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Groups > comp.ai.philosophy > #18797 > unrolled thread
| Started by | peteolcott <Here@Home> |
|---|---|
| First post | 2019-08-08 12:15 -0500 |
| Last post | 2019-09-11 17:31 -0700 |
| Articles | 20 on this page of 244 — 17 participants |
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Provably unprovable eliminates incompleteness peteolcott <Here@Home> - 2019-08-08 12:15 -0500
Re: Provably unprovable eliminates incompleteness Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-08-08 21:28 +0100
Re: Provably unprovable eliminates incompleteness Dhu on Gate <campbell@neotext.ca> - 2019-09-08 06:06 +0000
Re: Provably unprovable eliminates incompleteness Stuart Redmann <DerTopper@web.de> - 2019-09-09 11:30 +0200
Re: Provably unprovable eliminates incompleteness peteolcott <Here@Home> - 2019-09-09 10:01 -0500
Re: Provably unprovable eliminates incompleteness Mr Flibble <flibble@i42.removethisbit.co.uk> - 2019-09-09 21:38 +0100
Re: Provably unprovable eliminates incompleteness peteolcott <Here@Home> - 2019-09-09 18:05 -0500
Re: Provably unprovable eliminates incompleteness David Brown <david.brown@hesbynett.no> - 2019-09-10 09:01 +0200
Re: Provably unprovable eliminates incompleteness peteolcott <Here@Home> - 2019-09-10 10:38 -0500
Re: Provably unprovable eliminates incompleteness David Brown <david.brown@hesbynett.no> - 2019-09-10 22:01 +0200
Re: Provably unprovable eliminates incompleteness [ Infinitesimal numbers ] peteolcott <Here@Home> - 2019-09-10 20:12 -0500
Re: Provably unprovable eliminates incompleteness Reinhardt Behm <rbehm@hushmail.com> - 2019-09-11 15:31 +0800
Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) peteolcott <Here@Home> - 2019-09-11 12:04 -0500
Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) Reinhardt Behm <rbehm@hushmail.com> - 2019-09-12 10:20 +0800
Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) peteolcott <Here@Home> - 2019-09-11 23:51 -0500
Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) Keith Thompson <kst-u@mib.org> - 2019-09-12 01:22 -0700
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> - 2019-09-12 21:39 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Deceptive Strawman by Ben peteolcott <Here@Home> - 2019-09-13 11:48 -0500
Re: Cantor was wrong about cardinality --- APOLOGY peteolcott <Here@Home> - 2019-09-14 13:09 -0500
Re: Cantor was wrong about cardinality --- APOLOGY peteolcott <Here@Home> - 2019-09-14 19:24 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-14 21:33 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-16 10:17 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? André G. Isaak <agisaak@gm.invalid> - 2019-09-16 09:26 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 10:36 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 10:06 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 11:22 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 10:38 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 11:59 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 11:10 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 13:27 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 12:34 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 14:19 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 13:35 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 14:51 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 14:15 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 16:18 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 16:00 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Keith Thompson <kst-u@mib.org> - 2019-09-16 15:29 -0700
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently 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). peteolcott <Here@Home> - 2019-09-17 11:39 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (some concurrence). Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2019-09-17 18:38 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (some concurrence). peteolcott <Here@Home> - 2019-09-17 13:17 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (some concurrence). Keith Thompson <kst-u@mib.org> - 2019-09-17 11:13 -0700
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (some concurrence). peteolcott <Here@Home> - 2019-09-17 13:29 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (Concurrence) peteolcott <Here@Home> - 2019-09-17 11:20 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (Concurrence) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 10:56 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (Concurrence) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 11:29 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2019-09-17 18:32 +0100
Re: Cantor was wrong about cardinality --- (Correction) peteolcott <Here@Home> - 2019-09-17 12:49 -0500
Re: Cantor was wrong about cardinality --- (Correction) Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2019-09-17 19:59 +0100
Re: Cantor was wrong about cardinality --- (Correction) peteolcott <Here@Home> - 2019-09-17 14:09 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Keith Thompson <kst-u@mib.org> - 2019-09-17 11:42 -0700
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-17 14:01 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-17 17:51 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-17 15:58 -0500
Re: Cantor was wrong about cardinality --- (Mike Terry Concurrence ] peteolcott <Here@Home> - 2019-09-19 11:46 -0500
Re: Cantor was wrong about cardinality --- [ Keith Thompson Concurrence ] peteolcott <Here@Home> - 2019-09-19 11:52 -0500
Re: Cantor was wrong about cardinality --- [ Keith Thompson Concurrence ] peteolcott <Here@Home> - 2019-09-19 11:55 -0500
Re: Cantor was wrong about cardinality --- [ Keith Thompson Concurrence ] Keith Thompson <kst-u@mib.org> - 2019-09-19 14:44 -0700
Re: Cantor was wrong about cardinality --- [ Keith Thompson Concurrence ] peteolcott <Here@Home> - 2019-09-19 17:31 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 11:02 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-16 20:01 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-16 22:39 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-17 22:23 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? Keith Thompson <kst-u@mib.org> - 2019-09-18 01:39 -0700
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-18 09:59 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? André G. Isaak <agisaak@gm.invalid> - 2019-09-18 09:21 -0600
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-18 10:41 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? Keith Thompson <kst-u@mib.org> - 2019-09-18 13:55 -0700
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-18 16:17 -0500
Re: Cantor was wrong about cardinality --- [ CONCURRENCE ] peteolcott <Here@Home> - 2019-09-18 11:10 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-12 09:05 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-12 10:49 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-12 10:02 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 00:18 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-12 23:24 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 10:01 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-13 09:13 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-13 09:56 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 11:40 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-13 11:03 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 12:48 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-13 12:25 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-13 22:35 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-14 16:42 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 11:28 -0500
Re: Cantor was wrong about cardinality --- COMPLETE peteolcott <Here@Home> - 2019-09-14 11:34 -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) BEN LIED !!! peteolcott <Here@Home> - 2019-09-14 00:55 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-16 10:26 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-16 18:24 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-17 22:19 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 21:23 -0600
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-17 22:31 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 22:04 -0600
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-17 23:21 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 22:40 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 00:27 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 23:55 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 00:59 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 00:12 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 10:15 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 09:35 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 11:00 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 10:14 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 11:26 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 10:42 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 13:10 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 12:36 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 13:49 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 12:59 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 14:12 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 13:30 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 14:45 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 14:14 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 15:29 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 14:38 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 15:56 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 20:38 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 22:03 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 21:33 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-19 10:35 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-19 11:42 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-19 12:43 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-19 10:24 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-17 23:32 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 23:21 -0600
