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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 3 of 17 — ← Prev page 1 2 [3] 4 5 … 17 Next page →
| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-18 11:00 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <E7adncjwsIYayR_AnZ2dnUU7-IXNnZ2d@giganews.com> |
| In reply to | #211004 |
On 9/18/2019 10:35 AM, André G. Isaak wrote:
> 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.
>
I fully understand the common misconception that points are dimensionless
and on that basis supersede, override and correct that misconception and
say that POINTS DO NOT HAVE ZERO WIDTH THEY HAVE INFINITESIMAL WIDTH.
>> 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.
It is either blend together or discrete you can't have it both ways. If a
point ever specifies a distinct boundary then every point ALWAYS specifies
a distinct boundary, otherwise points NEVER specify distinct boundaries.
Even though defined otherwise this interval: [0,1] is one point longer than
this interval: (0, 1]
>
>>>
>>>>> 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.
>
Infinitesimals without an offset have a bijection to real numbers.
Infinitesimals with an offset specify the gaps between the real numbers.
An Infinitesimal is as if this: 1.0 / ∞ had a numerical value.
> 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)
It is only provably false when using Real numbers. It is defined as true when using Infinitesimals.
> or (b) there is some real number which is
> larger than zero and smaller than 0.0[1] (which contradicts your claim).
>
> André
This is defined as impossible because single points on the number line are defined as indivisible.
>
>>>>> 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
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| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2019-09-18 10:14 -0600 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <qltl5c$j8v$1@dont-email.me> |
| In reply to | #211006 |
On 2019-09-18 10:00 a.m., peteolcott wrote:
> On 9/18/2019 10:35 AM, André G. Isaak wrote:
>> 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.
>>
>
> I fully understand the common misconception that points are dimensionless
> and on that basis supersede, override and correct that misconception and
> say that POINTS DO NOT HAVE ZERO WIDTH THEY HAVE INFINITESIMAL WIDTH.
This isn't a misconception. It's how point is *defined*. It's how point
has always been defined. If you want to talk about something else that
isn't a point that's your prerogative, but don't call it a 'point'.
>>> 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.
>
> It is either blend together or discrete you can't have it both ways. If a
> point ever specifies a distinct boundary then every point ALWAYS specifies
> a distinct boundary, otherwise points NEVER specify distinct boundaries.
I can't even fathom what 'blend together' means.
> Even though defined otherwise this interval: [0,1] is one point longer than
> this interval: (0, 1]
>
>>
>>>>
>>>>>> 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.
>>
>
> Infinitesimals without an offset have a bijection to real numbers.
AFAICT your infinitesimals without an offset simply are the real numbers.
> Infinitesimals with an offset specify the gaps between the real numbers.
And do you have any evidence to support the existence of such gaps?
> An Infinitesimal is as if this: 1.0 / ∞ had a numerical value.
>
>> 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)
>
> It is only provably false when using Real numbers. It is defined as true
> when using Infinitesimals.
You can't simply define this as true if your system actually includes
the reals as a subset since the reals obey they axioms of the reals and
the above is not consistent with those axioms.
If you want to claim that this is true in your system, you'd need to
actually provide axioms for your system and show how it follows from
those axioms and that those axioms are actually consistent.
BTW you still haven't answered my question about arithmetic from another
thread.
What is 0.0[5] • 0.0[5] ?
What is 3.0[2] / 3.0[0]
>> or (b) there is some real number which is larger than zero and smaller
>> than 0.0[1] (which contradicts your claim).
>>
>> André
>
> This is defined as impossible because single points on the number line
> are defined as indivisible.
Its not the points which are being divided. Its the line segment between
the points. Points are zero dimensional. Lines are 1 dimensional. Lines
can be divided, points cannot be.
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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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-18 11:26 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <QeOdndC6qeIGxx_AnZ2dnUU7-RfNnZ2d@giganews.com> |
| In reply to | #211008 |
On 9/18/2019 11:14 AM, André G. Isaak wrote:
> On 2019-09-18 10:00 a.m., peteolcott wrote:
>> On 9/18/2019 10:35 AM, André G. Isaak wrote:
>>> 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.
>>>
>>
>> I fully understand the common misconception that points are dimensionless
>> and on that basis supersede, override and correct that misconception and
>> say that POINTS DO NOT HAVE ZERO WIDTH THEY HAVE INFINITESIMAL WIDTH.
>
> This isn't a misconception. It's how point is *defined*. It's how point has always been defined. If you want to talk about something else that isn't a point that's your prerogative, but don't call it a 'point'.
Definitions are incorrect when they result in inconsistency.
>
>>>> 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.
>>
>> It is either blend together or discrete you can't have it both ways. If a
>> point ever specifies a distinct boundary then every point ALWAYS specifies
>> a distinct boundary, otherwise points NEVER specify distinct boundaries.
>
> I can't even fathom what 'blend together' means.
>
>> Even though defined otherwise this interval: [0,1] is one point longer than
>> this interval: (0, 1]
>>
>>>
>>>>>
>>>>>>> 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.
>>>
>>
>> Infinitesimals without an offset have a bijection to real numbers.
>
> AFAICT your infinitesimals without an offset simply are the real numbers.
>
>> Infinitesimals with an offset specify the gaps between the real numbers.
>
> And do you have any evidence to support the existence of such gaps?
The point immediately adjacent to 0.0 cannot be expressed as a Real number,
it can be expression as this Infinitesimal: 0.0[1].
>
>> An Infinitesimal is as if this: 1.0 / ∞ had a numerical value.
>>
>>> 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)
>>
>> It is only provably false when using Real numbers. It is defined as true when using Infinitesimals.
>
> You can't simply define this as true if your system actually includes the reals as a subset since the reals obey they axioms of the reals and the above is not consistent with those axioms.
>
> If you want to claim that this is true in your system, you'd need to actually provide axioms for your system and show how it follows from those axioms and that those axioms are actually consistent.
>
> BTW you still haven't answered my question about arithmetic from another thread.
>
> What is 0.0[5] • 0.0[5] ?
0.0[25]
>
> What is 3.0[2] / 3.0[0]
[2] is indivisible by [0] so disallowed.
>
>>> or (b) there is some real number which is larger than zero and smaller than 0.0[1] (which contradicts your claim).
>>>
>>> André
>>
>> This is defined as impossible because single points on the number line are defined as indivisible.
>
> Its not the points which are being divided. Its the line segment between the points. Points are zero dimensional. Lines are 1 dimensional. Lines can be divided, points cannot be.
