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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 |
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Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) peteolcott <Here@Home> - 2019-09-11 12:05 -0500
Re: Provably unprovable eliminates incompleteness (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
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| From | Mike Terry <news.dead.person.stones@darjeeling.plus.com> |
|---|---|
| Date | 2019-09-17 18:32 +0100 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <rpadnarwpbo4hRzAnZ2dnUU78cPNnZ2d@brightview.co.uk> |
| In reply to | #458515 |
On 17/09/2019 05:38, Keith Thompson wrote:
> André G. Isaak <agisaak@gm.invalid> writes:
> [...]
>> Yeah, the account I offered in my earlier post was kind of mangled. My bad.
>>
>> The main point I was trying to get across with that post was that the
>> real numbers in a given interval can't be ordered as smallest, second
>> smallest, third smallest, etc. I shouldn't really have claimed that
>> there was no 'ordering' among reals - that was a rather unfortunate
>> choice of terms, but rather that there is nothing akin to a successor
>> relation in terms of which they can be ordered.
>>
>> It is possible for an interval to have a smallest element, but it is
>> certainly not guaranteed. In a closed interval, there will be a smallest
>> element but only because the interval is defined as including its
>> greatest lower bound. For an open interval, this is not the case.
>>
>> I wanted to get him away from trying to derive the open interval from
>> the closed interval and instead to focus on the defining properties of
>> the intervals simply because he is determined to impose something akin
>> to a successor relation on his intervals despite the fact that no such
>> relation exists.
>>
>> You're correct, though, that this probably isn't going to actually
>> clarify things for Pete.
>
> Thanks for the clarification.
>
> Here's my attempt to refute the idea that (A,B] has a least element.
> I'm not trying to argue with Pete (though I expect he'll see this and
> respond to it).
>
> Briefly, if (0,1] has a least member, what about the result of dividing
> that member by 2?
>
> In more detail, (0,1] is the half-open interval consisting of
> the real numbers between 0 and 1, including 1 but excluding 0.
> Assume that this interval has a least element, an element x such
> that for each member m of the interval, either x<m or x=m.
>
> Let y=x/2. If we're dealing with real numbers, then 0<y<x, and y
> is a member of the interval and is less than x -- contradicting
> the original assertion that x is the least element. Conclusion:
> the interval (a set of real numbers) has no least element.
>
> Arbitrarily inventing a new notation, let _(0,1]_ be a set of, um,
> something or other. It includes all the real numbers in (0,1],
> and it may also include some other things, each of which is greater
> than 0, less than or equal to 1, and somewhere on the "number line"
> between 0 and 1. Some of the non-real elements might be points,
> or infinitesimals. I'll assert that for any two members of *this*
> set, exactly one of a<b, a=b, a>b is true.
>
> I can imagine that *this* set has a least element. For example
> we can define "iota" as an infinitesimal, something that is not a
> real number but that has a location on the number line, directly
> adjacent to the real number 0. It is greater than 0 and less than
> any positive real number.
>
> For the same argument I used to demonstrate that x is not the least
> member of the interval *not* to apply to iota, one of the following
> must be true:
>
> - iota/2 <= 0
> - iota/2 >= iota
> - iota/2 is not defined
>
> Though we can define something like this, I'm having difficulty
> thinking of a way to define it that is consistent *and useful*.
> We need the non-real members of the interval to behave radically
> differently under common arithmetic operations. Not that there's
> anything wrong with that, but it seems to require inventing a set
> that's even denser than the reals, but is nevertheless in some
> vague sense discrete rather than continuous. As far as I can tell
> (and it's very likely I've missed something), most proposed systems
> of infinitesimals do not have this property.
There are properly defined concepts such as hyper-real numbers, and
Conway's sureal numbers:
<https://en.wikipedia.org/wiki/Hyperreal_number>
<https://en.wikipedia.org/wiki/Surreal_number>
and in both cases, the equivalent of your iota is divisible by 2 in a
reasonable way, and so as you correctly reason, there is still no "next
point" following a given point. Also, it should be obvious that both of
these systems of numbers are uncountable, as they both contain a subset
isomorphic to the set of real numbers.
IMO, just about everyone on this thread has lost track of the focus of
PO's claim. As I generally don't consider engaging with PO a useful
activity, I'll just post once, in case any other posters find my
analysis helpful. (Probably this is way too deep in the thread to make
any difference, but hey...)
The /focus/ of PO's claim is not whether or not the reals always have a
next point; it's that THE REALS ARE COUNTABLE. His (childlike)
intuition behind this is that reals are like points on a line, which in
turn are like tiny apples in a line, and you just count them "one, two,
three ..." in some obvious order (left to right) and then they're
counted. Um, that's it. His current position is the result of being
randomly jostled around by posters pointing out obvious flaws in this
idea and demanding clarifications (in a previous thread a year or so ago).
So his current position is in effect to DEFINE SOME NEW NOTATION where
the notation SUGGESTS a notion of "point adjacency" so that (LOCALLY at
least) points look like the integers, nicely discreet with a next/prev
relationship, and (in a "local" sense) countable.
Can such a scheme be made to work? Yes, IF we just focus on the
ordering aspect, ignoring stuff like addition and multiplication
operations, and all the other properties of the real numbers which make
them what they are, mathematically. [Of course, PO couldn't explain
this or understand the consequences of doing this...]
Here's a simple way. A totally ordered structure POR defined as
POR = R x Z [direct set product of Reals with Integers]
with ordering: (r,m) < (s,n) iff r<s or (r=s and m<n)
So identifying real number 3 with POR element (3,0), there is a next
element after (3,0), namely (3,1) and also a prior element (3,-1).
This is probably something close to PO's current thinking, but is it
"actually what PO is thinking"? That's a meaningless question, because
PO lacks the intellectual capacity to coherently think through what he's
proposing at this kind of level, or to see the consequences of his
childlike intuitions.
The KEY POINT is that if PO intends something like POR above, then
OBVIOUSLY POR IS UNCOUNTABLE, since it contains a copy of the reals.
I.e. with POR, we identify a real number r with (r,0) and this is a
one-one correspondence. There can't be less elements in POR than in R.
In effect, within POR, as we vary m, we get a set of "points" in POR
(0,m) which are ALL infinitesimally close to 0. (Closer than 0 is to any
other real number.) We've effectively replaced each real number with
its own personal (countable) set of nearby infinitesimal numbers,
without changing how many reals there are. We might word this as saying
that the POR ordering is "locally" discrete (and countable), but still
globally POR is uncountable.
This is hopeless for PO, but there's no way out for him. He /wants/
that by fixing r=0 and varying m in the (r,m) notation, the different
(0,m) cover ALL the real numbers which would make them ("globally")
countable - but this is incoherent, because THE OBVIOUS QUESTION is if
this is to be the case, what is the m making (0,m) equal to some
DIFFERENT real, say (1,0)?
EVERY answer to this is stupid, e.g. m=4 implies that there are just
three PO-points strictly between reals 0 and 1:
(0,0) [real number 0]
(0,1)
(0,2)
(0,3)
(0,4) = (1,0) [real number 1]
but we can all list more than 3 different reals between 0 and 1. So
this approach simply doesn't hang together.
What has PO ACTUALLY SAID on this? In a way it doesn't matter, because
he's incapable of thinking through the issues at this level, so he just
makes up whatever answer occurs in his head when asked, and then that
becomes what he'll defend for the next few days/months/years.
A year or so ago I asked him how far along the real number line you can
get by starting from real 0 (POR element (0,0) and varying the
infinitesimal component m in the POR element (0,m). Can you reach the
real 20? What about 0.000000001? etc. Initially no answer, but
eventually he suggested (0,m) covers the whole real line. This just
doesn't work, as explained above, although PO refused to identify the
actual m that made (0,m) = (1,0). [I think he realised every possible
answer was stupid, but didn't see that refusing to answer was equally
stupid.]
Recently in this thread someone (Ben?) asked exactly an equivalent
question, but it never really got followed up. This time PO replied
that ("as far as he could tell"!)
(r,m) = (s,n) if and only if both r=s and m=n.
The total opposite of his answer a year ago. Well, I guess he'd had
time to forget his previous answer, but he did leave a get-out that he
basically didn't know for sure what he intended! :) As I said, the
truth is that PO's answers are meaningless, just whatever occurs to him
at the time he's asked, without any understanding of implications.
