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Groups > sci.math > #645432 > unrolled thread
| Started by | olcott <polcott333@gmail.com> |
|---|---|
| First post | 2026-06-17 16:14 -0500 |
| Last post | 2026-06-23 09:26 +0300 |
| Articles | 20 on this page of 347 — 10 participants |
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Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-17 16:14 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-18 14:35 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-19 10:23 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-19 07:46 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Alan Mackenzie <acm@muc.de> - 2026-06-19 20:28 +0000
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-19 15:50 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Alan Mackenzie <acm@muc.de> - 2026-06-19 21:05 +0000
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-19 16:24 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-19 15:57 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-19 18:30 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-19 22:27 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-20 09:20 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-19 21:35 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-19 22:27 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-19 23:04 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-20 09:29 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-20 09:22 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-19 21:40 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-20 11:05 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-20 14:02 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics dbush <dbush.mobile@gmail.com> - 2026-06-20 15:17 -0400
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-20 12:30 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-20 15:45 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-20 15:03 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics dbush <dbush.mobile@gmail.com> - 2026-06-20 16:17 -0400
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-20 16:03 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics dbush <dbush.mobile@gmail.com> - 2026-06-20 17:17 -0400
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-21 13:02 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-21 12:57 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-21 18:51 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics dbush <dbush.mobile@gmail.com> - 2026-06-21 20:16 -0400
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-22 10:13 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-22 08:13 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-22 11:01 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-22 13:12 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-22 12:28 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-23 08:39 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-25 08:43 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-26 09:17 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-26 07:59 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-27 10:16 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-21 12:48 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-21 13:36 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics phoenix <j63840576@gmail.com> - 2026-06-21 12:54 -0600
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-22 09:23 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-20 08:50 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Alan Mackenzie <acm@muc.de> - 2026-06-20 15:34 +0000
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-20 10:47 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Alan Mackenzie <acm@muc.de> - 2026-06-20 16:08 +0000
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-20 11:37 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-07-11 22:52 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-21 13:11 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-21 18:55 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-22 09:27 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-22 07:05 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-23 08:43 +0300
Re: Ross A. Finlayson, readings in (some of the) --- One-two punch Destroys Liars olcott <polcott333@gmail.com> - 2026-06-23 09:38 -0500
Re: Ross A. Finlayson, readings in (some of the) --- One-two punch Destroys Liars Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-23 08:53 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-20 09:51 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-21 14:04 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-21 16:39 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics phoenix <j63840576@gmail.com> - 2026-06-21 16:36 -0600
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-21 18:15 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics phoenix <j63840576@gmail.com> - 2026-06-21 18:32 -0600
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-21 19:44 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-22 10:46 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-22 10:16 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-23 08:49 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-25 08:47 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-26 09:23 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-26 08:02 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-27 10:19 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics polcott <polcott333@gmail.com> - 2026-06-27 10:34 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-21 21:27 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-22 00:22 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-21 21:16 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics André G. Isaak <agisaak@gm.invalid> - 2026-06-21 18:05 -0600
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-21 19:14 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-20 10:50 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-20 09:41 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-21 13:17 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-21 18:58 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-22 09:41 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-22 07:09 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-23 08:55 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-25 08:58 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-26 09:34 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-26 08:05 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-27 10:27 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics polcott <polcott333@gmail.com> - 2026-06-27 10:36 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-28 11:04 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-19 22:25 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-20 09:18 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics dbush <dbush.mobile@gmail.com> - 2026-06-20 10:36 -0400
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-20 09:54 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics dbush <dbush.mobile@gmail.com> - 2026-06-20 10:57 -0400
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-20 10:22 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics dbush <dbush.mobile@gmail.com> - 2026-06-20 11:23 -0400
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-20 10:44 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics dbush <dbush.mobile@gmail.com> - 2026-06-20 11:48 -0400
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-20 09:45 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics dbush <dbush.mobile@gmail.com> - 2026-06-20 16:20 -0400
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-20 09:29 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-20 11:45 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-20 09:47 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-20 11:57 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics dbush <dbush.mobile@gmail.com> - 2026-06-20 13:13 -0400
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-20 10:21 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-20 10:19 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-20 12:33 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics dbush <dbush.mobile@gmail.com> - 2026-06-20 13:36 -0400
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-20 12:13 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Alan Mackenzie <acm@muc.de> - 2026-06-20 19:48 +0000
Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction olcott <polcott333@gmail.com> - 2026-06-20 16:00 -0500
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction dbush <dbush.mobile@gmail.com> - 2026-06-20 17:19 -0400
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction olcott <polcott333@gmail.com> - 2026-06-20 16:30 -0500
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction dbush <dbush.mobile@gmail.com> - 2026-06-20 17:34 -0400
Disjunction introduction --- new premise from out of no where olcott <polcott333@gmail.com> - 2026-06-20 17:26 -0500
Re: Disjunction introduction --- new premise from out of no where dbush <dbush.mobile@gmail.com> - 2026-06-20 20:11 -0400
Re: Disjunction introduction --- new premise from out of no where olcott <polcott333@gmail.com> - 2026-06-20 19:26 -0500
Re: Disjunction introduction --- new premise from out of no where dbush <dbush.mobile@gmail.com> - 2026-06-20 20:29 -0400
Re: Disjunction introduction --- new premise from out of no where olcott <polcott333@gmail.com> - 2026-06-20 20:06 -0500
Re: Disjunction introduction --- new premise from out of no where dbush <dbush.mobile@gmail.com> - 2026-06-20 21:28 -0400
Re: Disjunction introduction --- new premise from out of no where olcott <polcott333@gmail.com> - 2026-06-20 20:32 -0500
Re: Disjunction introduction --- new premise from out of no where dbush <dbush.mobile@gmail.com> - 2026-06-20 21:38 -0400
Re: Disjunction introduction --- new premise from out of no where olcott <polcott333@gmail.com> - 2026-06-20 20:48 -0500
Re: Disjunction introduction --- new premise from out of no where dbush <dbush.mobile@gmail.com> - 2026-06-20 21:51 -0400
Re: Disjunction introduction --- new premise from out of no where "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-06-25 12:54 -0700
Re: Disjunction introduction --- new premise from out of no where olcott <polcott333@gmail.com> - 2026-06-25 16:01 -0500
Re: Disjunction introduction --- new premise from out of no where olcott <polcott333@gmail.com> - 2026-06-25 16:05 -0500
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction Alan Mackenzie <acm@muc.de> - 2026-06-20 21:43 +0000
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction olcott <polcott333@gmail.com> - 2026-06-20 17:47 -0500
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction Alan Mackenzie <acm@muc.de> - 2026-06-21 11:26 +0000
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction olcott <polcott333@gmail.com> - 2026-06-21 13:42 -0500
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction phoenix <j63840576@gmail.com> - 2026-06-21 12:53 -0600
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction Alan Mackenzie <acm@muc.de> - 2026-06-21 20:04 +0000
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction olcott <polcott333@gmail.com> - 2026-06-21 15:42 -0500
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction André G. Isaak <agisaak@gm.invalid> - 2026-06-21 15:08 -0600
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-21 18:02 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge André G. Isaak <agisaak@gm.invalid> - 2026-06-21 18:02 -0600
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge -- Kristen Welker olcott <polcott333@gmail.com> - 2026-06-21 19:12 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge -- Kristen Welker dbush <dbush.mobile@gmail.com> - 2026-06-21 20:20 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-06-22 09:49 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-22 07:10 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-06-23 09:06 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-23 09:48 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-23 08:53 -0700
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-06-24 13:00 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-24 15:26 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-06-25 10:21 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-25 11:14 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-06-26 09:39 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-26 08:10 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-26 09:20 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-26 08:45 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-26 09:57 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-26 09:24 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-26 12:08 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-26 12:22 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-26 13:25 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-26 12:39 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-26 13:42 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-26 12:53 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-26 14:02 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge André G. Isaak <agisaak@gm.invalid> - 2026-06-26 12:14 -0600
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-26 13:48 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-26 14:51 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-26 14:07 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-26 15:17 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-26 14:38 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-26 15:55 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-26 17:01 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-26 18:08 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-26 17:58 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-26 19:18 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-26 19:05 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-26 20:23 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-26 19:48 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-26 21:11 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-26 20:39 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-26 21:51 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-26 21:00 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge polcott <polcott333@gmail.com> - 2026-06-27 08:34 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-06-27 11:05 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge polcott <polcott333@gmail.com> - 2026-06-27 10:47 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-27 15:37 -0700
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 17:47 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-27 19:24 -0700
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 22:21 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-27 19:25 -0700
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-06-28 11:22 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-06-28 11:17 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-06-27 10:48 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge polcott <polcott333@gmail.com> - 2026-06-27 10:45 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-06-28 11:38 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-06-27 10:35 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge polcott <polcott333@gmail.com> - 2026-06-27 10:43 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-27 14:01 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 13:27 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-27 14:29 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 13:38 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-27 14:39 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 14:01 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-27 15:04 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 14:16 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-27 15:23 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 14:40 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-27 15:54 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 15:04 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-27 16:11 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 15:17 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-27 16:22 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 15:27 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-27 16:30 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-27 16:36 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 15:52 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-27 16:59 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 16:24 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-27 17:50 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 17:11 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-27 18:15 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 17:18 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-27 18:21 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 17:29 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-27 18:33 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 17:44 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-27 18:53 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 18:27 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-27 19:33 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 18:59 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge dbush <dbush.mobile@gmail.com> - 2026-06-27 21:13 -0400
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 20:33 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-06-28 12:38 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-06-28 12:31 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-28 22:12 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-06-29 09:23 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-29 08:38 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-06-30 10:48 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-30 08:43 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-07-01 10:01 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-07-01 10:09 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-06-30 11:43 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-30 09:22 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-07-01 10:13 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-07-01 10:13 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-07-02 09:44 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-07-02 09:45 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-07-02 08:16 -0700
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-07-02 11:47 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-07-03 12:15 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-07-03 11:41 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-07-03 10:23 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-07-03 10:34 -0700
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-07-03 13:17 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-07-03 13:36 -0700
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-07-03 18:14 -0700
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-07-04 10:02 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-07-04 09:58 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-07-04 08:24 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-07-06 13:13 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-07-06 12:51 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-07-08 10:29 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-07-03 12:39 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-07-03 11:43 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-07-04 10:22 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-07-04 08:29 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Alan Mackenzie <acm@muc.de> - 2026-07-04 14:07 +0000
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-07-04 11:38 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Alan Mackenzie <acm@muc.de> - 2026-07-04 17:42 +0000
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-07-06 10:10 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-07-06 08:51 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-07-08 10:35 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-07-08 22:12 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-07-09 10:51 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-06-28 11:38 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge André G. Isaak <agisaak@gm.invalid> - 2026-06-27 13:40 -0600
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-27 14:46 -0500
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Mikko <mikko.levanto@iki.fi> - 2026-06-28 11:32 +0300
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge Alan Mackenzie <acm@muc.de> - 2026-06-22 12:47 +0000
Re: Readings in (some of the) foundations of mathematics --- tree of knowledge olcott <polcott333@gmail.com> - 2026-06-22 09:30 -0500
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction Mikko <mikko.levanto@iki.fi> - 2026-06-22 10:23 +0300
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction olcott <polcott333@gmail.com> - 2026-06-22 09:44 -0500
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction Alan Mackenzie <acm@muc.de> - 2026-06-22 15:22 +0000
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction olcott <polcott333@gmail.com> - 2026-06-22 10:36 -0500
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-22 12:07 -0700
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction olcott <polcott333@gmail.com> - 2026-06-22 14:21 -0500
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction Mikko <mikko.levanto@iki.fi> - 2026-06-23 09:15 +0300
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction olcott <polcott333@gmail.com> - 2026-06-24 16:31 -0500
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction Mikko <mikko.levanto@iki.fi> - 2026-06-25 10:49 +0300
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction Mikko <mikko.levanto@iki.fi> - 2026-07-09 10:55 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-21 13:26 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-21 13:23 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-21 19:00 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-22 10:40 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-22 10:12 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Alan Mackenzie <acm@muc.de> - 2026-06-22 15:48 +0000
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-22 11:23 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Alan Mackenzie <acm@muc.de> - 2026-06-22 18:42 +0000
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-22 13:59 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Alan Mackenzie <acm@muc.de> - 2026-06-22 19:50 +0000
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-22 15:06 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Alan Mackenzie <acm@muc.de> - 2026-06-22 20:38 +0000
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-22 16:01 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-22 16:55 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-22 21:00 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-22 23:14 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-22 21:31 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-23 09:22 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-23 08:51 -0700
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs olcott <polcott333@gmail.com> - 2026-06-23 11:54 -0500
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-23 10:32 -0700
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-23 10:58 -0700
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs olcott <polcott333@gmail.com> - 2026-06-23 13:24 -0500
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-27 07:26 -0700
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs olcott <polcott333@gmail.com> - 2026-06-23 13:20 -0500
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs Mikko <mikko.levanto@iki.fi> - 2026-06-24 13:13 +0300
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs olcott <polcott333@gmail.com> - 2026-06-24 16:33 -0500
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs phoenix <j63840576@gmail.com> - 2026-06-24 18:28 -0600
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs Mikko <mikko.levanto@iki.fi> - 2026-06-25 10:29 +0300
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs olcott <polcott333@gmail.com> - 2026-06-25 11:16 -0500
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs Mikko <mikko.levanto@iki.fi> - 2026-06-26 09:45 +0300
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs olcott <polcott333@gmail.com> - 2026-06-26 08:15 -0500
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs Mikko <mikko.levanto@iki.fi> - 2026-06-27 11:13 +0300
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-27 07:25 -0700
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs polcott <polcott333@gmail.com> - 2026-06-27 10:53 -0500
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs Mikko <mikko.levanto@iki.fi> - 2026-06-28 12:51 +0300
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-30 06:23 -0700
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs olcott <polcott333@gmail.com> - 2026-06-30 09:53 -0500
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-30 10:36 -0700
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-30 19:47 -0700
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs olcott <polcott333@gmail.com> - 2026-06-30 22:01 -0500
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-07-01 05:13 -0700
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs olcott <polcott333@gmail.com> - 2026-07-01 09:59 -0500
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-07-01 10:00 -0700
DAG of all general knowledge that can be expressed in Language olcott <polcott333@gmail.com> - 2026-07-01 12:57 -0500
Re: DAG of all general knowledge that can be expressed in Language Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-07-01 12:31 -0700
Re: DAG of all general knowledge that can be expressed in Language "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-07-01 12:37 -0700
Re: DAG of all general knowledge that can be expressed in Language Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-07-01 13:16 -0700
Re: DAG of all general knowledge that can be expressed in Language "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-07-01 18:59 -0700
Re: DAG of all general knowledge that can be expressed in Language olcott <polcott333@gmail.com> - 2026-07-01 14:51 -0500
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs Python <python@cccp.invalid> - 2026-06-23 21:04 +0000
Re: Ross A. Finlayson, readings in (some of the) --- cycles in directed graphs Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-23 19:25 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-22 21:16 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-22 21:28 -0700
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-22 15:08 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-23 09:17 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-23 09:26 +0300
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| From | dbush <dbush.mobile@gmail.com> |
|---|---|
| Date | 2026-06-27 18:15 -0400 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111pi1u$380jl$2@dont-email.me> |
| In reply to | #645777 |
On 6/27/2026 6:11 PM, olcott wrote:
> On 6/27/2026 4:50 PM, dbush wrote:
>> On 6/27/2026 5:24 PM, olcott wrote:
>>> On 6/27/2026 3:59 PM, dbush wrote:
>>>> On 6/27/2026 4:52 PM, olcott wrote:
>>>>> On 6/27/2026 3:30 PM, dbush wrote:
>>>>>> On 6/27/2026 4:27 PM, olcott wrote:
>>>>>>> On 6/27/2026 3:22 PM, dbush wrote:
>>>>>>>> On 6/27/2026 4:17 PM, olcott wrote:
>>>>>>>>> On 6/27/2026 3:11 PM, dbush wrote:
>>>>>>>>>> On 6/27/2026 4:04 PM, olcott wrote:
>>>>>>>>>>> On 6/27/2026 2:54 PM, dbush wrote:
>>>>>>>>>>>> On 6/27/2026 3:40 PM, olcott wrote:
>>>>>>>>>>>>> On 6/27/2026 2:23 PM, dbush wrote:
>>>>>>>>>>>>>> On 6/27/2026 3:16 PM, olcott wrote:
>>>>>>>>>>>>>>> On 6/27/2026 2:04 PM, dbush wrote:
>>>>>>>>>>>>>>>> On 6/27/2026 3:01 PM, olcott wrote:
>>>>>>>>>>>>>>>>> On 6/27/2026 1:39 PM, dbush wrote:
>>>>>>>>>>>>>>>>>> On 6/27/2026 2:38 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:29 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:27 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:01 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 11:43 AM, polcott wrote:
>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:35 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>> On 26/06/2026 16:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>> On 6/26/2026 1:39 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>> On 25/06/2026 19:14, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/25/2026 2:21 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 24/06/2026 23:26, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/24/2026 5:00 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 23/06/2026 17:48, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/23/2026 1:06 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 15:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/22/2026 1:49 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 02:02, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 4:08 PM, André G. Isaak
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-21 14:42, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 3:04 PM, Alan
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> [ Followup-To: set ]
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 6:26 AM, Alan
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I just found the term:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounding in a proof theoretic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> atomic base" yesterday.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You can find any number of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> terms. That doesn't mean you're
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> capable of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding them.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The above is the key reason why
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> under PTS Gödel 1931 incompleteness
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fails.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I don't believe you. You have no
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> respect for or understanding of the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> truth. If you really want to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> persuade anybody that PTS somehow
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> causes
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's theorem not to hold, then
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> cite an academic expert who'll have
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some credibility.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If they are mere gibberish words
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> to you then you will not understand.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You don't understand Proof-
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theoritic Semantics, and you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> certainly don't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand Gödel's Theorem,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> neither the theorem itself nor any
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> proof of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> it.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It is a verified fact that Gödel's
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> G is ungrounded
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in the atomic base of PA. That you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> do not understand
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> what: "grounded in the atomic base"
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> means is less
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> than no rebuttal at all.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounded in the atomic base of PA"
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> is an expression used only by you,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> and it is one which you have never
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> explicitly defined, so the fault
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> here certainly doesn't lie with
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Alan. It's certainly not a 'verified
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fact' when you haven't even
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> adequately explained what it is that
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> you mean.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> All of knowledge expressed in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> language is structured as a tree of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantic relations specified
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> syntactically between finite strings.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> What makes you believe semantic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> relations that can be structured as
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> a tree are sufficient to contain all
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> knowledge that is exressed in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some language?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The CycL language and the Cyc Project.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> They use a tree structure for concepts.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> But why would one try to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> put knowledge in a tree structure?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It must at least be a directed acyclic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> graph or
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> the proof gets stuck in an infinite loop
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> and never
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> completes.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> How can any ordering of knowledge prevent
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> getting stuck in a loop
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> when looking for a proof?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> By looking upward in a type hierarchy.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> If you mean not looking elsewhere that may
>>>>>>>>>>>>>>>>>>>>>>>>>>>> indeed prevent loops.
>>>>>>>>>>>>>>>>>>>>>>>>>>>> In most cases that also prevents finding the
>>>>>>>>>>>>>>>>>>>>>>>>>>>> proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> Truth Conditional Semantics (TCS) <is>
>>>>>>>>>>>>>>>>>>>>>>>>>>> incoherent
>>>>>>>>>>>>>>>>>>>>>>>>>>> compared to Proof Theoretic Semantics (PTS).
>>>>>>>>>>>>>>>>>>>>>>>>>>> Essentially
>>>>>>>>>>>>>>>>>>>>>>>>>>> PTS just coherently connects the semantic
>>>>>>>>>>>>>>>>>>>>>>>>>>> meanings
>>>>>>>>>>>>>>>>>>>>>>>>>>> expressed in language together into one
>>>>>>>>>>>>>>>>>>>>>>>>>>> coherent body
>>>>>>>>>>>>>>>>>>>>>>>>>>> of general knowledge. It does this without
>>>>>>>>>>>>>>>>>>>>>>>>>>> undecidability
>>>>>>>>>>>>>>>>>>>>>>>>>>> or mathematical incompleteness.