>
> 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
>>>>>
>>>>
>>>>
>>>
>>>
>>
>>
>
>
--
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 10:42 -0600 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <qltmqm$tlt$1@dont-email.me> |
| In reply to | #211009 |
On 2019-09-18 10:26 a.m., peteolcott wrote: > On 9/18/2019 11:14 AM, André G. Isaak wrote: >> On 2019-09-18 10:00 a.m., peteolcott wrote: <snip> >>> I fully understand the common misconception that points are >>> dimensionless >>> and on that basis supersede, override and correct that misconception and >>> say that POINTS DO NOT HAVE ZERO WIDTH THEY HAVE INFINITESIMAL WIDTH. >> >> This isn't a misconception. It's how point is *defined*. It's how >> point has always been defined. If you want to talk about something >> else that isn't a point that's your prerogative, but don't call it a >> 'point'. > > Definitions are incorrect when they result in inconsistency. Nothing about the definition of point leads to an inconsistency. If you believe otherwise, show the inconsistency. >> >>>>> 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. >>> >>> It is either blend together or discrete you can't have it both ways. >>> If a >>> point ever specifies a distinct boundary then every point ALWAYS >>> specifies >>> a distinct boundary, otherwise points NEVER specify distinct boundaries. >> >> I can't even fathom what 'blend together' means. >> >>> Even though defined otherwise this interval: [0,1] is one point >>> longer than >>> this interval: (0, 1] >>> >>>> >>>>>> >>>>>>>> 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. >>>> >>> >>> Infinitesimals without an offset have a bijection to real numbers. >> >> AFAICT your infinitesimals without an offset simply are the real numbers. >> >>> Infinitesimals with an offset specify the gaps between the real numbers. >> >> And do you have any evidence to support the existence of such gaps? > > The point immediately adjacent to 0.0 cannot be expressed as a Real number, > it can be expression as this Infinitesimal: 0.0[1]. That's not evidence for a gap. That's simply you claiming there is a gap and then giving a name to something that you claim fills this alleged gap. But unless you can show the gap exists, there's nothing to fill. >> >>> An Infinitesimal is as if this: 1.0 / ∞ had a numerical value. >>> >>>> 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) >>> >>> It is only provably false when using Real numbers. It is defined as >>> true when using Infinitesimals. >> >> You can't simply define this as true if your system actually includes >> the reals as a subset since the reals obey they axioms of the reals >> and the above is not consistent with those axioms. >> >> If you want to claim that this is true in your system, you'd need to >> actually provide axioms for your system and show how it follows from >> those axioms and that those axioms are actually consistent. >> >> BTW you still haven't answered my question about arithmetic from >> another thread. >> >> What is 0.0[5] • 0.0[5] ? > > 0.0[25] This, of course, makes absolutely no sense. When you multiply two numbers both of which are less than 1.0, you expect the product to be *less* than either multiplicand, not greater. >> >> What is 3.0[2] / 3.0[0] > [2] is indivisible by [0] so disallowed. So then what is [3.0] / [2.0] ? If your infinitesimals are non-zero (as you claim), then both 3.0[0] and 2.0[0] should correspond to some numbers 0.0[x] and 0.0[3x], but since 3 isn't divisible by 2 are you claiming this division is also disallowed? You need to actually provide an explanation of how arithmetic works (or, rather, fails to work) in your system rather than simply providing examples like you did in your prior post. André -- To email remove 'invalid' & replace 'gm' with well known Google mail service.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-18 13:10 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <LLadnSPuRsKG7h_AnZ2dnUU7-IHNnZ2d@giganews.com> |
| In reply to | #211013 |
On 9/18/2019 11:42 AM, André G. Isaak wrote: > On 2019-09-18 10:26 a.m., peteolcott wrote: >> On 9/18/2019 11:14 AM, André G. Isaak wrote: >>> On 2019-09-18 10:00 a.m., peteolcott wrote: > > <snip> > >>>> I fully understand the common misconception that points are dimensionless >>>> and on that basis supersede, override and correct that misconception and >>>> say that POINTS DO NOT HAVE ZERO WIDTH THEY HAVE INFINITESIMAL WIDTH. >>> >>> This isn't a misconception. It's how point is *defined*. It's how point has always been defined. If you want to talk about something else that isn't a point that's your prerogative, but don't call it a 'point'. >> >> Definitions are incorrect when they result in inconsistency. > > Nothing about the definition of point leads to an inconsistency. If you believe otherwise, show the inconsistency. > >>> >>>>>> 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. >>>> >>>> It is either blend together or discrete you can't have it both ways. If a >>>> point ever specifies a distinct boundary then every point ALWAYS specifies >>>> a distinct boundary, otherwise points NEVER specify distinct boundaries. >>> >>> I can't even fathom what 'blend together' means. >>> >>>> Even though defined otherwise this interval: [0,1] is one point longer than >>>> this interval: (0, 1] >>>> >>>>> >>>>>>> >>>>>>>>> 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. >>>>> >>>> >>>> Infinitesimals without an offset have a bijection to real numbers. >>> >>> AFAICT your infinitesimals without an offset simply are the real numbers. >>> >>>> Infinitesimals with an offset specify the gaps between the real numbers. >>> >>> And do you have any evidence to support the existence of such gaps? >> >> The point immediately adjacent to 0.0 cannot be expressed as a Real number, >> it can be expression as this Infinitesimal: 0.0[1]. > > That's not evidence for a gap. That's simply you claiming there is a gap and then giving a name to something that you claim fills this alleged gap. But unless you can show the gap exists, there's nothing to fill. When you try to use Real numbers to locate the point X that is immediately adjacent to 0.0 there is always a measurable distance between 0.0 and X. A measurable distance is ALWAYS a gap, hence Real numbers always have gaps. Infinitesimals fill in all these gaps thus forming a superset of Reals. -- 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 12:36 -0600 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <qlttfu$8jh$1@dont-email.me> |
| In reply to | #211032 |
On 2019-09-18 12:10 p.m., peteolcott wrote: > On 9/18/2019 11:42 AM, André G. Isaak wrote: >> On 2019-09-18 10:26 a.m., peteolcott wrote: >>> On 9/18/2019 11:14 AM, André G. Isaak wrote: >>>> On 2019-09-18 10:00 a.m., peteolcott wrote: <snip> >>>>> Infinitesimals with an offset specify the gaps between the real >>>>> numbers. >>>> >>>> And do you have any evidence to support the existence of such gaps? >>> >>> The point immediately adjacent to 0.0 cannot be expressed as a Real >>> number, >>> it can be expression as this Infinitesimal: 0.0[1]. >> >> That's not evidence for a gap. That's simply you claiming there is a >> gap and then giving a name to something that you claim fills this >> alleged gap. But unless you can show the gap exists, there's nothing >> to fill. > When you try to use Real numbers to locate the point X that is > immediately adjacent to 0.0 there is always a measurable distance > between 0.0 and X. A measurable distance is ALWAYS a gap, hence > Real numbers always have gaps. Infinitesimals fill in all these gaps > thus forming a superset of Reals. That 'gap' is full of real numbers (2^ℵ₀ of them). Which means that X isn't actually adjacent