Anyhow, the above is is the KEY POINT to be settled, in my opinion.
Whether or not the infinitesimals (0,m) in PO's alleged system are to be :
A) purely "local" to real number 0, or
B) cover the whole real line somehow
(or even just some finite portion of it)
(A) OBVIOUSLY can't be used to argue the reals are countable, since each
real has its own personal set of infinitessimals that don't interact
with those of any other real.
(B) OBVIOUSLY doesn't work as explained above. (Although the
explanation would be beyond PO I think, so he might argue forever
denying this.)
Chasing PO round and round about whether (a,b] has a least element is a
side issue, it seems to me. [It makes him look stupid, but misses the
focus on his original claim that he has proved the reals are countable.]
Regards,
Mike.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-17 12:49 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- (Correction) |
| Message-ID | <cdWdnc6FjJMQgRzAnZ2dnUU7-VfNnZ2d@giganews.com> |
| In reply to | #458571 |
On 9/17/2019 12:32 PM, Mike Terry wrote: > On 17/09/2019 05:38, Keith Thompson wrote: >> André G. Isaak <agisaak@gm.invalid> writes: >> [...] >>> Yeah, the account I offered in my earlier post was kind of mangled. My bad. >>> >>> The main point I was trying to get across with that post was that the >>> real numbers in a given interval can't be ordered as smallest, second >>> smallest, third smallest, etc. I shouldn't really have claimed that >>> there was no 'ordering' among reals - that was a rather unfortunate >>> choice of terms, but rather that there is nothing akin to a successor >>> relation in terms of which they can be ordered. >>> >>> It is possible for an interval to have a smallest element, but it is >>> certainly not guaranteed. In a closed interval, there will be a smallest >>> element but only because the interval is defined as including its >>> greatest lower bound. For an open interval, this is not the case. >>> >>> I wanted to get him away from trying to derive the open interval from >>> the closed interval and instead to focus on the defining properties of >>> the intervals simply because he is determined to impose something akin >>> to a successor relation on his intervals despite the fact that no such >>> relation exists. >>> >>> You're correct, though, that this probably isn't going to actually >>> clarify things for Pete. >> >> Thanks for the clarification. >> >> Here's my attempt to refute the idea that (A,B] has a least element. >> I'm not trying to argue with Pete (though I expect he'll see this and >> respond to it). >> >> Briefly, if (0,1] has a least member, what about the result of dividing >> that member by 2? >> >> In more detail, (0,1] is the half-open interval consisting of >> the real numbers between 0 and 1, including 1 but excluding 0. >> Assume that this interval has a least element, an element x such >> that for each member m of the interval, either x<m or x=m. >> >> Let y=x/2. If we're dealing with real numbers, then 0<y<x, and y >> is a member of the interval and is less than x -- contradicting >> the original assertion that x is the least element. Conclusion: >> the interval (a set of real numbers) has no least element. >> >> Arbitrarily inventing a new notation, let _(0,1]_ be a set of, um, >> something or other. It includes all the real numbers in (0,1], >> and it may also include some other things, each of which is greater >> than 0, less than or equal to 1, and somewhere on the "number line" >> between 0 and 1. Some of the non-real elements might be points, >> or infinitesimals. I'll assert that for any two members of *this* >> set, exactly one of a<b, a=b, a>b is true. >> >> I can imagine that *this* set has a least element. For example >> we can define "iota" as an infinitesimal, something that is not a >> real number but that has a location on the number line, directly >> adjacent to the real number 0. It is greater than 0 and less than >> any positive real number. >> >> For the same argument I used to demonstrate that x is not the least >> member of the interval *not* to apply to iota, one of the following >> must be true: >> >> - iota/2 <= 0 >> - iota/2 >= iota >> - iota/2 is not defined >> >> Though we can define something like this, I'm having difficulty >> thinking of a way to define it that is consistent *and useful*. >> We need the non-real members of the interval to behave radically >> differently under common arithmetic operations. Not that there's >> anything wrong with that, but it seems to require inventing a set >> that's even denser than the reals, but is nevertheless in some >> vague sense discrete rather than continuous. As far as I can tell >> (and it's very likely I've missed something), most proposed systems >> of infinitesimals do not have this property. > > There are properly defined concepts such as hyper-real numbers, and Conway's sureal numbers: > > <https://en.wikipedia.org/wiki/Hyperreal_number> > <https://en.wikipedia.org/wiki/Surreal_number> > > and in both cases, the equivalent of your iota is divisible by 2 in a reasonable way, and so as you correctly reason, there is still no "next point" following a given point. Also, it should be obvious that both of these systems of numbers are uncountable, > as they both contain a subset isomorphic to the set of real numbers. > > IMO, just about everyone on this thread has lost track of the focus of PO's claim. As I generally don't consider engaging with PO a useful activity, I'll just post once, in case any other posters find my analysis helpful. (Probably this is way too deep > in the thread to make any difference, but hey...) > > The /focus/ of PO's claim is not whether or not the reals always have a next point; it's that THE REALS ARE COUNTABLE. This focus has now been adapted to conform to Keith Thompson's summation (shown above). I have been going back and forth on this aspect for a few days. The points on a number line correspond to Infinitesimal numbers that are a superset of the Reals. These Infinitesimal numbers are countable with integers. [0.0[1], ∞[0]] are all of the points on the number line corresponding to numbers greater than zero. This includes all of the real numbers as a subset. There is no bijection between Infinitesimals with an offset and reals. -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | Mike Terry <news.dead.person.stones@darjeeling.plus.com> |
|---|---|
| Date | 2019-09-17 19:59 +0100 |
| Subject | Re: Cantor was wrong about cardinality --- (Correction) |
| Message-ID | <DdednZDlVoyRsBzAnZ2dnUU78d3NnZ2d@brightview.co.uk> |
| In reply to | #458575 |
On 17/09/2019 18:49, peteolcott wrote: > On 9/17/2019 12:32 PM, Mike Terry wrote: >> On 17/09/2019 05:38, Keith Thompson wrote: >>> André G. Isaak <agisaak@gm.invalid> writes: >>> [...] >>>> Yeah, the account I offered in my earlier post was kind of mangled. >>>> My bad. >>>> >>>> The main point I was trying to get across with that post was that the >>>> real numbers in a given interval can't be ordered as smallest, second >>>> smallest, third smallest, etc. I shouldn't really have claimed that >>>> there was no 'ordering' among reals - that was a rather unfortunate >>>> choice of terms, but rather that there is nothing akin to a successor >>>> relation in terms of which they can be ordered. >>>> >>>> It is possible for an interval to have a smallest element, but it is >>>> certainly not guaranteed. In a closed interval, there will be a >>>> smallest >>>> element but only because the interval is defined as including its >>>> greatest lower bound. For an open interval, this is not the case. >>>> >>>> I wanted to get him away from trying to derive the open interval from >>>> the closed interval and instead to focus on the defining properties of >>>> the intervals simply because he is determined to impose something akin >>>> to a successor relation on his intervals despite the fact that no such >>>> relation exists. >>>> >>>> You're correct, though, that this probably isn't going to actually >>>> clarify things for Pete. >>> >>> Thanks for the clarification. >>> >>> Here's my attempt to refute the idea that (A,B] has a least element. >>> I'm not trying to argue with Pete (though I expect he'll see this and >>> respond to it). >>> >>> Briefly, if (0,1] has a least member, what about the result of dividing >>> that member by 2? >>> >>> In more detail, (0,1] is the half-open interval consisting of >>> the real numbers between 0 and 1, including 1 but excluding 0. >>> Assume that this interval has a least element, an element x such >>> that for each member m of the interval, either x<m or x=m. >>> >>> Let y=x/2. If we're dealing with real numbers, then 0<y<x, and y >>> is a member of the interval and is less than x -- contradicting >>> the original assertion that x is the least element. Conclusion: >>> the interval (a set of real numbers) has no least element. >>> >>> Arbitrarily inventing a new notation, let _(0,1]_ be a set of, um, >>> something or other. It includes all the real numbers in (0,1], >>> and it may also include some other things, each of which is greater >>> than 0, less than or equal to 1, and somewhere on the "number line" >>> between 0 and 1. Some of the non-real elements might be points, >>> or infinitesimals. I'll assert that for any two members of *this* >>> set, exactly one of a<b, a=b, a>b is true. >>> >>> I can imagine that *this* set has a least element. For example >>> we can define "iota" as an infinitesimal, something that is not a >>> real number but that has a location on the number line, directly >>> adjacent to the real number 0. It is greater than 0 and less than >>> any positive real number. >>> >>> For the same argument I used to demonstrate that x is not the least >>> member of the interval *not* to apply to