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-19 17:13 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-19 19:49 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) 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 (CONCISE RESTATEMENT) Don Stockbauer <donstockbauer@hotmail.com> - 2019-09-27 07:24 -0700
Re: Provably unprovable eliminates incompleteness Juha Nieminen <nospam@thanks.invalid> - 2019-09-12 22:07 +0000
Re: Provably unprovable eliminates incompleteness peteolcott <Here@Home> - 2019-09-12 18:53 -0500
Re: Provably unprovable eliminates incompleteness Keith Thompson <kst-u@mib.org> - 2019-09-12 17:47 -0700
Re: Provably unprovable eliminates incompleteness Juha Nieminen <nospam@thanks.invalid> - 2019-09-13 08:14 +0000
Re: Provably unprovable eliminates incompleteness peteolcott <Here@Home> - 2019-09-13 09:35 -0500
Re: Provably unprovable eliminates incompleteness Mr Flibble <flibble@i42.removethisbit.co.uk> - 2019-09-13 15:55 +0100
Re: Provably unprovable eliminates incompleteness--- [ Cantor was wrong about cardinality ] peteolcott <Here@Home> - 2019-09-14 13:35 -0500
Re: Provably unprovable eliminates incompleteness--- [ Cantor was wrong about cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-14 22:52 +0100
Re: Provably unprovable eliminates incompleteness--- [ Cantor was wrong about cardinality ] peteolcott <Here@Home> - 2019-09-14 17:18 -0500
Re: Provably unprovable eliminates incompleteness--- [ Cantor was wrong about cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-14 23:30 +0100
Re: Provably unprovable eliminates incompleteness--- [ Cantor was wrong about cardinality ] peteolcott <Here@Home> - 2019-09-14 17:42 -0500
Re: Provably unprovable eliminates incompleteness--- [ Cantor was wrong about cardinality ] Jeff Barnett <jbb@notatt.com> - 2019-09-14 18:12 -0600
Re: Provably unprovable eliminates incompleteness Juha Nieminen <nospam@thanks.invalid> - 2019-09-15 08:09 +0000
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-16 09:50 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-16 17:13 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-16 11:24 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-16 17:30 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-16 12:08 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-16 18:15 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-16 13:35 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-16 20:02 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-16 14:48 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-16 22:26 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-16 18:11 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-16 18:12 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Juha Nieminen <nospam@thanks.invalid> - 2019-09-17 08:11 +0000
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-17 11:48 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] David Brown <david.brown@hesbynett.no> - 2019-09-17 21:10 +0200
Re: Provably unprovable eliminates incompleteness [ Keith and Mike basis for Infinitesimal numbers ] peteolcott <Here@Home> - 2019-09-17 14:30 -0500
Re: Provably unprovable eliminates incompleteness [ Keith and Mike basis for Infinitesimal numbers ] peteolcott <Here@Home> - 2019-09-17 14:33 -0500
Re: Provably unprovable eliminates incompleteness [ Keith and Mike basis for Infinitesimal numbers ] Keith Thompson <kst-u@mib.org> - 2019-09-17 13:28 -0700
Re: Provably unprovable eliminates incompleteness [ Keith and Mike basis for Infinitesimal numbers ] peteolcott <Here@Home> - 2019-09-17 15:31 -0500
Re: Provably unprovable eliminates incompleteness [ Keith and Mike basis for Infinitesimal numbers ] Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2019-09-17 22:16 +0100
Re: Provably unprovable eliminates incompleteness [ Keith and Mike basis for Infinitesimal numbers ] peteolcott <Here@Home> - 2019-09-17 16:42 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Juha Nieminen <nospam@thanks.invalid> - 2019-09-17 08:05 +0000
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-17 11:45 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-17 18:14 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-17 12:36 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2019-09-17 19:45 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-17 20:09 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-17 14:14 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2019-09-17 18:59 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Keith Thompson <kst-u@mib.org> - 2019-09-17 11:10 -0700
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-17 13:48 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Keith Thompson <kst-u@mib.org> - 2019-09-17 12:01 -0700
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Juha Nieminen <nospam@thanks.invalid> - 2019-09-18 07:35 +0000
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-18 10:02 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-18 22:25 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-18 16:41 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-18 22:56 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Juha Nieminen <nospam@thanks.invalid> - 2019-09-18 21:34 +0000
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-18 16:42 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Juha Nieminen <nospam@thanks.invalid> - 2019-09-19 17:54 +0000
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-19 13:26 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Juha Nieminen <nospam@thanks.invalid> - 2019-09-20 08:06 +0000
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 13:37 -0700
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-19 16:07 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 15:04 -0700
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-19 17:32 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Juha Nieminen <nospam@thanks.invalid> - 2019-09-20 08:10 +0000
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 13:33 -0700
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-19 16:05 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 15:48 -0700
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-19 18:00 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] David Brown <david.brown@hesbynett.no> - 2019-09-20 08:52 +0200
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Siri Cruise <chine.bleu@yahoo.com> - 2019-09-20 00:10 -0700
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] David Brown <david.brown@hesbynett.no> - 2019-09-20 10:11 +0200
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Juha Nieminen <nospam@thanks.invalid> - 2019-09-20 08:25 +0000
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-16 09:54 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] André G. Isaak <agisaak@gm.invalid> - 2019-09-16 09:15 -0600
Re: Provably unprovable eliminates incompleteness Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 13:13 -0700
Re: Provably unprovable eliminates incompleteness Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 12:54 -0700
Re: Provably unprovable eliminates incompleteness peteolcott <Here@Home> - 2019-09-19 15:09 -0500
Re: Provably unprovable eliminates incompleteness Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 12:45 -0700
Re: Provably unprovable eliminates incompleteness Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 12:34 -0700
Re: Provably unprovable eliminates incompleteness [ COMPLETE-2 ] peteolcott <Here@Home> - 2019-09-19 14:58 -0500
Re: Provably unprovable eliminates incompleteness [ COMPLETE-2 ] Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 15:57 -0700
Re: Provably unprovable eliminates incompleteness [ COMPLETE-2 ] peteolcott <Here@Home> - 2019-09-19 18:10 -0500
Re: Provably unprovable eliminates incompleteness [ COMPLETE-2 ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-20 13:18 +0100
Re: Provably unprovable eliminates incompleteness Louis Valence <lvalence@example.com> - 2019-09-09 17:43 -0300
Re: Provably unprovable eliminates incompleteness Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 12:28 -0700
Re: Provably unprovable eliminates incompleteness peteolcott <Here@Home> - 2019-09-19 15:04 -0500
Re: Provably unprovable eliminates incompleteness peteolcott <Here@Home> - 2019-09-09 09:57 -0500
Re: Provably unprovable eliminates incompleteness Keith Thompson <kst-u@mib.org> - 2019-09-09 13:30 -0700
Re: Provably unprovable eliminates incompleteness temakoutera62@gmail.com - 2019-09-11 17:31 -0700
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-16 14:51 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <4qqdnSmfIroweuLAnZ2dnUU7-cmdnZ2d@giganews.com> |
| In reply to | #19211 |
On 9/16/2019 2:35 PM, André G. Isaak wrote:
> On 2019-09-16 1:19 p.m., peteolcott wrote:
>> On 9/16/2019 1:34 PM, André G. Isaak wrote:
>>> On 2019-09-16 12:27 p.m., peteolcott wrote:
>>>> On 9/16/2019 12:10 PM, André G. Isaak wrote:
>>>>> On 2019-09-16 10:59 a.m., peteolcott wrote:
>>>>>> On 9/16/2019 11:38 AM, André G. Isaak wrote:
>>>>>>> On 2019-09-16 10:22 a.m., peteolcott wrote:
>>>>>>>> On 9/16/2019 11:06 AM, André G. Isaak wrote:
>>>>>>>>> On 2019-09-16 9:36 a.m., peteolcott wrote:
>>>>>>>>>> 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.
>>>>>>>>>
>>>>>>>>> There's nothing in the definition of the real numbers nor in the definition of interval that is stated in terms of 'geometric points', nor do either of these things make any reference to 'geometric points'.
>>>>>>>>>
>>>>>>>>> The inconsistency which arises here lies in your particular conception of the number line and/or of 'geometric points'.
>>>>>>>>>
>>>>>>>>> The problem here is that you fail to recognize the fact that the real numbers are continuous, whereas your definitions of infinitesimals and/or geometric points are discrete.
>>>>>>>>>
>>>>>>>>> André
>>>>>>>>>
>>>>>>>>
>>>>>>>> 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.
>>>>>>>>
>>>>>>>> Since [0, 100) clearly specifies every point on the number line between
>>>>>>>> 0 and 100 except 100 it is specifying the point immediately before 100
>>>>>>>> as the last point of this interval with no points in-between this point
>>>>>>>> and 100. THAT IS WHAT IT SAYS.
