Groups | Search | Server Info | Keyboard shortcuts | Login | Register [http] [https] [nntp] [nntps]
Groups > comp.theory > #19819 > unrolled thread
| Started by | peteolcott <Here@Home> |
|---|---|
| First post | 2019-09-11 18:32 -0500 |
| Last post | 2019-09-21 09:45 -0500 |
| Articles | 20 on this page of 163 — 11 participants |
Back to article view | Back to comp.theory
This discussion starts older than the indexed window; earlier articles aren't shown. The article labeled Started by
below is the oldest one visible, not the original post.
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-11 18:32 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) temakoutera62@gmail.com - 2019-09-11 17:28 -0700
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-11 20:03 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-11 20:59 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-11 22:16 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-12 11:14 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Deceptive Strawman by Ben peteolcott <Here@Home> - 2019-09-12 21:00 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Deceptive Strawman by Ben David Kleinecke <dkleinecke@gmail.com> - 2019-09-12 19:11 -0700
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Deceptive Strawman by Ben peteolcott <Here@Home> - 2019-09-12 21:39 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Deceptive Strawman by Ben peteolcott <Here@Home> - 2019-09-13 11:48 -0500
Re: Cantor was wrong about cardinality --- APOLOGY peteolcott <Here@Home> - 2019-09-14 13:09 -0500
Re: Cantor was wrong about cardinality --- APOLOGY peteolcott <Here@Home> - 2019-09-14 16:06 -0500
Re: Cantor was wrong about cardinality --- APOLOGY peteolcott <Here@Home> - 2019-09-14 19:24 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-14 21:33 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-16 10:17 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? André G. Isaak <agisaak@gm.invalid> - 2019-09-16 09:26 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 10:36 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 10:06 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 11:22 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 10:38 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 11:59 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 11:10 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 13:27 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 12:34 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 14:19 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 13:35 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 14:51 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 14:15 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 16:18 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 16:00 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Keith Thompson <kst-u@mib.org> - 2019-09-16 15:29 -0700
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Andy Walker <anw@cuboid.co.uk> - 2019-09-16 23:44 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 18:14 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Andy Walker <anw@cuboid.co.uk> - 2019-09-17 17:04 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-17 12:06 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-17 02:57 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Jeff Barnett <jbb@notatt.com> - 2019-09-16 23:34 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 14:25 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 16:21 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 16:01 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Keith Thompson <kst-u@mib.org> - 2019-09-16 15:46 -0700
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 18:07 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Me <invalid@invalid.de> - 2019-09-17 01:09 +0200
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2019-09-17 02:33 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 20:53 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 21:55 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Keith Thompson <kst-u@mib.org> - 2019-09-16 19:21 -0700
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently André G. Isaak <agisaak@gm.invalid> - 2019-09-16 21:40 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Keith Thompson <kst-u@mib.org> - 2019-09-16 21:38 -0700
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (some concurrence). peteolcott <Here@Home> - 2019-09-17 00:24 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (some concurrence). peteolcott <Here@Home> - 2019-09-17 11:39 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (some concurrence). Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2019-09-17 18:38 +0100
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (some concurrence). peteolcott <Here@Home> - 2019-09-17 13:17 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (some concurrence). Keith Thompson <kst-u@mib.org> - 2019-09-17 11:13 -0700
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (some concurrence). peteolcott <Here@Home> - 2019-09-17 13:29 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (Concurrence) peteolcott <Here@Home> - 2019-09-17 11:20 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (Concurrence) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 10:56 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently (Concurrence) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 11:29 -0600
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2019-09-17 18:32 +0100
Re: Cantor was wrong about cardinality --- (Correction) peteolcott <Here@Home> - 2019-09-17 12:49 -0500