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>> Looking for a proof does not need any
>>>>>>>>>>>>>>>>>>>>>>>>>> semantics so it is not obvious
>>>>>>>>>>>>>>>>>>>>>>>>>> how switching to another semantics could
>>>>>>>>>>>>>>>>>>>>>>>>>> improve it.
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>> In proof theoretic semantics an expression only
>>>>>>>>>>>>>>>>>>>>>>>>> gains
>>>>>>>>>>>>>>>>>>>>>>>>> semantic meaning by finding a proof.
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>> It should be obvious that finding a proof does
>>>>>>>>>>>>>>>>>>>>>>>> not happen before
>>>>>>>>>>>>>>>>>>>>>>>> looking for a proof.
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> If there is no sequence of inference steps in Q from
>>>>>>>>>>>>>>>>>>>>>>> ~∃x x=S(x) to the axioms of Q
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> There are, but that sequence is infinite
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> If there is no FINITE sequence of inference steps
>>>>>>>>>>>>>>>>>>>>> in Q from ~∃x x=S(x) to the axioms of Q then ~∃x
>>>>>>>>>>>>>>>>>>>>> x=S(x)
>>>>>>>>>>>>>>>>>>>>> is ungrounded in the PTS atomic base of Q.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> i.e., ~∃x x=S(x) is unprovable is Q, as is commonly
>>>>>>>>>>>>>>>>>>>> known.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> Is it commonly known that ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> Which has the semantic meaning "no number is equal to
>>>>>>>>>>>>>>>>>> its successor" as per the definition of Q.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> Since there are no steps in Q that affirm ~∃x x=S(x)
>>>>>>>>>>>>>>>>> in Q it is an open question in Q and not a confirmed
>>>>>>>>>>>>>>>>> statement in Q.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> In other words, unproven as is commonly known.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Yet never gets to undecidable or in any sense of incomplete.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> False, as by definition, Q is incomplete because ~∃x
>>>>>>>>>>>>>> x=S(x) is unprovable / out-of-scope / not semantically
>>>>>>>>>>>>>> grounded in Q.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> Proof theoretic semantics DOES NOT DO IT THAT WAY !!!
>>>>>>>>>>>>
>>>>>>>>>>>> And because PTS claims the semantically valid sentence in Q
>>>>>>>>>>>> "no number is equal to its successor" is not semantically
>>>>>>>>>>>> valid, it must be discarded as useless.
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> No you are just not bothering to pay 100% totally
>>>>>>>>>>> complete attention to every single word.
>>>>>>>>>>>
>>>>>>>>>>> Wittgenstein
>>>>>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>>>>>> system' means: the opposite has been proved
>>>>>>>>>>> in Russell's system
>>>>>>>>>>>
>>>>>>>>>>> Proof Theoretic Semantics has almost gotten there.
>>>>>>>>>>> For the most part they stop at semantically grounded
>>>>>>>>>>
>>>>>>>>>> i.e. proven
>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> and never quite get all the way to True.
>>>>>>>>>>>
>>>>>>>>>>> For Wittgenstein's slight extension of the PTS
>>>>>>>>>>> notion ~∃x x=S(x) is untrue
>>>>>>>>>>
>>>>>>>>>> i.e. unproven
>>>>>>>>>>
>>>>>>>>>>> in Q and true in PA.
>>>>>>>>>>> I have been saying it that way long before I ever
>>>>>>>>>>> heard of Wittgenstein.
>>>>>>>>>>
>>>>>>>>>> So again, you're saying the same thing as everyone else but
>>>>>>>>>> with different words.
>>>>>>>>>
>>>>>>>>> So everyone says that ~∃x x=S(x)
>>>>>>>>
>>>>>>>> which has the semantic meaning "no number is equal to its
>>>>>>>> successor" as per the definition of Q
>>>>>>>>
>>>>>>>>> is simply untrue
>>>>>>>>
>>>>>>>> i.e. unprovable
>>>>>>>>
>>>>>>>>> in Q and does nor derive either undecidability or
>>>>>>>>> incompleteness?
>>>>>>>>
>>>>>>>> It does derive incompleteness, as by definition Q is incomplete
>>>>>>>> because "no number is equal to its successor" is unprovable in Q.
>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>> Wittgenstein (1937)
>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>> system' means: the opposite has been proved
>>>>>>> in Russell's system
>>>>>>
>>>>>> So what term would you use to describe a sentence that has an
>>>>>> infinite sequence of inference steps between it and the axioms of
>>>>>> the system?
>>>>>>
>>>>>
>>>>> untrue and unfalse.
>>>>
>>>> What about the more general case, i.e. a term for a statement that
>>>> has *any* sequence of inference steps, either finite or infinite,
>>>> between it and the axioms of the system? And what would the
>>>> negation of such a statement be called?
>>>>
>>>>
>>>
>>> The truth value of the Goldbach conjecture
>>> may have an infinite number of steps thus
>>> would be unknowable and not a member of the
>>> body of knowledge that can be expressed in
>>> language. Negation has no effect on expressions
>>> that are neither true no false.
>>>
>>> Every finite string including gibberish has the
>>> truth value of: {True, False, Neither}.
>>>
>>> Finite strings are a superset of expressions
>>> of language.
>>>
>>
>> That's not the definition I asked for.
>>
>> What term would you use for a statement that has *any* sequence of
>> inference steps, either finite or infinite, between it and the axioms
>> of the system?
>>
>
> "true on the basis of meaning expressed in language"
> reliably computable for the entire body of GENERAL knowledge.
> Is the limit of the topic of all my posts.
>
> Infinite inference steps are off topic.
Why are you so reluctant to provide a simple term?
It seems you're attempting to engage in Newspeak.
https://en.wikipedia.org/wiki/Newspeak
>
>> What term would you use for the negation of the above statement?
>
> Does not have any proof finite or infinite?
> That would be untrue and possibly nonsense.
>
>
>
[toc] | [prev] | [next] | [standalone]
| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-27 17:18 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111pi7i$38dlj$1@dont-email.me> |
| In reply to | #645778 |
On 6/27/2026 5:15 PM, dbush wrote:
> On 6/27/2026 6:11 PM, olcott wrote:
>> On 6/27/2026 4:50 PM, dbush wrote:
>>> On 6/27/2026 5:24 PM, olcott wrote:
>>>> On 6/27/2026 3:59 PM, dbush wrote:
>>>>> On 6/27/2026 4:52 PM, olcott wrote:
>>>>>> On 6/27/2026 3:30 PM, dbush wrote:
>>>>>>> On 6/27/2026 4:27 PM, olcott wrote:
>>>>>>>> On 6/27/2026 3:22 PM, dbush wrote:
>>>>>>>>> On 6/27/2026 4:17 PM, olcott wrote:
>>>>>>>>>> On 6/27/2026 3:11 PM, dbush wrote:
>>>>>>>>>>> On 6/27/2026 4:04 PM, olcott wrote:
>>>>>>>>>>>> On 6/27/2026 2:54 PM, dbush wrote:
>>>>>>>>>>>>> On 6/27/2026 3:40 PM, olcott wrote:
>>>>>>>>>>>>>> On 6/27/2026 2:23 PM, dbush wrote:
>>>>>>>>>>>>>>> On 6/27/2026 3:16 PM, olcott wrote:
>>>>>>>>>>>>>>>> On 6/27/2026 2:04 PM, dbush wrote:
>>>>>>>>>>>>>>>>> On 6/27/2026 3:01 PM, olcott wrote:
>>>>>>>>>>>>>>>>>> On 6/27/2026 1:39 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:38 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:29 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:27 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:01 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 11:43 AM, polcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:35 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>> On 26/06/2026 16:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/26/2026 1:39 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>> On 25/06/2026 19:14, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/25/2026 2:21 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 24/06/2026 23:26, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/24/2026 5:00 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 23/06/2026 17:48, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/23/2026 1:06 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 15:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/22/2026 1:49 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 02:02, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 4:08 PM, André G. Isaak
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-21 14:42, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 3:04 PM, Alan
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> [ Followup-To: set ]
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 6:26 AM, Alan
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I just found the term:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounding in a proof
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theoretic atomic base" yesterday.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You can find any number of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> terms. That doesn't mean you're
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> capable of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding them.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The above is the key reason why
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> under PTS Gödel 1931 incompleteness
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fails.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I don't believe you. You have no
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> respect for or understanding of the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> truth. If you really want to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> persuade anybody that PTS somehow
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> causes
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's theorem not to hold, then
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> cite an academic expert who'll have
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some credibility.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If they are mere gibberish words
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> to you then you will not
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You don't understand Proof-
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theoritic Semantics, and you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> certainly don't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand Gödel's Theorem,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> neither the theorem itself nor
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> any proof of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> it.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It is a verified fact that Gödel's
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> G is ungrounded
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in the atomic base of PA. That you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> do not understand
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> what: "grounded in the atomic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> base" means is less
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> than no rebuttal at all.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounded in the atomic base of PA"
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> is an expression used only by you,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> and it is one which you have never
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> explicitly defined, so the fault
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> here certainly doesn't lie with
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Alan. It's certainly not a
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 'verified fact' when you haven't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> even adequately explained what it
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> is that you mean.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> All of knowledge expressed in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> language is structured as a tree of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantic relations specified
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> syntactically between finite strings.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> What makes you believe semantic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> relations that can be structured as
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> a tree are sufficient to contain all
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> knowledge that is exressed in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some language?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The CycL language and the Cyc Project.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> They use a tree structure for concepts.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> But why would one try to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> put knowledge in a tree structure?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It must at least be a directed acyclic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> graph or
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> the proof gets stuck in an infinite loop
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> and never
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> completes.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> How can any ordering of knowledge prevent
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> getting stuck in a loop
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> when looking for a proof?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> By looking upward in a type hierarchy.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If you mean not looking elsewhere that may
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> indeed prevent loops.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In most cases that also prevents finding
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> the proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> Truth Conditional Semantics (TCS) <is>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> incoherent
>>>>>>>>>>>>>>>>>>>>>>>>>>>> compared to Proof Theoretic Semantics (PTS).
>>>>>>>>>>>>>>>>>>>>>>>>>>>> Essentially
>>>>>>>>>>>>>>>>>>>>>>>>>>>> PTS just coherently connects the semantic
>>>>>>>>>>>>>>>>>>>>>>>>>>>> meanings
>>>>>>>>>>>>>>>>>>>>>>>>>>>> expressed in language together into one
>>>>>>>>>>>>>>>>>>>>>>>>>>>> coherent body
>>>>>>>>>>>>>>>>>>>>>>>>>>>> of general knowledge. It does this without
>>>>>>>>>>>>>>>>>>>>>>>>>>>> undecidability
>>>>>>>>>>>>>>>>>>>>>>>>>>>> or mathematical incompleteness.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> Looking for a proof does not need any
>>>>>>>>>>>>>>>>>>>>>>>>>>> semantics so it is not obvious
>>>>>>>>>>>>>>>>>>>>>>>>>>> how switching to another semantics could
>>>>>>>>>>>>>>>>>>>>>>>>>>> improve it.
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>> In proof theoretic semantics an expression
>>>>>>>>>>>>>>>>>>>>>>>>>> only gains
>>>>>>>>>>>>>>>>>>>>>>>>>> semantic meaning by finding a proof.
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>> It should be obvious that finding a proof does
>>>>>>>>>>>>>>>>>>>>>>>>> not happen before
>>>>>>>>>>>>>>>>>>>>>>>>> looking for a proof.
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>> If there is no sequence of inference steps in Q
>>>>>>>>>>>>>>>>>>>>>>>> from
>>>>>>>>>>>>>>>>>>>>>>>> ~∃x x=S(x) to the axioms of Q
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> There are, but that sequence is infinite
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> If there is no FINITE sequence of inference steps
>>>>>>>>>>>>>>>>>>>>>> in Q from ~∃x x=S(x) to the axioms of Q then ~∃x
>>>>>>>>>>>>>>>>>>>>>> x=S(x)
>>>>>>>>>>>>>>>>>>>>>> is ungrounded in the PTS atomic base of Q.
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> i.e., ~∃x x=S(x) is unprovable is Q, as is commonly
>>>>>>>>>>>>>>>>>>>>> known.
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> Is it commonly known that ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> Which has the semantic meaning "no number is equal to
>>>>>>>>>>>>>>>>>>> its successor" as per the definition of Q.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> Since there are no steps in Q that affirm ~∃x x=S(x)
>>>>>>>>>>>>>>>>>> in Q it is an open question in Q and not a confirmed
>>>>>>>>>>>>>>>>>> statement in Q.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> In other words, unproven as is commonly known.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Yet never gets to undecidable or in any sense of
>>>>>>>>>>>>>>>> incomplete.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> False, as by definition, Q is incomplete because ~∃x
>>>>>>>>>>>>>>> x=S(x) is unprovable / out-of-scope / not semantically
>>>>>>>>>>>>>>> grounded in Q.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Proof theoretic semantics DOES NOT DO IT THAT WAY !!!
>>>>>>>>>>>>>
>>>>>>>>>>>>> And because PTS claims the semantically valid sentence in Q
>>>>>>>>>>>>> "no number is equal to its successor" is not semantically
>>>>>>>>>>>>> valid, it must be discarded as useless.
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> No you are just not bothering to pay 100% totally
>>>>>>>>>>>> complete attention to every single word.
>>>>>>>>>>>>
>>>>>>>>>>>> Wittgenstein
>>>>>>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>>>>>>> system' means: the opposite has been proved
>>>>>>>>>>>> in Russell's system
>>>>>>>>>>>>
>>>>>>>>>>>> Proof Theoretic Semantics has almost gotten there.
>>>>>>>>>>>> For the most part they stop at semantically grounded
>>>>>>>>>>>
>>>>>>>>>>> i.e. proven
>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> and never quite get all the way to True.
>>>>>>>>>>>>
>>>>>>>>>>>> For Wittgenstein's slight extension of the PTS
>>>>>>>>>>>> notion ~∃x x=S(x) is untrue
>>>>>>>>>>>
>>>>>>>>>>> i.e. unproven
>>>>>>>>>>>
>>>>>>>>>>>> in Q and true in PA.
>>>>>>>>>>>> I have been saying it that way long before I ever
>>>>>>>>>>>> heard of Wittgenstein.
>>>>>>>>>>>
>>>>>>>>>>> So again, you're saying the same thing as everyone else but
>>>>>>>>>>> with different words.
>>>>>>>>>>
>>>>>>>>>> So everyone says that ~∃x x=S(x)
>>>>>>>>>
>>>>>>>>> which has the semantic meaning "no number is equal to its
>>>>>>>>> successor" as per the definition of Q
>>>>>>>>>
>>>>>>>>>> is simply untrue
>>>>>>>>>
>>>>>>>>> i.e. unprovable
>>>>>>>>>
>>>>>>>>>> in Q and does nor derive either undecidability or
>>>>>>>>>> incompleteness?
>>>>>>>>>
>>>>>>>>> It does derive incompleteness, as by definition Q is incomplete
>>>>>>>>> because "no number is equal to its successor" is unprovable in Q.
>>>>>>>>>
>>>>>>>>>
>>>>>>>>
>>>>>>>> Wittgenstein (1937)
>>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>>> system' means: the opposite has been proved
>>>>>>>> in Russell's system
>>>>>>>
>>>>>>> So what term would you use to describe a sentence that has an
>>>>>>> infinite sequence of inference steps between it and the axioms of
>>>>>>> the system?
>>>>>>>
>>>>>>
>>>>>> untrue and unfalse.
>>>>>
>>>>> What about the more general case, i.e. a term for a statement that
>>>>> has *any* sequence of inference steps, either finite or infinite,
>>>>> between it and the axioms of the system? And what would the
>>>>> negation of such a statement be called?
>>>>>
>>>>>
>>>>
>>>> The truth value of the Goldbach conjecture
>>>> may have an infinite number of steps thus
>>>> would be unknowable and not a member of the
>>>> body of knowledge that can be expressed in
>>>> language. Negation has no effect on expressions
>>>> that are neither true no false.
>>>>
>>>> Every finite string including gibberish has the
>>>> truth value of: {True, False, Neither}.
>>>>
>>>> Finite strings are a superset of expressions
>>>> of language.
>>>>
>>>
>>> That's not the definition I asked for.
>>>
>>> What term would you use for a statement that has *any* sequence of
>>> inference steps, either finite or infinite, between it and the axioms
>>> of the system?
>>>
>>
>> "true on the basis of meaning expressed in language"
>> reliably computable for the entire body of GENERAL knowledge.
>> Is the limit of the topic of all my posts.
>>
>> Infinite inference steps are off topic.
>
> Why are you so reluctant to provide a simple term?
>
OFF-TOPIC <is> THE TERM.
> It seems you're attempting to engage in Newspeak.
>
> https://en.wikipedia.org/wiki/Newspeak
>
>>
>>> What term would you use for the negation of the above statement?
>>
>> Does not have any proof finite or infinite?
>> That would be untrue and possibly nonsense.
>>
>>
>>
>
--
Copyright 2026 Olcott
My 28 year goal has been to make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
The complete structure of this system is now defined.
The entire body of knowledge expressed in language is
comprised of two types of relations between finite strings:
(a) *Axioms* Expressions of language that are stipulated to be true.
My system bridges the analytic/synthetic distinction by
expressly encoding all empirical "atomic facts" in a formal
language such as CycL of the Cyc project.
(b) *Inference Rules* Expressions of language that are semantically
entailed syntactically from (a) and/or (b).