>>>>>>>
>>>>>>> No. That is how you are interpreting it. Nowhere does the above mention anything about this interval having a 'last point'.
>>>>>>
>>>>>> THAT IS WHAT THIS STIPULATES:
>>>>>> >> 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.
>>>>>>
>>>>>> If it has ALL of the points between A and B besides A and B, then it must have the point
>>>>>> immediately after A and the point immediately before B.
>>>>>
>>>>> 'Immediately before' and 'immediately after' are not relations that exist for the real numbers.
>>>>
>>>> Hence the correspondence between real numbers and open intervals on
>>>> the number line is inconsistent.
>>>>
>>>> An open interval specifies its first point as immediately after
>>>> a specified point with no points in-between.
>>>
>>> Show me any definition of open interval which is worded as above (one including the word 'immediate' and the 'no points in-between' part).
>>>
>>> Did you actually read the page I referred you to which explains infima and suprema? In particular, the section entitled "Relation to maximum and minimum elements"?
>>>
>>> André
>>>
>>
>> <true by definition of open interval>
>> If you chop off the first point of this: [0, 1]
>> You have this: (0, 1] which is defined as having exactly one less point.
>
> This isn't true 'by definition', not by any definition of which I am aware. Please show a definition which claims that (A,B] has exactly one point less than [A,B].
>
>> We know that numbers on a number line are in ascending order therefore
>> there is a first point of the second interval and it is immediately
>> adjacent to 0.
>
> Being in 'ascending order' tells us nothing about there being a first point, let alone one immediately adjacent to zero. Again, you are ascribing properties of discrete systems such as the integers to non-discrete systems like the reals.
>
When you have the points 0 to 1 inclusive and remove the first one
then you necessarily must have a next one.
> Again, did you read the link I provided? It discusses the example of the negative reals, pointing out that while they are bounded by zero, there is no greatest negative real (i.e. no negative real immediately adjacent to zero).
>
>> </true by definition of open interval>
>> Since this is true on the basis of the definitions of the concepts
>> when it contradicts other definitions this contradiction does not
>> form a rebuttal, instead it proves that definitions are inconsistent
>> with each other.
>
> Except you keep adding stuff which you claim is part of the definitions when it is not.
>
> André
>
--
Copyright 2019 Pete Olcott All rights reserved
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein
[toc] | [prev] | [next] | [standalone]
| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2019-09-16 14:15 -0600 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <qloqhf$1kn$1@dont-email.me> |
| In reply to | #19214 |
On 2019-09-16 1:51 p.m., peteolcott wrote:
> On 9/16/2019 2:35 PM, André G. Isaak wrote:
>> On 2019-09-16 1:19 p.m., peteolcott wrote:
>>> On 9/16/2019 1:34 PM, André G. Isaak wrote:
>>>> On 2019-09-16 12:27 p.m., peteolcott wrote:
>>>>> On 9/16/2019 12:10 PM, André G. Isaak wrote:
>>>>>> On 2019-09-16 10:59 a.m., peteolcott wrote:
>>>>>>> On 9/16/2019 11:38 AM, André G. Isaak wrote:
>>>>>>>> On 2019-09-16 10:22 a.m., peteolcott wrote:
>>>>>>>>> On 9/16/2019 11:06 AM, André G. Isaak wrote:
>>>>>>>>>> On 2019-09-16 9:36 a.m., peteolcott wrote:
>>>>>>>>>>> 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.
>>>>>>>>>>
>>>>>>>>>> There's nothing in the definition of the real numbers nor in
>>>>>>>>>> the definition of interval that is stated in terms of
>>>>>>>>>> 'geometric points', nor do either of these things make any
>>>>>>>>>> reference to 'geometric points'.
>>>>>>>>>>
>>>>>>>>>> The inconsistency which arises here lies in your particular
>>>>>>>>>> conception of the number line and/or of 'geometric points'.
>>>>>>>>>>
>>>>>>>>>> The problem here is that you fail to recognize the fact that
>>>>>>>>>> the real numbers are continuous, whereas your definitions of
>>>>>>>>>> infinitesimals and/or geometric points are discrete.
>>>>>>>>>>
>>>>>>>>>> André
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> 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.
>>>>>>>>>
>>>>>>>>> Since [0, 100) clearly specifies every point on the number line
>>>>>>>>> between
>>>>>>>>> 0 and 100 except 100 it is specifying the point immediately
>>>>>>>>> before 100
>>>>>>>>> as the last point of this interval with no points in-between
>>>>>>>>> this point
>>>>>>>>> and 100. THAT IS WHAT IT SAYS.
>>>>>>>>
>>>>>>>> No. That is how you are interpreting it. Nowhere does the above
>>>>>>>> mention anything about this interval having a 'last point'.
>>>>>>>
>>>>>>> THAT IS WHAT THIS STIPULATES:
>>>>>>> >> 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.
>>>>>>>
>>>>>>> If it has ALL of the points between A and B besides A and B, then
>>>>>>> it must have the point
>>>>>>> immediately after A and the point immediately before B.
>>>>>>
>>>>>> 'Immediately before' and 'immediately after' are not relations
>>>>>> that exist for the real numbers.
>>>>>
>>>>> Hence the correspondence between real numbers and open intervals on
>>>>> the number line is inconsistent.
>>>>>
>>>>> An open interval specifies its first point as immediately after
>>>>> a specified point with no points in-between.
>>>>
>>>> Show me any definition of open interval which is worded as above
>>>> (one including the word 'immediate' and the 'no points in-between'
>>>> part).
>>>>
>>>> Did you actually read the page I referred you to which explains
>>>> infima and suprema? In particular, the section entitled "Relation to
>>>> maximum and minimum elements"?
>>>>
>>>> André
>>>>
>>>
>>> <true by definition of open interval>
>>> If you chop off the first point of this: [0, 1]
>>> You have this: (0, 1] which is defined as having exactly one less
>>> point.
>>
>> This isn't true 'by definition', not by any definition of which I am
>> aware. Please show a definition which claims that (A,B] has exactly
>> one point less than [A,B].
>>
>>> We know that numbers on a number line are in ascending order
>>> therefore
>>> there is a first point of the second interval and it is immediately
>>> adjacent to 0.
>>
>> Being in 'ascending order' tells us nothing about there being a first
>> point, let alone one immediately adjacent to zero. Again, you are
>> ascribing properties of discrete systems such as the integers to
>> non-discrete systems like the reals.
>>
>
>
> When you have the points 0 to 1 inclusive and remove the first one
> then you necessarily must have a next one.
Yes, you've said this many times. What you *haven't* done is actually
demonstrated that this is true. To do that you need to demonstrate that
this somehow follows from the axioms which define the real numbers. Pick
any axiomatization and show how this follows from those axioms. You
won't be able to, because it doesn't.
The reals are densely ordered. Between any two real numbers, there is
always an infinite number of real numbers. That means that for any real
number X, either X = 0, or there exist an infinite number of reals which
lie between 0 and X (or X and zero if X is negative).
You still haven't answered what was a rather simple yes/no question I
asked: did you read the link I provided.
If you did, did you understand the section which discussed the
non-existence of the largest negative real?
André
>
>> Again, did you read the link I provided? It discusses the example of
>> the negative reals, pointing out that while they are bounded by zero,
>> there is no greatest negative real (i.e. no negative real immediately
>> adjacent to zero).
>>
>>> </true by definition of open interval>
>>> Since this is true on the basis of the definitions of the concepts
>>> when it contradicts other definitions this contradiction does not
>>> form a rebuttal, instead it proves that definitions are inconsistent
>>> with each other.
>>
>> Except you keep adding stuff which you claim is part of the
>> definitions when it is not.