to zero. This is because there *does* *not* *exist* any number which is adjacent to zero, because 'adjacency' is only something that can be defined for discrete systems. Any non-negative point X is either equal to 0 or greater than zero. If X is greater than zero by even the most minuscule amount, then there will be 2^ℵ₀ additional points between zero X. I've restored the questions on arithmetic which you snipped. It would be nice if you actually addressed them. >> What is 0.0[5] • 0.0[5] ? > > 0.0[25] This, of course, makes absolutely no sense. When you multiply two numbers both of which are less than 1.0, you expect the product to be *less* than either multiplicand, not greater. ADDENDUM: If the above makes no sense to you, you might want to consider what 0.0[5] • 5.0[0] is in your system. >> >> What is 3.0[2] / 3.0[0] > [2] is indivisible by [0] so disallowed. So then what is [3.0] / [2.0] ? If your infinitesimals are non-zero (as you claim), then both 3.0[0] and 2.0[0] should correspond to some numbers 0.0[x] and 0.0[3x], but since 3 isn't divisible by 2 are you claiming this division is also disallowed? André -- To email remove 'invalid' & replace 'gm' with well known Google mail service.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-18 13:49 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <sLSdnaLH0vu14R_AnZ2dnUU7-IXNnZ2d@giganews.com> |
| In reply to | #211033 |
On 9/18/2019 1:36 PM, André G. Isaak wrote: > On 2019-09-18 12:10 p.m., peteolcott wrote: >> On 9/18/2019 11:42 AM, André G. Isaak wrote: >>> On 2019-09-18 10:26 a.m., peteolcott wrote: >>>> On 9/18/2019 11:14 AM, André G. Isaak wrote: >>>>> On 2019-09-18 10:00 a.m., peteolcott wrote: > > <snip> > >>>>>> Infinitesimals with an offset specify the gaps between the real numbers. >>>>> >>>>> And do you have any evidence to support the existence of such gaps? >>>> >>>> The point immediately adjacent to 0.0 cannot be expressed as a Real number, >>>> it can be expression as this Infinitesimal: 0.0[1]. >>> >>> That's not evidence for a gap. That's simply you claiming there is a gap and then giving a name to something that you claim fills this alleged gap. But unless you can show the gap exists, there's nothing to fill. >> When you try to use Real numbers to locate the point X that is >> immediately adjacent to 0.0 there is always a measurable distance >> between 0.0 and X. A measurable distance is ALWAYS a gap, hence >> Real numbers always have gaps. Infinitesimals fill in all these gaps >> thus forming a superset of Reals. > > That 'gap' is full of real numbers (2^ℵ₀ of them). Which means that X isn't actually adjacent to zero. If there are no points adjacent to zero then there is a gap that can be filled in with Infinitesimals. If there is no gap then there is a point adjacent to zero. -- 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 12:59 -0600 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <qltur4$gs9$1@dont-email.me> |
| In reply to | #211035 |
On 2019-09-18 12:49 p.m., peteolcott wrote: > On 9/18/2019 1:36 PM, André G. Isaak wrote: >> On 2019-09-18 12:10 p.m., peteolcott wrote: >>> On 9/18/2019 11:42 AM, André G. Isaak wrote: >>>> On 2019-09-18 10:26 a.m., peteolcott wrote: >>>>> On 9/18/2019 11:14 AM, André G. Isaak wrote: >>>>>> On 2019-09-18 10:00 a.m., peteolcott wrote: >> >> <snip> >> >>>>>>> Infinitesimals with an offset specify the gaps between the real >>>>>>> numbers. >>>>>> >>>>>> And do you have any evidence to support the existence of such gaps? >>>>> >>>>> The point immediately adjacent to 0.0 cannot be expressed as a Real >>>>> number, >>>>> it can be expression as this Infinitesimal: 0.0[1]. >>>> >>>> That's not evidence for a gap. That's simply you claiming there is a >>>> gap and then giving a name to something that you claim fills this >>>> alleged gap. But unless you can show the gap exists, there's nothing >>>> to fill. >>> When you try to use Real numbers to locate the point X that is >>> immediately adjacent to 0.0 there is always a measurable distance >>> between 0.0 and X. A measurable distance is ALWAYS a gap, hence >>> Real numbers always have gaps. Infinitesimals fill in all these gaps >>> thus forming a superset of Reals. >> >> That 'gap' is full of real numbers (2^ℵ₀ of them). Which means that X >> isn't actually adjacent to zero. > > If there are no points adjacent to zero then there is a gap that can be > filled in with Infinitesimals. If there is no gap then there is a point > adjacent to zero. I've restored the questions on arithmetic which you snipped. It would be nice if you actually addressed them. >> What is 0.0[5] • 0.0[5] ? > > 0.0[25] This, of course, makes absolutely no sense. When you multiply two numbers both of which are less than 1.0, you expect the product to be *less* than either multiplicand, not greater. ADDENDUM: If the above makes no sense to you, you might want to consider what 0.0[5] • 5.0[0] is in your system. >> >> What is 3.0[2] / 3.0[0] > [2] is indivisible by [0] so disallowed. So then what is [3.0] / [2.0] ? If your infinitesimals are non-zero (as you claim), then both 3.0[0] and 2.0[0] should correspond to some numbers 0.0[x] and 0.0[3x], but since 3 isn't divisible by 2 are you claiming this division is also disallowed? André -- To email remove 'invalid' & replace 'gm' with well known Google mail service.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-18 14:12 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <2sidnR-s6Ko2HB_AnZ2dnUU7-IudnZ2d@giganews.com> |
| In reply to | #211036 |
On 9/18/2019 1:59 PM, André G. Isaak wrote: > On 2019-09-18 12:49 p.m., peteolcott wrote: >> On 9/18/2019 1:36 PM, André G. Isaak wrote: >>> On 2019-09-18 12:10 p.m., peteolcott wrote: >>>> On 9/18/2019 11:42 AM, André G. Isaak wrote: >>>>> On 2019-09-18 10:26 a.m., peteolcott wrote: >>>>>> On 9/18/2019 11:14 AM, André G. Isaak wrote: >>>>>>> On 2019-09-18 10:00 a.m., peteolcott wrote: >>> >>> <snip> >>> >>>>>>>> Infinitesimals with an offset specify the gaps between the real numbers. >>>>>>> >>>>>>> And do you have any evidence to support the existence of such gaps? >>>>>> >>>>>> The point immediately adjacent to 0.0 cannot be expressed as a Real number, >>>>>> it can be expression as this Infinitesimal: 0.0[1]. >>>>> >>>>> That's not evidence for a gap. That's simply you claiming there is a gap and then giving a name to something that you claim fills this alleged gap. But unless you can show the gap exists, there's nothing to fill. >>>> When you try to use Real numbers to locate the point X that is >>>> immediately adjacent to 0.0 there is always a measurable distance >>>> between 0.0 and X. A measurable distance is ALWAYS a gap, hence >>>> Real numbers always have gaps. Infinitesimals fill in all these gaps >>>> thus forming a superset of Reals. >>> >>> That 'gap' is full of real numbers (2^ℵ₀ of them). Which means that X isn't actually adjacent to zero. >> >> If there are no points adjacent to zero then there is a gap that can be >> filled in with Infinitesimals. If there is no gap then there is a point >> adjacent to zero. > > > I've restored the questions on arithmetic which you snipped. It would be nice if you actually addressed them. > The above is prerequisite to anything else. I insist that we actually get to the point instead of perpetually talking in circles. -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | Fred <Fred@nowhere.invalid> |
|---|---|
| Date | 2019-09-18 20:25 +0100 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <qlu0c4$mjv$1@news.albasani.net> |
| In reply to | #211037 |