[toc] | [prev] | [next] | [standalone]
| From | dbush <dbush.mobile@gmail.com> |
|---|---|
| Date | 2026-06-27 18:21 -0400 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111pid9$380jl$3@dont-email.me> |
| In reply to | #645779 |
On 6/27/2026 6:18 PM, olcott wrote:
> On 6/27/2026 5:15 PM, dbush wrote:
>> On 6/27/2026 6:11 PM, olcott wrote:
>>> On 6/27/2026 4:50 PM, dbush wrote:
>>>> On 6/27/2026 5:24 PM, olcott wrote:
>>>>> On 6/27/2026 3:59 PM, dbush wrote:
>>>>>> On 6/27/2026 4:52 PM, olcott wrote:
>>>>>>> On 6/27/2026 3:30 PM, dbush wrote:
>>>>>>>> On 6/27/2026 4:27 PM, olcott wrote:
>>>>>>>>> On 6/27/2026 3:22 PM, dbush wrote:
>>>>>>>>>> On 6/27/2026 4:17 PM, olcott wrote:
>>>>>>>>>>> On 6/27/2026 3:11 PM, dbush wrote:
>>>>>>>>>>>> On 6/27/2026 4:04 PM, olcott wrote:
>>>>>>>>>>>>> On 6/27/2026 2:54 PM, dbush wrote:
>>>>>>>>>>>>>> On 6/27/2026 3:40 PM, olcott wrote:
>>>>>>>>>>>>>>> On 6/27/2026 2:23 PM, dbush wrote:
>>>>>>>>>>>>>>>> On 6/27/2026 3:16 PM, olcott wrote:
>>>>>>>>>>>>>>>>> On 6/27/2026 2:04 PM, dbush wrote:
>>>>>>>>>>>>>>>>>> On 6/27/2026 3:01 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:39 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:38 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:29 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:27 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:01 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 11:43 AM, polcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:35 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>> On 26/06/2026 16:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/26/2026 1:39 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 25/06/2026 19:14, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/25/2026 2:21 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 24/06/2026 23:26, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/24/2026 5:00 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 23/06/2026 17:48, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/23/2026 1:06 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 15:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/22/2026 1:49 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 02:02, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 4:08 PM, André G.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Isaak wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-21 14:42, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 3:04 PM, Alan
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> [ Followup-To: set ]
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 6:26 AM, Alan
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I just found the term:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounding in a proof
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theoretic atomic base"
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> yesterday.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You can find any number of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> terms. That doesn't mean
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> you're capable of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding them.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The above is the key reason why
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> under PTS Gödel 1931
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> incompleteness
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fails.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I don't believe you. You have
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> no respect for or understanding
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> of the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> truth. If you really want to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> persuade anybody that PTS
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> somehow causes
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's theorem not to hold,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> then cite an academic expert
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> who'll have
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some credibility.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If they are mere gibberish
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> words to you then you will not
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You don't understand Proof-
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theoritic Semantics, and you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> certainly don't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand Gödel's Theorem,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> neither the theorem itself nor
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> any proof of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> it.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It is a verified fact that
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's G is ungrounded
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in the atomic base of PA. That
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> you do not understand
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> what: "grounded in the atomic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> base" means is less
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> than no rebuttal at all.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounded in the atomic base of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PA" is an expression used only by
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> you, and it is one which you have
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> never explicitly defined, so the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fault here certainly doesn't lie
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> with Alan. It's certainly not a
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 'verified fact' when you haven't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> even adequately explained what it
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> is that you mean.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> All of knowledge expressed in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> language is structured as a tree of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantic relations specified
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> syntactically between finite strings.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> What makes you believe semantic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> relations that can be structured as
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> a tree are sufficient to contain all
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> knowledge that is exressed in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some language?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The CycL language and the Cyc Project.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> They use a tree structure for
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> concepts. But why would one try to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> put knowledge in a tree structure?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It must at least be a directed acyclic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> graph or
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> the proof gets stuck in an infinite
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> loop and never
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> completes.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> How can any ordering of knowledge
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> prevent getting stuck in a loop
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> when looking for a proof?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> By looking upward in a type hierarchy.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If you mean not looking elsewhere that may
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> indeed prevent loops.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In most cases that also prevents finding
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> the proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Truth Conditional Semantics (TCS) <is>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> incoherent
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> compared to Proof Theoretic Semantics
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> (PTS). Essentially
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PTS just coherently connects the semantic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> meanings
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> expressed in language together into one
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> coherent body
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> of general knowledge. It does this without
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> undecidability
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> or mathematical incompleteness.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> Looking for a proof does not need any
>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantics so it is not obvious
>>>>>>>>>>>>>>>>>>>>>>>>>>>> how switching to another semantics could
>>>>>>>>>>>>>>>>>>>>>>>>>>>> improve it.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> In proof theoretic semantics an expression
>>>>>>>>>>>>>>>>>>>>>>>>>>> only gains
>>>>>>>>>>>>>>>>>>>>>>>>>>> semantic meaning by finding a proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>> It should be obvious that finding a proof does
>>>>>>>>>>>>>>>>>>>>>>>>>> not happen before
>>>>>>>>>>>>>>>>>>>>>>>>>> looking for a proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>> If there is no sequence of inference steps in Q
>>>>>>>>>>>>>>>>>>>>>>>>> from
>>>>>>>>>>>>>>>>>>>>>>>>> ~∃x x=S(x) to the axioms of Q
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>> There are, but that sequence is infinite
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> If there is no FINITE sequence of inference steps
>>>>>>>>>>>>>>>>>>>>>>> in Q from ~∃x x=S(x) to the axioms of Q then ~∃x
>>>>>>>>>>>>>>>>>>>>>>> x=S(x)
>>>>>>>>>>>>>>>>>>>>>>> is ungrounded in the PTS atomic base of Q.
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> i.e., ~∃x x=S(x) is unprovable is Q, as is
>>>>>>>>>>>>>>>>>>>>>> commonly known.
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> Is it commonly known that ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> Which has the semantic meaning "no number is equal
>>>>>>>>>>>>>>>>>>>> to its successor" as per the definition of Q.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> Since there are no steps in Q that affirm ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>> in Q it is an open question in Q and not a confirmed
>>>>>>>>>>>>>>>>>>> statement in Q.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> In other words, unproven as is commonly known.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> Yet never gets to undecidable or in any sense of
>>>>>>>>>>>>>>>>> incomplete.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> False, as by definition, Q is incomplete because ~∃x
>>>>>>>>>>>>>>>> x=S(x) is unprovable / out-of-scope / not semantically
>>>>>>>>>>>>>>>> grounded in Q.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Proof theoretic semantics DOES NOT DO IT THAT WAY !!!
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> And because PTS claims the semantically valid sentence in
>>>>>>>>>>>>>> Q "no number is equal to its successor" is not
>>>>>>>>>>>>>> semantically valid, it must be discarded as useless.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> No you are just not bothering to pay 100% totally
>>>>>>>>>>>>> complete attention to every single word.
>>>>>>>>>>>>>
>>>>>>>>>>>>> Wittgenstein
>>>>>>>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>>>>>>>> system' means: the opposite has been proved
>>>>>>>>>>>>> in Russell's system
>>>>>>>>>>>>>
>>>>>>>>>>>>> Proof Theoretic Semantics has almost gotten there.
>>>>>>>>>>>>> For the most part they stop at semantically grounded
>>>>>>>>>>>>
>>>>>>>>>>>> i.e. proven
>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> and never quite get all the way to True.
>>>>>>>>>>>>>
>>>>>>>>>>>>> For Wittgenstein's slight extension of the PTS
>>>>>>>>>>>>> notion ~∃x x=S(x) is untrue
>>>>>>>>>>>>
>>>>>>>>>>>> i.e. unproven
>>>>>>>>>>>>
>>>>>>>>>>>>> in Q and true in PA.
>>>>>>>>>>>>> I have been saying it that way long before I ever
>>>>>>>>>>>>> heard of Wittgenstein.
>>>>>>>>>>>>
>>>>>>>>>>>> So again, you're saying the same thing as everyone else but
>>>>>>>>>>>> with different words.
>>>>>>>>>>>
>>>>>>>>>>> So everyone says that ~∃x x=S(x)
>>>>>>>>>>
>>>>>>>>>> which has the semantic meaning "no number is equal to its
>>>>>>>>>> successor" as per the definition of Q
>>>>>>>>>>
>>>>>>>>>>> is simply untrue
>>>>>>>>>>
>>>>>>>>>> i.e. unprovable
>>>>>>>>>>
>>>>>>>>>>> in Q and does nor derive either undecidability or
>>>>>>>>>>> incompleteness?
>>>>>>>>>>
>>>>>>>>>> It does derive incompleteness, as by definition Q is
>>>>>>>>>> incomplete because "no number is equal to its successor" is
>>>>>>>>>> unprovable in Q.
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> Wittgenstein (1937)
>>>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>>>> system' means: the opposite has been proved
>>>>>>>>> in Russell's system
>>>>>>>>
>>>>>>>> So what term would you use to describe a sentence that has an
>>>>>>>> infinite sequence of inference steps between it and the axioms
>>>>>>>> of the system?
>>>>>>>>
>>>>>>>
>>>>>>> untrue and unfalse.
>>>>>>
>>>>>> What about the more general case, i.e. a term for a statement that
>>>>>> has *any* sequence of inference steps, either finite or infinite,
>>>>>> between it and the axioms of the system? And what would the
>>>>>> negation of such a statement be called?
>>>>>>
>>>>>>
>>>>>
>>>>> The truth value of the Goldbach conjecture
>>>>> may have an infinite number of steps thus
>>>>> would be unknowable and not a member of the
>>>>> body of knowledge that can be expressed in
>>>>> language. Negation has no effect on expressions
>>>>> that are neither true no false.
>>>>>
>>>>> Every finite string including gibberish has the
>>>>> truth value of: {True, False, Neither}.
>>>>>
>>>>> Finite strings are a superset of expressions
>>>>> of language.
>>>>>
>>>>
>>>> That's not the definition I asked for.
>>>>
>>>> What term would you use for a statement that has *any* sequence of
>>>> inference steps, either finite or infinite, between it and the
>>>> axioms of the system?
>>>>
>>>
>>> "true on the basis of meaning expressed in language"
>>> reliably computable for the entire body of GENERAL knowledge.
>>> Is the limit of the topic of all my posts.
>>>
>>> Infinite inference steps are off topic.
>>
>> Why are you so reluctant to provide a simple term?
>>
>
> OFF-TOPIC <is> THE TERM.
So we've established that "off-topic" means "a statement that has *any*
sequence of inference steps, either finite or infinite, between it and
the axioms of the system".
So going back to Wittgenstein, using his terminology and the term you
provided:
A system is incomplete if it contains a statement that is off-topic but
not true.
>
>> It seems you're attempting to engage in Newspeak.
>>
>> https://en.wikipedia.org/wiki/Newspeak
>>
>>>
>>>> What term would you use for the negation of the above statement?
>>>
>>> Does not have any proof finite or infinite?
>>> That would be untrue and possibly nonsense.
>>>
>>>
>>>
>>
>
>
[toc] | [prev] | [next] | [standalone]
| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-27 17:29 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111pisi$38i0n$1@dont-email.me> |
| In reply to | #645780 |
On 6/27/2026 5:21 PM, dbush wrote:
> On 6/27/2026 6:18 PM, olcott wrote:
>> On 6/27/2026 5:15 PM, dbush wrote:
>>> On 6/27/2026 6:11 PM, olcott wrote:
>>>> On 6/27/2026 4:50 PM, dbush wrote:
>>>>> On 6/27/2026 5:24 PM, olcott wrote:
>>>>>> On 6/27/2026 3:59 PM, dbush wrote:
>>>>>>> On 6/27/2026 4:52 PM, olcott wrote:
>>>>>>>> On 6/27/2026 3:30 PM, dbush wrote:
>>>>>>>>> On 6/27/2026 4:27 PM, olcott wrote:
>>>>>>>>>> On 6/27/2026 3:22 PM, dbush wrote:
>>>>>>>>>>> On 6/27/2026 4:17 PM, olcott wrote:
>>>>>>>>>>>> On 6/27/2026 3:11 PM, dbush wrote:
>>>>>>>>>>>>> On 6/27/2026 4:04 PM, olcott wrote:
>>>>>>>>>>>>>> On 6/27/2026 2:54 PM, dbush wrote:
>>>>>>>>>>>>>>> On 6/27/2026 3:40 PM, olcott wrote:
>>>>>>>>>>>>>>>> On 6/27/2026 2:23 PM, dbush wrote:
>>>>>>>>>>>>>>>>> On 6/27/2026 3:16 PM, olcott wrote:
>>>>>>>>>>>>>>>>>> On 6/27/2026 2:04 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>> On 6/27/2026 3:01 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:39 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:38 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:29 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:27 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:01 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 11:43 AM, polcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:35 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>> On 26/06/2026 16:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/26/2026 1:39 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 25/06/2026 19:14, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/25/2026 2:21 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 24/06/2026 23:26, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/24/2026 5:00 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 23/06/2026 17:48, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/23/2026 1:06 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 15:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/22/2026 1:49 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 02:02, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 4:08 PM, André G.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Isaak wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-21 14:42, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 3:04 PM, Alan
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> [ Followup-To: set ]
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 6:26 AM, Alan
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I just found the term:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounding in a proof
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theoretic atomic base"
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> yesterday.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You can find any number of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> terms. That doesn't mean
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> you're capable of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding them.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The above is the key reason
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> why under PTS Gödel 1931
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> incompleteness
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fails.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I don't believe you. You have
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> no respect for or understanding
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> of the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> truth. If you really want to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> persuade anybody that PTS
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> somehow causes
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's theorem not to hold,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> then cite an academic expert
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> who'll have
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some credibility.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If they are mere gibberish
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> words to you then you will not
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You don't understand Proof-
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theoritic Semantics, and you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> certainly don't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand Gödel's Theorem,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> neither the theorem itself nor
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> any proof of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> it.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It is a verified fact that
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's G is ungrounded
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in the atomic base of PA. That
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> you do not understand
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> what: "grounded in the atomic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> base" means is less
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> than no rebuttal at all.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounded in the atomic base of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PA" is an expression used only by
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> you, and it is one which you have
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> never explicitly defined, so the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fault here certainly doesn't lie
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> with Alan. It's certainly not a
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 'verified fact' when you haven't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> even adequately explained what it
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> is that you mean.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> All of knowledge expressed in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> language is structured as a tree
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> of semantic relations specified
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> syntactically between finite strings.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> What makes you believe semantic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> relations that can be structured as
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> a tree are sufficient to contain
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> all knowledge that is exressed in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some language?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The CycL language and the Cyc Project.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> They use a tree structure for
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> concepts. But why would one try to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> put knowledge in a tree structure?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It must at least be a directed acyclic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> graph or
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> the proof gets stuck in an infinite
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> loop and never
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> completes.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> How can any ordering of knowledge
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> prevent getting stuck in a loop
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> when looking for a proof?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> By looking upward in a type hierarchy.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If you mean not looking elsewhere that
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> may indeed prevent loops.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In most cases that also prevents finding
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> the proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Truth Conditional Semantics (TCS) <is>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> incoherent
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> compared to Proof Theoretic Semantics
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> (PTS). Essentially
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PTS just coherently connects the semantic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> meanings
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> expressed in language together into one
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> coherent body
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> of general knowledge. It does this without
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> undecidability
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> or mathematical incompleteness.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Looking for a proof does not need any
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantics so it is not obvious
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> how switching to another semantics could
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> improve it.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> In proof theoretic semantics an expression
>>>>>>>>>>>>>>>>>>>>>>>>>>>> only gains
>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantic meaning by finding a proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> It should be obvious that finding a proof
>>>>>>>>>>>>>>>>>>>>>>>>>>> does not happen before
>>>>>>>>>>>>>>>>>>>>>>>>>>> looking for a proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>> If there is no sequence of inference steps in
>>>>>>>>>>>>>>>>>>>>>>>>>> Q from
>>>>>>>>>>>>>>>>>>>>>>>>>> ~∃x x=S(x) to the axioms of Q
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>> There are, but that sequence is infinite
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>> If there is no FINITE sequence of inference steps
>>>>>>>>>>>>>>>>>>>>>>>> in Q from ~∃x x=S(x) to the axioms of Q then ~∃x
>>>>>>>>>>>>>>>>>>>>>>>> x=S(x)
>>>>>>>>>>>>>>>>>>>>>>>> is ungrounded in the PTS atomic base of Q.
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> i.e., ~∃x x=S(x) is unprovable is Q, as is
>>>>>>>>>>>>>>>>>>>>>>> commonly known.
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> Is it commonly known that ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> Which has the semantic meaning "no number is equal
>>>>>>>>>>>>>>>>>>>>> to its successor" as per the definition of Q.
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> Since there are no steps in Q that affirm ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>>> in Q it is an open question in Q and not a confirmed
>>>>>>>>>>>>>>>>>>>> statement in Q.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> In other words, unproven as is commonly known.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> Yet never gets to undecidable or in any sense of
>>>>>>>>>>>>>>>>>> incomplete.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> False, as by definition, Q is incomplete because ~∃x
>>>>>>>>>>>>>>>>> x=S(x) is unprovable / out-of-scope / not semantically
>>>>>>>>>>>>>>>>> grounded in Q.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Proof theoretic semantics DOES NOT DO IT THAT WAY !!!
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> And because PTS claims the semantically valid sentence in
>>>>>>>>>>>>>>> Q "no number is equal to its successor" is not
>>>>>>>>>>>>>>> semantically valid, it must be discarded as useless.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> No you are just not bothering to pay 100% totally
>>>>>>>>>>>>>> complete attention to every single word.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Wittgenstein
>>>>>>>>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>>>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>>>>>>>>> system' means: the opposite has been proved
>>>>>>>>>>>>>> in Russell's system
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Proof Theoretic Semantics has almost gotten there.
>>>>>>>>>>>>>> For the most part they stop at semantically grounded
>>>>>>>>>>>>>
>>>>>>>>>>>>> i.e. proven
>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> and never quite get all the way to True.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> For Wittgenstein's slight extension of the PTS
>>>>>>>>>>>>>> notion ~∃x x=S(x) is untrue
>>>>>>>>>>>>>
>>>>>>>>>>>>> i.e. unproven
>>>>>>>>>>>>>
>>>>>>>>>>>>>> in Q and true in PA.
>>>>>>>>>>>>>> I have been saying it that way long before I ever
>>>>>>>>>>>>>> heard of Wittgenstein.
>>>>>>>>>>>>>
>>>>>>>>>>>>> So again, you're saying the same thing as everyone else but
>>>>>>>>>>>>> with different words.
>>>>>>>>>>>>
>>>>>>>>>>>> So everyone says that ~∃x x=S(x)
>>>>>>>>>>>
>>>>>>>>>>> which has the semantic meaning "no number is equal to its
>>>>>>>>>>> successor" as per the definition of Q
>>>>>>>>>>>
>>>>>>>>>>>> is simply untrue
>>>>>>>>>>>
>>>>>>>>>>> i.e. unprovable
>>>>>>>>>>>
>>>>>>>>>>>> in Q and does nor derive either undecidability or
>>>>>>>>>>>> incompleteness?
>>>>>>>>>>>
>>>>>>>>>>> It does derive incompleteness, as by definition Q is
>>>>>>>>>>> incomplete because "no number is equal to its successor" is
>>>>>>>>>>> unprovable in Q.
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Wittgenstein (1937)
>>>>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>>>>> system' means: the opposite has been proved
>>>>>>>>>> in Russell's system
>>>>>>>>>
>>>>>>>>> So what term would you use to describe a sentence that has an
>>>>>>>>> infinite sequence of inference steps between it and the axioms
>>>>>>>>> of the system?
>>>>>>>>>
>>>>>>>>
>>>>>>>> untrue and unfalse.
>>>>>>>
>>>>>>> What about the more general case, i.e. a term for a statement
>>>>>>> that has *any* sequence of inference steps, either finite or
>>>>>>> infinite, between it and the axioms of the system? And what
>>>>>>> would the negation of such a statement be called?
>>>>>>>
>>>>>>>
>>>>>>
>>>>>> The truth value of the Goldbach conjecture
>>>>>> may have an infinite number of steps thus
>>>>>> would be unknowable and not a member of the
>>>>>> body of knowledge that can be expressed in
>>>>>> language. Negation has no effect on expressions
>>>>>> that are neither true no false.
>>>>>>
>>>>>> Every finite string including gibberish has the
>>>>>> truth value of: {True, False, Neither}.
>>>>>>
>>>>>> Finite strings are a superset of expressions
>>>>>> of language.
>>>>>>
>>>>>
>>>>> That's not the definition I asked for.
>>>>>
>>>>> What term would you use for a statement that has *any* sequence of
>>>>> inference steps, either finite or infinite, between it and the
>>>>> axioms of the system?
>>>>>
>>>>
>>>> "true on the basis of meaning expressed in language"
>>>> reliably computable for the entire body of GENERAL knowledge.
>>>> Is the limit of the topic of all my posts.
>>>>
>>>> Infinite inference steps are off topic.
>>>
>>> Why are you so reluctant to provide a simple term?
>>>
>>
>> OFF-TOPIC <is> THE TERM.
>
> So we've established that "off-topic" means "a statement that has *any*
> sequence of inference steps, either finite or infinite, between it and
> the axioms of the system".
>
The ones that have infinite steps out outside the
body of knowledge and off topic for that reason.
--
Copyright 2026 Olcott
My 28 year goal has been to make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
The complete structure of this system is now defined.
The entire body of knowledge expressed in language is
comprised of two types of relations between finite strings:
(a) *Axioms* Expressions of language that are stipulated to be true.
My system bridges the analytic/synthetic distinction by
expressly encoding all empirical "atomic facts" in a formal
language such as CycL of the Cyc project.
(b) *Inference Rules* Expressions of language that are semantically
entailed syntactically from (a) and/or (b).