>>
>> André
>>
>
>
--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.
[toc] | [prev] | [next] | [standalone]
| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-16 16:18 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <7cOdnTrXI5WJYeLAnZ2dnUU7-c3NnZ2d@giganews.com> |
| In reply to | #19215 |
On 9/16/2019 3:15 PM, André G. Isaak wrote:
> On 2019-09-16 1:51 p.m., peteolcott wrote:
>> On 9/16/2019 2:35 PM, André G. Isaak wrote:
>>> On 2019-09-16 1:19 p.m., peteolcott wrote:
>>>> On 9/16/2019 1:34 PM, André G. Isaak wrote:
>>>>> On 2019-09-16 12:27 p.m., peteolcott wrote:
>>>>>> On 9/16/2019 12:10 PM, André G. Isaak wrote:
>>>>>>> On 2019-09-16 10:59 a.m., peteolcott wrote:
>>>>>>>> On 9/16/2019 11:38 AM, André G. Isaak wrote:
>>>>>>>>> On 2019-09-16 10:22 a.m., peteolcott wrote:
>>>>>>>>>> On 9/16/2019 11:06 AM, André G. Isaak wrote:
>>>>>>>>>>> On 2019-09-16 9:36 a.m., peteolcott wrote:
>>>>>>>>>>>> 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.
>>>>>>>>>>>
>>>>>>>>>>> There's nothing in the definition of the real numbers nor in the definition of interval that is stated in terms of 'geometric points', nor do either of these things make any reference to 'geometric points'.
>>>>>>>>>>>
>>>>>>>>>>> The inconsistency which arises here lies in your particular conception of the number line and/or of 'geometric points'.
>>>>>>>>>>>
>>>>>>>>>>> The problem here is that you fail to recognize the fact that the real numbers are continuous, whereas your definitions of infinitesimals and/or geometric points are discrete.
>>>>>>>>>>>
>>>>>>>>>>> André
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> 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.
>>>>>>>>>>
>>>>>>>>>> Since [0, 100) clearly specifies every point on the number line between
>>>>>>>>>> 0 and 100 except 100 it is specifying the point immediately before 100
>>>>>>>>>> as the last point of this interval with no points in-between this point
>>>>>>>>>> and 100. THAT IS WHAT IT SAYS.
>>>>>>>>>
>>>>>>>>> No. That is how you are interpreting it. Nowhere does the above mention anything about this interval having a 'last point'.
>>>>>>>>
>>>>>>>> THAT IS WHAT THIS STIPULATES:
>>>>>>>> >> 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.
>>>>>>>>
>>>>>>>> If it has ALL of the points between A and B besides A and B, then it must have the point
>>>>>>>> immediately after A and the point immediately before B.
>>>>>>>
>>>>>>> 'Immediately before' and 'immediately after' are not relations that exist for the real numbers.
>>>>>>
>>>>>> Hence the correspondence between real numbers and open intervals on
>>>>>> the number line is inconsistent.
>>>>>>
>>>>>> An open interval specifies its first point as immediately after
>>>>>> a specified point with no points in-between.
>>>>>
>>>>> Show me any definition of open interval which is worded as above (one including the word 'immediate' and the 'no points in-between' part).
>>>>>
>>>>> Did you actually read the page I referred you to which explains infima and suprema? In particular, the section entitled "Relation to maximum and minimum elements"?
>>>>>
>>>>> André
>>>>>
>>>>
>>>> <true by definition of open interval>
>>>> If you chop off the first point of this: [0, 1]
>>>> You have this: (0, 1] which is defined as having exactly one less point.
>>>
>>> This isn't true 'by definition', not by any definition of which I am aware. Please show a definition which claims that (A,B] has exactly one point less than [A,B].
>>>
>>>> We know that numbers on a number line are in ascending order therefore
>>>> there is a first point of the second interval and it is immediately
>>>> adjacent to 0.
>>>
>>> Being in 'ascending order' tells us nothing about there being a first point, let alone one immediately adjacent to zero. Again, you are ascribing properties of discrete systems such as the integers to non-discrete systems like the reals.
>>>
>>
>>
>> When you have the points 0 to 1 inclusive and remove the first one
>> then you necessarily must have a next one.
>
> Yes, you've said this many times. What you *haven't* done is actually demonstrated that this is true. To do that you need to demonstrate that this somehow follows from the axioms which define the real numbers.
Even though it contradicts the axioms of real numbers it follows
from the definition of open interval.
If you have every point in an interval besides the first one
then you necessarily must have the next one as the true first one.
If we can imagine the square root of negative -1 which is impossible,
we can certainly imagine infinitesimal differences in length.
--
Copyright 2019 Pete Olcott All rights reserved
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein
[toc] | [prev] | [next] | [standalone]
| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2019-09-16 16:00 -0600 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <qlp0lc$7t2$1@dont-email.me> |
| In reply to | #19217 |
On 2019-09-16 3:18 p.m., peteolcott wrote:
> On 9/16/2019 3:15 PM, André G. Isaak wrote:
>> On 2019-09-16 1:51 p.m., peteolcott wrote:
>>> On 9/16/2019 2:35 PM, André G. Isaak wrote:
>>>> On 2019-09-16 1:19 p.m., peteolcott wrote:
>>>>> On 9/16/2019 1:34 PM, André G. Isaak wrote:
>>>>>> On 2019-09-16 12:27 p.m., peteolcott wrote:
>>>>>>> On 9/16/2019 12:10 PM, André G. Isaak wrote:
>>>>>>>> On 2019-09-16 10:59 a.m., peteolcott wrote:
>>>>>>>>> On 9/16/2019 11:38 AM, André G. Isaak wrote:
>>>>>>>>>> On 2019-09-16 10:22 a.m., peteolcott wrote:
>>>>>>>>>>> On 9/16/2019 11:06 AM, André G. Isaak wrote:
>>>>>>>>>>>> On 2019-09-16 9:36 a.m., peteolcott wrote:
>>>>>>>>>>>>> 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.
>>>>>>>>>>>>
>>>>>>>>>>>> There's nothing in the definition of the real numbers nor in
>>>>>>>>>>>> the definition of interval that is stated in terms of
>>>>>>>>>>>> 'geometric points', nor do either of these things make any
>>>>>>>>>>>> reference to 'geometric points'.
>>>>>>>>>>>>
>>>>>>>>>>>> The inconsistency which arises here lies in your particular
>>>>>>>>>>>> conception of the number line and/or of 'geometric points'.
>>>>>>>>>>>>
>>>>>>>>>>>> The problem here is that you fail to recognize the fact that
>>>>>>>>>>>> the real numbers are continuous, whereas your definitions of
>>>>>>>>>>>> infinitesimals and/or geometric points are discrete.
>>>>>>>>>>>>
>>>>>>>>>>>> André
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> 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.
>>>>>>>>>>>
>>>>>>>>>>> Since [0, 100) clearly specifies every point on the number
>>>>>>>>>>> line between
>>>>>>>>>>> 0 and 100 except 100 it is specifying the point immediately
>>>>>>>>>>> before 100
>>>>>>>>>>> as the last point of this interval with no points in-between
>>>>>>>>>>> this point
>>>>>>>>>>> and 100. THAT IS WHAT IT SAYS.
>>>>>>>>>>
>>>>>>>>>> No. That is how you are interpreting it. Nowhere does the
>>>>>>>>>> above mention anything about this interval having a 'last point'.
>>>>>>>>>
>>>>>>>>> THAT IS WHAT THIS STIPULATES:
>>>>>>>>> >> 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.
>>>>>>>>>
>>>>>>>>> If it has ALL of the points between A and B besides A and B,
>>>>>>>>> then it must have the point
>>>>>>>>> immediately after A and the point immediately before B.
