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Groups > comp.theory > #19819 > unrolled thread
| Started by | peteolcott <Here@Home> |
|---|---|
| First post | 2019-09-11 18:32 -0500 |
| Last post | 2019-09-21 09:45 -0500 |
| Articles | 20 on this page of 163 — 11 participants |
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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) temakoutera62@gmail.com - 2019-09-11 17:28 -0700
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) peteolcott <Here@Home> - 2019-09-11 20:59 -0500
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) peteolcott <Here@Home> - 2019-09-12 11:14 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Deceptive Strawman by Ben peteolcott <Here@Home> - 2019-09-12 21:00 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Deceptive Strawman by Ben David Kleinecke <dkleinecke@gmail.com> - 2019-09-12 19:11 -0700
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 peteolcott <Here@Home> - 2019-09-13 11:48 -0500
Re: Cantor was wrong about cardinality --- APOLOGY peteolcott <Here@Home> - 2019-09-14 13:09 -0500
Re: Cantor was wrong about cardinality --- APOLOGY peteolcott <Here@Home> - 2019-09-14 16:06 -0500
Re: Cantor was wrong about cardinality --- APOLOGY peteolcott <Here@Home> - 2019-09-14 19:24 -0500
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? 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 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 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 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 Andy Walker <anw@cuboid.co.uk> - 2019-09-16 23:44 +0100
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 Andy Walker <anw@cuboid.co.uk> - 2019-09-17 17:04 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-17 12:06 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-17 02:57 +0100
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 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 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). 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 (Concurrence) peteolcott <Here@Home> - 2019-09-17 11:20 -0500
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 --- 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 peteolcott <Here@Home> - 2019-09-17 17:51 -0500
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 --- (Mike Terry Concurrence ] peteolcott <Here@Home> - 2019-09-19 11:46 -0500
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 ] 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 --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 11:02 -0500
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? peteolcott <Here@Home> - 2019-09-16 22:39 -0500
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? 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 --- [ CONCURRENCE ] peteolcott <Here@Home> - 2019-09-18 11:10 -0500
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) peteolcott <Here@Home> - 2019-09-13 11:28 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Andy Walker <anw@cuboid.co.uk> - 2019-09-13 17:34 +0100
Re: Cantor was wrong about cardinality --- COMPLETE peteolcott <Here@Home> - 2019-09-14 11:34 -0500
Re: Cantor was wrong about cardinality --- COMPLETE Andy Walker <anw@cuboid.co.uk> - 2019-09-14 23:44 +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 Didn't Goof) Andy Walker <anw@cuboid.co.uk> - 2019-09-16 11:52 +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 peteolcott <Here@Home> - 2019-09-14 19:36 -0500
Re: Cantor was wrong about cardinality --- COMPLETE peteolcott <Here@Home> - 2019-09-16 10:21 -0500
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) peteolcott <Here@Home> - 2019-09-13 13:03 -0500
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) André G. Isaak <agisaak@gm.invalid> - 2019-09-14 22:00 -0600
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) peteolcott <Here@Home> - 2019-09-16 10:26 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-16 18:24 -0500
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) 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) Andy Walker <anw@cuboid.co.uk> - 2019-09-19 15:27 +0100
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-19 10:35 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-19 11:42 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-19 12:43 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-19 10:24 -0500
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) peteolcott <Here@Home> - 2019-09-19 17:13 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-19 19:49 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) David Kleinecke <dkleinecke@gmail.com> - 2019-09-19 20:37 -0700
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Andy Walker <anw@cuboid.co.uk> - 2019-09-20 09:12 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-20 12:55 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Andy Walker <anw@cuboid.co.uk> - 2019-09-21 11:13 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Kaz Kylheku <847-115-0292@kylheku.com> - 2019-09-21 14:55 +0000
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Andy Walker <anw@cuboid.co.uk> - 2019-09-21 16:30 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-21 15:22 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-22 12:09 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-20 13:08 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-21 09:45 -0500