[toc] | [prev] | [next] | [standalone]
| From | dbush <dbush.mobile@gmail.com> |
|---|---|
| Date | 2026-06-27 18:33 -0400 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111pj4j$38fsa$2@dont-email.me> |
| In reply to | #645781 |
On 6/27/2026 6:29 PM, olcott wrote:
> On 6/27/2026 5:21 PM, dbush wrote:
>> On 6/27/2026 6:18 PM, olcott wrote:
>>> On 6/27/2026 5:15 PM, dbush wrote:
>>>> On 6/27/2026 6:11 PM, olcott wrote:
>>>>> On 6/27/2026 4:50 PM, dbush wrote:
>>>>>> On 6/27/2026 5:24 PM, olcott wrote:
>>>>>>> On 6/27/2026 3:59 PM, dbush wrote:
>>>>>>>> On 6/27/2026 4:52 PM, olcott wrote:
>>>>>>>>> On 6/27/2026 3:30 PM, dbush wrote:
>>>>>>>>>> On 6/27/2026 4:27 PM, olcott wrote:
>>>>>>>>>>> On 6/27/2026 3:22 PM, dbush wrote:
>>>>>>>>>>>> On 6/27/2026 4:17 PM, olcott wrote:
>>>>>>>>>>>>> On 6/27/2026 3:11 PM, dbush wrote:
>>>>>>>>>>>>>> On 6/27/2026 4:04 PM, olcott wrote:
>>>>>>>>>>>>>>> On 6/27/2026 2:54 PM, dbush wrote:
>>>>>>>>>>>>>>>> On 6/27/2026 3:40 PM, olcott wrote:
>>>>>>>>>>>>>>>>> On 6/27/2026 2:23 PM, dbush wrote:
>>>>>>>>>>>>>>>>>> On 6/27/2026 3:16 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:04 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>> On 6/27/2026 3:01 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:39 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:38 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:29 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:27 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:01 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 11:43 AM, polcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:35 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 26/06/2026 16:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/26/2026 1:39 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 25/06/2026 19:14, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/25/2026 2:21 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 24/06/2026 23:26, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/24/2026 5:00 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 23/06/2026 17:48, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/23/2026 1:06 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 15:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/22/2026 1:49 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 02:02, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 4:08 PM, André G.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Isaak wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-21 14:42, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 3:04 PM, Alan
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> [ Followup-To: set ]
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 6:26 AM, Alan
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I just found the term:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounding in a proof
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theoretic atomic base"
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> yesterday.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You can find any number of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> terms. That doesn't mean
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> you're capable of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding them.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The above is the key reason
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> why under PTS Gödel 1931
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> incompleteness
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fails.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I don't believe you. You have
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> no respect for or
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding of the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> truth. If you really want to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> persuade anybody that PTS
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> somehow causes
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's theorem not to hold,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> then cite an academic expert
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> who'll have
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some credibility.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If they are mere gibberish
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> words to you then you will
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> not understand.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You don't understand Proof-
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theoritic Semantics, and you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> certainly don't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand Gödel's Theorem,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> neither the theorem itself nor
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> any proof of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> it.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It is a verified fact that
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's G is ungrounded
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in the atomic base of PA. That
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> you do not understand
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> what: "grounded in the atomic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> base" means is less
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> than no rebuttal at all.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounded in the atomic base of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PA" is an expression used only
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> by you, and it is one which you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> have never explicitly defined,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> so the fault here certainly
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> doesn't lie with Alan. It's
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> certainly not a 'verified fact'
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> when you haven't even adequately
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> explained what it is that you mean.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> All of knowledge expressed in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> language is structured as a tree
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> of semantic relations specified
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> syntactically between finite
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> strings.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> What makes you believe semantic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> relations that can be structured as
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> a tree are sufficient to contain
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> all knowledge that is exressed in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some language?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The CycL language and the Cyc Project.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> They use a tree structure for
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> concepts. But why would one try to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> put knowledge in a tree structure?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It must at least be a directed
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> acyclic graph or
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> the proof gets stuck in an infinite
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> loop and never
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> completes.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> How can any ordering of knowledge
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> prevent getting stuck in a loop
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> when looking for a proof?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> By looking upward in a type hierarchy.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If you mean not looking elsewhere that
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> may indeed prevent loops.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In most cases that also prevents finding
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> the proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Truth Conditional Semantics (TCS) <is>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> incoherent
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> compared to Proof Theoretic Semantics
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> (PTS). Essentially
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PTS just coherently connects the semantic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> meanings
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> expressed in language together into one
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> coherent body
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> of general knowledge. It does this
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> without undecidability
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> or mathematical incompleteness.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Looking for a proof does not need any
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantics so it is not obvious
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> how switching to another semantics could
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> improve it.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In proof theoretic semantics an expression
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> only gains
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantic meaning by finding a proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> It should be obvious that finding a proof
>>>>>>>>>>>>>>>>>>>>>>>>>>>> does not happen before
>>>>>>>>>>>>>>>>>>>>>>>>>>>> looking for a proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> If there is no sequence of inference steps in
>>>>>>>>>>>>>>>>>>>>>>>>>>> Q from
>>>>>>>>>>>>>>>>>>>>>>>>>>> ~∃x x=S(x) to the axioms of Q
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>> There are, but that sequence is infinite
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>> If there is no FINITE sequence of inference steps
>>>>>>>>>>>>>>>>>>>>>>>>> in Q from ~∃x x=S(x) to the axioms of Q then
>>>>>>>>>>>>>>>>>>>>>>>>> ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>>>>>>>> is ungrounded in the PTS atomic base of Q.
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>> i.e., ~∃x x=S(x) is unprovable is Q, as is
>>>>>>>>>>>>>>>>>>>>>>>> commonly known.
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> Is it commonly known that ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> Which has the semantic meaning "no number is equal
>>>>>>>>>>>>>>>>>>>>>> to its successor" as per the definition of Q.
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> Since there are no steps in Q that affirm ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>>>> in Q it is an open question in Q and not a confirmed
>>>>>>>>>>>>>>>>>>>>> statement in Q.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> In other words, unproven as is commonly known.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> Yet never gets to undecidable or in any sense of
>>>>>>>>>>>>>>>>>>> incomplete.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> False, as by definition, Q is incomplete because ~∃x
>>>>>>>>>>>>>>>>>> x=S(x) is unprovable / out-of-scope / not semantically
>>>>>>>>>>>>>>>>>> grounded in Q.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> Proof theoretic semantics DOES NOT DO IT THAT WAY !!!
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> And because PTS claims the semantically valid sentence
>>>>>>>>>>>>>>>> in Q "no number is equal to its successor" is not
>>>>>>>>>>>>>>>> semantically valid, it must be discarded as useless.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> No you are just not bothering to pay 100% totally
>>>>>>>>>>>>>>> complete attention to every single word.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Wittgenstein
>>>>>>>>>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>>>>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>>>>>>>>>> system' means: the opposite has been proved
>>>>>>>>>>>>>>> in Russell's system
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Proof Theoretic Semantics has almost gotten there.
>>>>>>>>>>>>>>> For the most part they stop at semantically grounded
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> i.e. proven
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> and never quite get all the way to True.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> For Wittgenstein's slight extension of the PTS
>>>>>>>>>>>>>>> notion ~∃x x=S(x) is untrue
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> i.e. unproven
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> in Q and true in PA.
>>>>>>>>>>>>>>> I have been saying it that way long before I ever
>>>>>>>>>>>>>>> heard of Wittgenstein.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> So again, you're saying the same thing as everyone else
>>>>>>>>>>>>>> but with different words.
>>>>>>>>>>>>>
>>>>>>>>>>>>> So everyone says that ~∃x x=S(x)
>>>>>>>>>>>>
>>>>>>>>>>>> which has the semantic meaning "no number is equal to its
>>>>>>>>>>>> successor" as per the definition of Q
>>>>>>>>>>>>
>>>>>>>>>>>>> is simply untrue
>>>>>>>>>>>>
>>>>>>>>>>>> i.e. unprovable
>>>>>>>>>>>>
>>>>>>>>>>>>> in Q and does nor derive either undecidability or
>>>>>>>>>>>>> incompleteness?
>>>>>>>>>>>>
>>>>>>>>>>>> It does derive incompleteness, as by definition Q is
>>>>>>>>>>>> incomplete because "no number is equal to its successor" is
>>>>>>>>>>>> unprovable in Q.
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> Wittgenstein (1937)
>>>>>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>>>>>> system' means: the opposite has been proved
>>>>>>>>>>> in Russell's system
>>>>>>>>>>
>>>>>>>>>> So what term would you use to describe a sentence that has an
>>>>>>>>>> infinite sequence of inference steps between it and the axioms
>>>>>>>>>> of the system?
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> untrue and unfalse.
>>>>>>>>
>>>>>>>> What about the more general case, i.e. a term for a statement
>>>>>>>> that has *any* sequence of inference steps, either finite or
>>>>>>>> infinite, between it and the axioms of the system? And what
>>>>>>>> would the negation of such a statement be called?
>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>> The truth value of the Goldbach conjecture
>>>>>>> may have an infinite number of steps thus
>>>>>>> would be unknowable and not a member of the
>>>>>>> body of knowledge that can be expressed in
>>>>>>> language. Negation has no effect on expressions
>>>>>>> that are neither true no false.
>>>>>>>
>>>>>>> Every finite string including gibberish has the
>>>>>>> truth value of: {True, False, Neither}.
>>>>>>>
>>>>>>> Finite strings are a superset of expressions
>>>>>>> of language.
>>>>>>>
>>>>>>
>>>>>> That's not the definition I asked for.
>>>>>>
>>>>>> What term would you use for a statement that has *any* sequence of
>>>>>> inference steps, either finite or infinite, between it and the
>>>>>> axioms of the system?
>>>>>>
>>>>>
>>>>> "true on the basis of meaning expressed in language"
>>>>> reliably computable for the entire body of GENERAL knowledge.
>>>>> Is the limit of the topic of all my posts.
>>>>>
>>>>> Infinite inference steps are off topic.
>>>>
>>>> Why are you so reluctant to provide a simple term?
>>>>
>>>
>>> OFF-TOPIC <is> THE TERM.
>>
>> So we've established that "off-topic" means "a statement that has
>> *any* sequence of inference steps, either finite or infinite, between
>> it and the axioms of the system".
>>
>
> The ones that have infinite steps out outside the
> body of knowledge and off topic for that reason.
>
So you're claiming the sentence "no number is equal to its successor" in
Q is "off topic".
Rejected, as it is a semantically valid sentence in the language of Q.
[toc] | [prev] | [next] | [standalone]
| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-27 17:44 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111pjnq$38pef$1@dont-email.me> |
| In reply to | #645783 |
On 6/27/2026 5:33 PM, dbush wrote:
> On 6/27/2026 6:29 PM, olcott wrote:
>> On 6/27/2026 5:21 PM, dbush wrote:
>>> On 6/27/2026 6:18 PM, olcott wrote:
>>>> On 6/27/2026 5:15 PM, dbush wrote:
>>>>> On 6/27/2026 6:11 PM, olcott wrote:
>>>>>> On 6/27/2026 4:50 PM, dbush wrote:
>>>>>>> On 6/27/2026 5:24 PM, olcott wrote:
>>>>>>>> On 6/27/2026 3:59 PM, dbush wrote:
>>>>>>>>> On 6/27/2026 4:52 PM, olcott wrote:
>>>>>>>>>> On 6/27/2026 3:30 PM, dbush wrote:
>>>>>>>>>>> On 6/27/2026 4:27 PM, olcott wrote:
>>>>>>>>>>>> On 6/27/2026 3:22 PM, dbush wrote:
>>>>>>>>>>>>> On 6/27/2026 4:17 PM, olcott wrote:
>>>>>>>>>>>>>> On 6/27/2026 3:11 PM, dbush wrote:
>>>>>>>>>>>>>>> On 6/27/2026 4:04 PM, olcott wrote:
>>>>>>>>>>>>>>>> On 6/27/2026 2:54 PM, dbush wrote:
>>>>>>>>>>>>>>>>> On 6/27/2026 3:40 PM, olcott wrote:
>>>>>>>>>>>>>>>>>> On 6/27/2026 2:23 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>> On 6/27/2026 3:16 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:04 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 3:01 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:39 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:38 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:29 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:27 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:01 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 11:43 AM, polcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:35 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 26/06/2026 16:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/26/2026 1:39 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 25/06/2026 19:14, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/25/2026 2:21 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 24/06/2026 23:26, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/24/2026 5:00 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 23/06/2026 17:48, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/23/2026 1:06 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 15:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/22/2026 1:49 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 02:02, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 4:08 PM, André G.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Isaak wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-21 14:42, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 3:04 PM, Alan
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> [ Followup-To: set ]
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 6:26 AM, Alan
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I just found the term:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounding in a proof
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theoretic atomic base"
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> yesterday.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You can find any number of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> terms. That doesn't mean
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> you're capable of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding them.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The above is the key reason
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> why under PTS Gödel 1931
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> incompleteness
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fails.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I don't believe you. You
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> have no respect for or
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding of the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> truth. If you really want to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> persuade anybody that PTS
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> somehow causes
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's theorem not to hold,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> then cite an academic expert
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> who'll have
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some credibility.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If they are mere gibberish
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> words to you then you will
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> not understand.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You don't understand Proof-
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theoritic Semantics, and you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> certainly don't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand Gödel's Theorem,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> neither the theorem itself
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> nor any proof of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> it.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It is a verified fact that
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's G is ungrounded
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in the atomic base of PA. That
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> you do not understand
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> what: "grounded in the atomic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> base" means is less
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> than no rebuttal at all.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounded in the atomic base of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PA" is an expression used only
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> by you, and it is one which you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> have never explicitly defined,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> so the fault here certainly
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> doesn't lie with Alan. It's
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> certainly not a 'verified fact'
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> when you haven't even
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> adequately explained what it is
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> that you mean.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> All of knowledge expressed in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> language is structured as a tree
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> of semantic relations specified
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> syntactically between finite
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> strings.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> What makes you believe semantic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> relations that can be structured as
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> a tree are sufficient to contain
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> all knowledge that is exressed in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some language?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The CycL language and the Cyc
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Project.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> They use a tree structure for
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> concepts. But why would one try to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> put knowledge in a tree structure?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It must at least be a directed
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> acyclic graph or
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> the proof gets stuck in an infinite
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> loop and never
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> completes.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> How can any ordering of knowledge
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> prevent getting stuck in a loop
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> when looking for a proof?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> By looking upward in a type hierarchy.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If you mean not looking elsewhere that
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> may indeed prevent loops.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In most cases that also prevents
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> finding the proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Truth Conditional Semantics (TCS) <is>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> incoherent
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> compared to Proof Theoretic Semantics
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> (PTS). Essentially
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PTS just coherently connects the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantic meanings
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> expressed in language together into one
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> coherent body
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> of general knowledge. It does this
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> without undecidability
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> or mathematical incompleteness.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Looking for a proof does not need any
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantics so it is not obvious
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> how switching to another semantics could
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> improve it.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In proof theoretic semantics an expression
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> only gains
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantic meaning by finding a proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It should be obvious that finding a proof
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> does not happen before
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> looking for a proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> If there is no sequence of inference steps
>>>>>>>>>>>>>>>>>>>>>>>>>>>> in Q from
>>>>>>>>>>>>>>>>>>>>>>>>>>>> ~∃x x=S(x) to the axioms of Q
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> There are, but that sequence is infinite
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>> If there is no FINITE sequence of inference steps
>>>>>>>>>>>>>>>>>>>>>>>>>> in Q from ~∃x x=S(x) to the axioms of Q then
>>>>>>>>>>>>>>>>>>>>>>>>>> ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>>>>>>>>> is ungrounded in the PTS atomic base of Q.
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>> i.e., ~∃x x=S(x) is unprovable is Q, as is
>>>>>>>>>>>>>>>>>>>>>>>>> commonly known.
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>> Is it commonly known that ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> Which has the semantic meaning "no number is
>>>>>>>>>>>>>>>>>>>>>>> equal to its successor" as per the definition of Q.
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> Since there are no steps in Q that affirm ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>>>>> in Q it is an open question in Q and not a confirmed
>>>>>>>>>>>>>>>>>>>>>> statement in Q.
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> In other words, unproven as is commonly known.
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> Yet never gets to undecidable or in any sense of
>>>>>>>>>>>>>>>>>>>> incomplete.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> False, as by definition, Q is incomplete because ~∃x
>>>>>>>>>>>>>>>>>>> x=S(x) is unprovable / out-of-scope / not
>>>>>>>>>>>>>>>>>>> semantically grounded in Q.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> Proof theoretic semantics DOES NOT DO IT THAT WAY !!!
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> And because PTS claims the semantically valid sentence
>>>>>>>>>>>>>>>>> in Q "no number is equal to its successor" is not
>>>>>>>>>>>>>>>>> semantically valid, it must be discarded as useless.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> No you are just not bothering to pay 100% totally
>>>>>>>>>>>>>>>> complete attention to every single word.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Wittgenstein
>>>>>>>>>>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>>>>>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>>>>>>>>>>> system' means: the opposite has been proved
>>>>>>>>>>>>>>>> in Russell's system
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Proof Theoretic Semantics has almost gotten there.
>>>>>>>>>>>>>>>> For the most part they stop at semantically grounded
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> i.e. proven
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> and never quite get all the way to True.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> For Wittgenstein's slight extension of the PTS
>>>>>>>>>>>>>>>> notion ~∃x x=S(x) is untrue
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> i.e. unproven
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> in Q and true in PA.
>>>>>>>>>>>>>>>> I have been saying it that way long before I ever
>>>>>>>>>>>>>>>> heard of Wittgenstein.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> So again, you're saying the same thing as everyone else
>>>>>>>>>>>>>>> but with different words.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> So everyone says that ~∃x x=S(x)
>>>>>>>>>>>>>
>>>>>>>>>>>>> which has the semantic meaning "no number is equal to its
>>>>>>>>>>>>> successor" as per the definition of Q
>>>>>>>>>>>>>
>>>>>>>>>>>>>> is simply untrue
>>>>>>>>>>>>>
>>>>>>>>>>>>> i.e. unprovable
>>>>>>>>>>>>>
>>>>>>>>>>>>>> in Q and does nor derive either undecidability or
>>>>>>>>>>>>>> incompleteness?
>>>>>>>>>>>>>
>>>>>>>>>>>>> It does derive incompleteness, as by definition Q is
>>>>>>>>>>>>> incomplete because "no number is equal to its successor" is
>>>>>>>>>>>>> unprovable in Q.
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> Wittgenstein (1937)
>>>>>>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>>>>>>> system' means: the opposite has been proved
>>>>>>>>>>>> in Russell's system
>>>>>>>>>>>
>>>>>>>>>>> So what term would you use to describe a sentence that has an
>>>>>>>>>>> infinite sequence of inference steps between it and the
>>>>>>>>>>> axioms of the system?
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> untrue and unfalse.
>>>>>>>>>
>>>>>>>>> What about the more general case, i.e. a term for a statement
>>>>>>>>> that has *any* sequence of inference steps, either finite or
>>>>>>>>> infinite, between it and the axioms of the system? And what
>>>>>>>>> would the negation of such a statement be called?
>>>>>>>>>
>>>>>>>>>
>>>>>>>>
>>>>>>>> The truth value of the Goldbach conjecture
>>>>>>>> may have an infinite number of steps thus
>>>>>>>> would be unknowable and not a member of the
>>>>>>>> body of knowledge that can be expressed in
>>>>>>>> language. Negation has no effect on expressions
>>>>>>>> that are neither true no false.
>>>>>>>>
>>>>>>>> Every finite string including gibberish has the
>>>>>>>> truth value of: {True, False, Neither}.
>>>>>>>>
>>>>>>>> Finite strings are a superset of expressions
>>>>>>>> of language.
>>>>>>>>
>>>>>>>
>>>>>>> That's not the definition I asked for.
>>>>>>>
>>>>>>> What term would you use for a statement that has *any* sequence
>>>>>>> of inference steps, either finite or infinite, between it and the
>>>>>>> axioms of the system?
>>>>>>>
>>>>>>
>>>>>> "true on the basis of meaning expressed in language"
>>>>>> reliably computable for the entire body of GENERAL knowledge.
>>>>>> Is the limit of the topic of all my posts.
>>>>>>
>>>>>> Infinite inference steps are off topic.
>>>>>
>>>>> Why are you so reluctant to provide a simple term?
>>>>>
>>>>
>>>> OFF-TOPIC <is> THE TERM.
>>>
>>> So we've established that "off-topic" means "a statement that has
>>> *any* sequence of inference steps, either finite or infinite, between
>>> it and the axioms of the system".
>>>
>>
>> The ones that have infinite steps out outside the
>> body of knowledge and off topic for that reason.
>>
>
> So you're claiming the sentence "no number is equal to its successor" in
> Q is "off topic".
>
> Rejected, as it is a semantically valid sentence in the language of Q.
If there is no finite sequence of inference steps
between x and the axioms of Q then PTS stipulates
that x is not semantically valid in Q.
--
Copyright 2026 Olcott
My 28 year goal has been to make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
The complete structure of this system is now defined.
The entire body of knowledge expressed in language is
comprised of two types of relations between finite strings:
(a) *Axioms* Expressions of language that are stipulated to be true.