>>>>>>>>
>>>>>>>> 'Immediately before' and 'immediately after' are not relations
>>>>>>>> that exist for the real numbers.
>>>>>>>
>>>>>>> Hence the correspondence between real numbers and open intervals on
>>>>>>> the number line is inconsistent.
>>>>>>>
>>>>>>> An open interval specifies its first point as immediately after
>>>>>>> a specified point with no points in-between.
>>>>>>
>>>>>> Show me any definition of open interval which is worded as above
>>>>>> (one including the word 'immediate' and the 'no points in-between'
>>>>>> part).
>>>>>>
>>>>>> Did you actually read the page I referred you to which explains
>>>>>> infima and suprema? In particular, the section entitled "Relation
>>>>>> to maximum and minimum elements"?
>>>>>>
>>>>>> André
>>>>>>
>>>>>
>>>>> <true by definition of open interval>
>>>>> If you chop off the first point of this: [0, 1]
>>>>> You have this: (0, 1] which is defined as having exactly one
>>>>> less point.
>>>>
>>>> This isn't true 'by definition', not by any definition of which I am
>>>> aware. Please show a definition which claims that (A,B] has exactly
>>>> one point less than [A,B].
>>>>
>>>>> We know that numbers on a number line are in ascending order
>>>>> therefore
>>>>> there is a first point of the second interval and it is immediately
>>>>> adjacent to 0.
>>>>
>>>> Being in 'ascending order' tells us nothing about there being a
>>>> first point, let alone one immediately adjacent to zero. Again, you
>>>> are ascribing properties of discrete systems such as the integers to
>>>> non-discrete systems like the reals.
>>>>
>>>
>>>
>>> When you have the points 0 to 1 inclusive and remove the first one
>>> then you necessarily must have a next one.
>>
>> Yes, you've said this many times. What you *haven't* done is actually
>> demonstrated that this is true. To do that you need to demonstrate
>> that this somehow follows from the axioms which define the real numbers.
>
> Even though it contradicts the axioms of real numbers it follows
> from the definition of open interval.
>
> If you have every point in an interval besides the first one
> then you necessarily must have the next one as the true first one.
Intervals are sets. In case you're not familiar with basic set theory,
{A, B}, {B, A}, and {A, A, B} are all the exact same set. There is no
order defined over the elements of a set. If you want to talk about the
first element or second element of a set, you'd actually need to define
some sort of ordering relation. For countable sets, this is trivially
easy to do. For uncountable sets, though, it is not. So unless you can
propose some way of actually determining the order of the elements in a
nondenumerably infinite set, this argument doesn't hold up.
For example, in the interval (0, 1) can you explain how you would
determine the position of (e.g.) 0.25? Assume it is the nth element in
this set. solve for n.
> If we can imagine the square root of negative -1 which is impossible,
What makes you say it is impossible? Sure, it isn't defined for the
reals, but it is defined for the complex numbers.
> we can certainly imagine infinitesimal differences in length.
Imagining things, though, isn't sufficient. You need to actually
rigorously define them in a way which is consistent with the number
system in which you are working.
André
--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.
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| From | Keith Thompson <kst-u@mib.org> |
|---|---|
| Date | 2019-09-16 15:29 -0700 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <lna7b3an9s.fsf@kst-u.example.com> |
| In reply to | #19220 |
André G. Isaak <agisaak@gm.invalid> writes:
[...]
> Intervals are sets. In case you're not familiar with basic set theory,
> {A, B}, {B, A}, and {A, A, B} are all the exact same set. There is no
> order defined over the elements of a set. If you want to talk about the
> first element or second element of a set, you'd actually need to define
> some sort of ordering relation. For countable sets, this is trivially
> easy to do. For uncountable sets, though, it is not. So unless you can
> propose some way of actually determining the order of the elements in a
> nondenumerably infinite set, this argument doesn't hold up.
>
> For example, in the interval (0, 1) can you explain how you would
> determine the position of (e.g.) 0.25? Assume it is the nth element in
> this set. solve for n.
So by "ordering", you mean, for example, a mapping between the
members of the set and the positive integers? Of course no such
mapping is possible for the real interval (0, 1). We can define
an ordering that lets us determine n, but it will miss some members
of the interval (in fact <handwave> almost all of them</handwave>).
I would have thought of an "ordering" as a relationship such that
for any two members X and Y of the set, exactly one of X<Y, X=Y,
or X>Y is true -- which is certainly the case for sets of reals.
But that kind of ordering doesn't help us determine the "first"
member of the set.
I agree with your point. I just wanted to be clear about the
terminology.
--
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 | 2019-09-16 18:14 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <y-GdnUGSA6Skih3AnZ2dnUU7-VednZ2d@giganews.com> |
| In reply to | #19220 |
On 9/16/2019 5:44 PM, Andy Walker wrote: > On 16/09/2019 23:00, André G. Isaak wrote: >> [...] If you want to >> talk about the first element or second element of a set, you'd >> actually need to define some sort of ordering relation. For countable >> sets, this is trivially easy to do. For uncountable sets, though, it >> is not. So unless you can propose some way of actually determining >> the order of the elements in a nondenumerably infinite set, this >> argument doesn't hold up. > > May be worth noting [out of interest rather than because there > is any hope of explaining things to our resident troll] that although > any set of *rationals* has a *first* member [in any of the many ways > by which the rationals are countable] it does not necessarily have a > *least* member. So the non-denumerability of the reals is not a > necessary part of discussions about his misunderstandings. > > Specifically, there is no least strictly positive rational. > Proof left as an exercise. > When we remove the beginning 0.5 from [0.0, 2.0] we derive [0.5, 2.0] with a first point of 0.5. So when we remove a single point from the beginning of [0.0, 1.0] we end up with (0.0, 1.0] with beginning point that cannot be named. WE DON'T END UP WITHOUT A BEGINNING POINT. -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | Jeff Barnett <jbb@notatt.com> |
|---|---|
| Date | 2019-09-16 23:34 -0600 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <qlpr96$lr$1@dont-email.me> |
| In reply to | #19217 |
peteolcott wrote on 9/16/2019 3:18 PM:
> On 9/16/2019 3:15 PM, André G. Isaak wrote:
>> On 2019-09-16 1:51 p.m., peteolcott wrote:
>>> On 9/16/2019 2:35 PM, André G. Isaak wrote:
>>>> On 2019-09-16 1:19 p.m., peteolcott wrote:
>>>>> On 9/16/2019 1:34 PM, André G. Isaak wrote:
>>>>>> On 2019-09-16 12:27 p.m., peteolcott wrote:
>>>>>>> On 9/16/2019 12:10 PM, André G. Isaak wrote:
>>>>>>>> On 2019-09-16 10:59 a.m., peteolcott wrote:
>>>>>>>>> On 9/16/2019 11:38 AM, André G. Isaak wrote:
>>>>>>>>>> On 2019-09-16 10:22 a.m., peteolcott wrote:
>>>>>>>>>>> On 9/16/2019 11:06 AM, André G. Isaak wrote:
>>>>>>>>>>>> On 2019-09-16 9:36 a.m., peteolcott wrote:
>>>>>>>>>>>>> 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.
>>>>>>>>>>>>
>>>>>>>>>>>> There's nothing in the definition of the real numbers nor in
>>>>>>>>>>>> the definition of interval that is stated in terms of
>>>>>>>>>>>> 'geometric points', nor do either of these things make any
>>>>>>>>>>>> reference to 'geometric points'.
>>>>>>>>>>>>
>>>>>>>>>>>> The inconsistency which arises here lies in your particular
>>>>>>>>>>>> conception of the number line and/or of 'geometric points'.