On 18/09/2019 20:12, peteolcott wrote: > On 9/18/2019 1:59 PM, André G. Isaak wrote: >> On 2019-09-18 12:49 p.m., peteolcott wrote: >>> If there are no points adjacent to zero then there is a gap that can be >>> filled in with Infinitesimals. If there is no gap then there is a point >>> adjacent to zero. I would have thought it was just the opposite: if x is adjacent to 0 then the interval (0,x) is all gap. >> I've restored the questions on arithmetic which you snipped. It would >> be nice if you actually addressed them. >> > > The above is prerequisite to anything else. > I insist that we actually get to the point instead of perpetually > talking in circles. >
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| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2019-09-18 13:30 -0600 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <qlu0ln$tdf$1@dont-email.me> |
| In reply to | #211037 |
On 2019-09-18 1:12 p.m., peteolcott wrote: > On 9/18/2019 1:59 PM, André G. Isaak wrote: >> On 2019-09-18 12:49 p.m., peteolcott wrote: >>> On 9/18/2019 1:36 PM, André G. Isaak wrote: >>>> On 2019-09-18 12:10 p.m., peteolcott wrote: >>>>> On 9/18/2019 11:42 AM, André G. Isaak wrote: >>>>>> On 2019-09-18 10:26 a.m., peteolcott wrote: >>>>>>> On 9/18/2019 11:14 AM, André G. Isaak wrote: >>>>>>>> On 2019-09-18 10:00 a.m., peteolcott wrote: >>>> >>>> <snip> >>>> >>>>>>>>> Infinitesimals with an offset specify the gaps between the real >>>>>>>>> numbers. >>>>>>>> >>>>>>>> And do you have any evidence to support the existence of such gaps? >>>>>>> >>>>>>> The point immediately adjacent to 0.0 cannot be expressed as a >>>>>>> Real number, >>>>>>> it can be expression as this Infinitesimal: 0.0[1]. >>>>>> >>>>>> That's not evidence for a gap. That's simply you claiming there is >>>>>> a gap and then giving a name to something that you claim fills >>>>>> this alleged gap. But unless you can show the gap exists, there's >>>>>> nothing to fill. >>>>> When you try to use Real numbers to locate the point X that is >>>>> immediately adjacent to 0.0 there is always a measurable distance >>>>> between 0.0 and X. A measurable distance is ALWAYS a gap, hence >>>>> Real numbers always have gaps. Infinitesimals fill in all these gaps >>>>> thus forming a superset of Reals. >>>> >>>> That 'gap' is full of real numbers (2^ℵ₀ of them). Which means that >>>> X isn't actually adjacent to zero. >>> >>> If there are no points adjacent to zero then there is a gap that can be >>> filled in with Infinitesimals. If there is no gap then there is a point >>> adjacent to zero. >> >> >> I've restored the questions on arithmetic which you snipped. It would >> be nice if you actually addressed them. >> > > The above is prerequisite to anything else. > I insist that we actually get to the point instead of perpetually > talking in circles. I've already addressed the above several times. There is no number adjacent to zero. Any 'gap' in the line is filled with other real numbers. There is nothing whatsoever in the axioms which define the reals which supports your premise that 'either there is a gap or there is a number adjacent to zero'. Real numbers are dense. There are no gaps in the real number system. Your faulty premise is based on your inability to conceive of a dense ordering. Until you've actually shown that your infinitesimal system is actually a coherent system in which arithmetic can be performed, it is not possible to discuss what your infinitesimals can or cannot accomplish. So I've restored the questions on arithmetic which you snipped. It would be nice if you actually addressed them. >> What is 0.0[5] • 0.0[5] ? > > 0.0[25] This, of course, makes absolutely no sense. When you multiply two numbers both of which are less than 1.0, you expect the product to be *less* than either multiplicand, not greater. ADDENDUM: If the above makes no sense to you, you might want to consider what 0.0[5] • 5.0[0] is in your system. >> >> What is 3.0[2] / 3.0[0] > [2] is indivisible by [0] so disallowed. So then what is [3.0] / [2.0] ? If your infinitesimals are non-zero (as you claim), then both 3.0[0] and 2.0[0] should correspond to some numbers 0.0[x] and 0.0[3x], but since 3 isn't divisible by 2 are you claiming this division is also disallowed? André -- To email remove 'invalid' & replace 'gm' with well known Google mail service.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-18 14:45 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <iZ-dnRV6mJ3GFB_AnZ2dnUU7-cPNnZ2d@giganews.com> |
| In reply to | #211039 |
On 9/18/2019 2:30 PM, André G. Isaak wrote: > On 2019-09-18 1:12 p.m., peteolcott wrote: >> On 9/18/2019 1:59 PM, André G. Isaak wrote: >>> On 2019-09-18 12:49 p.m., peteolcott wrote: >>>> On 9/18/2019 1:36 PM, André G. Isaak wrote: >>>>> On 2019-09-18 12:10 p.m., peteolcott wrote: >>>>>> On 9/18/2019 11:42 AM, André G. Isaak wrote: >>>>>>> On 2019-09-18 10:26 a.m., peteolcott wrote: >>>>>>>> On 9/18/2019 11:14 AM, André G. Isaak wrote: >>>>>>>>> On 2019-09-18 10:00 a.m., peteolcott wrote: >>>>> >>>>> <snip> >>>>> >>>>>>>>>> Infinitesimals with an offset specify the gaps between the real numbers. >>>>>>>>> >>>>>>>>> And do you have any evidence to support the existence of such gaps? >>>>>>>> >>>>>>>> The point immediately adjacent to 0.0 cannot be expressed as a Real number, >>>>>>>> it can be expression as this Infinitesimal: 0.0[1]. >>>>>>> >>>>>>> That's not evidence for a gap. That's simply you claiming there is a gap and then giving a name to something that you claim fills this alleged gap. But unless you can show the gap exists, there's nothing to fill. >>>>>> When you try to use Real numbers to locate the point X that is >>>>>> immediately adjacent to 0.0 there is always a measurable distance >>>>>> between 0.0 and X. A measurable distance is ALWAYS a gap, hence >>>>>> Real numbers always have gaps. Infinitesimals fill in all these gaps >>>>>> thus forming a superset of Reals. >>>>> >>>>> That 'gap' is full of real numbers (2^ℵ₀ of them). Which means that X isn't actually adjacent to zero. >>>> >>>> If there are no points adjacent to zero then there is a gap that can be >>>> filled in with Infinitesimals. If there is no gap then there is a point >>>> adjacent to zero. >>> >>> >>> I've restored the questions on arithmetic which you snipped. It would be nice if you actually addressed them. >>> >> >> The above is prerequisite to anything else. >> I insist that we actually get to the point instead of perpetually >> talking in circles. > > I've already addressed the above several times. There is no number adjacent to zero. Any 'gap' in the line is filled with other real numbers. Each one having their own gap. If we divide the distance between 0.1 and 0.0 an infinite number of times we reach the point immediately adjacent to 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 | 2019-09-18 14:14 -0600 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <qlu37l$e9q$1@dont-email.me> |
| In reply to | #211040 |