iota, one of the following >>> must be true: >>> >>> - iota/2 <= 0 >>> - iota/2 >= iota >>> - iota/2 is not defined >>> >>> Though we can define something like this, I'm having difficulty >>> thinking of a way to define it that is consistent *and useful*. >>> We need the non-real members of the interval to behave radically >>> differently under common arithmetic operations. Not that there's >>> anything wrong with that, but it seems to require inventing a set >>> that's even denser than the reals, but is nevertheless in some >>> vague sense discrete rather than continuous. As far as I can tell >>> (and it's very likely I've missed something), most proposed systems >>> of infinitesimals do not have this property. >> >> There are properly defined concepts such as hyper-real numbers, and >> Conway's sureal numbers: >> >> <https://en.wikipedia.org/wiki/Hyperreal_number> >> <https://en.wikipedia.org/wiki/Surreal_number> >> >> and in both cases, the equivalent of your iota is divisible by 2 in a >> reasonable way, and so as you correctly reason, there is still no >> "next point" following a given point. Also, it should be obvious that >> both of these systems of numbers are uncountable, as they both contain >> a subset isomorphic to the set of real numbers. >> >> IMO, just about everyone on this thread has lost track of the focus of >> PO's claim. As I generally don't consider engaging with PO a useful >> activity, I'll just post once, in case any other posters find my >> analysis helpful. (Probably this is way too deep in the thread to >> make any difference, but hey...) >> >> The /focus/ of PO's claim is not whether or not the reals always have >> a next point; it's that THE REALS ARE COUNTABLE. > > This focus has now been adapted to conform to Keith Thompson's > summation (shown above). > I have been going back and forth on this aspect for a few days. > > The points on a number line correspond to Infinitesimal numbers > that are a superset of the Reals. These Infinitesimal numbers > are countable with integers. You have not defined "a number line" or how your "infinitesimals" work in sufficient detail for anything you say to be meaingful. You have not shown that your Infinitesimal numbers are countable, and I doubt you even understand what that really means. It's all beyond you! > > [0.0[1], ∞[0]] are all of the points on the number line corresponding > to numbers greater than zero. This includes all of the real numbers > as a subset. > > There is no bijection between Infinitesimals with an offset and reals. If you understood what all those terms meant, you would realise that there's a basic contradiction in your claims: - Infinitesimals are countable, - Reals are a subset of Infinitesimals - Reals have no bijection to Infinitesimals Regards, Mike.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-17 14:09 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- (Correction) |
| Message-ID | <MoOdnTJRr_rashzAnZ2dnUU7-YvNnZ2d@giganews.com> |
| In reply to | #458588 |
On 9/17/2019 1:59 PM, Mike Terry wrote: > On 17/09/2019 18:49, peteolcott wrote: >> On 9/17/2019 12:32 PM, Mike Terry wrote: >>> On 17/09/2019 05:38, Keith Thompson wrote: >>>> André G. Isaak <agisaak@gm.invalid> writes: >>>> [...] >>>>> Yeah, the account I offered in my earlier post was kind of mangled. >>>>> My bad. >>>>> >>>>> The main point I was trying to get across with that post was that the >>>>> real numbers in a given interval can't be ordered as smallest, second >>>>> smallest, third smallest, etc. I shouldn't really have claimed that >>>>> there was no 'ordering' among reals - that was a rather unfortunate >>>>> choice of terms, but rather that there is nothing akin to a successor >>>>> relation in terms of which they can be ordered. >>>>> >>>>> It is possible for an interval to have a smallest element, but it is >>>>> certainly not guaranteed. In a closed interval, there will be a >>>>> smallest >>>>> element but only because the interval is defined as including its >>>>> greatest lower bound. For an open interval, this is not the case. >>>>> >>>>> I wanted to get him away from trying to derive the open interval from >>>>> the closed interval and instead to focus on the defining properties of >>>>> the intervals simply because he is determined to impose something akin >>>>> to a successor relation on his intervals despite the fact that no such >>>>> relation exists. >>>>> >>>>> You're correct, though, that this probably isn't going to actually >>>>> clarify things for Pete. >>>> >>>> Thanks for the clarification. >>>> >>>> Here's my attempt to refute the idea that (A,B] has a least element. >>>> I'm not trying to argue with Pete (though I expect he'll see this and >>>> respond to it). >>>> >>>> Briefly, if (0,1] has a least member, what about the result of dividing >>>> that member by 2? >>>> >>>> In more detail, (0,1] is the half-open interval consisting of >>>> the real numbers between 0 and 1, including 1 but excluding 0. >>>> Assume that this interval has a least element, an element x such >>>> that for each member m of the interval, either x<m or x=m. >>>> >>>> Let y=x/2. If we're dealing with real numbers, then 0<y<x, and y >>>> is a member of the interval and is less than x -- contradicting >>>> the original assertion that x is the least element. Conclusion: >>>> the interval (a set of real numbers) has no least element. >>>> >>>> Arbitrarily inventing a new notation, let _(0,1]_ be a set of, um, >>>> something or other. It includes all the real numbers in (0,1], >>>> and it may also include some other things, each of which is greater >>>> than 0, less than or equal to 1, and somewhere on the "number line" >>>> between 0 and 1. Some of the non-real elements might be points, >>>> or infinitesimals. I'll assert that for any two members of *this* >>>> set, exactly one of a<b, a=b, a>b is true. >>>> >>>> I can imagine that *this* set has a least element. For example >>>> we can define "iota" as an infinitesimal, something that is not a >>>> real number but that has a location on the number line, directly >>>> adjacent to the real number 0. It is greater than 0 and less than >>>> any positive real number. >>>> >>>> For the same argument I used to demonstrate that x is not the least >>>> member of the interval *not* to apply to iota, one of the following >>>> must be true: >>>> >>>> - iota/2 <= 0 >>>> - iota/2 >= iota >>>> - iota/2 is not defined >>>> >>>> Though we can define something like this, I'm having difficulty >>>> thinking of a way to define it that is consistent *and useful*. >>>> We need the non-real members of the interval to behave radically >>>> differently under common arithmetic operations. Not that there's >>>> anything wrong with that, but it seems to require inventing a set >>>> that's even denser than the reals, but is nevertheless in some >>>> vague sense discrete rather than continuous. As far as I can tell >>>> (and it's very likely I've missed something), most proposed systems >>>> of infinitesimals do not have this property. >>> >>> There are properly defined concepts such as hyper-real numbers, and >>> Conway's sureal numbers: >>> >>> <https://en.wikipedia.org/wiki/Hyperreal_number> >>> <https://en.wikipedia.org/wiki/Surreal_number> >>> >>> and in both cases, the equivalent of your iota is divisible by 2 in a >>> reasonable way, and so as you correctly reason, there is still no >>> "next point" following a given point. Also, it should be obvious that >>> both of these systems of numbers are uncountable, as they both contain >>> a subset isomorphic to the set of real numbers. >>> >>> IMO, just about everyone on this thread has lost track of the focus of >>> PO's claim. As I generally don't consider engaging with PO a useful >>> activity, I'll just post once, in case any other posters find my >>> analysis helpful. (Probably this is way too deep in the thread to >>> make any difference, but hey...) >>> >>> The /focus/ of PO's claim is not whether or not the reals always have >>> a next point; it's that THE REALS ARE COUNTABLE. >> >> This focus has now been adapted to conform to Keith Thompson's >> summation (shown above). >> I have been going back and forth on this aspect for a few days. >> >> The points on a number line correspond to Infinitesimal numbers >> that are a superset of the Reals. These Infinitesimal numbers >> are countable with integers. > > You have not defined "a number line" or how your "infinitesimals" work in sufficient detail for anything you say to be meaingful. > > You have not shown that your Infinitesimal numbers are countable, and I doubt you even understand what that really means. It's all beyond you! > >> >> [0.0[1], ∞[0]] are all of the points on the number line corresponding >> to numbers greater than zero. This includes all of the real numbers >> as a subset. >> >> There is no bijection between Infinitesimals with an offset and reals. > > If you understood what all those terms meant, you would realise that there's a basic contradiction in your claims: > - Infinitesimals are countable, This is proven by their definition. > - Reals are a subset of Infinitesimals Reals cannot express the point on the number line immediately adjacent to 0.0 Infinitesimals can express this point as 0.0[1]. > - Reals have no bijection to Infinitesimals Reals have a bijection to the subset of Infinitesimals that are Reals. > > Regards, > Mike. > -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | Keith Thompson <kst-u@mib.org> |
|---|---|
| Date | 2019-09-17 11:42 -0700 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <ln8sqm932m.fsf@kst-u.example.com> |
| In reply to | #458571 |
Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:
> On 17/09/2019 05:38, Keith Thompson wrote:
>> André G. Isaak <agisaak@gm.invalid> writes:
>> [...]