Re: Cantor was wrong about cardinality --- (Correction) Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2019-09-17 19:59 +0100
Re: Cantor was wrong about cardinality --- (Correction) peteolcott <Here@Home> - 2019-09-17 14:09 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently Keith Thompson <kst-u@mib.org> - 2019-09-17 11:42 -0700
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-17 14:01 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-17 17:51 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-17 15:58 -0500
Re: Cantor was wrong about cardinality --- (Mike Terry Concurrence ] peteolcott <Here@Home> - 2019-09-19 11:46 -0500
Re: Cantor was wrong about cardinality --- [ Keith Thompson Concurrence ] peteolcott <Here@Home> - 2019-09-19 11:52 -0500
Re: Cantor was wrong about cardinality --- [ Keith Thompson Concurrence ] peteolcott <Here@Home> - 2019-09-19 11:55 -0500
Re: Cantor was wrong about cardinality --- [ Keith Thompson Concurrence ] Keith Thompson <kst-u@mib.org> - 2019-09-19 14:44 -0700
Re: Cantor was wrong about cardinality --- [ Keith Thompson Concurrence ] peteolcott <Here@Home> - 2019-09-19 17:31 -0500
Re: Cantor was wrong about cardinality --- Reals are defined inconsistently peteolcott <Here@Home> - 2019-09-16 11:02 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-16 20:01 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-16 22:39 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-17 22:23 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? Keith Thompson <kst-u@mib.org> - 2019-09-18 01:39 -0700
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-18 09:59 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? André G. Isaak <agisaak@gm.invalid> - 2019-09-18 09:21 -0600
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-18 10:41 -0500
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? Keith Thompson <kst-u@mib.org> - 2019-09-18 13:55 -0700
Re: Cantor was wrong about cardinality --- APOLOGY clear yet? peteolcott <Here@Home> - 2019-09-18 16:17 -0500
Re: Cantor was wrong about cardinality --- [ CONCURRENCE ] peteolcott <Here@Home> - 2019-09-18 11:10 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-12 09:05 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-12 10:49 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-12 10:02 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 00:18 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-12 23:24 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 10:01 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-13 09:13 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-13 09:56 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 11:40 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-13 11:03 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 12:48 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-13 12:25 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-13 22:35 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-14 16:42 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 11:28 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Andy Walker <anw@cuboid.co.uk> - 2019-09-13 17:34 +0100
Re: Cantor was wrong about cardinality --- COMPLETE peteolcott <Here@Home> - 2019-09-14 11:34 -0500
Re: Cantor was wrong about cardinality --- COMPLETE Andy Walker <anw@cuboid.co.uk> - 2019-09-14 23:44 +0100
Re: Cantor was wrong about cardinality --- COMPLETE (Andy Goofed) peteolcott <Here@Home> - 2019-09-14 18:00 -0500
Re: Cantor was wrong about cardinality --- COMPLETE (Andy Didn't Goof) Andy Walker <anw@cuboid.co.uk> - 2019-09-16 11:52 +0100
Re: Cantor was wrong about cardinality --- COMPLETE (Andy really did Goof) peteolcott <Here@Home> - 2019-09-16 09:42 -0500
Re: Cantor was wrong about cardinality --- COMPLETE (Andy really did Goof) André G. Isaak <agisaak@gm.invalid> - 2019-09-16 09:13 -0600
Re: Cantor was wrong about cardinality --- COMPLETE peteolcott <Here@Home> - 2019-09-14 19:36 -0500
Re: Cantor was wrong about cardinality --- COMPLETE peteolcott <Here@Home> - 2019-09-16 10:21 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 11:55 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-13 13:03 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-13 12:34 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-14 16:38 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) André G. Isaak <agisaak@gm.invalid> - 2019-09-14 22:00 -0600
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) BEN LIED !!! peteolcott <Here@Home> - 2019-09-17 11:33 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-16 10:26 -0500
Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) peteolcott <Here@Home> - 2019-09-16 18:24 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-17 22:19 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 21:23 -0600
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-17 22:31 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 22:04 -0600