My system bridges the analytic/synthetic distinction by
expressly encoding all empirical "atomic facts" in a formal
language such as CycL of the Cyc project.
(b) *Inference Rules* Expressions of language that are semantically
entailed syntactically from (a) and/or (b).
[toc] | [prev] | [next] | [standalone]
| From | dbush <dbush.mobile@gmail.com> |
|---|---|
| Date | 2026-06-27 18:53 -0400 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111pk9i$380jl$5@dont-email.me> |
| In reply to | #645786 |
On 6/27/2026 6:44 PM, olcott wrote:
> On 6/27/2026 5:33 PM, dbush wrote:
>> On 6/27/2026 6:29 PM, olcott wrote:
>>> On 6/27/2026 5:21 PM, dbush wrote:
>>>> On 6/27/2026 6:18 PM, olcott wrote:
>>>>> On 6/27/2026 5:15 PM, dbush wrote:
>>>>>> On 6/27/2026 6:11 PM, olcott wrote:
>>>>>>> On 6/27/2026 4:50 PM, dbush wrote:
>>>>>>>> On 6/27/2026 5:24 PM, olcott wrote:
>>>>>>>>> On 6/27/2026 3:59 PM, dbush wrote:
>>>>>>>>>> On 6/27/2026 4:52 PM, olcott wrote:
>>>>>>>>>>> On 6/27/2026 3:30 PM, dbush wrote:
>>>>>>>>>>>> On 6/27/2026 4:27 PM, olcott wrote:
>>>>>>>>>>>>> On 6/27/2026 3:22 PM, dbush wrote:
>>>>>>>>>>>>>> On 6/27/2026 4:17 PM, olcott wrote:
>>>>>>>>>>>>>>> On 6/27/2026 3:11 PM, dbush wrote:
>>>>>>>>>>>>>>>> On 6/27/2026 4:04 PM, olcott wrote:
>>>>>>>>>>>>>>>>> On 6/27/2026 2:54 PM, dbush wrote:
>>>>>>>>>>>>>>>>>> On 6/27/2026 3:40 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:23 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>> On 6/27/2026 3:16 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:04 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 3:01 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:39 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:38 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:29 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:27 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:01 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 11:43 AM, polcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:35 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 26/06/2026 16:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/26/2026 1:39 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 25/06/2026 19:14, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/25/2026 2:21 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 24/06/2026 23:26, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/24/2026 5:00 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 23/06/2026 17:48, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/23/2026 1:06 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 15:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/22/2026 1:49 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 02:02, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 4:08 PM, André G.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Isaak wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-21 14:42, olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 3:04 PM, Alan
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> [ Followup-To: set ]
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 6:26 AM, Alan
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I just found the term:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounding in a proof
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theoretic atomic base"
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> yesterday.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You can find any number of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> terms. That doesn't mean
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> you're capable of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding them.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The above is the key reason
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> why under PTS Gödel 1931
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> incompleteness
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fails.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I don't believe you. You
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> have no respect for or
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding of the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> truth. If you really want
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> to persuade anybody that PTS
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> somehow causes
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's theorem not to hold,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> then cite an academic expert
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> who'll have
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some credibility.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If they are mere gibberish
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> words to you then you will
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> not understand.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You don't understand Proof-
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theoritic Semantics, and you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> certainly don't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand Gödel's Theorem,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> neither the theorem itself
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> nor any proof of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> it.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It is a verified fact that
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's G is ungrounded
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in the atomic base of PA.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> That you do not understand
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> what: "grounded in the atomic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> base" means is less
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> than no rebuttal at all.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounded in the atomic base
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> of PA" is an expression used
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> only by you, and it is one
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> which you have never
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> explicitly defined, so the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fault here certainly doesn't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> lie with Alan. It's certainly
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> not a 'verified fact' when you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> haven't even adequately
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> explained what it is that you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> mean.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> All of knowledge expressed in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> language is structured as a
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> tree of semantic relations
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> specified syntactically between
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> finite strings.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> What makes you believe semantic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> relations that can be structured as
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> a tree are sufficient to contain
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> all knowledge that is exressed in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some language?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The CycL language and the Cyc
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Project.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> They use a tree structure for
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> concepts. But why would one try to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> put knowledge in a tree structure?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It must at least be a directed
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> acyclic graph or
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> the proof gets stuck in an infinite
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> loop and never
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> completes.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> How can any ordering of knowledge
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> prevent getting stuck in a loop
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> when looking for a proof?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> By looking upward in a type hierarchy.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If you mean not looking elsewhere that
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> may indeed prevent loops.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In most cases that also prevents
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> finding the proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Truth Conditional Semantics (TCS) <is>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> incoherent
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> compared to Proof Theoretic Semantics
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> (PTS). Essentially
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PTS just coherently connects the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantic meanings
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> expressed in language together into one
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> coherent body
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> of general knowledge. It does this
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> without undecidability
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> or mathematical incompleteness.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Looking for a proof does not need any
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantics so it is not obvious
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> how switching to another semantics could
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> improve it.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In proof theoretic semantics an
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> expression only gains
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantic meaning by finding a proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It should be obvious that finding a proof
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> does not happen before
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> looking for a proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If there is no sequence of inference steps
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in Q from
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> ~∃x x=S(x) to the axioms of Q
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> There are, but that sequence is infinite
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> If there is no FINITE sequence of inference
>>>>>>>>>>>>>>>>>>>>>>>>>>> steps
>>>>>>>>>>>>>>>>>>>>>>>>>>> in Q from ~∃x x=S(x) to the axioms of Q then
>>>>>>>>>>>>>>>>>>>>>>>>>>> ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>>>>>>>>>> is ungrounded in the PTS atomic base of Q.
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>> i.e., ~∃x x=S(x) is unprovable is Q, as is
>>>>>>>>>>>>>>>>>>>>>>>>>> commonly known.
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>> Is it commonly known that ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>> Which has the semantic meaning "no number is
>>>>>>>>>>>>>>>>>>>>>>>> equal to its successor" as per the definition of Q.
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> Since there are no steps in Q that affirm ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>>>>>> in Q it is an open question in Q and not a confirmed
>>>>>>>>>>>>>>>>>>>>>>> statement in Q.
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> In other words, unproven as is commonly known.
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> Yet never gets to undecidable or in any sense of
>>>>>>>>>>>>>>>>>>>>> incomplete.
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> False, as by definition, Q is incomplete because ~∃x
>>>>>>>>>>>>>>>>>>>> x=S(x) is unprovable / out-of-scope / not
>>>>>>>>>>>>>>>>>>>> semantically grounded in Q.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> Proof theoretic semantics DOES NOT DO IT THAT WAY !!!
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> And because PTS claims the semantically valid sentence
>>>>>>>>>>>>>>>>>> in Q "no number is equal to its successor" is not
>>>>>>>>>>>>>>>>>> semantically valid, it must be discarded as useless.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> No you are just not bothering to pay 100% totally
>>>>>>>>>>>>>>>>> complete attention to every single word.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> Wittgenstein
>>>>>>>>>>>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>>>>>>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>>>>>>>>>>>> system' means: the opposite has been proved
>>>>>>>>>>>>>>>>> in Russell's system
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> Proof Theoretic Semantics has almost gotten there.
>>>>>>>>>>>>>>>>> For the most part they stop at semantically grounded
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> i.e. proven
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> and never quite get all the way to True.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> For Wittgenstein's slight extension of the PTS
>>>>>>>>>>>>>>>>> notion ~∃x x=S(x) is untrue
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> i.e. unproven
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> in Q and true in PA.
>>>>>>>>>>>>>>>>> I have been saying it that way long before I ever
>>>>>>>>>>>>>>>>> heard of Wittgenstein.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> So again, you're saying the same thing as everyone else
>>>>>>>>>>>>>>>> but with different words.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> So everyone says that ~∃x x=S(x)
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> which has the semantic meaning "no number is equal to its
>>>>>>>>>>>>>> successor" as per the definition of Q
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> is simply untrue
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> i.e. unprovable
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> in Q and does nor derive either undecidability or
>>>>>>>>>>>>>>> incompleteness?
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> It does derive incompleteness, as by definition Q is
>>>>>>>>>>>>>> incomplete because "no number is equal to its successor"
>>>>>>>>>>>>>> is unprovable in Q.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> Wittgenstein (1937)
>>>>>>>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>>>>>>>> system' means: the opposite has been proved
>>>>>>>>>>>>> in Russell's system
>>>>>>>>>>>>
>>>>>>>>>>>> So what term would you use to describe a sentence that has
>>>>>>>>>>>> an infinite sequence of inference steps between it and the
>>>>>>>>>>>> axioms of the system?
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> untrue and unfalse.
>>>>>>>>>>
>>>>>>>>>> What about the more general case, i.e. a term for a statement
>>>>>>>>>> that has *any* sequence of inference steps, either finite or
>>>>>>>>>> infinite, between it and the axioms of the system? And what
>>>>>>>>>> would the negation of such a statement be called?
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> The truth value of the Goldbach conjecture
>>>>>>>>> may have an infinite number of steps thus
>>>>>>>>> would be unknowable and not a member of the
>>>>>>>>> body of knowledge that can be expressed in
>>>>>>>>> language. Negation has no effect on expressions
>>>>>>>>> that are neither true no false.
>>>>>>>>>
>>>>>>>>> Every finite string including gibberish has the
>>>>>>>>> truth value of: {True, False, Neither}.
>>>>>>>>>
>>>>>>>>> Finite strings are a superset of expressions
>>>>>>>>> of language.
>>>>>>>>>
>>>>>>>>
>>>>>>>> That's not the definition I asked for.
>>>>>>>>
>>>>>>>> What term would you use for a statement that has *any* sequence
>>>>>>>> of inference steps, either finite or infinite, between it and
>>>>>>>> the axioms of the system?
>>>>>>>>
>>>>>>>
>>>>>>> "true on the basis of meaning expressed in language"
>>>>>>> reliably computable for the entire body of GENERAL knowledge.
>>>>>>> Is the limit of the topic of all my posts.
>>>>>>>
>>>>>>> Infinite inference steps are off topic.
>>>>>>
>>>>>> Why are you so reluctant to provide a simple term?
>>>>>>
>>>>>
>>>>> OFF-TOPIC <is> THE TERM.
>>>>
>>>> So we've established that "off-topic" means "a statement that has
>>>> *any* sequence of inference steps, either finite or infinite,
>>>> between it and the axioms of the system".
>>>>
>>>
>>> The ones that have infinite steps out outside the
>>> body of knowledge and off topic for that reason.
>>>
>>
>> So you're claiming the sentence "no number is equal to its successor"
>> in Q is "off topic".
>>
>> Rejected, as it is a semantically valid sentence in the language of Q.
>
> If there is no finite sequence of inference steps
> between x and the axioms of Q then PTS stipulates
> that x is not semantically valid in Q.
And since x = "no number is equal to its successor" is semantically
valid as per the definition of Q, PTS must be discarded as useless.
[toc] | [prev] | [next] | [standalone]
| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-27 18:27 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111pm98$39c4o$1@dont-email.me> |
| In reply to | #645789 |
On 6/27/2026 5:53 PM, dbush wrote:
> On 6/27/2026 6:44 PM, olcott wrote:
>> On 6/27/2026 5:33 PM, dbush wrote:
>>> On 6/27/2026 6:29 PM, olcott wrote:
>>>> On 6/27/2026 5:21 PM, dbush wrote:
>>>>> On 6/27/2026 6:18 PM, olcott wrote:
>>>>>> On 6/27/2026 5:15 PM, dbush wrote:
>>>>>>> On 6/27/2026 6:11 PM, olcott wrote:
>>>>>>>> On 6/27/2026 4:50 PM, dbush wrote:
>>>>>>>>> On 6/27/2026 5:24 PM, olcott wrote:
>>>>>>>>>> On 6/27/2026 3:59 PM, dbush wrote:
>>>>>>>>>>> On 6/27/2026 4:52 PM, olcott wrote:
>>>>>>>>>>>> On 6/27/2026 3:30 PM, dbush wrote:
>>>>>>>>>>>>> On 6/27/2026 4:27 PM, olcott wrote:
>>>>>>>>>>>>>> On 6/27/2026 3:22 PM, dbush wrote:
>>>>>>>>>>>>>>> On 6/27/2026 4:17 PM, olcott wrote:
>>>>>>>>>>>>>>>> On 6/27/2026 3:11 PM, dbush wrote:
>>>>>>>>>>>>>>>>> On 6/27/2026 4:04 PM, olcott wrote:
>>>>>>>>>>>>>>>>>> On 6/27/2026 2:54 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>> On 6/27/2026 3:40 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:23 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 3:16 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:04 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 3:01 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:39 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:38 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:29 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:27 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:01 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 11:43 AM, polcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:35 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 26/06/2026 16:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/26/2026 1:39 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 25/06/2026 19:14, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/25/2026 2:21 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 24/06/2026 23:26, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/24/2026 5:00 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 23/06/2026 17:48, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/23/2026 1:06 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 15:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/22/2026 1:49 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 02:02, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 4:08 PM, André G.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Isaak wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-21 14:42, olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 3:04 PM, Alan
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> [ Followup-To: set ]
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 6:26 AM, Alan
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I just found the term:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounding in a proof
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theoretic atomic base"
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> yesterday.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You can find any number
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> of terms. That doesn't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> mean you're capable of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding them.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The above is the key
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> reason why under PTS Gödel
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1931 incompleteness
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fails.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I don't believe you. You
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> have no respect for or
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding of the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> truth. If you really want
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> to persuade anybody that
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PTS somehow causes
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's theorem not to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> hold, then cite an academic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> expert who'll have
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some credibility.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If they are mere gibberish
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> words to you then you will
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> not understand.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You don't understand Proof-
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theoritic Semantics, and
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> you certainly don't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand Gödel's Theorem,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> neither the theorem itself
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> nor any proof of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> it.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It is a verified fact that
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's G is ungrounded
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in the atomic base of PA.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> That you do not understand
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> what: "grounded in the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> atomic base" means is less
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> than no rebuttal at all.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounded in the atomic base
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> of PA" is an expression used
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> only by you, and it is one
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> which you have never
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> explicitly defined, so the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fault here certainly doesn't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> lie with Alan. It's certainly
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> not a 'verified fact' when
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> you haven't even adequately
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> explained what it is that you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> mean.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> All of knowledge expressed in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> language is structured as a
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> tree of semantic relations
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> specified syntactically
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> between finite strings.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> What makes you believe semantic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> relations that can be
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> structured as
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> a tree are sufficient to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> contain all knowledge that is
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> exressed in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some language?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The CycL language and the Cyc
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Project.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> They use a tree structure for
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> concepts. But why would one try to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> put knowledge in a tree structure?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It must at least be a directed
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> acyclic graph or
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> the proof gets stuck in an
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> infinite loop and never
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> completes.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> How can any ordering of knowledge
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> prevent getting stuck in a loop
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> when looking for a proof?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> By looking upward in a type hierarchy.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If you mean not looking elsewhere
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> that may indeed prevent loops.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In most cases that also prevents
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> finding the proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Truth Conditional Semantics (TCS) <is>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> incoherent
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> compared to Proof Theoretic Semantics
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> (PTS). Essentially
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PTS just coherently connects the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantic meanings
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> expressed in language together into
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> one coherent body
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> of general knowledge. It does this
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> without undecidability
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> or mathematical incompleteness.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Looking for a proof does not need any
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantics so it is not obvious
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> how switching to another semantics
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> could improve it.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In proof theoretic semantics an
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> expression only gains
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantic meaning by finding a proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It should be obvious that finding a proof
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> does not happen before
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> looking for a proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If there is no sequence of inference steps
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in Q from
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> ~∃x x=S(x) to the axioms of Q
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> There are, but that sequence is infinite
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> If there is no FINITE sequence of inference
>>>>>>>>>>>>>>>>>>>>>>>>>>>> steps
>>>>>>>>>>>>>>>>>>>>>>>>>>>> in Q from ~∃x x=S(x) to the axioms of Q then
>>>>>>>>>>>>>>>>>>>>>>>>>>>> ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>>>>>>>>>>> is ungrounded in the PTS atomic base of Q.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> i.e., ~∃x x=S(x) is unprovable is Q, as is
>>>>>>>>>>>>>>>>>>>>>>>>>>> commonly known.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>> Is it commonly known that ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>> Which has the semantic meaning "no number is
>>>>>>>>>>>>>>>>>>>>>>>>> equal to its successor" as per the definition
>>>>>>>>>>>>>>>>>>>>>>>>> of Q.
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>> Since there are no steps in Q that affirm ~∃x
>>>>>>>>>>>>>>>>>>>>>>>> x=S(x)
>>>>>>>>>>>>>>>>>>>>>>>> in Q it is an open question in Q and not a
>>>>>>>>>>>>>>>>>>>>>>>> confirmed
>>>>>>>>>>>>>>>>>>>>>>>> statement in Q.
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> In other words, unproven as is commonly known.
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> Yet never gets to undecidable or in any sense of
>>>>>>>>>>>>>>>>>>>>>> incomplete.
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> False, as by definition, Q is incomplete because
>>>>>>>>>>>>>>>>>>>>> ~∃x x=S(x) is unprovable / out-of-scope / not
>>>>>>>>>>>>>>>>>>>>> semantically grounded in Q.
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> Proof theoretic semantics DOES NOT DO IT THAT WAY !!!
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> And because PTS claims the semantically valid
>>>>>>>>>>>>>>>>>>> sentence in Q "no number is equal to its successor"
>>>>>>>>>>>>>>>>>>> is not semantically valid, it must be discarded as
>>>>>>>>>>>>>>>>>>> useless.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> No you are just not bothering to pay 100% totally
>>>>>>>>>>>>>>>>>> complete attention to every single word.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> Wittgenstein
>>>>>>>>>>>>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>>>>>>>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>>>>>>>>>>>>> system' means: the opposite has been proved
>>>>>>>>>>>>>>>>>> in Russell's system
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> Proof Theoretic Semantics has almost gotten there.
>>>>>>>>>>>>>>>>>> For the most part they stop at semantically grounded
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> i.e. proven
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> and never quite get all the way to True.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> For Wittgenstein's slight extension of the PTS
>>>>>>>>>>>>>>>>>> notion ~∃x x=S(x) is untrue
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> i.e. unproven
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> in Q and true in PA.
>>>>>>>>>>>>>>>>>> I have been saying it that way long before I ever
>>>>>>>>>>>>>>>>>> heard of Wittgenstein.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> So again, you're saying the same thing as everyone else
>>>>>>>>>>>>>>>>> but with different words.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> So everyone says that ~∃x x=S(x)
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> which has the semantic meaning "no number is equal to its
>>>>>>>>>>>>>>> successor" as per the definition of Q
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> is simply untrue
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> i.e. unprovable
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> in Q and does nor derive either undecidability or
>>>>>>>>>>>>>>>> incompleteness?
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> It does derive incompleteness, as by definition Q is
>>>>>>>>>>>>>>> incomplete because "no number is equal to its successor"
>>>>>>>>>>>>>>> is unprovable in Q.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Wittgenstein (1937)
>>>>>>>>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>>>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>>>>>>>>> system' means: the opposite has been proved
>>>>>>>>>>>>>> in Russell's system
>>>>>>>>>>>>>
>>>>>>>>>>>>> So what term would you use to describe a sentence that has
>>>>>>>>>>>>> an infinite sequence of inference steps between it and the
>>>>>>>>>>>>> axioms of the system?
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> untrue and unfalse.
>>>>>>>>>>>
>>>>>>>>>>> What about the more general case, i.e. a term for a statement
>>>>>>>>>>> that has *any* sequence of inference steps, either finite or
>>>>>>>>>>> infinite, between it and the axioms of the system? And what
>>>>>>>>>>> would the negation of such a statement be called?
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> The truth value of the Goldbach conjecture
>>>>>>>>>> may have an infinite number of steps thus
>>>>>>>>>> would be unknowable and not a member of the
>>>>>>>>>> body of knowledge that can be expressed in
>>>>>>>>>> language. Negation has no effect on expressions
>>>>>>>>>> that are neither true no false.
>>>>>>>>>>
>>>>>>>>>> Every finite string including gibberish has the
>>>>>>>>>> truth value of: {True, False, Neither}.
>>>>>>>>>>
>>>>>>>>>> Finite strings are a superset of expressions
>>>>>>>>>> of language.