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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 | #19996 |
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 | #19998 |
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 | #20000 |
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 | #20001 |
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 | Andy Walker <anw@cuboid.co.uk> |
|---|---|
| Date | 2019-09-19 15:27 +0100 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <qm038g$14fb$1@gioia.aioe.org> |
| In reply to | #20000 |
On 19/09/2019 03:38, André G. Isaak wrote: > 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. I think you're right, and Pete misunderstands an important difference between maths and computing. We see this in his version of what infinity is [and which seems to him to be the only possible way to explain/define infinity]. He sees maths as being a process, in which you start from somewhere and take steps, like constructing a proof, to get to some endpoint, or at least to get near to it [eg in the sense of numerical analysis]. He doesn't understand that the real numbers just "are", they don't have to be built one by one. OTOH the construction of the surreals does at least partly meet Pete's needs. Starting from nothing, you can construct 0 [from the empty set, containing no numbers (as you don't yet have any)], and then 1 and -1 [from the empty set and 0], and then 2, 1/2, -1/2 and -2 [from 1, 0 and -1] and so on. But you never reach all the reals that way, let alone infinity and infinitesimals, just as Achilles never catches the tortoise if you insist on him taking it step by step; the "paradox" can be resolved in the same way. > 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. If you do try to construct the reals by a sequential process, you run into the problem that any computable process creating points produces a set of measure zero, whereas of course an interval of reals of length L has measure L. It's Pete's process that is making all those gaps! Uncountably many reals are sitting there, out of his reach. > 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'? What is a "gap"? But Pete's infinitesimals don't solve his problems anyway. Whether the reals are countable or uncountable, numbers of form x[i] where x is a real and i an integer have an obvious bijection with the reals, and the process can't be used to fill in the gaps. -- Andy Walker, Nottingham.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-19 10:35 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <ePidnXnjxsujPR7AnZ2dnUU7-anNnZ2d@giganews.com> |
| In reply to | #20003 |
On 9/19/2019 9:27 AM, Andy Walker wrote: > On 19/09/2019 03:38, André G. Isaak wrote: >> 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. > > I think you're right, and Pete misunderstands an important > difference between maths and computing. We see this in his version > of what infinity is [and which seems to him to be the only possible > way to explain/define infinity]. He sees maths as being a process, > in which you start from somewhere and take steps, like constructing > a proof, to get to some endpoint, or at least to get near to it [eg > in the sense of numerical analysis]. He doesn't understand that the > real numbers just "are", they don't have to be built one by one. > > OTOH the construction of the surreals does at least partly > meet Pete's needs. Starting from nothing, you can construct 0 [from > the empty set, containing no numbers (as you don't yet have any)], > and then 1 and -1 [from the empty set and 0], and then 2, 1/2, -1/2 > and -2 [from 1, 0 and -1] and so on. But you never reach all the > reals that way, let alone infinity and infinitesimals, just as > Achilles never catches the tortoise if you insist on him taking it > step by step; the "paradox" can be resolved in the same way. > >> 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. > > If you do try to construct the reals by a sequential process, > you run into the problem that any computable process creating points > produces a set of measure zero, whereas of course an interval of > reals of length L has measure L. It's Pete's process that is making > all those gaps! Uncountably many reals are sitting there, out of his > reach. > >> 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'? > > What is a "gap"? But Pete's infinitesimals don't solve his > problems anyway. Whether the reals are countable or uncountable, > numbers of form x[i] where x is a real and i an integer have an > obvious bijection with the reals, and the process can't be used to > fill in the gaps. > [0.0[1], 1.0[0]] specifies all of the adjacent points in this interval: (0, 1]. 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. [0.0[1], ∞] or (0, ∞] specifies all of the points > 0.0 on the number line. The first point of the open interval is immediately adjacent to 0.0 because every point from 0.0 to ∞ is included in that interval besides 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 | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-19 11:42 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <leGdnbr22-JCMh7AnZ2dnUU7-SPNnZ2d@giganews.com> |
| In reply to | #20000 |
On 9/19/2019 11:00 AM, Eram semper recta wrote:
> On Wednesday, 18 September 2019 22:38:22 UTC-4, 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.