On 2019-09-18 1:45 p.m., peteolcott wrote: > On 9/18/2019 2:30 PM, André G. Isaak wrote: >> On 2019-09-18 1:12 p.m., peteolcott wrote: >>> On 9/18/2019 1:59 PM, André G. Isaak wrote: >>>> On 2019-09-18 12:49 p.m., peteolcott wrote: >>>>> On 9/18/2019 1:36 PM, André G. Isaak wrote: >>>>>> On 2019-09-18 12:10 p.m., peteolcott wrote: >>>>>>> On 9/18/2019 11:42 AM, André G. Isaak wrote: >>>>>>>> On 2019-09-18 10:26 a.m., peteolcott wrote: >>>>>>>>> On 9/18/2019 11:14 AM, André G. Isaak wrote: >>>>>>>>>> On 2019-09-18 10:00 a.m., peteolcott wrote: >>>>>> >>>>>> <snip> >>>>>> >>>>>>>>>>> Infinitesimals with an offset specify the gaps between the >>>>>>>>>>> real numbers. >>>>>>>>>> >>>>>>>>>> And do you have any evidence to support the existence of such >>>>>>>>>> gaps? >>>>>>>>> >>>>>>>>> The point immediately adjacent to 0.0 cannot be expressed as a >>>>>>>>> Real number, >>>>>>>>> it can be expression as this Infinitesimal: 0.0[1]. >>>>>>>> >>>>>>>> That's not evidence for a gap. That's simply you claiming there >>>>>>>> is a gap and then giving a name to something that you claim >>>>>>>> fills this alleged gap. But unless you can show the gap exists, >>>>>>>> there's nothing to fill. >>>>>>> When you try to use Real numbers to locate the point X that is >>>>>>> immediately adjacent to 0.0 there is always a measurable distance >>>>>>> between 0.0 and X. A measurable distance is ALWAYS a gap, hence >>>>>>> Real numbers always have gaps. Infinitesimals fill in all these gaps >>>>>>> thus forming a superset of Reals. >>>>>> >>>>>> That 'gap' is full of real numbers (2^ℵ₀ of them). Which means >>>>>> that X isn't actually adjacent to zero. >>>>> >>>>> If there are no points adjacent to zero then there is a gap that >>>>> can be >>>>> filled in with Infinitesimals. If there is no gap then there is a >>>>> point >>>>> adjacent to zero. >>>> >>>> >>>> I've restored the questions on arithmetic which you snipped. It >>>> would be nice if you actually addressed them. >>>> >>> >>> The above is prerequisite to anything else. >>> I insist that we actually get to the point instead of perpetually >>> talking in circles. >> >> I've already addressed the above several times. There is no number >> adjacent to zero. Any 'gap' in the line is filled with other real >> numbers. > > Each one having their own gap. Which, again, is filled with real numbers. That is what 'densely ordered' means. > If we divide the distance between 0.1 and 0.0 an infinite number > of times we reach the point immediately adjacent to 0.0. No nonzero number divided by 2 equals 0. Dividing an infinite number of times gives a *limit* of zero. Do you understand what a limit is? I'm assuming not since you don't seem to understand what an open interval is. If you claim that 0.0[1] is 'adjacent' to zero, but that 0.0[1] is not equal to zero, then by your logic there is a gap of length 0.0[1] between zero and your 'adjacent' point. The only line segment which cannot be further subdivided is one of length zero (i.e. not a line segment at all but rather a point). Until you've actually shown that your infinitesimal system is actually a coherent system in which arithmetic can be performed, it is not possible to discuss what your infinitesimals can or cannot accomplish. So I've restored the questions on arithmetic which you snipped. It would be nice if you actually addressed them. >> What is 0.0[5] • 0.0[5] ? > > 0.0[25] This, of course, makes absolutely no sense. When you multiply two numbers both of which are less than 1.0, you expect the product to be *less* than either multiplicand, not greater. ADDENDUM: If the above makes no sense to you, you might want to consider what 0.0[5] • 5.0[0] is in your system. >> >> What is 3.0[2] / 3.0[0] > [2] is indivisible by [0] so disallowed. So then what is [3.0] / [2.0] ? If your infinitesimals are non-zero (as you claim), then both 3.0[0] and 2.0[0] should correspond to some numbers 0.0[x] and 0.0[3x], but since 3 isn't divisible by 2 are you claiming this division is also disallowed? André -- To email remove 'invalid' & replace 'gm' with well known Google mail service.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-18 15:29 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <cKmdnYeIz84zDh_AnZ2dnUU7-eGdnZ2d@giganews.com> |
| In reply to | #211041 |
On 9/18/2019 3:14 PM, André G. Isaak wrote: > On 2019-09-18 1:45 p.m., peteolcott wrote: >> On 9/18/2019 2:30 PM, André G. Isaak wrote: >>> On 2019-09-18 1:12 p.m., peteolcott wrote: >>>> On 9/18/2019 1:59 PM, André G. Isaak wrote: >>>>> On 2019-09-18 12:49 p.m., peteolcott wrote: >>>>>> On 9/18/2019 1:36 PM, André G. Isaak wrote: >>>>>>> On 2019-09-18 12:10 p.m., peteolcott wrote: >>>>>>>> On 9/18/2019 11:42 AM, André G. Isaak wrote: >>>>>>>>> On 2019-09-18 10:26 a.m., peteolcott wrote: >>>>>>>>>> On 9/18/2019 11:14 AM, André G. Isaak wrote: >>>>>>>>>>> On 2019-09-18 10:00 a.m., peteolcott wrote: >>>>>>> >>>>>>> <snip> >>>>>>> >>>>>>>>>>>> Infinitesimals with an offset specify the gaps between the real numbers. >>>>>>>>>>> >>>>>>>>>>> And do you have any evidence to support the existence of such gaps? >>>>>>>>>> >>>>>>>>>> The point immediately adjacent to 0.0 cannot be expressed as a Real number, >>>>>>>>>> it can be expression as this Infinitesimal: 0.0[1]. >>>>>>>>> >>>>>>>>> That's not evidence for a gap. That's simply you claiming there is a gap and then giving a name to something that you claim fills this alleged gap. But unless you can show the gap exists, there's nothing to fill. >>>>>>>> When you try to use Real numbers to locate the point X that is >>>>>>>> immediately adjacent to 0.0 there is always a measurable distance >>>>>>>> between 0.0 and X. A measurable distance is ALWAYS a gap, hence >>>>>>>> Real numbers always have gaps. Infinitesimals fill in all these gaps >>>>>>>> thus forming a superset of Reals. >>>>>>> >>>>>>> That 'gap' is full of real numbers (2^ℵ₀ of them). Which means that X isn't actually adjacent to zero. >>>>>> >>>>>> If there are no points adjacent to zero then there is a gap that can be >>>>>> filled in with Infinitesimals. If there is no gap then there is a point >>>>>> adjacent to zero. >>>>> >>>>> >>>>> I've restored the questions on arithmetic which you snipped. It would be nice if you actually addressed them. >>>>> >>>> >>>> The above is prerequisite to anything else. >>>> I insist that we actually get to the point instead of perpetually >>>> talking in circles. >>> >>> I've already addressed the above several times. There is no number adjacent to zero. Any 'gap' in the line is filled with other real numbers. >> >> Each one having their own gap. > > Which, again, is filled with real numbers. That is what 'densely ordered' means. >> Each one having their own gap. >> Each one having their own gap. >> Each one having their own gap. >> Each one having their own gap. >> Each one having their own gap. >> Each one having their own gap. >> Each one having their own gap. >> Each one having their own gap. >> Each one having their own gap. >> Each one having their own gap. >> Each one having their own gap. >> Each one having their own gap. >> Each one having their own gap. >> Each one having their own gap. >> Each one having their own gap. >> Each one having their own gap. >> Each one having their own gap. -- 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 14:38 -0600 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <qlu4l5$mue$1@dont-email.me> |
| In reply to | #211042 |
On 2019-09-18 2:29 p.m., peteolcott wrote: > On 9/18/2019 3:14 PM, André G. Isaak wrote: >> On 2019-09-18 1:45 p.m., peteolcott wrote: >>> On 9/18/2019 2:30 PM, André G. Isaak wrote: >>>> On 2019-09-18 1:12 p.m., peteolcott wrote: >>>>> On 9/18/2019 1:59 PM, André G. Isaak wrote: >>>>>> On 2019-09-18 12:49 p.m., peteolcott wrote: >>>>>>> On 9/18/2019 1:36 PM, André G. Isaak wrote: >>>>>>>> On 2019-09-18 12:10 p.m., peteolcott wrote: >>>>>>>>> On 9/18/2019 11:42 AM, André G. Isaak wrote: >>>>>>>>>> On 2019-09-18 10:26 a.m., peteolcott wrote: >>>>>>>>>>> On 9/18/2019 11:14 AM, André G. Isaak wrote: >>>>>>>>>>>> On 2019-09-18 10:00 a.m., peteolcott wrote: >>>>>>>> >>>>>>>> <snip> >>>>>>>> >>>>>>>>>>>>> Infinitesimals with an offset specify the gaps between the >>>>>>>>>>>>> real numbers. >>>>>>>>>>>> >>>>>>>>>>>> And do you have any evidence to support the existence of >>>>>>>>>>>> such gaps? >>>>>>>>>>> >>>>>>>>>>> The point immediately adjacent to 0.0 cannot be expressed as >>>>>>>>>>> a Real number, >>>>>>>>>>> it can be expression as this Infinitesimal: 0.0[1]. >>>>>>>>>> >>>>>>>>>> That's not evidence for a gap. That's simply you claiming >>>>>>>>>> there is a gap