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> That's not the definition I asked for.
>>>>>>>>>
>>>>>>>>> What term would you use for a statement that has *any* sequence
>>>>>>>>> of inference steps, either finite or infinite, between it and
>>>>>>>>> the axioms of the system?
>>>>>>>>>
>>>>>>>>
>>>>>>>> "true on the basis of meaning expressed in language"
>>>>>>>> reliably computable for the entire body of GENERAL knowledge.
>>>>>>>> Is the limit of the topic of all my posts.
>>>>>>>>
>>>>>>>> Infinite inference steps are off topic.
>>>>>>>
>>>>>>> Why are you so reluctant to provide a simple term?
>>>>>>>
>>>>>>
>>>>>> OFF-TOPIC <is> THE TERM.
>>>>>
>>>>> So we've established that "off-topic" means "a statement that has
>>>>> *any* sequence of inference steps, either finite or infinite,
>>>>> between it and the axioms of the system".
>>>>>
>>>>
>>>> The ones that have infinite steps out outside the
>>>> body of knowledge and off topic for that reason.
>>>>
>>>
>>> So you're claiming the sentence "no number is equal to its successor"
>>> in Q is "off topic".
>>>
>>> Rejected, as it is a semantically valid sentence in the language of Q.
>>
>> If there is no finite sequence of inference steps
>> between x and the axioms of Q then PTS stipulates
>> that x is not semantically valid in Q.
>
> And since x = "no number is equal to its successor" is semantically
> valid as per the definition of Q, PTS must be discarded as useless.
>
So you also have no idea what Stipulative definition is.
A stipulative definition is a type of definition in which
a new or currently existing term is given a new specific meaning
https://en.wikipedia.org/wiki/Stipulative_definition
--
Copyright 2026 Olcott
My 28 year goal has been to make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
The complete structure of this system is now defined.
The entire body of knowledge expressed in language is
comprised of two types of relations between finite strings:
(a) *Axioms* Expressions of language that are stipulated to be true.
My system bridges the analytic/synthetic distinction by
expressly encoding all empirical "atomic facts" in a formal
language such as CycL of the Cyc project.
(b) *Inference Rules* Expressions of language that are semantically
entailed syntactically from (a) and/or (b).
[toc] | [prev] | [next] | [standalone]
| From | dbush <dbush.mobile@gmail.com> |
|---|---|
| Date | 2026-06-27 19:33 -0400 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111pmjq$38fsa$4@dont-email.me> |
| In reply to | #645791 |
On 6/27/2026 7:27 PM, olcott wrote:
> On 6/27/2026 5:53 PM, dbush wrote:
>> On 6/27/2026 6:44 PM, olcott wrote:
>>> On 6/27/2026 5:33 PM, dbush wrote:
>>>> On 6/27/2026 6:29 PM, olcott wrote:
>>>>> On 6/27/2026 5:21 PM, dbush wrote:
>>>>>> On 6/27/2026 6:18 PM, olcott wrote:
>>>>>>> On 6/27/2026 5:15 PM, dbush wrote:
>>>>>>>> On 6/27/2026 6:11 PM, olcott wrote:
>>>>>>>>> On 6/27/2026 4:50 PM, dbush wrote:
>>>>>>>>>> On 6/27/2026 5:24 PM, olcott wrote:
>>>>>>>>>>> On 6/27/2026 3:59 PM, dbush wrote:
>>>>>>>>>>>> On 6/27/2026 4:52 PM, olcott wrote:
>>>>>>>>>>>>> On 6/27/2026 3:30 PM, dbush wrote:
>>>>>>>>>>>>>> On 6/27/2026 4:27 PM, olcott wrote:
>>>>>>>>>>>>>>> On 6/27/2026 3:22 PM, dbush wrote:
>>>>>>>>>>>>>>>> On 6/27/2026 4:17 PM, olcott wrote:
>>>>>>>>>>>>>>>>> On 6/27/2026 3:11 PM, dbush wrote:
>>>>>>>>>>>>>>>>>> On 6/27/2026 4:04 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:54 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>> On 6/27/2026 3:40 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:23 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 3:16 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:04 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 3:01 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:39 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:38 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:29 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:27 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:01 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 11:43 AM, polcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:35 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 26/06/2026 16:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/26/2026 1:39 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 25/06/2026 19:14, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/25/2026 2:21 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 24/06/2026 23:26, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/24/2026 5:00 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 23/06/2026 17:48, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/23/2026 1:06 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 15:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/22/2026 1:49 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 02:02, olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 4:08 PM, André
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> G. Isaak wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-21 14:42, olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 3:04 PM, Alan
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> [ Followup-To: set ]
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 6:26 AM,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Alan Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I just found the term:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounding in a proof
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theoretic atomic base"
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> yesterday.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You can find any number
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> of terms. That doesn't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> mean you're capable of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding them.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The above is the key
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> reason why under PTS
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel 1931 incompleteness
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fails.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I don't believe you. You
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> have no respect for or
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding of the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> truth. If you really want
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> to persuade anybody that
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PTS somehow causes
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's theorem not to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> hold, then cite an
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> academic expert who'll have
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some credibility.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If they are mere
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> gibberish words to you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> then you will not
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You don't understand
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Proof- theoritic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Semantics, and you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> certainly don't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand Gödel's
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Theorem, neither the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theorem itself nor any
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> proof of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> it.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It is a verified fact that
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's G is ungrounded
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in the atomic base of PA.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> That you do not understand
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> what: "grounded in the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> atomic base" means is less
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> than no rebuttal at all.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounded in the atomic base
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> of PA" is an expression used
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> only by you, and it is one
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> which you have never
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> explicitly defined, so the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fault here certainly doesn't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> lie with Alan. It's
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> certainly not a 'verified
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fact' when you haven't even
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> adequately explained what it
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> is that you mean.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> All of knowledge expressed in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> language is structured as a
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> tree of semantic relations
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> specified syntactically
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> between finite strings.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> What makes you believe
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantic relations that can be
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> structured as
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> a tree are sufficient to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> contain all knowledge that is
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> exressed in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some language?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The CycL language and the Cyc
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Project.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> They use a tree structure for
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> concepts. But why would one try to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> put knowledge in a tree structure?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It must at least be a directed
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> acyclic graph or
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> the proof gets stuck in an
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> infinite loop and never
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> completes.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> How can any ordering of knowledge
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> prevent getting stuck in a loop
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> when looking for a proof?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> By looking upward in a type hierarchy.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If you mean not looking elsewhere
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> that may indeed prevent loops.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In most cases that also prevents
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> finding the proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Truth Conditional Semantics (TCS)
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <is> incoherent
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> compared to Proof Theoretic Semantics
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> (PTS). Essentially
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PTS just coherently connects the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantic meanings
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> expressed in language together into
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> one coherent body
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> of general knowledge. It does this
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> without undecidability
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> or mathematical incompleteness.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Looking for a proof does not need any
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantics so it is not obvious
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> how switching to another semantics
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> could improve it.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In proof theoretic semantics an
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> expression only gains
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantic meaning by finding a proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It should be obvious that finding a
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> proof does not happen before
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> looking for a proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If there is no sequence of inference
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> steps in Q from
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> ~∃x x=S(x) to the axioms of Q
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> There are, but that sequence is infinite
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If there is no FINITE sequence of inference
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> steps
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in Q from ~∃x x=S(x) to the axioms of Q
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> then ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> is ungrounded in the PTS atomic base of Q.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> i.e., ~∃x x=S(x) is unprovable is Q, as is
>>>>>>>>>>>>>>>>>>>>>>>>>>>> commonly known.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> Is it commonly known that ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>> Which has the semantic meaning "no number is
>>>>>>>>>>>>>>>>>>>>>>>>>> equal to its successor" as per the definition
>>>>>>>>>>>>>>>>>>>>>>>>>> of Q.
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>> Since there are no steps in Q that affirm ~∃x
>>>>>>>>>>>>>>>>>>>>>>>>> x=S(x)
>>>>>>>>>>>>>>>>>>>>>>>>> in Q it is an open question in Q and not a
>>>>>>>>>>>>>>>>>>>>>>>>> confirmed
>>>>>>>>>>>>>>>>>>>>>>>>> statement in Q.
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>> In other words, unproven as is commonly known.
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> Yet never gets to undecidable or in any sense of
>>>>>>>>>>>>>>>>>>>>>>> incomplete.
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> False, as by definition, Q is incomplete because
>>>>>>>>>>>>>>>>>>>>>> ~∃x x=S(x) is unprovable / out-of-scope / not
>>>>>>>>>>>>>>>>>>>>>> semantically grounded in Q.
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> Proof theoretic semantics DOES NOT DO IT THAT WAY !!!
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> And because PTS claims the semantically valid
>>>>>>>>>>>>>>>>>>>> sentence in Q "no number is equal to its successor"
>>>>>>>>>>>>>>>>>>>> is not semantically valid, it must be discarded as
>>>>>>>>>>>>>>>>>>>> useless.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> No you are just not bothering to pay 100% totally
>>>>>>>>>>>>>>>>>>> complete attention to every single word.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> Wittgenstein
>>>>>>>>>>>>>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>>>>>>>>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>>>>>>>>>>>>>> system' means: the opposite has been proved
>>>>>>>>>>>>>>>>>>> in Russell's system
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> Proof Theoretic Semantics has almost gotten there.
>>>>>>>>>>>>>>>>>>> For the most part they stop at semantically grounded
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> i.e. proven
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> and never quite get all the way to True.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> For Wittgenstein's slight extension of the PTS
>>>>>>>>>>>>>>>>>>> notion ~∃x x=S(x) is untrue
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> i.e. unproven
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> in Q and true in PA.
>>>>>>>>>>>>>>>>>>> I have been saying it that way long before I ever
>>>>>>>>>>>>>>>>>>> heard of Wittgenstein.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> So again, you're saying the same thing as everyone
>>>>>>>>>>>>>>>>>> else but with different words.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> So everyone says that ~∃x x=S(x)
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> which has the semantic meaning "no number is equal to
>>>>>>>>>>>>>>>> its successor" as per the definition of Q
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> is simply untrue
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> i.e. unprovable
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> in Q and does nor derive either undecidability or
>>>>>>>>>>>>>>>>> incompleteness?
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> It does derive incompleteness, as by definition Q is
>>>>>>>>>>>>>>>> incomplete because "no number is equal to its successor"
>>>>>>>>>>>>>>>> is unprovable in Q.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Wittgenstein (1937)
>>>>>>>>>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>>>>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>>>>>>>>>> system' means: the opposite has been proved
>>>>>>>>>>>>>>> in Russell's system
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> So what term would you use to describe a sentence that has
>>>>>>>>>>>>>> an infinite sequence of inference steps between it and the
>>>>>>>>>>>>>> axioms of the system?
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> untrue and unfalse.
>>>>>>>>>>>>
>>>>>>>>>>>> What about the more general case, i.e. a term for a
>>>>>>>>>>>> statement that has *any* sequence of inference steps, either
>>>>>>>>>>>> finite or infinite, between it and the axioms of the
>>>>>>>>>>>> system? And what would the negation of such a statement be
>>>>>>>>>>>> called?
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> The truth value of the Goldbach conjecture
>>>>>>>>>>> may have an infinite number of steps thus
>>>>>>>>>>> would be unknowable and not a member of the
>>>>>>>>>>> body of knowledge that can be expressed in
>>>>>>>>>>> language. Negation has no effect on expressions
>>>>>>>>>>> that are neither true no false.
>>>>>>>>>>>
>>>>>>>>>>> Every finite string including gibberish has the
>>>>>>>>>>> truth value of: {True, False, Neither}.
>>>>>>>>>>>
>>>>>>>>>>> Finite strings are a superset of expressions
>>>>>>>>>>> of language.
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> That's not the definition I asked for.
>>>>>>>>>>
>>>>>>>>>> What term would you use for a statement that has *any*
>>>>>>>>>> sequence of inference steps, either finite or infinite,
>>>>>>>>>> between it and the axioms of the system?
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> "true on the basis of meaning expressed in language"
>>>>>>>>> reliably computable for the entire body of GENERAL knowledge.
>>>>>>>>> Is the limit of the topic of all my posts.
>>>>>>>>>
>>>>>>>>> Infinite inference steps are off topic.
>>>>>>>>
>>>>>>>> Why are you so reluctant to provide a simple term?
>>>>>>>>
>>>>>>>
>>>>>>> OFF-TOPIC <is> THE TERM.
>>>>>>
>>>>>> So we've established that "off-topic" means "a statement that has
>>>>>> *any* sequence of inference steps, either finite or infinite,
>>>>>> between it and the axioms of the system".
>>>>>>
>>>>>
>>>>> The ones that have infinite steps out outside the
>>>>> body of knowledge and off topic for that reason.
>>>>>
>>>>
>>>> So you're claiming the sentence "no number is equal to its
>>>> successor" in Q is "off topic".
>>>>
>>>> Rejected, as it is a semantically valid sentence in the language of Q.
>>>
>>> If there is no finite sequence of inference steps
>>> between x and the axioms of Q then PTS stipulates
>>> that x is not semantically valid in Q.
>>
>> And since x = "no number is equal to its successor" is semantically
>> valid as per the definition of Q, PTS must be discarded as useless.
>>
>
> So you also have no idea what Stipulative definition is.
>
> A stipulative definition is a type of definition in which
> a new or currently existing term is given a new specific meaning
>
> https://en.wikipedia.org/wiki/Stipulative_definition
Not allowed, as "semantically valid" is already defined, and "no number
is equal to its successor" meets that definition.
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| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-27 18:59 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111po5g$39okg$1@dont-email.me> |
| In reply to | #645793 |
On 6/27/2026 6:33 PM, dbush wrote:
> On 6/27/2026 7:27 PM, olcott wrote:
>> On 6/27/2026 5:53 PM, dbush wrote:
>>> On 6/27/2026 6:44 PM, olcott wrote:
>>>> On 6/27/2026 5:33 PM, dbush wrote:
>>>>> On 6/27/2026 6:29 PM, olcott wrote:
>>>>>> On 6/27/2026 5:21 PM, dbush wrote:
>>>>>>> On 6/27/2026 6:18 PM, olcott wrote:
>>>>>>>> On 6/27/2026 5:15 PM, dbush wrote:
>>>>>>>>> On 6/27/2026 6:11 PM, olcott wrote:
>>>>>>>>>> On 6/27/2026 4:50 PM, dbush wrote:
>>>>>>>>>>> On 6/27/2026 5:24 PM, olcott wrote:
>>>>>>>>>>>> On 6/27/2026 3:59 PM, dbush wrote:
>>>>>>>>>>>>> On 6/27/2026 4:52 PM, olcott wrote:
>>>>>>>>>>>>>> On 6/27/2026 3:30 PM, dbush wrote:
>>>>>>>>>>>>>>> On 6/27/2026 4:27 PM, olcott wrote:
>>>>>>>>>>>>>>>> On 6/27/2026 3:22 PM, dbush wrote:
>>>>>>>>>>>>>>>>> On 6/27/2026 4:17 PM, olcott wrote:
>>>>>>>>>>>>>>>>>> On 6/27/2026 3:11 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>> On 6/27/2026 4:04 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:54 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 3:40 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:23 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 3:16 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:04 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 3:01 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:39 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:38 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:29 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:27 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 1:01 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 11:43 AM, polcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/27/2026 2:35 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 26/06/2026 16:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/26/2026 1:39 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 25/06/2026 19:14, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/25/2026 2:21 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 24/06/2026 23:26, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/24/2026 5:00 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 23/06/2026 17:48, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/23/2026 1:06 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 15:10, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/22/2026 1:49 AM, Mikko
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 02:02, olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 4:08 PM, André
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> G. Isaak wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-21 14:42, olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 3:04 PM, Alan
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> [ Followup-To: set ]
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 6:26 AM,
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Alan Mackenzie wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I just found the term:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounding in a proof
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theoretic atomic base"
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> yesterday.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You can find any number
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> of terms. That doesn't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> mean you're capable of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding them.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The above is the key
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> reason why under PTS
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel 1931 incompleteness
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fails.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I don't believe you. You
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> have no respect for or
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding of the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> truth. If you really
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> want to persuade anybody
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> that PTS somehow causes
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's theorem not to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> hold, then cite an
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> academic expert who'll have
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some credibility.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If they are mere
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> gibberish words to you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> then you will not
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You don't understand
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Proof- theoritic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Semantics, and you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> certainly don't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand Gödel's
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Theorem, neither the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theorem itself nor any
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> proof of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> it.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It is a verified fact that
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's G is ungrounded
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in the atomic base of PA.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> That you do not understand
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> what: "grounded in the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> atomic base" means is less
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> than no rebuttal at all.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounded in the atomic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> base of PA" is an
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> expression used only by
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> you, and it is one which
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> you have never explicitly
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> defined, so the fault here
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> certainly doesn't lie with
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Alan. It's certainly not a
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 'verified fact' when you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> haven't even adequately
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> explained what it is that
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> you mean.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> All of knowledge expressed
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in language is structured as
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> a tree of semantic relations
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> specified syntactically
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> between finite strings.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> What makes you believe
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantic relations that can
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> be structured as
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> a tree are sufficient to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> contain all knowledge that is
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> exressed in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some language?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The CycL language and the Cyc
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Project.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> They use a tree structure for
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> concepts. But why would one try to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> put knowledge in a tree structure?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It must at least be a directed
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> acyclic graph or
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> the proof gets stuck in an
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> infinite loop and never
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> completes.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> How can any ordering of knowledge
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> prevent getting stuck in a loop
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> when looking for a proof?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> By looking upward in a type
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> hierarchy.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If you mean not looking elsewhere
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> that may indeed prevent loops.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In most cases that also prevents
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> finding the proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Truth Conditional Semantics (TCS)
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <is> incoherent
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> compared to Proof Theoretic
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Semantics (PTS). Essentially
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PTS just coherently connects the
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantic meanings
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> expressed in language together into
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> one coherent body
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> of general knowledge. It does this
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> without undecidability
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> or mathematical incompleteness.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Looking for a proof does not need any
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantics so it is not obvious
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> how switching to another semantics
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> could improve it.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In proof theoretic semantics an
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> expression only gains
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> semantic meaning by finding a proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It should be obvious that finding a
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> proof does not happen before
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> looking for a proof.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If there is no sequence of inference
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> steps in Q from
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> ~∃x x=S(x) to the axioms of Q
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> There are, but that sequence is infinite
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If there is no FINITE sequence of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> inference steps
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in Q from ~∃x x=S(x) to the axioms of Q
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> then ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> is ungrounded in the PTS atomic base of Q.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> i.e., ~∃x x=S(x) is unprovable is Q, as is
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> commonly known.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> Is it commonly known that ~∃x x=S(x)
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> Which has the semantic meaning "no number is
>>>>>>>>>>>>>>>>>>>>>>>>>>> equal to its successor" as per the definition
>>>>>>>>>>>>>>>>>>>>>>>>>>> of Q.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>> Since there are no steps in Q that affirm ~∃x
>>>>>>>>>>>>>>>>>>>>>>>>>> x=S(x)
>>>>>>>>>>>>>>>>>>>>>>>>>> in Q it is an open question in Q and not a
>>>>>>>>>>>>>>>>>>>>>>>>>> confirmed
>>>>>>>>>>>>>>>>>>>>>>>>>> statement in Q.
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>> In other words, unproven as is commonly known.
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>> Yet never gets to undecidable or in any sense of
>>>>>>>>>>>>>>>>>>>>>>>> incomplete.
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> False, as by definition, Q is incomplete because
>>>>>>>>>>>>>>>>>>>>>>> ~∃x x=S(x) is unprovable / out-of-scope / not
>>>>>>>>>>>>>>>>>>>>>>> semantically grounded in Q.
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> Proof theoretic semantics DOES NOT DO IT THAT WAY !!!
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> And because PTS claims the semantically valid
>>>>>>>>>>>>>>>>>>>>> sentence in Q "no number is equal to its successor"
>>>>>>>>>>>>>>>>>>>>> is not semantically valid, it must be discarded as
>>>>>>>>>>>>>>>>>>>>> useless.