>
> That is utter rubbish. There is no such thing as a "real number" line. Do you even know what it means to reify a point on a line?
>
>
> https://www.linkedin.com/pulse/real-number-line-john-gabriel
>
>
>> There are no gaps.
>
> You can't even begin to talk about "gaps" because a line DOES NOT consist of points. A line is the DISTANCE between two points. The points are like road signs indicating how far the current point/location is from some origin.
>
> A point has NO size, dimension or extent. This is very well defined. The Ancient Greeks did not make any mistake in its definition. Only the morons who came after did not understand.
>
I disagree: [0,1] - (0,1] = {0} a single point on the number line.
Thus proving a difference (in length) of one point between the first
interval and the second interval.
In all of the years where the width of a single point has been defined
to be 0 it was defined incorrectly, the actual width is Infinitesimal.
Infinitesimals sometimes act as if they had zero width and other times
act as if they minimum width. This is similar to the dichotomy of the
particle versus wave theories of light.
> To learn more, study my free eBook:
>
> https://drive.google.com/file/d/1CIul68phzuOe6JZwsCuBuXUR8X-AkgEO/view
>
>> 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.
>
--
Copyright 2019 Pete Olcott All rights reserved
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein
[toc] | [prev] | [next] | [standalone]
| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-19 12:43 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <JLadnbYNEsCsIx7AnZ2dnUU7-THNnZ2d@giganews.com> |
| In reply to | #20006 |
On 9/19/2019 12:04 PM, Eram semper recta wrote:
> On Thursday, 19 September 2019 12:42:18 UTC-4, peteolcott wrote:
>> On 9/19/2019 11:00 AM, Eram semper recta wrote:
>>> On Wednesday, 18 September 2019 22:38:22 UTC-4, 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.
>>>
>>> That is utter rubbish. There is no such thing as a "real number" line. Do you even know what it means to reify a point on a line?
>>>
>>>
>>> https://www.linkedin.com/pulse/real-number-line-john-gabriel
>>>
>>>
>>>> There are no gaps.
>>>
>>> You can't even begin to talk about "gaps" because a line DOES NOT consist of points. A line is the DISTANCE between two points. The points are like road signs indicating how far the current point/location is from some origin.
>>>
>>> A point has NO size, dimension or extent. This is very well defined. The Ancient Greeks did not make any mistake in its definition. Only the morons who came after did not understand.
>>>
>>
>> I disagree: [0,1] - (0,1] = {0} a single point on the number line.
>
> No. {0} is not a point. It is meaningless because it is not a length. When we calibrate the *rational* number line with measures, "0" is just an invisible marker denoting no distance. All other points must be of distance > 0 and one must be able to calibrate the line with a measure, viz. number.
>
> In fact, you can't realise "0" at all unless you place a marker between distances that are greater than 0, for example:
>
> __|__
>
> The horizontal bar is the marker. There is no actual space between the two lines __ and __. As one they appear to be ____
>
>
You are getting confused by the notation.
[1, 2] - (1, 2] = {1} the single beginning point of the first interval
is excluded from the second interval making the second interval have one
less element than the first interval.
If you have the infinite set of every car that can possibly exist
besides the exact form of the 1987 Chevy Camaro LS then this set is
infinitesimally smaller than the set also having 1987 Chevy Camaro LS
even though both sets are infinite, the first set is defined to have
one less element than the second set.
>
>> Thus proving a difference (in length) of one point between the first
>> interval and the second interval.
>
> The origin is not a *distance*. It is merely a reference location/point.
>
>>
>> In all of the years where the width of a single point has been defined
>> to be 0 it was defined incorrectly, the actual width is Infinitesimal.
>
> Nonsense. There is no such thing as "infinitesimal".
>
>>
>> Infinitesimals sometimes act as if they had zero width and other times
>> act as if they minimum width. This is similar to the dichotomy of the
>> particle versus wave theories of light.
>
> Mythology.
>
>>
>>> To learn more, study my free eBook:
>>>
>>> https://drive.google.com/file/d/1CIul68phzuOe6JZwsCuBuXUR8X-AkgEO/view
>>>
>>>> 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.