>>> Yeah, the account I offered in my earlier post was kind of mangled. My bad.
>>>
>>> The main point I was trying to get across with that post was that the
>>> real numbers in a given interval can't be ordered as smallest, second
>>> smallest, third smallest, etc. I shouldn't really have claimed that
>>> there was no 'ordering' among reals - that was a rather unfortunate
>>> choice of terms, but rather that there is nothing akin to a successor
>>> relation in terms of which they can be ordered.
>>>
>>> It is possible for an interval to have a smallest element, but it is
>>> certainly not guaranteed. In a closed interval, there will be a smallest
>>> element but only because the interval is defined as including its
>>> greatest lower bound. For an open interval, this is not the case.
>>>
>>> I wanted to get him away from trying to derive the open interval from
>>> the closed interval and instead to focus on the defining properties of
>>> the intervals simply because he is determined to impose something akin
>>> to a successor relation on his intervals despite the fact that no such
>>> relation exists.
>>>
>>> You're correct, though, that this probably isn't going to actually
>>> clarify things for Pete.
>>
>> Thanks for the clarification.
>>
>> Here's my attempt to refute the idea that (A,B] has a least element.
>> I'm not trying to argue with Pete (though I expect he'll see this and
>> respond to it).
>>
>> Briefly, if (0,1] has a least member, what about the result of dividing
>> that member by 2?
>>
>> In more detail, (0,1] is the half-open interval consisting of
>> the real numbers between 0 and 1, including 1 but excluding 0.
>> Assume that this interval has a least element, an element x such
>> that for each member m of the interval, either x<m or x=m.
>>
>> Let y=x/2. If we're dealing with real numbers, then 0<y<x, and y
>> is a member of the interval and is less than x -- contradicting
>> the original assertion that x is the least element. Conclusion:
>> the interval (a set of real numbers) has no least element.
>>
>> Arbitrarily inventing a new notation, let _(0,1]_ be a set of, um,
>> something or other. It includes all the real numbers in (0,1],
>> and it may also include some other things, each of which is greater
>> than 0, less than or equal to 1, and somewhere on the "number line"
>> between 0 and 1. Some of the non-real elements might be points,
>> or infinitesimals. I'll assert that for any two members of *this*
>> set, exactly one of a<b, a=b, a>b is true.
>>
>> I can imagine that *this* set has a least element. For example
>> we can define "iota" as an infinitesimal, something that is not a
>> real number but that has a location on the number line, directly
>> adjacent to the real number 0. It is greater than 0 and less than
>> any positive real number.
>>
>> For the same argument I used to demonstrate that x is not the least
>> member of the interval *not* to apply to iota, one of the following
>> must be true:
>>
>> - iota/2 <= 0
>> - iota/2 >= iota
>> - iota/2 is not defined
>>
>> Though we can define something like this, I'm having difficulty
>> thinking of a way to define it that is consistent *and useful*.
>> We need the non-real members of the interval to behave radically
>> differently under common arithmetic operations. Not that there's
>> anything wrong with that, but it seems to require inventing a set
>> that's even denser than the reals, but is nevertheless in some
>> vague sense discrete rather than continuous. As far as I can tell
>> (and it's very likely I've missed something), most proposed systems
>> of infinitesimals do not have this property.
>
> There are properly defined concepts such as hyper-real numbers, and
> Conway's sureal numbers:
>
> <https://en.wikipedia.org/wiki/Hyperreal_number>
> <https://en.wikipedia.org/wiki/Surreal_number>
>
> and in both cases, the equivalent of your iota is divisible by 2 in a
> reasonable way, and so as you correctly reason, there is still no "next
> point" following a given point. Also, it should be obvious that both of
> these systems of numbers are uncountable, as they both contain a subset
> isomorphic to the set of real numbers.
It sounds to me (based on my extensive research consisting of a few
minutes on Wikipedia) that PO has invented something similar to dual
numbers <https://en.wikipedia.org/wiki/Dual_number>, except that the
coefficient of ε (epsilon) is an integer rather than a real number
and certain operations like division are (arbitrarily?) stated to
be undefined. Without directly speculating on PO's motivation,
I'll note that I see no use for this other than to make it possible
for an open interval over some superset of the reals to have a least
member, a more defensible variant of PO's original incorrect claim
that an open interval over the reals has a least member.
> IMO, just about everyone on this thread has lost track of the focus of
> PO's claim. As I generally don't consider engaging with PO a useful
> activity, I'll just post once, in case any other posters find my
> analysis helpful. (Probably this is way too deep in the thread to make
> any difference, but hey...)
>
> The /focus/ of PO's claim is not whether or not the reals always have a
> next point; it's that THE REALS ARE COUNTABLE.
Agreed. The reals are not countable, and Cantor's elegant proof of this
fact has been well known for well over a century. I've certainly seen
no refutation of it here.
[...]
--
Keith Thompson (The_Other_Keith) kst-u@mib.org <http://www.ghoti.net/~kst>
Will write code for food.
void Void(void) { Void(); } /* The recursive call of the void */
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-17 14:01 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <lNCdnVbg3usFsBzAnZ2dnUU7-ePNnZ2d@giganews.com> |
| In reply to | #458586 |
On 9/17/2019 1:42 PM, Keith Thompson wrote: > Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes: >> On 17/09/2019 05:38, Keith Thompson wrote: >>> André G. Isaak <agisaak@gm.invalid> writes: >>> [...] >>>> Yeah, the account I offered in my earlier post was kind of mangled. My bad. >>>> >>>> The main point I was trying to get across with that post was that the >>>> real numbers in a given interval can't be ordered as smallest, second >>>> smallest, third smallest, etc. I shouldn't really have claimed that >>>> there was no 'ordering' among reals - that was a rather unfortunate >>>> choice of terms, but rather that there is nothing akin to a successor >>>> relation in terms of which they can be ordered. >>>> >>>> It is possible for an interval to have a smallest element, but it is >>>> certainly not guaranteed. In a closed interval, there will be a smallest >>>> element but only because the interval is defined as including its >>>> greatest lower bound. For an open interval, this is not the case. >>>> >>>> I wanted to get him away from trying to derive the open interval from >>>> the closed interval and instead to focus on the defining properties of >>>> the intervals simply because he is determined to impose something akin >>>> to a successor relation on his intervals despite the fact that no such >>>> relation exists. >>>> >>>> You're correct, though, that this probably isn't going to actually >>>> clarify things for Pete. >>> >>> Thanks for the clarification. >>> >>> Here's my attempt to refute the idea that (A,B] has a least element. >>> I'm not trying to argue with Pete (though I expect he'll see this and >>> respond to it). >>> >>> Briefly, if (0,1] has a least member, what about the result of dividing >>> that member by 2? >>> >>> In more detail, (0,1] is the half-open interval consisting of >>> the real numbers between 0 and 1, including 1 but excluding 0. >>> Assume that this interval has a least element, an element x such >>> that for each member m of the interval, either x<m or x=m. >>> >>> Let y=x/2. If we're dealing with real numbers, then 0<y<x, and y >>> is a member of the interval and is less than x -- contradicting >>> the original assertion that x is the least element. Conclusion: >>> the interval (a set of real numbers) has no least element. >>> >>> Arbitrarily inventing a new notation, let _(0,1]_ be a set of, um, >>> something or other. It includes all the real numbers in (0,1], >>> and it may also include some other things, each of which is greater >>> than 0, less than or equal to 1, and somewhere on the "number line" >>> between 0 and 1. Some of the non-real elements might be points, >>> or infinitesimals. I'll assert that for any two members of *this* >>> set, exactly one of a<b, a=b, a>b is true. >>> >>> I can imagine that *this* set has a least element. For example >>> we can define "iota" as an infinitesimal, something that is not a >>> real number but that has a location on the number line, directly >>> adjacent to the real number 0. It is greater than 0 and less than >>> any positive real number. >>> >>> For the same argument I used to demonstrate that x is not the least >>> member of the interval *not* to apply to iota, one of the following >>> must be true: >>> >>> - iota/2 <= 0 >>> - iota/2 >= iota >>> - iota/2 is not defined >>> >>> Though we can define something like this, I'm having difficulty >>> thinking of a way to define it that is consistent *and useful*. >>> We need the non-real members of the interval to behave radically >>> differently under common arithmetic operations. Not that there's >>> anything wrong with that, but it seems to require inventing a set >>> that's even denser than the reals, but is nevertheless in some >>> vague sense discrete rather than continuous. As far as I can tell >>> (and it's very likely I've missed something), most proposed systems >>> of infinitesimals do not have this property. >> >> There are properly defined concepts such as hyper-real numbers, and >> Conway's sureal numbers: >> >> <https://en.wikipedia.org/wiki/Hyperreal_number> >> <https://en.wikipedia.org/wiki/Surreal_number> >> >> and in both cases, the equivalent of your iota is divisible