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-17 23:21 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 22:40 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 00:27 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 23:55 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 00:59 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 00:12 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 10:15 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 09:35 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 11:00 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 10:14 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 11:26 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 10:42 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 13:10 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 12:36 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 13:49 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 12:59 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 14:12 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 13:30 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 14:45 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 14:14 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 15:29 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 14:38 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 15:56 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 20:38 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-18 22:03 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) André G. Isaak <agisaak@gm.invalid> - 2019-09-18 21:33 -0600
Re: Provably unprovable eliminates incompleteness (NO GAPS) Andy Walker <anw@cuboid.co.uk> - 2019-09-19 15:27 +0100
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-19 10:35 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-19 11:42 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-19 12:43 -0500
Re: Provably unprovable eliminates incompleteness (NO GAPS) peteolcott <Here@Home> - 2019-09-19 10:24 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-17 23:32 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) André G. Isaak <agisaak@gm.invalid> - 2019-09-17 23:21 -0600
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-19 17:13 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-19 19:49 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) David Kleinecke <dkleinecke@gmail.com> - 2019-09-19 20:37 -0700
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Andy Walker <anw@cuboid.co.uk> - 2019-09-20 09:12 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-20 12:55 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Andy Walker <anw@cuboid.co.uk> - 2019-09-21 11:13 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Kaz Kylheku <847-115-0292@kylheku.com> - 2019-09-21 14:55 +0000
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Andy Walker <anw@cuboid.co.uk> - 2019-09-21 16:30 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-21 15:22 -0500
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-22 12:09 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2019-09-20 13:08 +0100
Re: Provably unprovable eliminates incompleteness (CONCISE RESTATEMENT) peteolcott <Here@Home> - 2019-09-21 09:45 -0500
Page 2 of 9 — ← Prev page 1 [2] 3 4 5 6 7 8 9 Next page →
| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-16 11:59 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <IYGdnYbXjMz_IuLAnZ2dnUU7-WfNnZ2d@giganews.com> |
| In reply to | #19907 |
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.
--
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 11:10 -0600 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <qlofm9$p6c$1@dont-email.me> |
| In reply to | #19908 |
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. This is something *you* are reading into the
above on the basis that you conceive of the reals as being discrete when
in fact they are not; it is not part of the actual claim you quote.
Before you argue about the real numbers, you real ought to learn what
the properties of the real numbers actually are. Clearly you have some
"intuitive" view of what they are, but your intuitions aren't actually
correct.
Why don't you read (not just skim) the following:
https://en.wikipedia.org/wiki/Infimum_and_supremum
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 13:27 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <tc-dnWrsh6KDSeLAnZ2dnUU7-IXNnZ2d@giganews.com> |
| In reply to | #19909 |
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.
--
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 12:34 -0600 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <qlokkc$q56$1@dont-email.me> |
| In reply to | #19910 |
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
--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.
[toc] | [prev] | [next] | [standalone]
| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-16 14:19 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <bPWdnWj5i-O9feLAnZ2dnUU7-T_NnZ2d@giganews.com> |
| In reply to | #19911 |
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.
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.
</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.
--
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 13:35 -0600 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <qloo6o$i4c$1@dont-email.me> |
| In reply to | #19912 |
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.
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 14:51 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <4qqdnSmfIroweuLAnZ2dnUU7-cmdnZ2d@giganews.com> |
| In reply to | #19913 |
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 | #19914 |
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 | #19915 |
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 | #19917 |
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.
[toc] | [prev] | [next] | [standalone]
| 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 | #19919 |
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 */
[toc] | [prev] | [next] | [standalone]
| From | Andy Walker <anw@cuboid.co.uk> |
|---|---|
| Date | 2019-09-16 23:44 +0100 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <qlp38h$pdt$1@gioia.aioe.org> |
| In reply to | #19919 |
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. -- Andy Walker, Nottingham.