>>>
>>
>>
>> --
>> Copyright 2019 Pete Olcott All rights reserved
>>
>> "Great spirits have always encountered violent
>> opposition from mediocre minds." Albert Einstein
>
--
Copyright 2019 Pete Olcott All rights reserved
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-19 10:24 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <W6edneonwuwrAB7AnZ2dnUU7-aPNnZ2d@giganews.com> |
| In reply to | #19972 |
On 9/19/2019 12:16 AM, Zelos Malum wrote: >> 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. > > It doesn't exist because whichever you claim it is, I can divide by 2 and voila! Even closer. > >> 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. > > Non-sequitor statements. > 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. If no Real number is immediately adjacent to 0.0 then some other type of number would fill the gap because it is clear that Real numbers do not correspond to all of the points on the number line because they always leave gaps. -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-17 23:32 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) |
| Message-ID | <xLOdnXgeYsLQLhzAnZ2dnUU7-eHNnZ2d@giganews.com> |
| In reply to | #19963 |
On 9/17/2019 11:15 PM, Ben Bacarisse wrote: > peteolcott <Here@Home> writes: > >> On 9/17/2019 9:55 PM, Ben Bacarisse wrote: >>> peteolcott <Here@Home> writes: > <cut> >>>> 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 > > Not on "the" number line (that's just the reals ordered by magnitude) > but on the PO-number line (as yet not well specified). Of course you > can define a set with such a point but that does not refute Cantor. > Sure it does because other number systems cannot account for ALL of the points on the number line. Since the ONLY number system that can account for ALL the points on the number line forms a bijection to ALL these points indexed by integers, there is a bijection between integers and ALL these points. Because infinity is a process rather than any value: Integer N = 0; HERE: N = N + 1; goto HERE; It is pretty silly to envision any infinite set that is "larger" than any other infinite set. >> 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. >> -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2019-09-17 23:21 -0600 |
| Subject | Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) |
| Message-ID | <qlsesd$saj$1@dont-email.me> |
| In reply to | #19969 |
On 2019-09-17 10:32 p.m., peteolcott wrote: > On 9/17/2019 11:15 PM, Ben Bacarisse wrote: >> peteolcott <Here@Home> writes: >> >>> On 9/17/2019 9:55 PM, Ben Bacarisse wrote: >>>> peteolcott <Here@Home> writes: >> <cut> >>>>> 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 >> >> Not on "the" number line (that's just the reals ordered by magnitude) >> but on the PO-number line (as yet not well specified). Of course you >> can define a set with such a point but that does not refute Cantor. >> > > Sure it does because other number systems cannot account for ALL > of the points on the number line. > > Since the ONLY number system that can account for ALL the points > on the number line forms a bijection to ALL these points indexed > by integers, there is a bijection between integers and ALL these points. How does this indexing work? If there is a bijection, what is that bijection? André -- To email remove 'invalid' & replace 'gm' with well known Google mail service.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-19 17:13 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) |
| Message-ID | <qIKdneGpdpbgYB7AnZ2dnUU7-Y3NnZ2d@giganews.com> |
| In reply to | #19969 |
On 9/19/2019 4:24 PM, Ben Bacarisse wrote: > peteolcott <Here@Home> writes: > >> On 9/18/2019 7:42 PM, Ben Bacarisse wrote: >>> peteolcott <Here@Home> writes: >>> >>>> On 9/18/2019 12:42 PM, Ben Bacarisse wrote: >>>>> peteolcott <Here@Home> writes: >>> <cut> >>>>>> Sure it does because other number systems cannot account for ALL >>>>>> of the points on the number line. >>>>> >>>>> Cantor proved the real are uncountable. The theorem stands even if they >>>>> don't "account for ALL of the points on the number line". The theorem >>>>> is about the reals. >>>> >>>> If Infinitesimals indexed by integers can count ALL of the numbers >>>> on the number line including the subset of numbers that are Reals >>>> then Cantor is refuted. >>> >>> Of course. In fact if you can do that you can simplify matters by >>> counting ONLY those that are reals. That's the daft thing: most of your >>> infinitesimals are just a side show. Throw away all of your >>> infinitesimals that are not reals and count the result (you can't). >>> >>>> That Integers cannot count the Real directly yet can count them >>>> indirectly does not actually indicate a difference in the "size" of >>>> the sets, it merely indicates an incompatibility between Reals and >>>> Integers. >>> >>> Well if this waffle means you can can't count the reals, you have not >>> refuted anything. >> >> As long as Integers can count ALL the points on a