by 2 in a >> reasonable way, and so as you correctly reason, there is still no "next >> point" following a given point. Also, it should be obvious that both of >> these systems of numbers are uncountable, as they both contain a subset >> isomorphic to the set of real numbers. > > It sounds to me (based on my extensive research consisting of a few > minutes on Wikipedia) that PO has invented something similar to dual > numbers <https://en.wikipedia.org/wiki/Dual_number>, except that the > coefficient of ε (epsilon) is an integer rather than a real number > and certain operations like division are (arbitrarily?) stated to > be undefined. It is not arbitrary. A single point on a number line is inherently indivisible. Also it does not have a width of zero, it has a width of infinitesimal. > Without directly speculating on PO's motivation, > I'll note that I see no use for this other than to make it possible > for an open interval over some superset of the reals to have a least > member, a more defensible variant of PO's original incorrect claim > that an open interval over the reals has a least member. > Yes and open interval over reals has no least member. Reals are not defined inconsistently. Infinitesimals have properties that Reals do not have. For about five days my position on elements of this aspect had not gelled. Your statement about the higher density of Infinitesimals than Reals gelled this position using conventional terminology. I was coming to the same conclusions prior to this by referring to the superset of Reals as Numbers. Now we more clearly know that Infinitesimals are proposed to be a superset of Reals. >> IMO, just about everyone on this thread has lost track of the focus of >> PO's claim. As I generally don't consider engaging with PO a useful >> activity, I'll just post once, in case any other posters find my >> analysis helpful. (Probably this is way too deep in the thread to make >> any difference, but hey...) >> >> The /focus/ of PO's claim is not whether or not the reals always have a >> next point; it's that THE REALS ARE COUNTABLE. > > Agreed. The reals are not countable, and Cantor's elegant proof of this > fact has been well known for well over a century. I've certainly seen > no refutation of it here. > > [...] > Yet when Infinitesimals are understood to be a superset of Reals and a bijection between Integers and Infinitesimals is shown though the notational convention of Infinitesimals, then within the analytical framework of Infinitesimals we have shown there is no difference in the cardinality of infinite sets. -- Copyright 2018, 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-17 17:51 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <CJidnUaUFIr9_hzAnZ2dnUU7-IfNnZ2d@giganews.com> |
| In reply to | #458586 |
On 9/17/2019 1:42 PM, Keith Thompson wrote: > Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes: >> On 17/09/2019 05:38, Keith Thompson wrote: >>> André G. Isaak <agisaak@gm.invalid> writes: >>> [...] >>>> Yeah, the account I offered in my earlier post was kind of mangled. My bad. >>>> >>>> The main point I was trying to get across with that post was that the >>>> real numbers in a given interval can't be ordered as smallest, second >>>> smallest, third smallest, etc. I shouldn't really have claimed that >>>> there was no 'ordering' among reals - that was a rather unfortunate >>>> choice of terms, but rather that there is nothing akin to a successor >>>> relation in terms of which they can be ordered. >>>> >>>> It is possible for an interval to have a smallest element, but it is >>>> certainly not guaranteed. In a closed interval, there will be a smallest >>>> element but only because the interval is defined as including its >>>> greatest lower bound. For an open interval, this is not the case. >>>> >>>> I wanted to get him away from trying to derive the open interval from >>>> the closed interval and instead to focus on the defining properties of >>>> the intervals simply because he is determined to impose something akin >>>> to a successor relation on his intervals despite the fact that no such >>>> relation exists. >>>> >>>> You're correct, though, that this probably isn't going to actually >>>> clarify things for Pete. >>> >>> Thanks for the clarification. >>> >>> Here's my attempt to refute the idea that (A,B] has a least element. >>> I'm not trying to argue with Pete (though I expect he'll see this and >>> respond to it). >>> >>> Briefly, if (0,1] has a least member, what about the result of dividing >>> that member by 2? >>> >>> In more detail, (0,1] is the half-open interval consisting of >>> the real numbers between 0 and 1, including 1 but excluding 0. >>> Assume that this interval has a least element, an element x such >>> that for each member m of the interval, either x<m or x=m. >>> >>> Let y=x/2. If we're dealing with real numbers, then 0<y<x, and y >>> is a member of the interval and is less than x -- contradicting >>> the original assertion that x is the least element. Conclusion: >>> the interval (a set of real numbers) has no least element. >>> >>> Arbitrarily inventing a new notation, let _(0,1]_ be a set of, um, >>> something or other. It includes all the real numbers in (0,1], >>> and it may also include some other things, each of which is greater >>> than 0, less than or equal to 1, and somewhere on the "number line" >>> between 0 and 1. Some of the non-real elements might be points, >>> or infinitesimals. I'll assert that for any two members of *this* >>> set, exactly one of a<b, a=b, a>b is true. >>> >>> I can imagine that *this* set has a least element. For example >>> we can define "iota" as an infinitesimal, something that is not a >>> real number but that has a location on the number line, directly >>> adjacent to the real number 0. It is greater than 0 and less than >>> any positive real number. >>> >>> For the same argument I used to demonstrate that x is not the least >>> member of the interval *not* to apply to iota, one of the following >>> must be true: >>> >>> - iota/2 <= 0 >>> - iota/2 >= iota >>> - iota/2 is not defined >>> >>> Though we can define something like this, I'm having difficulty >>> thinking of a way to define it that is consistent *and useful*. >>> We need the non-real members of the interval to behave radically >>> differently under common arithmetic operations. Not that there's >>> anything wrong with that, but it seems to require inventing a set >>> that's even denser than the reals, but is nevertheless in some >>> vague sense discrete rather than continuous. As far as I can tell >>> (and it's very likely I've missed something), most proposed systems >>> of infinitesimals do not have this property. >> >> There are properly defined concepts such as hyper-real numbers, and >> Conway's sureal numbers: >> >> <https://en.wikipedia.org/wiki/Hyperreal_number> >> <https://en.wikipedia.org/wiki/Surreal_number> >> >> and in both cases, the equivalent of your iota is divisible by 2 in a >> reasonable way, and so as you correctly reason, there is still no "next >> point" following a given point. Also, it should be obvious that both of >> these systems of numbers are uncountable, as they both contain a subset >> isomorphic to the set of real numbers. > > It sounds to me (based on my extensive research consisting of a few > minutes on Wikipedia) that PO has invented something similar to dual > numbers <https://en.wikipedia.org/wiki/Dual_number>, except that the Dual numbers have NOTHING to do with counting the number of points on a number line thus are NOTHING like Infinitesimal numbers. > coefficient of ε (epsilon) is an integer rather than a real number > and certain operations like division are (arbitrarily?) stated to > be undefined. Without directly speculating on PO's motivation, > I'll note that I see no use for this other than to make it possible > for an open interval over some superset of the reals to have a least > member, a more defensible variant of PO's original incorrect claim > that an open interval over the reals has a least member. > >> IMO, just about everyone on this thread has lost track of the focus of >> PO's claim. As I generally don't consider engaging with PO a useful >> activity, I'll just post once, in case any other posters find my >> analysis helpful. (Probably this is way too deep in the thread to make >> any difference, but hey...) >> >> The /focus/ of PO's claim is not whether or not the reals always have a >> next point; it's that THE REALS ARE COUNTABLE. > > Agreed. The reals are not countable, and Cantor's elegant proof of this > fact has been well known for well over a century. I've certainly seen > no refutation of it here. > > [...] > -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-17 15:58 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <n4ydnX0vVPJL1RzAnZ2dnUU7-RHNnZ2d@giganews.com> |
| In reply to | #458571 |
On 9/17/2019 12:32 PM, Mike Terry wrote: the focus on his original claim that he has proved the reals are countable.] > > Regards, > Mike. > My original assertion that Cantor was wrong about the differing cardinality of infinite sets is refuted on the basis that points on a number line are countable using the Infinitesimal number system thus Reals are countable. This is now qualified that Infinitesimal numbers are a superset of Reals thus there are more points on the number line than there are Reals. Although I had this same idea: Keith Thompson helped gell this idea into its more complete form. On 9/16/2019 11:38 PM, Keith Thompson wrote: > ...it seems to require inventing a set that's even denser than the reals... -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-19 11:46 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- (Mike Terry Concurrence ] |
| Message-ID | <leGdnbX22-J1LR7AnZ2dnUU7-SOdnZ2d@giganews.com> |
| In reply to | #458497 |
On 9/16/2019 8:33 PM, Mike Terry wrote:
> On 16/09/2019 23:46, Keith Thompson wrote:
>> André G. Isaak <agisaak@gm.invalid> writes:
>> [...]