and then giving a name to something that you >>>>>>>>>> claim fills this alleged gap. But unless you can show the gap >>>>>>>>>> exists, there's nothing to fill. >>>>>>>>> When you try to use Real numbers to locate the point X that is >>>>>>>>> immediately adjacent to 0.0 there is always a measurable distance >>>>>>>>> between 0.0 and X. A measurable distance is ALWAYS a gap, hence >>>>>>>>> Real numbers always have gaps. Infinitesimals fill in all these >>>>>>>>> gaps >>>>>>>>> thus forming a superset of Reals. >>>>>>>> >>>>>>>> That 'gap' is full of real numbers (2^ℵ₀ of them). Which means >>>>>>>> that X isn't actually adjacent to zero. >>>>>>> >>>>>>> If there are no points adjacent to zero then there is a gap that >>>>>>> can be >>>>>>> filled in with Infinitesimals. If there is no gap then there is a >>>>>>> point >>>>>>> adjacent to zero. >>>>>> >>>>>> >>>>>> I've restored the questions on arithmetic which you snipped. It >>>>>> would be nice if you actually addressed them. >>>>>> >>>>> >>>>> The above is prerequisite to anything else. >>>>> I insist that we actually get to the point instead of perpetually >>>>> talking in circles. >>>> >>>> I've already addressed the above several times. There is no number >>>> adjacent to zero. Any 'gap' in the line is filled with other real >>>> numbers. >>> >>> Each one having their own gap. >> >> Which, again, is filled with real numbers. That is what 'densely >> ordered' means. > > >> Each one having their own gap. <snip pointless repetition> If I were as childish as you are, I would simply repeat 'each of which is filled with real numbers' over and over again. But, I will try to explain this to you. There is no point on the real number line which is not occupied by a real number. That precludes there from being caps. Any two distinct points on the real number line are separated by an infinite number of points. That precludes the notion that any two points can be adjacent. The fact that you don't get this simply indicates that there are at least some properties of the reals which you don't understand. Now that you've had your little temper tantrum about the fact that you didn't think your points were being addressed (which they were), perhaps you'll actually address mine. Until you've actually shown that your infinitesimal system is actually a coherent system in which arithmetic can be performed, it is not possible to discuss what your infinitesimals can or cannot accomplish. So I've restored the questions on arithmetic which you snipped. It would be nice if you actually addressed them. >> What is 0.0[5] • 0.0[5] ? > > 0.0[25] This, of course, makes absolutely no sense. When you multiply two numbers both of which are less than 1.0, you expect the product to be *less* than either multiplicand, not greater. ADDENDUM: If the above makes no sense to you, you might want to consider what 0.0[5] • 5.0[0] is in your system. >> >> What is 3.0[2] / 3.0[0] > [2] is indivisible by [0] so disallowed. So then what is [3.0] / [2.0] ? If your infinitesimals are non-zero (as you claim), then both 3.0[0] and 2.0[0] should correspond to some numbers 0.0[x] and 0.0[3x], but since 3 isn't divisible by 2 are you claiming this division is also disallowed? -- To email remove 'invalid' & replace 'gm' with well known Google mail service.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-18 15:56 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <Kv6dnQdSMZRwBB_AnZ2dnUU7-WvNnZ2d@giganews.com> |
| In reply to | #211044 |
On 9/18/2019 3:38 PM, André G. Isaak wrote: > On 2019-09-18 2:29 p.m., peteolcott wrote: >> On 9/18/2019 3:14 PM, André G. Isaak wrote: >>> On 2019-09-18 1:45 p.m., peteolcott wrote: >>>> On 9/18/2019 2:30 PM, André G. Isaak wrote: >>>>> On 2019-09-18 1:12 p.m., peteolcott wrote: >>>>>> On 9/18/2019 1:59 PM, André G. Isaak wrote: >>>>>>> On 2019-09-18 12:49 p.m., peteolcott wrote: >>>>>>>> On 9/18/2019 1:36 PM, André G. Isaak wrote: >>>>>>>>> On 2019-09-18 12:10 p.m., peteolcott wrote: >>>>>>>>>> On 9/18/2019 11:42 AM, André G. Isaak wrote: >>>>>>>>>>> On 2019-09-18 10:26 a.m., peteolcott wrote: >>>>>>>>>>>> On 9/18/2019 11:14 AM, André G. Isaak wrote: >>>>>>>>>>>>> On 2019-09-18 10:00 a.m., peteolcott wrote: >>>>>>>>> >>>>>>>>> <snip> >>>>>>>>> >>>>>>>>>>>>>> Infinitesimals with an offset specify the gaps between the real numbers. >>>>>>>>>>>>> >>>>>>>>>>>>> And do you have any evidence to support the existence of such gaps? >>>>>>>>>>>> >>>>>>>>>>>> The point immediately adjacent to 0.0 cannot be expressed as a Real number, >>>>>>>>>>>> it can be expression as this Infinitesimal: 0.0[1]. >>>>>>>>>>> >>>>>>>>>>> That's not evidence for a gap. That's simply you claiming there is a gap and then giving a name to something that you claim fills this alleged gap. But unless you can show the gap exists, there's nothing to fill. >>>>>>>>>> When you try to use Real numbers to locate the point X that is >>>>>>>>>> immediately adjacent to 0.0 there is always a measurable distance >>>>>>>>>> between 0.0 and X. A measurable distance is ALWAYS a gap, hence >>>>>>>>>> Real numbers always have gaps. Infinitesimals fill in all these gaps >>>>>>>>>> thus forming a superset of Reals. >>>>>>>>> >>>>>>>>> That 'gap' is full of real numbers (2^ℵ₀ of them). Which means that X isn't actually adjacent to zero. >>>>>>>> >>>>>>>> If there are no points adjacent to zero then there is a gap that can be >>>>>>>> filled in with Infinitesimals. If there is no gap then there is a point >>>>>>>> adjacent to zero. >>>>>>> >>>>>>> >>>>>>> I've restored the questions on arithmetic which you snipped. It would be nice if you actually addressed them. >>>>>>> >>>>>> >>>>>> The above is prerequisite to anything else. >>>>>> I insist that we actually get to the point instead of perpetually >>>>>> talking in circles. >>>>> >>>>> I've already addressed the above several times. There is no number adjacent to zero. Any 'gap' in the line is filled with other real numbers. >>>> >>>> Each one having their own gap. >>> >>> Which, again, is filled with real numbers. That is what 'densely ordered' means. >> >> >> Each one having their own gap. > > <snip pointless repetition> > > If I were as childish as you are, I would simply repeat 'each of which is filled with real numbers' over and over again. NEVER FRIGGING FILLED BECAUSE EACH REAL BRINGS ITS OWN GAP UNTIL THIS PROCESS IS REPEATED AN INFINITE NUMBER OF TIMES THEN A POINT IMMEDIATELY ADJACENT TO 0.0 IS DERIVED THUS PROVING ME CORRECT. -- 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 20:38 -0600 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <qlupml$r9h$1@dont-email.me> |
| In reply to | #211047 |
