Groups | Search | Server Info | Keyboard shortcuts | Login | Register [http] [https] [nntp] [nntps]
Groups > sci.math > #457916 > unrolled thread
| Started by | peteolcott <Here@Home> |
|---|---|
| First post | 2019-09-11 12:05 -0500 |
| Last post | 2019-09-16 22:17 -0700 |
| Articles | 20 on this page of 169 — 9 participants |
Back to article view | Back to sci.math
This discussion starts older than the indexed window; earlier articles aren't shown. The article labeled Started by
below is the oldest one visible, not the original post.
Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) peteolcott <Here@Home> - 2019-09-11 12:05 -0500
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) 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 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 14:41 -0500
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 peteolcott <Here@Home> - 2019-09-14 15:45 -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> - 2019-09-14 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? 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 Transfinite Numbers <bursejan@gmail.com> - 2019-09-16 12:30 -0700
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 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 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). Zelos Malum <zelos.malum@gmail.com> - 2019-09-17 00:22 -0700
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 (some concurrence). Zelos Malum <zelos.malum@gmail.com> - 2019-09-17 22:23 -0700
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: 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 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> - 2019-09-14 21:38 -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> - 2019-09-14 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) peteolcott <Here@Home> - 2019-09-14 21:25 -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 (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 (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) peteolcott <Here@Home> - 2019-09-19 10:35 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) Eram semper recta <thenewcalculus@gmail.com> - 2019-09-19 09:00 -0700
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-19 11:42 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) Eram semper recta <thenewcalculus@gmail.com> - 2019-09-19 10:04 -0700
Re: Provably unprovable eliminates incompleteness (NO GAPS) Eram semper recta <thenewcalculus@gmail.com> - 2019-09-19 10:14 -0700
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-19 12:43 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) Eram semper recta <thenewcalculus@gmail.com> - 2019-09-19 15:03 -0700
Re: Provably unprovable eliminates incompleteness (NO GAPS) Zelos Malum <zelos.malum@gmail.com> - 2019-09-18 22:16 -0700
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) peteolcott <Here@Home> - 2019-09-21 09:45 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-21 15:22 -0500
Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) Zelos Malum <zelos.malum@gmail.com> - 2019-09-11 22:23 -0700
Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) peteolcott <Here@Home> - 2019-09-12 01:07 -0500
Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) Zelos Malum <zelos.malum@gmail.com> - 2019-09-12 22:24 -0700
Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) peteolcott <Here@Home> - 2019-09-13 00:34 -0500
Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) Zelos Malum <zelos.malum@gmail.com> - 2019-09-15 22:31 -0700
Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) peteolcott <Here@Home> - 2019-09-16 09:55 -0500
Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) Zelos Malum <zelos.malum@gmail.com> - 2019-09-16 22:17 -0700
Page 8 of 9 — ← Prev page 1 2 3 4 5 6 7 [8] 9 Next page →
| 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 | #458739 |
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.
[toc] | [prev] | [next] | [standalone]
| 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 | #458746 |
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
[toc] | [prev] | [next] | [standalone]
| 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 | #458747 |
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.
[toc] | [prev] | [next] | [standalone]
| 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 | #458748 |
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
[toc] | [prev] | [next] | [standalone]
| 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 | #458752 |
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.
[toc] | [prev] | [next] | [standalone]
| 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 | #458773 |
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
[toc] | [prev] | [next] | [standalone]
| 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 | #458781 |
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.
[toc] | [prev] | [next] | [standalone]
| 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 | #458773 |
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
[toc] | [prev] | [next] | [standalone]
| From | Eram semper recta <thenewcalculus@gmail.com> |
|---|---|
| Date | 2019-09-19 09:00 -0700 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <2e21f35b-8376-44ec-be5a-37a8b37c9705@googlegroups.com> |
| In reply to | #458773 |
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. 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.
[toc] | [prev] | [next] | [standalone]
| 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 | #458827 |
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 | Eram semper recta <thenewcalculus@gmail.com> |
|---|---|
| Date | 2019-09-19 10:04 -0700 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <af3a4e3e-7d71-487f-bee5-c4276448ba6f@googlegroups.com> |
| In reply to | #458829 |
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 ____
> 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
[toc] | [prev] | [next] | [standalone]
| From | Eram semper recta <thenewcalculus@gmail.com> |
|---|---|
| Date | 2019-09-19 10:14 -0700 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <9dbbe0f9-fea2-4c44-9164-37060d46db78@googlegroups.com> |
| In reply to | #458834 |
On Thursday, 19 September 2019 13:04:18 UTC-4, 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
vertical bar
> is the marker. There is no actual space between the two lines __ and __. As one they appear to be ____
>
>
>
> > 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
[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 | #458834 |
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
[toc] | [prev] | [next] | [standalone]
| From | Eram semper recta <thenewcalculus@gmail.com> |
|---|---|
| Date | 2019-09-19 15:03 -0700 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <5f93fbd1-3a38-4b35-9a6f-b8397852b8e3@googlegroups.com> |
| In reply to | #458836 |
On Thursday, 19 September 2019 13:43:26 UTC-4, peteolcott wrote:
> 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.
No.
> [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.
First of all you have to define what [1,2] means. You can't say it is comprised of infinitely many numbers because you have no means of *listing* these.
Secondly, you assume that each number corresponds to a point which is utter nonsense since you cannot tell how many points are in any given interval.
>
> If you have the infinite set
There is no such thing as an "infinite set" or "infinite" anything.
> 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
[toc] | [prev] | [next] | [standalone]
| From | Zelos Malum <zelos.malum@gmail.com> |
|---|---|
| Date | 2019-09-18 22:16 -0700 |
| Subject | Re: Provably unprovable eliminates incompleteness (NO GAPS) |
| Message-ID | <b6624692-a073-4429-8f55-22cc5d4f0d27@googlegroups.com> |
| In reply to | #458651 |
>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.
[toc] | [prev] | [next] | [standalone]
| 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 | #458791 |
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
[toc] | [prev] | [next] | [standalone]
| 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 | #458635 |
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
[toc] | [prev] | [next] | [standalone]
| 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 | #458643 |
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.
[toc] | [prev] | [next] | [standalone]
| 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 | #458643 |
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
[toc] | [prev] | [next] | [standalone]
| 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 | #458856 |
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
[toc] | [prev] | [next] | [standalone]
Page 8 of 9 — ← Prev page 1 2 3 4 5 6 7 [8] 9 Next page →
Back to top | Article view | sci.math
csiph-web