[toc] | [prev] | [next] | [standalone]
| 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 | #19922 |
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
[toc] | [prev] | [next] | [standalone]
| From | Andy Walker <anw@cuboid.co.uk> |
|---|---|
| Date | 2019-09-17 17:04 +0100 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <qlr06p$1od5$1@gioia.aioe.org> |
| In reply to | #19925 |
On 17/09/2019 00:14, peteolcott wrote: >> 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. Wake up, Pete. My article was about *rational* numbers, not the [conventionally defined] reals and intervals thereof. > 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. There is no rational number that cannot be named. It's not like Voldemort. There are uncountably many reals that cannot be named, but you can't tell us what they are, by definition. If there was such a beginning point, it would not only be named, it would even be computable. > WE DON'T END UP WITHOUT A BEGINNING POINT. In the conventional reals and rationals, that is exactly how we end up. If you want to define a different collection of numbers, you need to say first what its axioms are; and if you want Olcott numbers to behave in any way similarly to conventional numbers, you had better explain how to do elementary arithmetic with them. Which you have failed to do thus far, esp in relation to division. For at least the third time, I commend unto you a study of surreal numbers, which include conventional reals, but also infinite and infinitesimal numbers, conventional arithmetic, much else besides, and an obvious generalisation to games, which makes them useful as well as quirky. -- Andy Walker, Nottingham.
[toc] | [prev] | [next] | [standalone]
| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-17 12:06 -0500 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <McidnUShkM47jxzAnZ2dnUU7-QWdnZ2d@giganews.com> |
| In reply to | #19939 |
On 9/17/2019 11:04 AM, Andy Walker wrote: > On 17/09/2019 00:14, peteolcott wrote: >>> 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. > > Wake up, Pete. My article was about *rational* numbers, not > the [conventionally defined] reals and intervals thereof. > >> 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. > > There is no rational number that cannot be named. It's not > like Voldemort. There are uncountably many reals that cannot be > named, but you can't tell us what they are, by definition. If there > was such a beginning point, it would not only be named, it would even > be computable. > >> WE DON'T END UP WITHOUT A BEGINNING POINT. > > In the conventional reals and rationals, that is exactly how > we end up. If you want to define a different collection of numbers, > you need to say first what its axioms are; and if you want Olcott > numbers to behave in any way similarly to conventional numbers, you > had better explain how to do elementary arithmetic with them. Which > you have failed to do thus far, esp in relation to division. > > For at least the third time, I commend unto you a study of > surreal numbers, which include conventional reals, but also infinite > and infinitesimal numbers, conventional arithmetic, much else besides, > and an obvious generalisation to games, which makes them useful as > well as quirky. > Keith Thompson agree with me: On 9/16/2019 11:38 PM, Keith Thompson wrote: > 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. -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
[toc] | [prev] | [next] | [standalone]
| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2019-09-17 02:57 +0100 |
| Subject | Re: Cantor was wrong about cardinality --- Reals are defined inconsistently |
| Message-ID | <878sqnbs78.fsf@bsb.me.uk> |
| In reply to | #19922 |
Andy Walker <anw@cuboid.co.uk> writes: > 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. The countability of the rationals is a red herring since in ZFC all non-empty sets (including uncountable ones) can be well-ordered, giving every such set a "first" member. Obviously PO is not talking about some arbitrary order because he refers an adjacent point on a (number) line. That refers to the usual ordering by magnitude (even when there is no adjacent point). Without the C in ZFC (the well-ordering theorem is equivalent to the axiom of choice) then maybe only countable sets can be well-ordered? Maybe someone can clarify? > Specifically, there is no least strictly positive rational. > Proof left as an exercise. -- Ben.
[toc] | [prev] | [next] | [standalone]
| 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 | #19917 |
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
[toc] | [prev] | [next] | [standalone]
| 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 | #19914 |
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.
[toc] | [prev] | [next] | [standalone]
| 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 | #19916 |
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
[toc] | [prev] | [next] | [standalone]
| 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 | #19918 |
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.
[toc] | [prev] | [next] | [standalone]
Page 2 of 9 — ← Prev page 1 [2] 3 4 5 6 7 8 9 Next page →
Back to top | Article view | comp.theory
csiph-web