>>>>>>>>>>>>
>>>>>>>>>>>> The problem here is that you fail to recognize the fact that
>>>>>>>>>>>> the real numbers are continuous, whereas your definitions of
>>>>>>>>>>>> infinitesimals and/or geometric points are discrete.
>>>>>>>>>>>>
>>>>>>>>>>>> André
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> 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.
>>>>>>>>>>>
>>>>>>>>>>> Since [0, 100) clearly specifies every point on the number
>>>>>>>>>>> line between
>>>>>>>>>>> 0 and 100 except 100 it is specifying the point immediately
>>>>>>>>>>> before 100
>>>>>>>>>>> as the last point of this interval with no points in-between
>>>>>>>>>>> this point
>>>>>>>>>>> and 100. THAT IS WHAT IT SAYS.
>>>>>>>>>>
>>>>>>>>>> No. That is how you are interpreting it. Nowhere does the
>>>>>>>>>> above mention anything about this interval having a 'last point'.
>>>>>>>>>
>>>>>>>>> THAT IS WHAT THIS STIPULATES:
>>>>>>>>> >> 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.
>>>>>>>>>
>>>>>>>>> If it has ALL of the points between A and B besides A and B,
>>>>>>>>> then it must have the point
>>>>>>>>> immediately after A and the point immediately before B.
>>>>>>>>
>>>>>>>> 'Immediately before' and 'immediately after' are not relations
>>>>>>>> that exist for the real numbers.
>>>>>>>
>>>>>>> Hence the correspondence between real numbers and open intervals on
>>>>>>> the number line is inconsistent.
>>>>>>>
>>>>>>> An open interval specifies its first point as immediately after
>>>>>>> a specified point with no points in-between.
>>>>>>
>>>>>> Show me any definition of open interval which is worded as above
>>>>>> (one including the word 'immediate' and the 'no points in-between'
>>>>>> part).
>>>>>>
>>>>>> Did you actually read the page I referred you to which explains
>>>>>> infima and suprema? In particular, the section entitled "Relation
>>>>>> to maximum and minimum elements"?
>>>>>>
>>>>>> André
>>>>>>
>>>>>
>>>>> <true by definition of open interval>
>>>>> If you chop off the first point of this: [0, 1]
>>>>> You have this: (0, 1] which is defined as having exactly one
>>>>> less point.
>>>>
>>>> This isn't true 'by definition', not by any definition of which I am
>>>> aware. Please show a definition which claims that (A,B] has exactly
>>>> one point less than [A,B].
>>>>
>>>>> We know that numbers on a number line are in ascending order
>>>>> therefore
>>>>> there is a first point of the second interval and it is immediately
>>>>> adjacent to 0.
>>>>
>>>> Being in 'ascending order' tells us nothing about there being a
>>>> first point, let alone one immediately adjacent to zero. Again, you
>>>> are ascribing properties of discrete systems such as the integers to
>>>> non-discrete systems like the reals.
>>>>
>>>
>>>
>>> When you have the points 0 to 1 inclusive and remove the first one
>>> then you necessarily must have a next one.
>>
>> Yes, you've said this many times. What you *haven't* done is actually
>> demonstrated that this is true. To do that you need to demonstrate
>> that this somehow follows from the axioms which define the real numbers.
>
> Even though it contradicts the axioms of real numbers it follows
> from the definition of open interval.
Moron, did you read the fucking paper? For the nth time, get your hands
out of your pants, concentrate, and answer th gentleman's simple
question? Why do you always, and I mean always, opt to remain an
ignoramus? You are failing questions that some kids in grade school can
handle.
--
Jeff Barnett
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| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2019-09-16 14:25 -0600 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <qlor3m$59j$1@dont-email.me> |
| In reply to | #19214 |
On 2019-09-16 1:51 p.m., peteolcott wrote:
> On 9/16/2019 2:35 PM, André G. Isaak wrote:
>> On 2019-09-16 1:19 p.m., peteolcott wrote:
>>> On 9/16/2019 1:34 PM, André G. Isaak wrote:
>>>> On 2019-09-16 12:27 p.m., peteolcott wrote:
>>>>> On 9/16/2019 12:10 PM, André G. Isaak wrote:
>>>>>> On 2019-09-16 10:59 a.m., peteolcott wrote:
>>>>>>> On 9/16/2019 11:38 AM, André G. Isaak wrote:
>>>>>>>> On 2019-09-16 10:22 a.m., peteolcott wrote:
>>>>>>>>> On 9/16/2019 11:06 AM, André G. Isaak wrote:
>>>>>>>>>> On 2019-09-16 9:36 a.m., peteolcott wrote:
>>>>>>>>>>> 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.
>>>>>>>>>>
>>>>>>>>>> There's nothing in the definition of the real numbers nor in
>>>>>>>>>> the definition of interval that is stated in terms of
>>>>>>>>>> 'geometric points', nor do either of these things make any
>>>>>>>>>> reference to 'geometric points'.
>>>>>>>>>>
>>>>>>>>>> The inconsistency which arises here lies in your particular
>>>>>>>>>> conception of the number line and/or of 'geometric points'.
>>>>>>>>>>
>>>>>>>>>> The problem here is that you fail to recognize the fact that
>>>>>>>>>> the real numbers are continuous, whereas your definitions of
>>>>>>>>>> infinitesimals and/or geometric points are discrete.
>>>>>>>>>>
>>>>>>>>>> André
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> 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.
>>>>>>>>>
>>>>>>>>> Since [0, 100) clearly specifies every point on the number line
>>>>>>>>> between
>>>>>>>>> 0 and 100 except 100 it is specifying the point immediately
>>>>>>>>> before 100
>>>>>>>>> as the last point of this interval with no points in-between
>>>>>>>>> this point
>>>>>>>>> and 100. THAT IS WHAT IT SAYS.
>>>>>>>>
>>>>>>>> No. That is how you are interpreting it. Nowhere does the above
>>>>>>>> mention anything about this interval having a 'last point'.
>>>>>>>
>>>>>>> THAT IS WHAT THIS STIPULATES:
>>>>>>> >> 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.
>>>>>>>
>>>>>>> If it has ALL of the points between A and B besides A and B, then
>>>>>>> it must have the point
>>>>>>> immediately after A and the point immediately before B.
>>>>>>
>>>>>> 'Immediately before' and 'immediately after' are not relations
>>>>>> that exist for the real numbers.
>>>>>
>>>>> Hence the correspondence between real numbers and open intervals on
>>>>> the number line is inconsistent.
>>>>>
>>>>> An open interval specifies its first point as immediately after
>>>>> a specified point with no points in-between.
>>>>
>>>> Show me any definition of open interval which is worded as above
>>>> (one including the word 'immediate' and the 'no points in-between'
>>>> part).
>>>>
>>>> Did you actually read the page I referred you to which explains
>>>> infima and suprema? In particular, the section entitled "Relation to
>>>> maximum and minimum elements"?
>>>>
>>>> André
>>>>
>>>
>>> <true by definition of open interval>
>>> If you chop off the first point of this: [0, 1]
>>> You have this: (0, 1] which is defined as having exactly one less
>>> point.
>>
>> This isn't true 'by definition', not by any definition of which I am
>> aware. Please show a definition which claims that (A,B] has exactly
>> one point less than [A,B].
>>
>>> We know that numbers on a number line are in ascending order
>>> therefore
>>> there is a first point of the second interval and it is immediately
>>> adjacent to 0.
>>
>> Being in 'ascending order' tells us nothing about there being a first
>> point, let alone one immediately adjacent to zero. Again, you are
>> ascribing properties of discrete systems such as the integers to
>> non-discrete systems like the reals.
>>
>
>
> When you have the points 0 to 1 inclusive and remove the first one
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.