On 2019-09-18 2:56 p.m., peteolcott wrote: > On 9/18/2019 3:38 PM, André G. Isaak wrote: >> On 2019-09-18 2:29 p.m., peteolcott wrote: >>> On 9/18/2019 3:14 PM, André G. Isaak wrote: >>>> On 2019-09-18 1:45 p.m., peteolcott wrote: >>>>> On 9/18/2019 2:30 PM, André G. Isaak wrote: >>>>>> On 2019-09-18 1:12 p.m., peteolcott wrote: >>>>>>> On 9/18/2019 1:59 PM, André G. Isaak wrote: >>>>>>>> On 2019-09-18 12:49 p.m., peteolcott wrote: >>>>>>>>> On 9/18/2019 1:36 PM, André G. Isaak wrote: >>>>>>>>>> On 2019-09-18 12:10 p.m., peteolcott wrote: >>>>>>>>>>> On 9/18/2019 11:42 AM, André G. Isaak wrote: >>>>>>>>>>>> On 2019-09-18 10:26 a.m., peteolcott wrote: >>>>>>>>>>>>> On 9/18/2019 11:14 AM, André G. Isaak wrote: >>>>>>>>>>>>>> On 2019-09-18 10:00 a.m., peteolcott wrote: >>>>>>>>>> >>>>>>>>>> <snip> >>>>>>>>>> >>>>>>>>>>>>>>> Infinitesimals with an offset specify the gaps between >>>>>>>>>>>>>>> the real numbers. >>>>>>>>>>>>>> >>>>>>>>>>>>>> And do you have any evidence to support the existence of >>>>>>>>>>>>>> such gaps? >>>>>>>>>>>>> >>>>>>>>>>>>> The point immediately adjacent to 0.0 cannot be expressed >>>>>>>>>>>>> as a Real number, >>>>>>>>>>>>> it can be expression as this Infinitesimal: 0.0[1]. >>>>>>>>>>>> >>>>>>>>>>>> That's not evidence for a gap. That's simply you claiming >>>>>>>>>>>> there is a gap and then giving a name to something that you >>>>>>>>>>>> claim fills this alleged gap. But unless you can show the >>>>>>>>>>>> gap exists, there's nothing to fill. >>>>>>>>>>> When you try to use Real numbers to locate the point X that is >>>>>>>>>>> immediately adjacent to 0.0 there is always a measurable >>>>>>>>>>> distance >>>>>>>>>>> between 0.0 and X. A measurable distance is ALWAYS a gap, hence >>>>>>>>>>> Real numbers always have gaps. Infinitesimals fill in all >>>>>>>>>>> these gaps >>>>>>>>>>> thus forming a superset of Reals. >>>>>>>>>> >>>>>>>>>> That 'gap' is full of real numbers (2^ℵ₀ of them). Which means >>>>>>>>>> that X isn't actually adjacent to zero. >>>>>>>>> >>>>>>>>> If there are no points adjacent to zero then there is a gap >>>>>>>>> that can be >>>>>>>>> filled in with Infinitesimals. If there is no gap then there is >>>>>>>>> a point >>>>>>>>> adjacent to zero. >>>>>>>> >>>>>>>> >>>>>>>> I've restored the questions on arithmetic which you snipped. It >>>>>>>> would be nice if you actually addressed them. >>>>>>>> >>>>>>> >>>>>>> The above is prerequisite to anything else. >>>>>>> I insist that we actually get to the point instead of perpetually >>>>>>> talking in circles. >>>>>> >>>>>> I've already addressed the above several times. There is no number >>>>>> adjacent to zero. Any 'gap' in the line is filled with other real >>>>>> numbers. >>>>> >>>>> Each one having their own gap. >>>> >>>> Which, again, is filled with real numbers. That is what 'densely >>>> ordered' means. >>> >>> >> Each one having their own gap. >> >> <snip pointless repetition> >> >> If I were as childish as you are, I would simply repeat 'each of which >> is filled with real numbers' over and over again. > > NEVER FRIGGING FILLED BECAUSE EACH REAL BRINGS ITS OWN GAP > UNTIL THIS PROCESS IS REPEATED AN INFINITE NUMBER OF TIMES > THEN A POINT IMMEDIATELY ADJACENT TO 0.0 IS DERIVED THUS > PROVING ME CORRECT. As I already noted, *every* point on the real number line represents a real number. There are no gaps. More importantly, there is no "process" being repeated. I think you are viewing this somewhat like the following: if we start with some real (say 1.0), we split the line in half at 0.5, leaving us with two gaps which need to be filled with 0.25 and 0.75 and so forth. But that is simply not the case. The real number line refers to an object which is already replete with points. While we can talk about constructing it by some process, that is entirely unnecessary. When we split 1.0 in half at 0.5, we now have a shorter segment which is *already* completely filled will points, which has no gaps, and was always that way. But, even if your argument were legitimate (it isn't), your 'infinitesimal' wouldn't solve it. If you claim that 0.0[1] is the least positive number, you are claiming that you have a point 0, followed by another point, 0.0[1] which are separated by a distance of 0.0[1]. What occupies that 0.0[1] length of line segment between zero and 0.0[1]? Is that not a 'gap'? André -- To email remove 'invalid' & replace 'gm' with well known Google mail service.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-18 22:03 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <uYednXFT3daQbR_AnZ2dnUU7-RPNnZ2d@giganews.com> |
| In reply to | #211062 |
On 9/18/2019 9:38 PM, André G. Isaak wrote: > On 2019-09-18 2:56 p.m., peteolcott wrote: >> On 9/18/2019 3:38 PM, André G. Isaak wrote: >>> On 2019-09-18 2:29 p.m., peteolcott wrote: >>>> On 9/18/2019 3:14 PM, André G. Isaak wrote: >>>>> On 2019-09-18 1:45 p.m., peteolcott wrote: >>>>>> On 9/18/2019 2:30 PM, André G. Isaak wrote: >>>>>>> On 2019-09-18 1:12 p.m., peteolcott wrote: >>>>>>>> On 9/18/2019 1:59 PM, André G. Isaak wrote: >>>>>>>>> On 2019-09-18 12:49 p.m., peteolcott wrote: >>>>>>>>>> On 9/18/2019 1:36 PM, André G. Isaak wrote: >>>>>>>>>>> On 2019-09-18 12:10 p.m., peteolcott wrote: >>>>>>>>>>>> On 9/18/2019 11:42 AM, André G. Isaak wrote: >>>>>>>>>>>>> On 2019-09-18 10:26 a.m., peteolcott wrote: >>>>>>>>>>>>>> On 9/18/2019 11:14 AM, André G. Isaak wrote: >>>>>>>>>>>>>>> On 2019-09-18 10:00 a.m., peteolcott wrote: >>>>>>>>>>> >>>>>>>>>>> <snip> >>>>>>>>>>> >>>>>>>>>>>>>>>> Infinitesimals with an offset specify the gaps between the real numbers. >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> And do you have any evidence to support the existence of such gaps? >>>>>>>>>>>>>> >>>>>>>>>>>>>> The point immediately adjacent to 0.0 cannot be expressed as a Real number, >>>>>>>>>>>>>> it can be expression as this Infinitesimal: 0.0[1]. >>>>>>>>>>>>> >>>>>>>>>>>>> That's not evidence for a gap. That's simply you claiming there is a gap and then giving a name to something that you claim fills this alleged gap. But unless you can show the gap exists, there's nothing to fill. >>>>>>>>>>>> When you try to use Real numbers to locate the point X that is >>>>>>>>>>>> immediately adjacent to 0.0 there is always a measurable distance >>>>>>>>>>>> between 0.0 and X. A measurable distance is ALWAYS a gap, hence >>>>>>>>>>>> Real numbers always have gaps. Infinitesimals fill in all these gaps >>>>>>>>>>>> thus forming a superset of Reals. >>>>>>>>>>> >>>>>>>>>>> That 'gap' is full of real numbers (2^ℵ₀ of them). Which means that X isn't actually adjacent to zero. >>>>>>>>>> >>>>>>>>>> If there are no points adjacent to zero then there is a gap that can be >>>>>>>>>> filled in with Infinitesimals. If there is no gap then there is a point >>>>>>>>>> adjacent to zero. >>>>>>>>> >>>>>>>>> >>>>>>>>> I've restored the questions on arithmetic which you snipped. It would be nice if you actually addressed them. >>>>>>>>> >>>>>>>> >>>>>>>> The above is prerequisite to anything else. >>>>>>>> I insist that we actually get to the point instead of perpetually >>>>>>>> talking in circles. >>>>>>> >>>>>>> I've already addressed the above several times. There is no number adjacent to zero. Any 'gap' in the line is filled with other real numbers. >>>>>> >>>>>> Each one having their own gap. >>>>> >>>>> Which, again, is filled with real numbers. That is what 'densely ordered' means. >>>> >>>> >> Each one having their own gap. >>> >>> <snip pointless repetition> >>> >>> If I were as childish as you are, I would simply repeat 'each of which is filled with real numbers' over and over again. >> >> NEVER FRIGGING FILLED BECAUSE EACH REAL BRINGS ITS OWN GAP >> UNTIL THIS PROCESS IS REPEATED AN INFINITE NUMBER OF TIMES >> THEN A POINT IMMEDIATELY ADJACENT TO 0.0 IS DERIVED THUS >> PROVING ME CORRECT. > > As I already noted, *every* point on the real number line represents a real number. There are no gaps. THERE IS THE INCONSISTENCY. (1) There cannot be a real number associated with the point immediately adjacent to 0.0 because real numbers must be divisible by 2 and single points are indivisible. (2) If there is no real number immediately adjacent to 0.0 then there is always a measurable gap after 0.0. By dividing the size of the gap by 2 it gets smaller and smaller yet never closes, there are always points in the gap that are unaccounted for. > More importantly, there is no "process" being repeated. I think you are viewing this somewhat like the following: if we start with some real (say 1.0), > we split the line in half at 0.5, leaving us with two gaps which need to be filled with 0.25 and 0.75 and so forth. > > But that is simply not the case. The real number line refers to an object which is already replete with points. While we can talk about constructing it by some process, that is entirely unnecessary. When we split 1.0 in half at 0.5, we now have a shorter > segment which is *already* completely filled will points, which has no gaps, and was always that way. > > But, even if your argument were legitimate (it isn't), your 'infinitesimal' wouldn't solve it. If you claim that 0.0[1] is the least positive number, you are claiming that you have a point 0, followed by another point, 0.0[1] which are separated by a > distance of 0.0[1]. What occupies that 0.0[1] length of line segment between zero and 0.0[1]? Is that not a 'gap'? > > 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 21:33 -0600 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <qlusv0$81r$1@dont-email.me> |