>>> I already replied to this post, but wanted to provide an alternative
>>> response in case you don't follow my earlier one.
>>>
>>> [A, B] is defined as all real numbers x such that A <= x <= B
>>>
>>> (A, B] is defined as all real numbers x such that A < x <= B
>>>
>>> What's important to realize here is that you don't get (A,B] by starting
>>> with [A,B] and 'removing the first point'. These are simply two
>>> different intervals. There is no operation involved in which you get one
>>> from the other.
>>
>> I disagree with that last statement, and would be interested in a
>> demonstration that I'm wrong about that.
>>
>> [A,B] is a set of real numbers. (A,B] can be constructed by taking
>> [A,B] and removing a single element, namely A, and similarily vice
>> versa. All members of (A,B] are members of [A,B], and there is
>> exactly one real number that is a member of [A,B] and not of (A,B].
>
> OK, so (A,B] = [A,B] - {A}, and that can be counted as an "operation in which we get (A,B] from [A,B].
[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.
> So you're right, but I sort of doubt that's exactly what Andre meant... (Just because it's obvious, and Andre seems reasonably clued up, so I'm sure
> he understands this.)
>
> I'm not sure what he was trying to say, but my guess would be that he's focussing on the basic /meaning/ of the intervals. The intervals are just sets of real numbers meeting the appropriate conditions. I.e. they are "static" mathematical objects, sort
> of "beyond time". Hmm, that's not clear. Specifically the open interval is NOT "some kind of "process" which starts with the closed interval, and THEN (step 2) removes the least element in the set". Summary: the intervals are SETS not PROCESSES.
>
> OTOH, I've no idea why Andre would think bringing that up would help PO. Andre should explain for himself of course, I'm just waffling :)
>
> One thing I think doesn't help anyone is calling A the "first" point in the interval, as this language in and of itself encourages the idea that there is a second, third, forth element and so on, when there isn't. (So "least" element is a bit better,
> keeping in mind that not all sets of real numbers have a least element.)
>
>>
>> Of course this doesn't imply that there's a "first element" of (A,B].
>> There clearly isn't. (Pete has correctly recognized a contradiction,
>> but has grabbed onto the wrong end of it.)
>
> So what is the contradiction you believe PO has recognised?
>
> Regards,
> Mike.
--
Copyright 2019 Pete Olcott All rights reserved
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-19 11:52 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- [ Keith Thompson Concurrence ] |
| Message-ID | <CuSdnb9275HCLx7AnZ2dnUU7-U_NnZ2d@giganews.com> |
| In reply to | #458487 |
On 9/16/2019 5:46 PM, Keith Thompson wrote:
> André G. Isaak <agisaak@gm.invalid> writes:
> [...]
>> I already replied to this post, but wanted to provide an alternative
>> response in case you don't follow my earlier one.
>>
>> [A, B] is defined as all real numbers x such that A <= x <= B
>>
>> (A, B] is defined as all real numbers x such that A < x <= B
>>
>> What's important to realize here is that you don't get (A,B] by starting
>> with [A,B] and 'removing the first point'. These are simply two
>> different intervals. There is no operation involved in which you get one
>> from the other.
>
> I disagree with that last statement, and would be interested in a
> demonstration that I'm wrong about that.
>
> [A,B] is a set of real numbers. (A,B] can be constructed by taking
> [A,B] and removing a single element, namely A, and similarily vice
> versa. All members of (A,B] are members of [A,B], and there is
> exactly one real number that is a member of [A,B] and not of (A,B].
>
[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.
> Of course this doesn't imply that there's a "first element" of (A,B].
> There clearly isn't. (Pete has correctly recognized a contradiction,
> but has grabbed onto the wrong end of it.)
>
We removed the first point of the interval leaving all of the
remaining points of the interval: [0,1] - (0,1] = {0}
When we encode (0,1] as [0[1], 1] we can not remove the
next point from this interval: [0[1],1] - {0[1]} = (0[1], 1]
--
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 11:55 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- [ Keith Thompson Concurrence ] |
| Message-ID | <CuSdnb5275GVLh7AnZ2dnUU7-U-dnZ2d@giganews.com> |
| In reply to | #458831 |
On 9/19/2019 11:52 AM, peteolcott wrote:
> On 9/16/2019 5:46 PM, Keith Thompson wrote:
>> André G. Isaak <agisaak@gm.invalid> writes:
>> [...]
>>> I already replied to this post, but wanted to provide an alternative
>>> response in case you don't follow my earlier one.
>>>
>>> [A, B] is defined as all real numbers x such that A <= x <= B
>>>
>>> (A, B] is defined as all real numbers x such that A < x <= B
>>>
>>> What's important to realize here is that you don't get (A,B] by starting
>>> with [A,B] and 'removing the first point'. These are simply two
>>> different intervals. There is no operation involved in which you get one
>>> from the other.
>>
>> I disagree with that last statement, and would be interested in a
>> demonstration that I'm wrong about that.
>>
>> [A,B] is a set of real numbers. (A,B] can be constructed by taking
>> [A,B] and removing a single element, namely A, and similarily vice
>> versa. All members of (A,B] are members of [A,B], and there is
>> exactly one real number that is a member of [A,B] and not of (A,B].
>>
>
> [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.
>
>> Of course this doesn't imply that there's a "first element" of (A,B].
>> There clearly isn't. (Pete has correctly recognized a contradiction,
>> but has grabbed onto the wrong end of it.)
>>
>
> We removed the first point of the interval leaving all of the
> remaining points of the interval: [0,1] - (0,1] = {0}
Correction:
> When we encode (0,1] as [0[1], 1] we can NOW remove the
> next point from this interval: [0[1],1] - {0[1]} = (0[1], 1]
>
--
Copyright 2019 Pete Olcott All rights reserved
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein
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| From | Keith Thompson <kst-u@mib.org> |
|---|---|
| Date | 2019-09-19 14:44 -0700 |
| Subject | Re: Cantor was wrong about cardinality --- [ Keith Thompson Concurrence ] |
| Message-ID | <ln5zlo6jwe.fsf@kst-u.example.com> |
| In reply to | #458831 |
peteolcott <Here@Home> writes:
[SNIP]
The phrase "Keith Thompson Concurrence" in the title of the parent
article is a lie.