André
--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-16 16:21 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <7cOdnTXXI5UjYeLAnZ2dnUU7-c2dnZ2d@giganews.com> |
| In reply to | #19216 |
On 9/16/2019 3:25 PM, André G. Isaak wrote:
> On 2019-09-16 1:51 p.m., peteolcott wrote:
>> On 9/16/2019 2:35 PM, André G. Isaak wrote:
>>> On 2019-09-16 1:19 p.m., peteolcott wrote:
>>>> On 9/16/2019 1:34 PM, André G. Isaak wrote:
>>>>> On 2019-09-16 12:27 p.m., peteolcott wrote:
>>>>>> On 9/16/2019 12:10 PM, André G. Isaak wrote:
>>>>>>> On 2019-09-16 10:59 a.m., peteolcott wrote:
>>>>>>>> On 9/16/2019 11:38 AM, André G. Isaak wrote:
>>>>>>>>> On 2019-09-16 10:22 a.m., peteolcott wrote:
>>>>>>>>>> On 9/16/2019 11:06 AM, André G. Isaak wrote:
>>>>>>>>>>> On 2019-09-16 9:36 a.m., peteolcott wrote:
>>>>>>>>>>>> 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.
>>>>>>>>>>>
>>>>>>>>>>> There's nothing in the definition of the real numbers nor in the definition of interval that is stated in terms of 'geometric points', nor do either of these things make any reference to 'geometric points'.
>>>>>>>>>>>
>>>>>>>>>>> The inconsistency which arises here lies in your particular conception of the number line and/or of 'geometric points'.
>>>>>>>>>>>
>>>>>>>>>>> The problem here is that you fail to recognize the fact that the real numbers are continuous, whereas your definitions of infinitesimals and/or geometric points are discrete.
>>>>>>>>>>>
>>>>>>>>>>> André
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> 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.
>>>>>>>>>>
>>>>>>>>>> Since [0, 100) clearly specifies every point on the number line between
>>>>>>>>>> 0 and 100 except 100 it is specifying the point immediately before 100
>>>>>>>>>> as the last point of this interval with no points in-between this point
>>>>>>>>>> and 100. THAT IS WHAT IT SAYS.
>>>>>>>>>
>>>>>>>>> No. That is how you are interpreting it. Nowhere does the above mention anything about this interval having a 'last point'.
>>>>>>>>
>>>>>>>> THAT IS WHAT THIS STIPULATES:
>>>>>>>> >> 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.
>>>>>>>>
>>>>>>>> If it has ALL of the points between A and B besides A and B, then it must have the point
>>>>>>>> immediately after A and the point immediately before B.
>>>>>>>
>>>>>>> 'Immediately before' and 'immediately after' are not relations that exist for the real numbers.
>>>>>>
>>>>>> Hence the correspondence between real numbers and open intervals on
>>>>>> the number line is inconsistent.
>>>>>>
>>>>>> An open interval specifies its first point as immediately after
>>>>>> a specified point with no points in-between.
>>>>>
>>>>> Show me any definition of open interval which is worded as above (one including the word 'immediate' and the 'no points in-between' part).
>>>>>
>>>>> Did you actually read the page I referred you to which explains infima and suprema? In particular, the section entitled "Relation to maximum and minimum elements"?
>>>>>
>>>>> André
>>>>>
>>>>
>>>> <true by definition of open interval>
>>>> If you chop off the first point of this: [0, 1]
>>>> You have this: (0, 1] which is defined as having exactly one less point.
>>>
>>> This isn't true 'by definition', not by any definition of which I am aware. Please show a definition which claims that (A,B] has exactly one point less than [A,B].
>>>
>>>> We know that numbers on a number line are in ascending order therefore
>>>> there is a first point of the second interval and it is immediately
>>>> adjacent to 0.
>>>
>>> Being in 'ascending order' tells us nothing about there being a first point, let alone one immediately adjacent to zero. Again, you are ascribing properties of discrete systems such as the integers to non-discrete systems like the reals.
>>>
>>
>>
>> When you have the points 0 to 1 inclusive and remove the first one
>
> 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.
>
> André
>
Length([A, B]) - Length((A, B]) = one geometric point which
is rounded down to zero in real numbers.
--
Copyright 2019 Pete Olcott All rights reserved
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein
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| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2019-09-16 16:01 -0600 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <qlp0nt$7t2$2@dont-email.me> |
| In reply to | #19218 |
On 2019-09-16 3:21 p.m., peteolcott wrote:
> On 9/16/2019 3:25 PM, André G. Isaak wrote:
>> On 2019-09-16 1:51 p.m., peteolcott wrote:
>>> On 9/16/2019 2:35 PM, André G. Isaak wrote:
>>>> On 2019-09-16 1:19 p.m., peteolcott wrote:
>>>>> On 9/16/2019 1:34 PM, André G. Isaak wrote:
>>>>>> On 2019-09-16 12:27 p.m., peteolcott wrote:
>>>>>>> On 9/16/2019 12:10 PM, André G. Isaak wrote:
>>>>>>>> On 2019-09-16 10:59 a.m., peteolcott wrote:
>>>>>>>>> On 9/16/2019 11:38 AM, André G. Isaak wrote:
>>>>>>>>>> On 2019-09-16 10:22 a.m., peteolcott wrote:
>>>>>>>>>>> On 9/16/2019 11:06 AM, André G. Isaak wrote:
>>>>>>>>>>>> On 2019-09-16 9:36 a.m., peteolcott wrote:
>>>>>>>>>>>>> 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.
>>>>>>>>>>>>
>>>>>>>>>>>> There's nothing in the definition of the real numbers nor in
>>>>>>>>>>>> the definition of interval that is stated in terms of
>>>>>>>>>>>> 'geometric points', nor do either of these things make any
>>>>>>>>>>>> reference to 'geometric points'.
>>>>>>>>>>>>
>>>>>>>>>>>> The inconsistency which arises here lies in your particular
>>>>>>>>>>>> conception of the number line and/or of 'geometric points'.
>>>>>>>>>>>>
>>>>>>>>>>>> The problem here is that you fail to recognize the fact that
>>>>>>>>>>>> the real numbers are continuous, whereas your definitions of
>>>>>>>>>>>> infinitesimals and/or geometric points are discrete.
>>>>>>>>>>>>
>>>>>>>>>>>> André
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> 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.
>>>>>>>>>>>
>>>>>>>>>>> Since [0, 100) clearly specifies every point on the number
>>>>>>>>>>> line between
>>>>>>>>>>> 0 and 100 except 100 it is specifying the point immediately
>>>>>>>>>>> before 100
>>>>>>>>>>> as the last point of this interval with no points in-between
>>>>>>>>>>> this point
>>>>>>>>>>> and 100. THAT IS WHAT IT SAYS.
>>>>>>>>>>
>>>>>>>>>> No. That is how you are interpreting it. Nowhere does the
>>>>>>>>>> above mention anything about this interval having a 'last point'.
>>>>>>>>>
>>>>>>>>> THAT IS WHAT THIS STIPULATES:
>>>>>>>>> >> 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.
>>>>>>>>>
>>>>>>>>> If it has ALL of the points between A and B besides A and B,
>>>>>>>>> then it must have the point
>>>>>>>>> immediately after A and the point immediately before B.
>>>>>>>>
>>>>>>>> 'Immediately before' and 'immediately after' are not relations
>>>>>>>> that exist for the real numbers.
>>>>>>>
>>>>>>> Hence the correspondence between real numbers and open intervals on
>>>>>>> the number line is inconsistent.
>>>>>>>
>>>>>>> An open interval specifies its first point as immediately after
>>>>>>> a specified point with no points in-between.