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> No you are just not bothering to pay 100% totally
>>>>>>>>>>>>>>>>>>>> complete attention to every single word.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> Wittgenstein
>>>>>>>>>>>>>>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>>>>>>>>>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>>>>>>>>>>>>>>> system' means: the opposite has been proved
>>>>>>>>>>>>>>>>>>>> in Russell's system
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> Proof Theoretic Semantics has almost gotten there.
>>>>>>>>>>>>>>>>>>>> For the most part they stop at semantically grounded
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> i.e. proven
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> and never quite get all the way to True.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> For Wittgenstein's slight extension of the PTS
>>>>>>>>>>>>>>>>>>>> notion ~∃x x=S(x) is untrue
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> i.e. unproven
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> in Q and true in PA.
>>>>>>>>>>>>>>>>>>>> I have been saying it that way long before I ever
>>>>>>>>>>>>>>>>>>>> heard of Wittgenstein.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> So again, you're saying the same thing as everyone
>>>>>>>>>>>>>>>>>>> else but with different words.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> So everyone says that ~∃x x=S(x)
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> which has the semantic meaning "no number is equal to
>>>>>>>>>>>>>>>>> its successor" as per the definition of Q
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> is simply untrue
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> i.e. unprovable
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> in Q and does nor derive either undecidability or
>>>>>>>>>>>>>>>>>> incompleteness?
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> It does derive incompleteness, as by definition Q is
>>>>>>>>>>>>>>>>> incomplete because "no number is equal to its
>>>>>>>>>>>>>>>>> successor" is unprovable in Q.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Wittgenstein (1937)
>>>>>>>>>>>>>>>> 'True in Russell's system' means, as was said:
>>>>>>>>>>>>>>>> proved in Russell's system; and 'false in Russell's
>>>>>>>>>>>>>>>> system' means: the opposite has been proved
>>>>>>>>>>>>>>>> in Russell's system
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> So what term would you use to describe a sentence that
>>>>>>>>>>>>>>> has an infinite sequence of inference steps between it
>>>>>>>>>>>>>>> and the axioms of the system?
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> untrue and unfalse.
>>>>>>>>>>>>>
>>>>>>>>>>>>> What about the more general case, i.e. a term for a
>>>>>>>>>>>>> statement that has *any* sequence of inference steps,
>>>>>>>>>>>>> either finite or infinite, between it and the axioms of the
>>>>>>>>>>>>> system? And what would the negation of such a statement be
>>>>>>>>>>>>> called?
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> The truth value of the Goldbach conjecture
>>>>>>>>>>>> may have an infinite number of steps thus
>>>>>>>>>>>> would be unknowable and not a member of the
>>>>>>>>>>>> body of knowledge that can be expressed in
>>>>>>>>>>>> language. Negation has no effect on expressions
>>>>>>>>>>>> that are neither true no false.
>>>>>>>>>>>>
>>>>>>>>>>>> Every finite string including gibberish has the
>>>>>>>>>>>> truth value of: {True, False, Neither}.
>>>>>>>>>>>>
>>>>>>>>>>>> Finite strings are a superset of expressions
>>>>>>>>>>>> of language.
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> That's not the definition I asked for.
>>>>>>>>>>>
>>>>>>>>>>> What term would you use for a statement that has *any*
>>>>>>>>>>> sequence of inference steps, either finite or infinite,
>>>>>>>>>>> between it and the axioms of the system?
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> "true on the basis of meaning expressed in language"
>>>>>>>>>> reliably computable for the entire body of GENERAL knowledge.
>>>>>>>>>> Is the limit of the topic of all my posts.
>>>>>>>>>>
>>>>>>>>>> Infinite inference steps are off topic.
>>>>>>>>>
>>>>>>>>> Why are you so reluctant to provide a simple term?
>>>>>>>>>
>>>>>>>>
>>>>>>>> OFF-TOPIC <is> THE TERM.
>>>>>>>
>>>>>>> So we've established that "off-topic" means "a statement that has
>>>>>>> *any* sequence of inference steps, either finite or infinite,
>>>>>>> between it and the axioms of the system".
>>>>>>>
>>>>>>
>>>>>> The ones that have infinite steps out outside the
>>>>>> body of knowledge and off topic for that reason.
>>>>>>
>>>>>
>>>>> So you're claiming the sentence "no number is equal to its
>>>>> successor" in Q is "off topic".
>>>>>
>>>>> Rejected, as it is a semantically valid sentence in the language of Q.
>>>>
>>>> If there is no finite sequence of inference steps
>>>> between x and the axioms of Q then PTS stipulates
>>>> that x is not semantically valid in Q.
>>>
>>> And since x = "no number is equal to its successor" is semantically
>>> valid as per the definition of Q, PTS must be discarded as useless.
>>>
>>
>> So you also have no idea what Stipulative definition is.
>>
>> A stipulative definition is a type of definition in which
>> a new or currently existing term is given a new specific meaning
>>
>> https://en.wikipedia.org/wiki/Stipulative_definition
>
> Not allowed, as "semantically valid" is already defined, and "no number
> is equal to its successor" meets that definition.
>
Proof Theoretic Semantics (PTS) supersedes and overrules this.
Truth Conditional Semantics (TCS) is only a point of view
it is not the infallible word of God. PTS is another
incompatible point of view.
--
Copyright 2026 Olcott
My 28 year goal has been to make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
The complete structure of this system is now defined.
The entire body of knowledge expressed in language is
comprised of two types of relations between finite strings:
(a) *Axioms* Expressions of language that are stipulated to be true.
My system bridges the analytic/synthetic distinction by
expressly encoding all empirical "atomic facts" in a formal
language such as CycL of the Cyc project.
(b) *Inference Rules* Expressions of language that are semantically
entailed syntactically from (a) and/or (b).
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| From | dbush <dbush.mobile@gmail.com> |
|---|---|
| Date | 2026-06-27 21:13 -0400 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111psga$3ak9b$2@dont-email.me> |
| In reply to | #645795 |
On 6/27/2026 7:59 PM, olcott wrote: > On 6/27/2026 6:33 PM, dbush wrote: >> On 6/27/2026 7:27 PM, olcott wrote: >>> On 6/27/2026 5:53 PM, dbush wrote: >>>> On 6/27/2026 6:44 PM, olcott wrote: >>>>> On 6/27/2026 5:33 PM, dbush wrote: >>>>>> >>>>>> So you're claiming the sentence "no number is equal to its >>>>>> successor" in Q is "off topic". >>>>>> >>>>>> Rejected, as it is a semantically valid sentence in the language >>>>>> of Q. >>>>> >>>>> If there is no finite sequence of inference steps >>>>> between x and the axioms of Q then PTS stipulates >>>>> that x is not semantically valid in Q. >>>> >>>> And since x = "no number is equal to its successor" is semantically >>>> valid as per the definition of Q, PTS must be discarded as useless. >>>> >>> >>> So you also have no idea what Stipulative definition is. >>> >>> A stipulative definition is a type of definition in which >>> a new or currently existing term is given a new specific meaning >>> >>> https://en.wikipedia.org/wiki/Stipulative_definition >> >> Not allowed, as "semantically valid" is already defined, and "no >> number is equal to its successor" meets that definition. >> > > Proof Theoretic Semantics (PTS) supersedes and overrules this. Then PTS is discarded as useless because it rejects "no number is equal to its successor".
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| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-27 20:33 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111ptm1$3aum9$1@dont-email.me> |
| In reply to | #645797 |
On 6/27/2026 8:13 PM, dbush wrote: > On 6/27/2026 7:59 PM, olcott wrote: >> On 6/27/2026 6:33 PM, dbush wrote: >>> On 6/27/2026 7:27 PM, olcott wrote: >>>> On 6/27/2026 5:53 PM, dbush wrote: >>>>> On 6/27/2026 6:44 PM, olcott wrote: >>>>>> On 6/27/2026 5:33 PM, dbush wrote: >>>>>>> >>>>>>> So you're claiming the sentence "no number is equal to its >>>>>>> successor" in Q is "off topic". >>>>>>> >>>>>>> Rejected, as it is a semantically valid sentence in the language >>>>>>> of Q. >>>>>> >>>>>> If there is no finite sequence of inference steps >>>>>> between x and the axioms of Q then PTS stipulates >>>>>> that x is not semantically valid in Q. >>>>> >>>>> And since x = "no number is equal to its successor" is semantically >>>>> valid as per the definition of Q, PTS must be discarded as useless. >>>>> >>>> >>>> So you also have no idea what Stipulative definition is. >>>> >>>> A stipulative definition is a type of definition in which >>>> a new or currently existing term is given a new specific meaning >>>> >>>> https://en.wikipedia.org/wiki/Stipulative_definition >>> >>> Not allowed, as "semantically valid" is already defined, and "no >>> number is equal to its successor" meets that definition. >>> >> >> Proof Theoretic Semantics (PTS) supersedes and overrules this. > > Then PTS is discarded as useless because it rejects "no number is equal > to its successor". > PTS is just the same as when a word is undefined then it has no meaning. TCS says that if an English word has no defined meaning in English then we will just pull one from Chinese. -- Copyright 2026 Olcott My 28 year goal has been to make "true on the basis of meaning expressed in language" reliably computable for the entire body of knowledge. The complete structure of this system is now defined. The entire body of knowledge expressed in language is comprised of two types of relations between finite strings: (a) *Axioms* Expressions of language that are stipulated to be true. My system bridges the analytic/synthetic distinction by expressly encoding all empirical "atomic facts" in a formal language such as CycL of the Cyc project. (b) *Inference Rules* Expressions of language that are semantically entailed syntactically from (a) and/or (b).
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| From | Mikko <mikko.levanto@iki.fi> |
|---|---|
| Date | 2026-06-28 12:38 +0300 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111qq37$3haak$1@dont-email.me> |
| In reply to | #645766 |
On 27/06/2026 23:04, olcott wrote: > On 6/27/2026 2:54 PM, dbush wrote: >> On 6/27/2026 3:40 PM, olcott wrote: >>> On 6/27/2026 2:23 PM, dbush wrote: >>>> On 6/27/2026 3:16 PM, olcott wrote: >>>>> On 6/27/2026 2:04 PM, dbush wrote: >>>>>> On 6/27/2026 3:01 PM, olcott wrote: >>>>>>> On 6/27/2026 1:39 PM, dbush wrote: >>>>>>>> On 6/27/2026 2:38 PM, olcott wrote: >>>>>>>>> On 6/27/2026 1:29 PM, dbush wrote: >>>>>>>>>> On 6/27/2026 2:27 PM, olcott wrote: >>>>>>>>>>> On 6/27/2026 1:01 PM, dbush wrote: >>>>>>>>>>>> On 6/27/2026 11:43 AM, polcott wrote: >>>>>>>>>>>>> On 6/27/2026 2:35 AM, Mikko wrote: >>>>>>>>>>>>>> On 26/06/2026 16:10, olcott wrote: >>>>>>>>>>>>>>> On 6/26/2026 1:39 AM, Mikko wrote: >>>>>>>>>>>>>>>> On 25/06/2026 19:14, olcott wrote: >>>>>>>>>>>>>>>>> On 6/25/2026 2:21 AM, Mikko wrote: >>>>>>>>>>>>>>>>>> On 24/06/2026 23:26, olcott wrote: >>>>>>>>>>>>>>>>>>> On 6/24/2026 5:00 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>> On 23/06/2026 17:48, olcott wrote: >>>>>>>>>>>>>>>>>>>>> On 6/23/2026 1:06 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 15:10, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>> On 6/22/2026 1:49 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 02:02, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 4:08 PM, André G. Isaak wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-21 14:42, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 3:04 PM, Alan Mackenzie wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> [ Followup-To: set ] >>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott <polcott333@gmail.com> >>>>>>>>>>>>>>>>>>>>>>>>>>>> wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 6:26 AM, Alan Mackenzie wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I just found the term: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounding in a proof theoretic atomic >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> base" yesterday. >>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You can find any number of terms. That >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> doesn't mean you're capable of >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding them. >>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>> The above is the key reason why under PTS >>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel 1931 incompleteness >>>>>>>>>>>>>>>>>>>>>>>>>>>>> fails. >>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>> I don't believe you. You have no respect >>>>>>>>>>>>>>>>>>>>>>>>>>>> for or understanding of the >>>>>>>>>>>>>>>>>>>>>>>>>>>> truth. If you really want to persuade >>>>>>>>>>>>>>>>>>>>>>>>>>>> anybody that PTS somehow causes >>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's theorem not to hold, then cite an >>>>>>>>>>>>>>>>>>>>>>>>>>>> academic expert who'll have >>>>>>>>>>>>>>>>>>>>>>>>>>>> some credibility. >>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>> If they are mere gibberish words to you >>>>>>>>>>>>>>>>>>>>>>>>>>>>> then you will not understand. >>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>> You don't understand Proof-theoritic >>>>>>>>>>>>>>>>>>>>>>>>>>>> Semantics, and you certainly don't >>>>>>>>>>>>>>>>>>>>>>>>>>>> understand Gödel's Theorem, neither the >>>>>>>>>>>>>>>>>>>>>>>>>>>> theorem itself nor any proof of >>>>>>>>>>>>>>>>>>>>>>>>>>>> it. >>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>> It is a verified fact that Gödel's G is >>>>>>>>>>>>>>>>>>>>>>>>>>> ungrounded >>>>>>>>>>>>>>>>>>>>>>>>>>> in the atomic base of PA. That you do not >>>>>>>>>>>>>>>>>>>>>>>>>>> understand >>>>>>>>>>>>>>>>>>>>>>>>>>> what: "grounded in the atomic base" means is >>>>>>>>>>>>>>>>>>>>>>>>>>> less >>>>>>>>>>>>>>>>>>>>>>>>>>> than no rebuttal at all. >>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>> "grounded in the atomic base of PA" is an >>>>>>>>>>>>>>>>>>>>>>>>>> expression used only by you, and it is one >>>>>>>>>>>>>>>>>>>>>>>>>> which you have never explicitly defined, so >>>>>>>>>>>>>>>>>>>>>>>>>> the fault here certainly doesn't lie with >>>>>>>>>>>>>>>>>>>>>>>>>> Alan. It's certainly not a 'verified fact' >>>>>>>>>>>>>>>>>>>>>>>>>> when you haven't even adequately explained >>>>>>>>>>>>>>>>>>>>>>>>>> what it is that you mean. >>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>> All of knowledge expressed in language is >>>>>>>>>>>>>>>>>>>>>>>>> structured as a tree of semantic relations >>>>>>>>>>>>>>>>>>>>>>>>> specified syntactically between finite strings. >>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>> What makes you believe semantic relations that >>>>>>>>>>>>>>>>>>>>>>>> can be structured as >>>>>>>>>>>>>>>>>>>>>>>> a tree are sufficient to contain all knowledge >>>>>>>>>>>>>>>>>>>>>>>> that is exressed in >>>>>>>>>>>>>>>>>>>>>>>> some language? >>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>> The CycL language and the Cyc Project. >>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>> They use a tree structure for concepts. But why >>>>>>>>>>>>>>>>>>>>>> would one try to >>>>>>>>>>>>>>>>>>>>>> put knowledge in a tree structure? >>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>> It must at least be a directed acyclic graph or >>>>>>>>>>>>>>>>>>>>> the proof gets stuck in an infinite loop and never >>>>>>>>>>>>>>>>>>>>> completes. >>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>> How can any ordering of knowledge prevent getting >>>>>>>>>>>>>>>>>>>> stuck in a loop >>>>>>>>>>>>>>>>>>>> when looking for a proof? >>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>> By looking upward in a type hierarchy. >>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>> If you mean not looking elsewhere that may indeed >>>>>>>>>>>>>>>>>> prevent loops. >>>>>>>>>>>>>>>>>> In most cases that also prevents finding the proof. >>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>> Truth Conditional Semantics (TCS) <is> incoherent >>>>>>>>>>>>>>>>> compared to Proof Theoretic Semantics (PTS). Essentially >>>>>>>>>>>>>>>>> PTS just coherently connects the semantic meanings >>>>>>>>>>>>>>>>> expressed in language together into one coherent body >>>>>>>>>>>>>>>>> of general knowledge. It does this without undecidability >>>>>>>>>>>>>>>>> or mathematical incompleteness. >>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> Looking for a proof does not need any semantics so it is >>>>>>>>>>>>>>>> not obvious >>>>>>>>>>>>>>>> how switching to another semantics could improve it. >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> In proof theoretic semantics an expression only gains >>>>>>>>>>>>>>> semantic meaning by finding a proof. >>>>>>>>>>>>>> >>>>>>>>>>>>>> It should be obvious that finding a proof does not happen >>>>>>>>>>>>>> before >>>>>>>>>>>>>> looking for a proof. >>>>>>>>>>>>>> >>>>>>>>>>>>> >>>>>>>>>>>>> If there is no sequence of inference steps in Q from >>>>>>>>>>>>> ~∃x x=S(x) to the axioms of Q >>>>>>>>>>>> >>>>>>>>>>>> There are, but that sequence is infinite >>>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> If there is no FINITE sequence of inference steps >>>>>>>>>>> in Q from ~∃x x=S(x) to the axioms of Q then ~∃x x=S(x) >>>>>>>>>>> is ungrounded in the PTS atomic base of Q. >>>>>>>>>> >>>>>>>>>> i.e., ~∃x x=S(x) is unprovable is Q, as is commonly known. >>>>>>>>>> >>>>>>>>> Is it commonly known that ~∃x x=S(x) >>>>>>>> >>>>>>>> Which has the semantic meaning "no number is equal to its >>>>>>>> successor" as per the definition of Q. >>>>>>>> >>>>>>> >>>>>>> Since there are no steps in Q that affirm ~∃x x=S(x) >>>>>>> in Q it is an open question in Q and not a confirmed >>>>>>> statement in Q. >>>>>> >>>>>> In other words, unproven as is commonly known. >>>>>> >>>>> Yet never gets to undecidable or in any sense of incomplete. >>>>> >>>> >>>> False, as by definition, Q is incomplete because ~∃x x=S(x) is >>>> unprovable / out-of-scope / not semantically grounded in Q. >>>> >>> >>> Proof theoretic semantics DOES NOT DO IT THAT WAY !!! >> >> And because PTS claims the semantically valid sentence in Q "no number >> is equal to its successor" is not semantically valid, it must be >> discarded as useless. > > No you are just not bothering to pay 100% totally > complete attention to every single word. > > Wittgenstein > 'True in Russell's system' means, as was said: > proved in Russell's system; and 'false in Russell's > system' means: the opposite has been proved > in Russell's system We reject Wittgenstain's statement as a violation of the current rules of the language game. > Proof Theoretic Semantics has almost gotten there. > For the most part they stop at semantically grounded > and never quite get all the way to True. We have almost gotten to the rejection of Proof Theoretic Semantics. Russell's solution to the problems in his system was to give up and focus to politics. -- Mikko
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| From | Mikko <mikko.levanto@iki.fi> |
|---|---|
| Date | 2026-06-28 12:31 +0300 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111qpm5$3h4i2$1@dont-email.me> |
| In reply to | #645763 |