--
Keith Thompson (The_Other_Keith) kst-u@mib.org <http://www.ghoti.net/~kst>
Will write code for food.
void Void(void) { Void(); } /* The recursive call of the void */
[toc] | [prev] | [next] | [standalone]
| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-19 17:31 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- [ Keith Thompson Concurrence ] |
| Message-ID | <d6udncuuBIQwnBnAnZ2dnUU7-a3NnZ2d@giganews.com> |
| In reply to | #458851 |
On 9/19/2019 4:44 PM, Keith Thompson wrote: > peteolcott <Here@Home> writes: > [SNIP] > > The phrase "Keith Thompson Concurrence" in the title of the parent > article is a lie. > That claim is proven to be counter-factual by the provided context. That I said you concur with me does not indicate that you totally agreed with every subtle nuance. The aspect that you did agree to forms the whole basis of my position: (1) Points on a number line immediately adjacent to each other can be conceived and differ by an infinitesimal value. On 9/16/2019 11:38 PM, Keith Thompson wrote: > I can imagine that *this* set has a least element. For example > we can define "iota" as an infinitesimal, something that is not a > real number but that has a location on the number line, directly > adjacent to the real number 0. It is greater than 0 and less than > any positive real number. (2) The intervals [A, B] - (A, B] = point: A. On 9/16/2019 5:46 PM, Keith Thompson wrote: > [A,B] is a set of real numbers. (A,B] can be constructed by taking > [A,B] and removing a single element, namely A, and similarily vice > versa. All members of (A,B] are members of [A,B], and there is > exactly one real number that is a member of [A,B] and not of (A,B]. From (1) and (2) the whole Infinitesimal number system is derived. -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-16 11:02 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <RI2dnZdu_JSPL-LAnZ2dnUU7-I2dnZ2d@giganews.com> |
| In reply to | #458435 |
On 9/16/2019 10:26 AM, André G. Isaak wrote:
> On 2019-09-16 9:17 a.m., peteolcott wrote:
>> On 9/15/2019 6:33 AM, Ben Bacarisse wrote:
>>> peteolcott <Here@Home> writes:
>>>
>>>> On 9/14/2019 10:09 PM, Ben Bacarisse wrote:
>>>>> peteolcott <Here@Home> writes:
>>>>>
>>>>>> On 9/14/2019 8:43 PM, Ben Bacarisse wrote:
>>>>>>> peteolcott <Here@Home> writes:
>>>>>>>
>>>>>>>> On 9/14/2019 6:47 PM, Ben Bacarisse wrote:
>>>>>>>>> peteolcott <Here@Home> writes:
>>>>>>>>>> On 9/14/2019 11:23 AM, Ben Bacarisse wrote:
>>>>>>>>>>> No. The common notational convention used for intervals shows that
>>>>>>>>>>> adjacent points do not exist. While [0, 1] has a least member, (0, 1]
>>>>>>>>>>> does not. You can remove the first number in [0, 1] to get (0, 1] but
>>>>>>>>>>> you can't remove the first and second.
>>>>>>>>>>
>>>>>>>>>> If we have a sequence of points [0.0, 1.0] on a number line and we remove
>>>>>>>>>> the first point (0.0, 1.0] then it is obvious that this second sequence is
>>>>>>>>>> exactly one geometric point shorter than the first sequence.
>>>>>>>>>
>>>>>>>>> The usual measure on intervals makes [0, 1] and (0, 1] the same length
>>>>>>>>> so you need to say PO-shorter, and then define what you mean by that.
>>>>>>>>> But this is a side issue.
>>>>>>>>
>>>>>>>> That is defining X = X + 1.
>>>>>>>
>>>>>>> No it isn't. It is defining length(X) = length(X U {0}) and this is
>>>>>>> indeed the usual definition when both X and X U {0} are intervals.
>>>>>>> Making an open interval closed does no change the length.
>>>>>>
>>>>>> (0, 1] versus (0, 1) as a range of whole %
>>>>>> The first one is 0% to 100%
>>>>>> The second one is 0% to 99%
>>>>>
>>>>> No, that would be (0, 0.99) (or [0, 0.99]).
>>>>
>>>> There are too many ways to correctly say this.
>>>> (0, 100] integers is 0 to 100
>>>> (0, 100) integers is 0 to 99
>>>
>>> No. Sadly, almost everything you post introduces new errors. There are
>>> no open intervals in the integers so using the () notation is silly.
>>>
>>> You can, of course, give it a meaning (as you have tried to do here) but
>>> even then you get it wrong. If the "open" end means the bound is
>>> excluded, then (0, 100] is 1 to 100 and (0, 100) is 1 to 99.
>>>
>>> Why switch to integers which have totally different properties?
>>>
>>>>>> You can't magically ignore an actual difference just because it is infinitesimal.
>>>>>> (0, 1] a length of exactly 1.0
>>>>>> (0, 1) a length infinitesimally shorter than 1.0.
>>>>>
>>>>> It's not magic. Length is defined that way for a reason. As I have
>>>>> said time and time again, just call your notion PO-length, define it
>>>>> properly, and we can decide if we care about it or not.
>>>>>
>>>>
>>>> 0.0 to 100.0 including the 100 is 0 to 100.0
>>>> 0.0 to 100.0 not including the 100 is 0 to
>>>> 99.999999999999999999999999999999999999999... on to infinity
>>>
>>> I propose Ben's law: Every sufficiently long mathematical crank thread
>>> ends up "debating" whether 0.999... = 1 or not.
>>>
>>> No. 99.999... = 100 and you stated that 100 is not included.
>>>
>>> There is no notation for the largest real in the open interval (0, 100),
>>> not because no one has come up with a notation for it (that's trivial,
>>> but the words will do) but because there is no such number.
>>>
>>
>> Do you realize that you just said that the number line is defined with gaps?
>> There is either a contiguous set of points between 0 and 100 or there are
>> gaps between 0 and 100, those are the only two choices.
>
>
> No. You are describing a number line with gaps. He is not. If there were indeed some largest real in the interval (0,100) (let's call it L), then what number lies between L and 100? According to you, none. But, it is axiomatic that for the real numbers,
> for every x and every y if z < y then there exists some z such that x < z < y. This proves (by reductio ad absurdum) that there is no such number L.
>
> André
>
>
Since it is clear that (0,100) specifies every geometric point on the number
line between 0 and 100 except for 0 and 100 then it does specify the geometric
point immediately after 0 and the geometric point immediately before 100 with
no geometric points in-between.
If these geometric points are defined to be real numbers and real numbers
are defined to always be divisible by 2, then something about the way that
these things are defined is inherently inconsistent.
--
Copyright 2019 Pete Olcott All rights reserved
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-16 20:01 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- APOLOGY clear yet? |
| Message-ID | <DOedncdA-ZLFrR3AnZ2dnUU7-W_NnZ2d@giganews.com> |
| In reply to | #458433 |
On 9/16/2019 7:49 PM, Ben Bacarisse wrote: > peteolcott <Here@Home> writes: > >> On 9/15/2019 6:33 AM, Ben Bacarisse wrote: >>> peteolcott <Here@Home> writes: > <cut> >>>> 0.0 to 100.0 including the 100 is 0 to 100.0 >>>> 0.0 to 100.0 not including the 100 is 0 to >>>> 99.999999999999999999999999999999999999999... on to infinity >>> >>> I propose Ben's law: Every sufficiently long mathematical crank thread >>> ends up "debating" whether 0.999... = 1 or not. >>> >>> No. 99.999... = 100 and you stated that 100 is not included. >>> >>> There is no notation for the largest real in the open interval (0, 100), >>> not because no one has come up with a notation for it (that's trivial, >>> but the words will do) but because there is no such number. >> >> Do you realize that you just said that the number line is defined with >> gaps? > > I realise I overestimated your reasoning powers and, frankly, I did > not think that possible given the low estimation I originally had. > >> There is either a contiguous set of points between 0 and 100 or there are >> gaps between 0 and 100, those are the only two choices. > > No, there are other options. The reals form a continuum, but contiguous > means touching or adjacent, and no two real are adjacent. Have you ever > considered getting an education? > >>> This fact follows trivially from the definition of an open set in a >>> metric space, or from the axioms of the reals, or from the first baby >>> steps in topology... Repeatedly referring to a provably non-existent >>> object is just silly. >> >> https://www.encyclopediaofmath.org/index.php/Interval_and_segment >> An interval (open interval) is a set of points on a line lying between >> two fixed points A and B, where A and B themselves are considered not >> to belong to the interval. >> >> This means that there is a first point and there is a last point of every >> open interval. > > No it does not. Nothing on that page lets you conclude that. If you > knew what an open set was... Oh, sorry, that's me suggesting education > again. > >> (0, 100) the first point is 0.0[1] and the last point is 100.0[-1]. > > Only if your points do not form a continuum. The real interval (0, 100) > does, and it has no first and no last number. [0, 100] has a first and last point and a first and last real number (0, 100) has a first and last point yet no first and last real number. When-so-ever we remove any portion from the front of the interval: [0, 100] It always has a first point. [0.5, 100] [0.1, 100] [0.0001, 100] [0.0000000000000001, 100] [0.000000000000...1, 100] (0, 100] -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-16 22:39 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- APOLOGY clear yet? |
| Message-ID | <l8ydnWn97MjFyB3AnZ2dnUU7-fHNnZ2d@giganews.com> |
| In reply to | #458494 |
On 9/16/2019 9:25 PM, Ben Bacarisse wrote:
> peteolcott <Here@Home> writes:
>
>> On 9/16/2019 7:49 PM, Ben Bacarisse wrote:
>>> peteolcott <Here@Home> writes:
> <cut>
>>>> (0, 100) the first point is 0.0[1] and the last point is 100.0[-1].