| In reply to | #211063 |
On 2019-09-18 9:03 p.m., peteolcott wrote: > On 9/18/2019 9:38 PM, André G. Isaak wrote: >> On 2019-09-18 2:56 p.m., peteolcott wrote: >>> On 9/18/2019 3:38 PM, André G. Isaak wrote: >>>> On 2019-09-18 2:29 p.m., peteolcott wrote: >>>>> On 9/18/2019 3:14 PM, André G. Isaak wrote: >>>>>> On 2019-09-18 1:45 p.m., peteolcott wrote: >>>>>>> On 9/18/2019 2:30 PM, André G. Isaak wrote: >>>>>>>> On 2019-09-18 1:12 p.m., peteolcott wrote: >>>>>>>>> On 9/18/2019 1:59 PM, André G. Isaak wrote: >>>>>>>>>> On 2019-09-18 12:49 p.m., peteolcott wrote: >>>>>>>>>>> On 9/18/2019 1:36 PM, André G. Isaak wrote: >>>>>>>>>>>> On 2019-09-18 12:10 p.m., peteolcott wrote: >>>>>>>>>>>>> On 9/18/2019 11:42 AM, André G. Isaak wrote: >>>>>>>>>>>>>> On 2019-09-18 10:26 a.m., peteolcott wrote: >>>>>>>>>>>>>>> On 9/18/2019 11:14 AM, André G. Isaak wrote: >>>>>>>>>>>>>>>> On 2019-09-18 10:00 a.m., peteolcott wrote: >>>>>>>>>>>> >>>>>>>>>>>> <snip> >>>>>>>>>>>> >>>>>>>>>>>>>>>>> Infinitesimals with an offset specify the gaps between >>>>>>>>>>>>>>>>> the real numbers. >>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> And do you have any evidence to support the existence of >>>>>>>>>>>>>>>> such gaps? >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> The point immediately adjacent to 0.0 cannot be expressed >>>>>>>>>>>>>>> as a Real number, >>>>>>>>>>>>>>> it can be expression as this Infinitesimal: 0.0[1]. >>>>>>>>>>>>>> >>>>>>>>>>>>>> That's not evidence for a gap. That's simply you claiming >>>>>>>>>>>>>> there is a gap and then giving a name to something that >>>>>>>>>>>>>> you claim fills this alleged gap. But unless you can show >>>>>>>>>>>>>> the gap exists, there's nothing to fill. >>>>>>>>>>>>> When you try to use Real numbers to locate the point X that is >>>>>>>>>>>>> immediately adjacent to 0.0 there is always a measurable >>>>>>>>>>>>> distance >>>>>>>>>>>>> between 0.0 and X. A measurable distance is ALWAYS a gap, >>>>>>>>>>>>> hence >>>>>>>>>>>>> Real numbers always have gaps. Infinitesimals fill in all >>>>>>>>>>>>> these gaps >>>>>>>>>>>>> thus forming a superset of Reals. >>>>>>>>>>>> >>>>>>>>>>>> That 'gap' is full of real numbers (2^ℵ₀ of them). Which >>>>>>>>>>>> means that X isn't actually adjacent to zero. >>>>>>>>>>> >>>>>>>>>>> If there are no points adjacent to zero then there is a gap >>>>>>>>>>> that can be >>>>>>>>>>> filled in with Infinitesimals. If there is no gap then there >>>>>>>>>>> is a point >>>>>>>>>>> adjacent to zero. >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> I've restored the questions on arithmetic which you snipped. >>>>>>>>>> It would be nice if you actually addressed them. >>>>>>>>>> >>>>>>>>> >>>>>>>>> The above is prerequisite to anything else. >>>>>>>>> I insist that we actually get to the point instead of perpetually >>>>>>>>> talking in circles. >>>>>>>> >>>>>>>> I've already addressed the above several times. There is no >>>>>>>> number adjacent to zero. Any 'gap' in the line is filled with >>>>>>>> other real numbers. >>>>>>> >>>>>>> Each one having their own gap. >>>>>> >>>>>> Which, again, is filled with real numbers. That is what 'densely >>>>>> ordered' means. >>>>> >>>>> >> Each one having their own gap. >>>> >>>> <snip pointless repetition> >>>> >>>> If I were as childish as you are, I would simply repeat 'each of >>>> which is filled with real numbers' over and over again. >>> >>> NEVER FRIGGING FILLED BECAUSE EACH REAL BRINGS ITS OWN GAP >>> UNTIL THIS PROCESS IS REPEATED AN INFINITE NUMBER OF TIMES >>> THEN A POINT IMMEDIATELY ADJACENT TO 0.0 IS DERIVED THUS >>> PROVING ME CORRECT. >> >> As I already noted, *every* point on the real number line represents a >> real number. There are no gaps. > > THERE IS THE INCONSISTENCY. > (1) There cannot be a real number associated with the point > immediately adjacent to 0.0 because real numbers must be divisible > by 2 and single points are indivisible. And, as I have repeatedly pointed out, 'adjacency' is simply not something which is even defined for the real numbers. You're working under the false assumption that there *is* a point adjacent to zero which is simply not the case. > (2) If there is no real number immediately adjacent to 0.0 then > there is always a measurable gap after 0.0. By dividing the > size of the gap by 2 it gets smaller and smaller yet never closes, > there are always points in the gap that are unaccounted for. > >> More importantly, there is no "process" being repeated. I think you >> are viewing this somewhat like the following: if we start with some >> real (say 1.0), we split the line in half at 0.5, leaving us with two >> gaps which need to be filled with 0.25 and 0.75 and so forth. >> >> But that is simply not the case. The real number line refers to an >> object which is already replete with points. While we can talk about >> constructing it by some process, that is entirely unnecessary. When we >> split 1.0 in half at 0.5, we now have a shorter segment which is >> *already* completely filled will points, which has no gaps, and was >> always that way. >> >> But, even if your argument were legitimate (it isn't), your >> 'infinitesimal' wouldn't solve it. If you claim that 0.0[1] is the >> least positive number, you are claiming that you have a point 0, >> followed by another point, 0.0[1] which are separated by a distance of >> 0.0[1]. What occupies that 0.0[1] length of line segment between zero >> and 0.0[1]? Is that not a 'gap'? And you failed to respond to the above. Your 'adjacency' argument would apply equally to your infinitesimals. Either your argument applies as much to your proposed solution as it does to the reals, or it doesn't apply at all. André -- To email remove 'invalid' & replace 'gm' with well known Google mail service.
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| From | Fred <Fred@nowhere.invalid> |
|---|---|
| Date | 2019-09-19 16:35 +0100 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <qm078l$647$1@news.albasani.net> |
| In reply to | #211063 |
On 19/09/2019 04:03, peteolcott wrote: > THERE IS THE INCONSISTENCY. > (1) There cannot be a real number associated with the point > immediately adjacent to 0.0 because real numbers must be divisible > by 2 and single points are indivisible. You are utterly lost. Suppose x is next after 0. If x is a number then there are questions about x/2 (also a number) that you refuse to address. If x is a point then it makes no sense to divide x by 2. "Number line" is (so far as I am aware) a school-room idiom wherein the nature of real numbers is not gone into much (for good reason). A sophisticated view of real numbers is *not* based on geometry. > (2) If there is no real number immediately adjacent to 0.0 then > there is always a measurable gap after 0.0. By dividing the > size of the gap by 2 it gets smaller and smaller yet never closes, > there are always points in the gap that are unaccounted for.
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