>>>>>>
>>>>>> Show me any definition of open interval which is worded as above
>>>>>> (one including the word 'immediate' and the 'no points in-between'
>>>>>> part).
>>>>>>
>>>>>> Did you actually read the page I referred you to which explains
>>>>>> infima and suprema? In particular, the section entitled "Relation
>>>>>> to maximum and minimum elements"?
>>>>>>
>>>>>> André
>>>>>>
>>>>>
>>>>> <true by definition of open interval>
>>>>> If you chop off the first point of this: [0, 1]
>>>>> You have this: (0, 1] which is defined as having exactly one
>>>>> less point.
>>>>
>>>> This isn't true 'by definition', not by any definition of which I am
>>>> aware. Please show a definition which claims that (A,B] has exactly
>>>> one point less than [A,B].
>>>>
>>>>> We know that numbers on a number line are in ascending order
>>>>> therefore
>>>>> there is a first point of the second interval and it is immediately
>>>>> adjacent to 0.
>>>>
>>>> Being in 'ascending order' tells us nothing about there being a
>>>> first point, let alone one immediately adjacent to zero. Again, you
>>>> are ascribing properties of discrete systems such as the integers to
>>>> non-discrete systems like the reals.
>>>>
>>>
>>>
>>> When you have the points 0 to 1 inclusive and remove the first one
>>
>> 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.
>>
>> André
>>
>
> Length([A, B]) - Length((A, B]) = one geometric point which
> is rounded down to zero in real numbers.
It isn't rounded down to zero. It is zero. What part of 'points are
dimensionless' is causing problems here?
André
--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.
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| From | Keith Thompson <kst-u@mib.org> |
|---|---|
| Date | 2019-09-16 15:46 -0700 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <ln5zlramg3.fsf@kst-u.example.com> |
| In reply to | #19216 |
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].
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.)
--
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 | 2019-09-16 18:07 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <ZrKdnUNV14IniB3AnZ2dnUU7-W3NnZ2d@giganews.com> |
| In reply to | #19223 |
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]. > > 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.) > There is a first element that cannot be expressed using the real number system. Most here have assumed that there is no first element entirely on the basis that it cannot be expressed in the real number system, none-the-less if the prior first point was removed there is NOT nothing left. When we remove the beginning 0.5 from [0.0, 2.0] we derive [0.5, 2.0] with a first point of 0.5. So when we remove a single point from the beginning of [0.0, 1.0] we end up with (0.0, 1.0] with beginning point that cannot be named. WE DON'T END UP WITHOUT A BEGINNING POINT. -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | Me <invalid@invalid.de> |
|---|---|
| Date | 2019-09-17 01:09 +0200 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <qlp4ni$1jem$1@news-cypress.fernuni-hagen.de> |
| In reply to | #19223 |
Am 17.09.2019 um 00:46 schrieb Keith Thompson:
> André G. Isaak <agisaak@gm.invalid> writes:
>>>> [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.
How so) After all, ther *is* a "relation" between [A,B] and (A,B] (for
all A,B e IR).
(A, B] = [A,B] \ min [A, B] .
So it seems to me that (contrary to André's claim) we may adopt an
operation to "get one from the other".
> [A,B] is a set of real numbers. (A,B] can be constructed by taking
> [A,B] and removing a single element, namely A
Exactly. Morevoer we do not even have to know A. It suffices that [A, B]
is a closed intervall (with more than a single element).
> 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].
Exactly, namely the mininimum of [A, B].
> 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.)
You are really generous.
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| From | Mike Terry <news.dead.person.stones@darjeeling.plus.com> |
|---|---|
| Date | 2019-09-17 02:33 +0100 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <0IidnQ4NqsdLqh3AnZ2dnUU78dfNnZ2d@brightview.co.uk> |
| In reply to | #19223 |
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]. 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.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-16 20:53 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <15WdnXUEJcTmoR3AnZ2dnUU7-bXNnZ2d@giganews.com> |
| In reply to | #19231 |
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]. 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.
>
So we are in total agreement up to here. I referred to it is a process only to make the meaning of each set more clear.
> 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.)
>
It seems clear that [1, 2] has both a first point: 1.0 and a least element.
Likewise for (1, 2], yet the real number associated with this first point might not exist:
1.00000...1 or 1.0 + 1/∞ is as close as I can figure.
>>
>> 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 | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2019-09-16 21:55 -0600 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <qlplg7$8im$1@dont-email.me> |
| In reply to | #19232 |
On 2019-09-16 7:53 p.m., peteolcott wrote:
> 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]. 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.
>>
>
> So we are in total agreement up to here. I referred to it is a process
> only to make the meaning of each set more clear.
>
>> 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.)
>>
>
> It seems clear that [1, 2] has both a first point: 1.0 and a least element.
> Likewise for (1, 2], yet the real number associated with this first
This second part is 'clear' only because you are conceptualizing the
reals as countable entities which they are not (that was the whole point
of Cantor's proof).
(1, 2] includes all reals which are less than or equal to 2 and whose
greatest lower bound is 1. That means that it includes all reals <= 2
which are arbitrarily close to, but not equal to, one. The important
word here is 'arbitrarily'. It doesn't matter how close to 1 you get --
unless you are actually equal to 1, you can always get closer to 1.
There is no 'smallest real greater than 1' anymore than there is any
'largest number'.
> point might not exist:
> 1.00000...1 or 1.0 + 1/∞ is as close as I can figure.
Neither of those are legitimate numbers. You can't specify a repeating
decimal with a final 1 since there is no final decimal place. And you
can't divide by ∞ because ∞ isn't a number.
André
--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.
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| From | Keith Thompson <kst-u@mib.org> |
|---|---|
| Date | 2019-09-16 19:21 -0700 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <ln1rwfaciq.fsf@kst-u.example.com> |
| In reply to | #19231 |
Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:
> On 16/09/2019 23:46, Keith Thompson wrote:
[...]
>> 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?
I don't believe there's any point in going into it. If you really
want to know what I was referring to, ask again and I'll answer
(though I'd have to look back through his posts, not something I'm
eager to do).
--
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 | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2019-09-16 21:40 -0600 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <qlpkkg$52p$1@dont-email.me> |
| In reply to | #19231 |
On 2019-09-16 7:33 p.m., 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]. 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 :)
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.
André
--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.
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| From | Keith Thompson <kst-u@mib.org> |
|---|---|
| Date | 2019-09-16 21:38 -0700 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <lnwoe78rlb.fsf@kst-u.example.com> |
| In reply to | #19235 |
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.
--
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 00:24 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (some concurrence). |
| Message-ID | <ta2dnSAhV7qH8x3AnZ2dnUU7-S_NnZ2d@giganews.com> |
| In reply to | #19237 |
On 9/16/2019 11:38 PM, 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?
>
If you stay stuck on only thinking of points on a number line in
terms of their correspondence to real numbers then every analysis
will seem to indicate there is no first point.
On 9/16/2019 8:33 PM, Mike Terry wrote:
> On 16/09/2019 23:46, 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].
>
> OK, so (A,B] = [A,B] - {A}, and that can be counted as an "operation
> in which we get (A,B] from [A,B]. So you're right...
If we can envision a single point difference between two intervals then
we only need to carry this same reasoning through to the subsequent
single point difference between two intervals to establish the basis
for the infinitesimal number system.
When we resort to looking at these things only through real numbers
it seems like a defined boundary suddenly turns blurry only because
real numbers make this boundary blurry.
> 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.
>
Sure there you go that is the idea.
> 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.
>
My ONLY purpose in this is to show that all infinities have the same
cardinality. As soon as I make any significant breakthrough my
work will be taken much more seriously and reviews will become enormously
more effective thus progressing my work much more efficiently.
--
Copyright 2019 Pete Olcott All rights reserved
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein
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