>> claim your definition of infinitesimals is actually consistent, you
>> need to address the above.
>>
>> André
>>
>
>
--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.
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