number line Cantor >> is refuted. > > Not unless "a number line" includes the reals. You have confusion derived from the conventional way that these things have been defined. Conventionally it is understood that there is a bijection between points on the number line and the set of Reals. Conventionally it has been understood that there is no such thing as points on the number line that are immediately adjacent to each other because Real numbers are always divisible. I have shown that these conventions result in the inconsistency of points on the number line that are not accounted for by Reals. Since there is no Real number immediately adjacent to 0.0 there is always a measurable gap between 0.0 and every element of the set of Real numbers. This gap has points on the number line that are not accounted for by any Real number. So although the number line does include every Real number it also includes the more dense set of Infinitesimal numbers which fill in the gaps left behind by the Reals. > You duck and dive about > this. I specifically asked about it when you briefly suggested you'd > like to learn something, and you specifically did not answer with any > clarity. > > And you are still ducking an diving. Show your bijection and I'll make > it better and publish it by ignoring your junk infinitesimals to present > a bijection from N to R. (You can't, of course, so I won't get to be > famous, but I can at least keep calling you out for boasting about > having things you can't actually show.) > -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-19 19:49 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) |
| Message-ID | <ZMmdnS2YxpCNvxnAnZ2dnUU7-SHNnZ2d@giganews.com> |
| In reply to | #20012 |
On 9/19/2019 7:05 PM, Ben Bacarisse wrote: > peteolcott <Here@Home> writes: > >> On 9/19/2019 4:24 PM, Ben Bacarisse wrote: >>> peteolcott <Here@Home> writes: > <cut> >>>> As long as Integers can count ALL the points on a number line Cantor >>>> is refuted. >>> >>> Not unless "a number line" includes the reals. > <cut> >> So although the number line does include every Real number it also >> includes the more dense set of Infinitesimal numbers which fill in >> the gaps left behind by the Reals. > > An answer! Just in case you are in an answering mood I'll try another: > > 0[1] is the first PO-point (and not a real) after 0. There are no real > points (I'll just call them reals) between 0 and 0[1]. The next point > on the number line (according to more than one post I've seen) is 0[2]. > Are there any reals between 0[1] and 0[2]? If so, how many? If not, > are there any between 0[2] and 0[3]? If so how many? If not, are there > any between 0[3] and 0[4]? Can you see where this is going? Can you > answer? By the way, it's fine to say you don't know. If you don't > know, please don't guess or bluff. > [A[n], A[n+1]] (where n is an integer) specifies two immediately adjacent points on a number line. There are no numbers of any kind what-so-ever Real or otherwise between any two immediately adjacent points on a number line. -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | David Kleinecke <dkleinecke@gmail.com> |
|---|---|
| Date | 2019-09-19 20:37 -0700 |
| Subject | Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) |
| Message-ID | <7a02304c-64f1-47ba-bf56-e2769a0684e9@googlegroups.com> |
| In reply to | #20014 |
On Thursday, September 19, 2019 at 5:49:26 PM UTC-7, peteolcott wrote: > On 9/19/2019 7:05 PM, Ben Bacarisse wrote: > > peteolcott <Here@Home> writes: > > > >> On 9/19/2019 4:24 PM, Ben Bacarisse wrote: > >>> peteolcott <Here@Home> writes: > > <cut> > >>>> As long as Integers can count ALL the points on a number line Cantor > >>>> is refuted. > >>> > >>> Not unless "a number line" includes the reals. > > <cut> > >> So although the number line does include every Real number it also > >> includes the more dense set of Infinitesimal numbers which fill in > >> the gaps left behind by the Reals. > > > > An answer! Just in case you are in an answering mood I'll try another: > > > > 0[1] is the first PO-point (and not a real) after 0. There are no real > > points (I'll just call them reals) between 0 and 0[1]. The next point > > on the number line (according to more than one post I've seen) is 0[2]. > > Are there any reals between 0[1] and 0[2]? If so, how many? If not, > > are there any between 0[2] and 0[3]? If so how many? If not, are there > > any between 0[3] and 0[4]? Can you see where this is going? Can you > > answer? By the way, it's fine to say you don't know. If you don't > > know, please don't guess or bluff. > > > > [A[n], A[n+1]] (where n is an integer) specifies two immediately adjacent > points on a number line. > > There are no numbers of any kind what-so-ever Real or otherwise > between any two immediately adjacent points on a number line. So your notion is that there are a countable infinity of PO-points between each pair of real numbers. That means there are an uncountable number of PO-points.