>>>
>>> Only if your points do not form a continuum. The real interval (0, 100)
>>> does, and it has no first and no last number.
>>
>> [0, 100] has a first and last point and a first and last real number
>
> Yes.
>
>> (0, 100) has a first and last point yet no first and last real number.
>
> No. For any real x in (0, 100) we can find smaller elements. In fact,
> infinitely many: xy for every y in (0, 1) is smaller than x.
There is a first point that cannot be identified with a real number.
It can ONLY be identified with the infinitesimal number: 0.0[1].
It only seems like there is no first point when trying to identify
this point with a real number.
>
>> When-so-ever we remove any portion from the front of the interval: [0, 100]
>> It always has a first point.
>
> No. It depends on whether the portion you remove is closed or only
> half-closed:
>
>> [0.5, 100]
>
> This results from removing [0, 0.5) and therefore does indeed have a
> least element. If you remove [0, 0.5] you get (0.5, 100] and, as you
> now know, that set does /not/ have a least element.
>
>> [0.1, 100]
>> [0.0001, 100]
>> [0.0000000000000001, 100]
>> [0.000000000000...1, 100]
>
> These, too, result from removing some half-open set.
>
>> (0, 100]
>
> This results from removing a closed set ({0}) and therefore does not
> have a least element.
>
> Don't you think you should pick a book about the real numbers now?
>
--
Copyright 2019 Pete Olcott All rights reserved
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-17 22:23 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- APOLOGY clear yet? |
| Message-ID | <q_GdnekMsYSOPhzAnZ2dnUU7-fOdnZ2d@giganews.com> |
| In reply to | #458508 |
On 9/17/2019 10:01 PM, Ben Bacarisse wrote:
> peteolcott <Here@Home> writes:
>
>> On 9/17/2019 8:26 AM, Ben Bacarisse wrote:
>>> peteolcott <Here@Home> writes:
>>>
>>>> On 9/16/2019 9:25 PM, Ben Bacarisse wrote:
>>>>> peteolcott <Here@Home> writes:
>>>>>
>>>>>> On 9/16/2019 7:49 PM, Ben Bacarisse wrote:
>>>>>>> peteolcott <Here@Home> writes:
>>>>> <cut>
>>>>>>>> (0, 100) the first point is 0.0[1] and the last point is 100.0[-1].
>>>>>>>
>>>>>>> Only if your points do not form a continuum. The real interval (0, 100)
>>>>>>> does, and it has no first and no last number.
>>>>>>
>>>>>> [0, 100] has a first and last point and a first and last real number
>>>>>
>>>>> Yes.
>>>>>
>>>>>> (0, 100) has a first and last point yet no first and last real number.
>>>>>
>>>>> No. For any real x in (0, 100) we can find smaller elements. In fact,
>>>>> infinitely many: xy for every y in (0, 1) is smaller than x.
>>>>
>>>> There is a first point that cannot be identified with a real number.
>>>> It can ONLY be identified with the infinitesimal number: 0.0[1].
>>>
>>> A notation you refused to explain when you saw the half-baked answers
>>> you gave to my initial questions were taking you into dangerous waters.
>>>
>>> You agree, it seems, that for the intervals everyone else cares about
>>> which are subsets or R, (0, 100) has no least element. Please confirm
>>> this. It matters. If you confirm it I can more on to ignoring
>>> PO-points and PO-intervals because no one cares about them until you
>>> can define.
>
> Are you really try to avoid confirming this, or have you just stopped
> reading what people write?
>
The conversation has moved way past this. Now my point is built
on the basis of Mike and Keith's agreement.
I SUMMED IT UP MORE CONCISELY HERE:
THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0
On 9/16/2019 11:38 PM, Keith Thompson wrote:
> we can define "iota" as an infinitesimal, something that is not a
> real number but that has a location on the number line, directly
> adjacent to the real number 0. It is greater than 0 and less than
> any positive real number.
WE CAN SUBTRACT A SINGLE POINT FROM AN INTERVAL DERIVING ANOTHER
INTERVAL THAT DOES NOT HAVE THIS POINT
On 9/16/2019 8:33 PM, Mike Terry wrote:
> OK, so (A,B] = [A,B] - {A}, and that can be counted as an "operation
in which we get (A,B] from [A,B].
Mike Terry agreed with this verbatim: [A, B] - {A} = (A, B]
Infinitesimal Numbers would encode (A, B] as [A[1], B]
We merely extrapolate the very next elaboration
[A[1], B] - {A[1]} = (A[1], B] encoded as [A[2], B]
This shows how Infinitesimal Numbers directly reference immediately
adjacent points on a number line.
0.0[0] = 0.0
0.0[1] = the next point after 0.0
0.0[2] = 2 points after 0.0
0.0[3] = 3 points after 0.0
--
Copyright 2019 Pete Olcott All rights reserved
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein
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| From | Keith Thompson <kst-u@mib.org> |
|---|---|
| Date | 2019-09-18 01:39 -0700 |
| Subject | Re: Cantor was wrong about cardinality --- APOLOGY clear yet? |
| Message-ID | <lnv9tq6lru.fsf@kst-u.example.com> |
| In reply to | #458636 |
peteolcott <Here@Home> writes:
[...]
> The conversation has moved way past this. Now my point is built
> on the basis of Mike and Keith's agreement.
Stop using my name to support your claims.
--
Keith Thompson (The_Other_Keith) kst-u@mib.org <http://www.ghoti.net/~kst>
Will write code for food.
void Void(void) { Void(); } /* The recursive call of the void */
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-18 09:59 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- APOLOGY clear yet? |
| Message-ID | <PYKdnZn3lqXG2x_AnZ2dnUU7-IWdnZ2d@giganews.com> |
| In reply to | #458669 |
On 9/18/2019 3:39 AM, Keith Thompson wrote: > peteolcott <Here@Home> writes: > [...] >> The conversation has moved way past this. Now my point is built >> on the basis of Mike and Keith's agreement. > > Stop using my name to support your claims. > HELL NO !!! THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0 On 9/16/2019 11:38 PM, Keith Thompson wrote: > we can define "iota" as an infinitesimal, something that is not a > real number but that has a location on the number line, directly > adjacent to the real number 0. It is greater than 0 and less than > any positive real number. -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2019-09-18 09:21 -0600 |
| Subject | Re: Cantor was wrong about cardinality --- APOLOGY clear yet? |
| Message-ID | <qlti2i$vm4$1@dont-email.me> |
| In reply to | #458690 |
On 2019-09-18 8:59 a.m., peteolcott wrote: > On 9/18/2019 3:39 AM, Keith Thompson wrote: >> peteolcott <Here@Home> writes: >> [...] >>> The conversation has moved way past this. Now my point is built >>> on the basis of Mike and Keith's agreement. >> >> Stop using my name to support your claims. >> > > HELL NO !!! > > THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0 > On 9/16/2019 11:38 PM, Keith Thompson wrote: > > we can define "iota" as an infinitesimal, something that is not a > > real number but that has a location on the number line, directly > > adjacent to the real number 0. It is greater than 0 and less than > > any positive real number. > This paragraph that you keep quoting was followed by additional material which you conveniently ignore. That material made it quite clear that he was *not* supporting your view. André -- To email remove 'invalid' & replace 'gm' with well known Google mail service.
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