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| From | Andy Walker <anw@cuboid.co.uk> |
|---|---|
| Date | 2019-09-20 09:12 +0100 |
| Subject | Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) |
| Message-ID | <qm21m3$1v5u$1@gioia.aioe.org> |
| In reply to | #20015 |
On 20/09/2019 04:37, David Kleinecke wrote: > So [PO's] notion is that there are a countable infinity of PO-points > between each pair of real numbers. That means there are an uncountable > number of PO-points. Step by step! As there are [at least] a countable infinity of [conventional] real numbers, what it means is that there is a rather simple bijection between the PO-points [a countable infinity per real] and the reals, and so the same cardinality of reals and PO-points. Pete disputes that there are as many reals as the rest of us believe; PO-points neither advance nor retard that disputation. Apart from filling Pete's need to have no gaps in his imagined world, and apart from making arithmetic impossible in general, PO-points are an irrelevance. Also, of course, not his invention, apart from his arm-wavery around them. -- Andy Walker, Nottingham.
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2019-09-20 12:55 +0100 |
| Subject | Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) |
| Message-ID | <878sqj41xl.fsf@bsb.me.uk> |
| In reply to | #20016 |
Andy Walker <anw@cuboid.co.uk> writes: > On 20/09/2019 04:37, David Kleinecke wrote: >> So [PO's] notion is that there are a countable infinity of PO-points >> between each pair of real numbers. That means there are an uncountable >> number of PO-points. > > Step by step! As there are [at least] a countable infinity > of [conventional] real numbers, what it means is that there is a > rather simple bijection between the PO-points [a countable infinity > per real] and the reals, Really? Has he actually stated that the "Real_Part" can be any real number? At least when he replied to my question about that he did not say so. His reply was | Any identifiable number can go into the "real" part. which is nothing if not classic crank -- it introduces new undefined term and uses scare quotes to suggest that the real part is not really for what the world calls real numbers. Depending on what an identifiable number is, it may turn out that the PO-points r[k] are indeed countable. Putting that aside, he's made the rookie crank mistake of being clear about some other things. For example: "0.0[1 to ∞] forms a bijection between the positive integers and all of the positive numbers through all of the positive points on the number line" and "the number line does include every Real number". So we don't need any "base" other than 0, but he's also said that there are no reals between any 0[k] and 0[k+1] and, because he's also agreed that there is no least positive real, none of the 0[k] (for k > 0) can be real themselves. It's all a self-contradictory mess. He either needs to do a Nam and "clarify" a lot of things, or he needs to start a new thread about how some other important theorem is wrong, this time taking care not to say anything too clearly. <cut> -- Ben.
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| From | Andy Walker <anw@cuboid.co.uk> |
|---|---|
| Date | 2019-09-21 11:13 +0100 |
| Subject | Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) |
| Message-ID | <qm4t44$vl2$1@gioia.aioe.org> |
| In reply to | #20017 |
On 20/09/2019 12:55, Ben Bacarisse wrote: [Me, in response to DK:] >> Step by step! As there are [at least] a countable infinity >> of [conventional] real numbers, what it means is that there is a >> rather simple bijection between the PO-points [a countable infinity >> per real] and the reals, > Really? Has he actually stated that the "Real_Part" can be any real > number? At least when he replied to my question about that he did not > say so. His reply was > | Any identifiable number can go into the "real" part. > which is nothing if not classic crank -- it introduces new undefined > term and uses scare quotes to suggest that the real part is not really > for what the world calls real numbers. Depending on what an > identifiable number is, it may turn out that the PO-points r[k] are > indeed countable. I think we're well beyond PO's pay grade. Given his views on the HP and on diagonal proofs, my suspicion is that he recognises only the computable reals. This is because the standard example of an uncomputable real is one whose nth digit is some representation of the halting state of the nth TM, eg when run with an empty tape; and in PO-land that is [no doubt] computable. So "identifiable" means "computable", and any other real number would be "empty rhetoric devoid blah-de-blah", on which he "calls BS". > Putting that aside, he's made the rookie crank mistake of being clear > about some other things. For example: > "0.0[1 to ∞] forms a bijection between the positive integers and all > of the positive numbers through all of the positive points on the > number line" > and > "the number line does include every Real number". > So we don't need any "base" other than 0, but he's also said that there > are no reals between any 0[k] and 0[k+1] and, because he's also agreed > that there is no least positive real, none of the 0[k] (for k > 0) can > be real themselves. It's all a self-contradictory mess. Yes; but that has never worried him! He latches on to some idea, and repeats and develops it as nauseam, without ever thinking about how it meshes with other parts of maths and computing. If you try to point out the contradictions, you are just indulging in "empty rhetoric ..." or "extraneous detail" or whatever. It's also classic troll behaviour, but my personal belief is that he is not capable of sustaining that. [...] [Later:] > Sorry, I meant to agree with this key point in my reply... >> ... Pete disputes that there are as many reals as the rest >> of us believe; PO-points neither advance nor retard that disputation. >> Apart from filling Pete's need to have no gaps in his imagined world, >> and apart from making arithmetic impossible in general, PO-points are >> an irrelevance. > Yes, any bijection between N and PO-points could be made better (and > simpler) just by throwing out all the extra points to leave a bijection > between N and R! True, but that's a different point [and there is indeed a (non-computable!) bijection between N and the computable reals]. Mine was that whether R is countable or uncountable doesn't affect the existence of a bijection between R and R-together-with-a- countable-infinity-of-infinitesimals. Only if R is finite do infinitesimals change that. But that would be rather an extreme view of the reals! -- Andy Walker, Nottingham.
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| From | Kaz Kylheku <847-115-0292@kylheku.com> |
|---|---|
| Date | 2019-09-21 14:55 +0000 |
| Subject | Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) |
| Message-ID | <20190921075046.324@kylheku.com> |
| In reply to | #20019 |
On 2019-09-21, Andy Walker <anw@cuboid.co.uk> wrote: > I think we're well beyond PO's pay grade. Given his views on > the HP and on diagonal proofs, my suspicion is that he recognises only > the computable reals. No; computable reals are rational numbers. The nonsense spewed by Olcott is false about those, irrespective of diagonalization. We don't require diagonalization to show that there is no smallest rational number above zero. Computer reals can always add another another bit to represent a smaller number either via exponent or mantissa. The problem here is that Peter Olcott is not very intelligent; for some reason he just has an interest in intellectual topics, which is not the same thing. For instance, the average car mechanic could come to understand this material better if his or her interest could be aroused. Or most fifth graders, for that matter.
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| From | Andy Walker <anw@cuboid.co.uk> |
|---|---|
| Date | 2019-09-21 16:30 +0100 |
| Subject | Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) |
| Message-ID | <qm5fn5$1k5l$1@gioia.aioe.org> |
| In reply to | #20021 |
On 21/09/2019 15:55, Kaz Kylheku wrote:
> On 2019-09-21, Andy Walker <anw@cuboid.co.uk> wrote:
>> I think we're well beyond PO's pay grade. Given his views on
>> the HP and on diagonal proofs, my suspicion is that he recognises only
>> the computable reals.
> No; computable reals are rational numbers. [...]
Comput*able* not comput*er* or comput*ing*. A real for which
there exists a computer program which, given n as parameter, will print
that real correct to n decimal places*. Sqrt(2), pi, e, ... are all
computable, as are all the standard trig functions [with computable
parameters], polynomials with computable coefficients, .... Anything
that is susceptible to the usual processes of numerical analysis. But
there are still only countably many of them, so almost all reals are
not computable. Nor describable, for reasonable meanings of that.
__
* More accurately, a TM which given n on its tape halts with the
real printed to n dp. Lots of equivalent definitions. A
computer program has to be prepared to use arbitrarily much
backing store [arbitrarily many USB sticks!] to do the work,
but that's "extraneous detail", to coin a phrase.
--
Andy Walker,
Nottingham.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-21 15:22 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) |
| Message-ID | <WPKdnX-fIbkSGxvAnZ2dnUU7-dXNnZ2d@giganews.com> |
| In reply to | #20019 |
On 9/21/2019 5:13 AM, Andy Walker wrote: > On 20/09/2019 12:55, Ben Bacarisse wrote: > [Me, in response to DK:] >>> Step by step! As there are [at least] a countable infinity >>> of [conventional] real numbers, what it means is that there is a >>> rather simple bijection between the PO-points [a countable infinity >>> per real] and the reals, >> Really? Has he actually stated that the "Real_Part" can be any real >> number? At least when he replied to my question about that he did not >> say so. His reply was >> | Any identifiable number can go into the "real" part. >> which is nothing if not classic crank -- it introduces new undefined >> term and uses scare quotes to suggest that the real part is not really >> for what the world calls real numbers. Depending on what an >> identifiable number is, it may turn out that the PO-points r[k] are >> indeed countable. > > I think we're well beyond PO's pay grade. Given his views on > the HP and on diagonal proofs, my suspicion is that he recognises only > the computable reals. This is because the standard example of an > uncomputable real is one whose nth digit is some representation of > the halting state of the nth TM, eg when run with an empty tape; and > in PO-land that is [no doubt] computable. So "identifiable" means > "computable", and any other real number would be "empty rhetoric > devoid blah-de-blah", on which he "calls BS". > >> Putting that aside, he's made the rookie crank mistake of being clear >> about some other things. For example: >> "0.0[1 to ∞] forms a bijection between the positive integers and all >> of the positive numbers through all of the positive points on the >> number line" >> and >> "the number line does include every Real number". >> So we don't need any "base" other than 0, but he's also said that there >> are no reals between any 0[k] and 0[k+1] and, because he's also agreed >> that there is no least positive real, none of the 0[k] (for k > 0) can >> be real themselves. It's all a self-contradictory mess. > > Yes; but that has never worried him! He latches on to some > idea, and repeats and develops it as nauseam, without ever thinking > about how it meshes with other parts of maths and computing. If you > try to point out the contradictions, you are just indulging in "empty > rhetoric ..." or "extraneous detail" or whatever. It's also classic > troll behaviour, but my personal belief is that he is not capable of > sustaining that. > > [...] > > [Later:] >> Sorry, I meant to agree with this key point in my reply... >>> ... Pete disputes that there are as many reals as the rest >>> of us believe; PO-points neither advance nor retard that disputation. >>> Apart from filling Pete's need to have no gaps in his imagined world, >>> and apart from making arithmetic impossible in general, PO-points are >>> an irrelevance. >> Yes, any bijection between N and PO-points could be made better (and >> simpler) just by throwing out all the extra points to leave a bijection >> between N and R! > > True, but that's a different point [and there is indeed a > (non-computable!) bijection between N and the computable reals]. > Mine was that whether R is countable or uncountable doesn't affect > the existence of a bijection between R and R-together-with-a- > countable-infinity-of-infinitesimals. Only if R is finite do > infinitesimals change that. But that would be rather an extreme > view of the reals! > The distance between zero and the point on the number line immediately after zero Number M = 1.0 / ∞; -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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