On 27/06/2026 22:40, olcott wrote: > On 6/27/2026 2:23 PM, dbush wrote: >> On 6/27/2026 3:16 PM, olcott wrote: >>> On 6/27/2026 2:04 PM, dbush wrote: >>>> On 6/27/2026 3:01 PM, olcott wrote: >>>>> On 6/27/2026 1:39 PM, dbush wrote: >>>>>> On 6/27/2026 2:38 PM, olcott wrote: >>>>>>> On 6/27/2026 1:29 PM, dbush wrote: >>>>>>>> On 6/27/2026 2:27 PM, olcott wrote: >>>>>>>>> On 6/27/2026 1:01 PM, dbush wrote: >>>>>>>>>> On 6/27/2026 11:43 AM, polcott wrote: >>>>>>>>>>> On 6/27/2026 2:35 AM, Mikko wrote: >>>>>>>>>>>> On 26/06/2026 16:10, olcott wrote: >>>>>>>>>>>>> On 6/26/2026 1:39 AM, Mikko wrote: >>>>>>>>>>>>>> On 25/06/2026 19:14, olcott wrote: >>>>>>>>>>>>>>> On 6/25/2026 2:21 AM, Mikko wrote: >>>>>>>>>>>>>>>> On 24/06/2026 23:26, olcott wrote: >>>>>>>>>>>>>>>>> On 6/24/2026 5:00 AM, Mikko wrote: >>>>>>>>>>>>>>>>>> On 23/06/2026 17:48, olcott wrote: >>>>>>>>>>>>>>>>>>> On 6/23/2026 1:06 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>> On 22/06/2026 15:10, olcott wrote: >>>>>>>>>>>>>>>>>>>>> On 6/22/2026 1:49 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 02:02, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 4:08 PM, André G. Isaak wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-21 14:42, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 3:04 PM, Alan Mackenzie wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> [ Followup-To: set ] >>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott <polcott333@gmail.com> >>>>>>>>>>>>>>>>>>>>>>>>>> wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 6:26 AM, Alan Mackenzie wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott <polcott333@gmail.com> >>>>>>>>>>>>>>>>>>>>>>>>>>>> wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> I just found the term: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounding in a proof theoretic atomic >>>>>>>>>>>>>>>>>>>>>>>>>>>>> base" yesterday. >>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>> You can find any number of terms. That >>>>>>>>>>>>>>>>>>>>>>>>>>>> doesn't mean you're capable of >>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding them. >>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>> The above is the key reason why under PTS >>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel 1931 incompleteness >>>>>>>>>>>>>>>>>>>>>>>>>>> fails. >>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>> I don't believe you. You have no respect for >>>>>>>>>>>>>>>>>>>>>>>>>> or understanding of the >>>>>>>>>>>>>>>>>>>>>>>>>> truth. If you really want to persuade anybody >>>>>>>>>>>>>>>>>>>>>>>>>> that PTS somehow causes >>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's theorem not to hold, then cite an >>>>>>>>>>>>>>>>>>>>>>>>>> academic expert who'll have >>>>>>>>>>>>>>>>>>>>>>>>>> some credibility. >>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>> If they are mere gibberish words to you then >>>>>>>>>>>>>>>>>>>>>>>>>>> you will not understand. >>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>> You don't understand Proof-theoritic >>>>>>>>>>>>>>>>>>>>>>>>>> Semantics, and you certainly don't >>>>>>>>>>>>>>>>>>>>>>>>>> understand Gödel's Theorem, neither the >>>>>>>>>>>>>>>>>>>>>>>>>> theorem itself nor any proof of >>>>>>>>>>>>>>>>>>>>>>>>>> it. >>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>> It is a verified fact that Gödel's G is ungrounded >>>>>>>>>>>>>>>>>>>>>>>>> in the atomic base of PA. That you do not >>>>>>>>>>>>>>>>>>>>>>>>> understand >>>>>>>>>>>>>>>>>>>>>>>>> what: "grounded in the atomic base" means is less >>>>>>>>>>>>>>>>>>>>>>>>> than no rebuttal at all. >>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>> "grounded in the atomic base of PA" is an >>>>>>>>>>>>>>>>>>>>>>>> expression used only by you, and it is one which >>>>>>>>>>>>>>>>>>>>>>>> you have never explicitly defined, so the fault >>>>>>>>>>>>>>>>>>>>>>>> here certainly doesn't lie with Alan. It's >>>>>>>>>>>>>>>>>>>>>>>> certainly not a 'verified fact' when you haven't >>>>>>>>>>>>>>>>>>>>>>>> even adequately explained what it is that you mean. >>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>> All of knowledge expressed in language is >>>>>>>>>>>>>>>>>>>>>>> structured as a tree of semantic relations >>>>>>>>>>>>>>>>>>>>>>> specified syntactically between finite strings. >>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>> What makes you believe semantic relations that can >>>>>>>>>>>>>>>>>>>>>> be structured as >>>>>>>>>>>>>>>>>>>>>> a tree are sufficient to contain all knowledge >>>>>>>>>>>>>>>>>>>>>> that is exressed in >>>>>>>>>>>>>>>>>>>>>> some language? >>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>> The CycL language and the Cyc Project. >>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>> They use a tree structure for concepts. But why >>>>>>>>>>>>>>>>>>>> would one try to >>>>>>>>>>>>>>>>>>>> put knowledge in a tree structure? >>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>> It must at least be a directed acyclic graph or >>>>>>>>>>>>>>>>>>> the proof gets stuck in an infinite loop and never >>>>>>>>>>>>>>>>>>> completes. >>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>> How can any ordering of knowledge prevent getting >>>>>>>>>>>>>>>>>> stuck in a loop >>>>>>>>>>>>>>>>>> when looking for a proof? >>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>> By looking upward in a type hierarchy. >>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> If you mean not looking elsewhere that may indeed >>>>>>>>>>>>>>>> prevent loops. >>>>>>>>>>>>>>>> In most cases that also prevents finding the proof. >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> Truth Conditional Semantics (TCS) <is> incoherent >>>>>>>>>>>>>>> compared to Proof Theoretic Semantics (PTS). Essentially >>>>>>>>>>>>>>> PTS just coherently connects the semantic meanings >>>>>>>>>>>>>>> expressed in language together into one coherent body >>>>>>>>>>>>>>> of general knowledge. It does this without undecidability >>>>>>>>>>>>>>> or mathematical incompleteness. >>>>>>>>>>>>>> >>>>>>>>>>>>>> Looking for a proof does not need any semantics so it is >>>>>>>>>>>>>> not obvious >>>>>>>>>>>>>> how switching to another semantics could improve it. >>>>>>>>>>>>> >>>>>>>>>>>>> In proof theoretic semantics an expression only gains >>>>>>>>>>>>> semantic meaning by finding a proof. >>>>>>>>>>>> >>>>>>>>>>>> It should be obvious that finding a proof does not happen >>>>>>>>>>>> before >>>>>>>>>>>> looking for a proof. >>>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> If there is no sequence of inference steps in Q from >>>>>>>>>>> ~∃x x=S(x) to the axioms of Q >>>>>>>>>> >>>>>>>>>> There are, but that sequence is infinite >>>>>>>>>> >>>>>>>>> >>>>>>>>> If there is no FINITE sequence of inference steps >>>>>>>>> in Q from ~∃x x=S(x) to the axioms of Q then ~∃x x=S(x) >>>>>>>>> is ungrounded in the PTS atomic base of Q. >>>>>>>> >>>>>>>> i.e., ~∃x x=S(x) is unprovable is Q, as is commonly known. >>>>>>>> >>>>>>> Is it commonly known that ~∃x x=S(x) >>>>>> >>>>>> Which has the semantic meaning "no number is equal to its >>>>>> successor" as per the definition of Q. >>>>>> >>>>> >>>>> Since there are no steps in Q that affirm ~∃x x=S(x) >>>>> in Q it is an open question in Q and not a confirmed >>>>> statement in Q. >>>> >>>> In other words, unproven as is commonly known. >>>> >>> Yet never gets to undecidable or in any sense of incomplete. >>> >> >> False, as by definition, Q is incomplete because ~∃x x=S(x) is >> unprovable / out-of-scope / not semantically grounded in Q. > > Proof theoretic semantics DOES NOT DO IT THAT WAY !!! Irrelevant. The statement that both ∃x x=S(x) and ~∃x x=S(x) are sentences of Q but neither is a rheorem or Q does not depend on any semantics. -- Mikko
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| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-28 22:12 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111snq5$1avt$1@dont-email.me> |
| In reply to | #645823 |
On 6/28/2026 4:31 AM, Mikko wrote: > On 27/06/2026 22:40, olcott wrote: >> On 6/27/2026 2:23 PM, dbush wrote: >>> On 6/27/2026 3:16 PM, olcott wrote: >>>> On 6/27/2026 2:04 PM, dbush wrote: >>>>> On 6/27/2026 3:01 PM, olcott wrote: >>>>>> On 6/27/2026 1:39 PM, dbush wrote: >>>>>>> On 6/27/2026 2:38 PM, olcott wrote: >>>>>>>> On 6/27/2026 1:29 PM, dbush wrote: >>>>>>>>> On 6/27/2026 2:27 PM, olcott wrote: >>>>>>>>>> On 6/27/2026 1:01 PM, dbush wrote: >>>>>>>>>>> On 6/27/2026 11:43 AM, polcott wrote: >>>>>>>>>>>> On 6/27/2026 2:35 AM, Mikko wrote: >>>>>>>>>>>>> On 26/06/2026 16:10, olcott wrote: >>>>>>>>>>>>>> On 6/26/2026 1:39 AM, Mikko wrote: >>>>>>>>>>>>>>> On 25/06/2026 19:14, olcott wrote: >>>>>>>>>>>>>>>> On 6/25/2026 2:21 AM, Mikko wrote: >>>>>>>>>>>>>>>>> On 24/06/2026 23:26, olcott wrote: >>>>>>>>>>>>>>>>>> On 6/24/2026 5:00 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>> On 23/06/2026 17:48, olcott wrote: >>>>>>>>>>>>>>>>>>>> On 6/23/2026 1:06 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>> On 22/06/2026 15:10, olcott wrote: >>>>>>>>>>>>>>>>>>>>>> On 6/22/2026 1:49 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 02:02, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 4:08 PM, André G. Isaak wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-21 14:42, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 3:04 PM, Alan Mackenzie wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> [ Followup-To: set ] >>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott <polcott333@gmail.com> >>>>>>>>>>>>>>>>>>>>>>>>>>> wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 6:26 AM, Alan Mackenzie wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott >>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I just found the term: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounding in a proof theoretic atomic >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> base" yesterday. >>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>> You can find any number of terms. That >>>>>>>>>>>>>>>>>>>>>>>>>>>>> doesn't mean you're capable of >>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding them. >>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>> The above is the key reason why under PTS >>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel 1931 incompleteness >>>>>>>>>>>>>>>>>>>>>>>>>>>> fails. >>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>> I don't believe you. You have no respect for >>>>>>>>>>>>>>>>>>>>>>>>>>> or understanding of the >>>>>>>>>>>>>>>>>>>>>>>>>>> truth. If you really want to persuade >>>>>>>>>>>>>>>>>>>>>>>>>>> anybody that PTS somehow causes >>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's theorem not to hold, then cite an >>>>>>>>>>>>>>>>>>>>>>>>>>> academic expert who'll have >>>>>>>>>>>>>>>>>>>>>>>>>>> some credibility. >>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>> If they are mere gibberish words to you then >>>>>>>>>>>>>>>>>>>>>>>>>>>> you will not understand. >>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>> You don't understand Proof-theoritic >>>>>>>>>>>>>>>>>>>>>>>>>>> Semantics, and you certainly don't >>>>>>>>>>>>>>>>>>>>>>>>>>> understand Gödel's Theorem, neither the >>>>>>>>>>>>>>>>>>>>>>>>>>> theorem itself nor any proof of >>>>>>>>>>>>>>>>>>>>>>>>>>> it. >>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>> It is a verified fact that Gödel's G is >>>>>>>>>>>>>>>>>>>>>>>>>> ungrounded >>>>>>>>>>>>>>>>>>>>>>>>>> in the atomic base of PA. That you do not >>>>>>>>>>>>>>>>>>>>>>>>>> understand >>>>>>>>>>>>>>>>>>>>>>>>>> what: "grounded in the atomic base" means is less >>>>>>>>>>>>>>>>>>>>>>>>>> than no rebuttal at all. >>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>> "grounded in the atomic base of PA" is an >>>>>>>>>>>>>>>>>>>>>>>>> expression used only by you, and it is one >>>>>>>>>>>>>>>>>>>>>>>>> which you have never explicitly defined, so the >>>>>>>>>>>>>>>>>>>>>>>>> fault here certainly doesn't lie with Alan. >>>>>>>>>>>>>>>>>>>>>>>>> It's certainly not a 'verified fact' when you >>>>>>>>>>>>>>>>>>>>>>>>> haven't even adequately explained what it is >>>>>>>>>>>>>>>>>>>>>>>>> that you mean. >>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>> All of knowledge expressed in language is >>>>>>>>>>>>>>>>>>>>>>>> structured as a tree of semantic relations >>>>>>>>>>>>>>>>>>>>>>>> specified syntactically between finite strings. >>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>> What makes you believe semantic relations that >>>>>>>>>>>>>>>>>>>>>>> can be structured as >>>>>>>>>>>>>>>>>>>>>>> a tree are sufficient to contain all knowledge >>>>>>>>>>>>>>>>>>>>>>> that is exressed in >>>>>>>>>>>>>>>>>>>>>>> some language? >>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>> The CycL language and the Cyc Project. >>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>> They use a tree structure for concepts. But why >>>>>>>>>>>>>>>>>>>>> would one try to >>>>>>>>>>>>>>>>>>>>> put knowledge in a tree structure? >>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>> It must at least be a directed acyclic graph or >>>>>>>>>>>>>>>>>>>> the proof gets stuck in an infinite loop and never >>>>>>>>>>>>>>>>>>>> completes. >>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>> How can any ordering of knowledge prevent getting >>>>>>>>>>>>>>>>>>> stuck in a loop >>>>>>>>>>>>>>>>>>> when looking for a proof? >>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>> By looking upward in a type hierarchy. >>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>> If you mean not looking elsewhere that may indeed >>>>>>>>>>>>>>>>> prevent loops. >>>>>>>>>>>>>>>>> In most cases that also prevents finding the proof. >>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> Truth Conditional Semantics (TCS) <is> incoherent >>>>>>>>>>>>>>>> compared to Proof Theoretic Semantics (PTS). Essentially >>>>>>>>>>>>>>>> PTS just coherently connects the semantic meanings >>>>>>>>>>>>>>>> expressed in language together into one coherent body >>>>>>>>>>>>>>>> of general knowledge. It does this without undecidability >>>>>>>>>>>>>>>> or mathematical incompleteness. >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> Looking for a proof does not need any semantics so it is >>>>>>>>>>>>>>> not obvious >>>>>>>>>>>>>>> how switching to another semantics could improve it. >>>>>>>>>>>>>> >>>>>>>>>>>>>> In proof theoretic semantics an expression only gains >>>>>>>>>>>>>> semantic meaning by finding a proof. >>>>>>>>>>>>> >>>>>>>>>>>>> It should be obvious that finding a proof does not happen >>>>>>>>>>>>> before >>>>>>>>>>>>> looking for a proof. >>>>>>>>>>>>> >>>>>>>>>>>> >>>>>>>>>>>> If there is no sequence of inference steps in Q from >>>>>>>>>>>> ~∃x x=S(x) to the axioms of Q >>>>>>>>>>> >>>>>>>>>>> There are, but that sequence is infinite >>>>>>>>>>> >>>>>>>>>> >>>>>>>>>> If there is no FINITE sequence of inference steps >>>>>>>>>> in Q from ~∃x x=S(x) to the axioms of Q then ~∃x x=S(x) >>>>>>>>>> is ungrounded in the PTS atomic base of Q. >>>>>>>>> >>>>>>>>> i.e., ~∃x x=S(x) is unprovable is Q, as is commonly known. >>>>>>>>> >>>>>>>> Is it commonly known that ~∃x x=S(x) >>>>>>> >>>>>>> Which has the semantic meaning "no number is equal to its >>>>>>> successor" as per the definition of Q. >>>>>>> >>>>>> >>>>>> Since there are no steps in Q that affirm ~∃x x=S(x) >>>>>> in Q it is an open question in Q and not a confirmed >>>>>> statement in Q. >>>>> >>>>> In other words, unproven as is commonly known. >>>>> >>>> Yet never gets to undecidable or in any sense of incomplete. >>>> >>> >>> False, as by definition, Q is incomplete because ~∃x x=S(x) is >>> unprovable / out-of-scope / not semantically grounded in Q. >> >> Proof theoretic semantics DOES NOT DO IT THAT WAY !!! > > Irrelevant. The statement that both ∃x x=S(x) and ~∃x x=S(x) are > sentences of Q but neither is a rheorem or Q does not depend on > any semantics. > The entire body of knowledge expressed in language can be represented as a semantic tautology in an acyclic directed graph. That knowledge is a DAG was my very thought on this subject more than 30 years ago. This single idea gets rid of all undecidability within the entire body of knowledge. -- Copyright 2026 Olcott My 28 year goal has been to make "true on the basis of meaning expressed in language" reliably computable for the entire body of knowledge. The complete structure of this system is now defined. The entire body of knowledge expressed in language is comprised of two types of relations between finite strings: (a) *Axioms* Expressions of language that are stipulated to be true. My system bridges the analytic/synthetic distinction by expressly encoding all empirical "atomic facts" in a formal language such as CycL of the Cyc project. (b) *Inference Rules* Expressions of language that are semantically entailed syntactically from (a) and/or (b).
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| From | Mikko <mikko.levanto@iki.fi> |
|---|---|
| Date | 2026-06-29 09:23 +0300 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111t31l$52jd$1@dont-email.me> |
| In reply to | #645828 |
On 29/06/2026 06:12, olcott wrote: > On 6/28/2026 4:31 AM, Mikko wrote: >> On 27/06/2026 22:40, olcott wrote: >>> On 6/27/2026 2:23 PM, dbush wrote: >>>> On 6/27/2026 3:16 PM, olcott wrote: >>>>> On 6/27/2026 2:04 PM, dbush wrote: >>>>>> On 6/27/2026 3:01 PM, olcott wrote: >>>>>>> On 6/27/2026 1:39 PM, dbush wrote: >>>>>>>> On 6/27/2026 2:38 PM, olcott wrote: >>>>>>>>> On 6/27/2026 1:29 PM, dbush wrote: >>>>>>>>>> On 6/27/2026 2:27 PM, olcott wrote: >>>>>>>>>>> On 6/27/2026 1:01 PM, dbush wrote: >>>>>>>>>>>> On 6/27/2026 11:43 AM, polcott wrote: >>>>>>>>>>>>> On 6/27/2026 2:35 AM, Mikko wrote: >>>>>>>>>>>>>> On 26/06/2026 16:10, olcott wrote: >>>>>>>>>>>>>>> On 6/26/2026 1:39 AM, Mikko wrote: >>>>>>>>>>>>>>>> On 25/06/2026 19:14, olcott wrote: >>>>>>>>>>>>>>>>> On 6/25/2026 2:21 AM, Mikko wrote: >>>>>>>>>>>>>>>>>> On 24/06/2026 23:26, olcott wrote: >>>>>>>>>>>>>>>>>>> On 6/24/2026 5:00 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>> On 23/06/2026 17:48, olcott wrote: >>>>>>>>>>>>>>>>>>>>> On 6/23/2026 1:06 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 15:10, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>> On 6/22/2026 1:49 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 02:02, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 4:08 PM, André G. Isaak wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-21 14:42, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 3:04 PM, Alan Mackenzie wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> [ Followup-To: set ] >>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott <polcott333@gmail.com> >>>>>>>>>>>>>>>>>>>>>>>>>>>> wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 6:26 AM, Alan Mackenzie wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I just found the term: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounding in a proof theoretic atomic >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> base" yesterday. >>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You can find any number of terms. That >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> doesn't mean you're capable of >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding them. >>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>> The above is the key reason why under PTS >>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel 1931 incompleteness >>>>>>>>>>>>>>>>>>>>>>>>>>>>> fails. >>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>> I don't believe you. You have no respect >>>>>>>>>>>>>>>>>>>>>>>>>>>> for or understanding of the >>>>>>>>>>>>>>>>>>>>>>>>>>>> truth. If you really want to persuade >>>>>>>>>>>>>>>>>>>>>>>>>>>> anybody that PTS somehow causes >>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's theorem not to hold, then cite an >>>>>>>>>>>>>>>>>>>>>>>>>>>> academic expert who'll have >>>>>>>>>>>>>>>>>>>>>>>>>>>> some credibility. >>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>> If they are mere gibberish words to you >>>>>>>>>>>>>>>>>>>>>>>>>>>>> then you will not understand. >>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>> You don't understand Proof-theoritic >>>>>>>>>>>>>>>>>>>>>>>>>>>> Semantics, and you certainly don't >>>>>>>>>>>>>>>>>>>>>>>>>>>> understand Gödel's Theorem, neither the >>>>>>>>>>>>>>>>>>>>>>>>>>>> theorem itself nor any proof of >>>>>>>>>>>>>>>>>>>>>>>>>>>> it. >>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>> It is a verified fact that Gödel's G is >>>>>>>>>>>>>>>>>>>>>>>>>>> ungrounded >>>>>>>>>>>>>>>>>>>>>>>>>>> in the atomic base of PA. That you do not >>>>>>>>>>>>>>>>>>>>>>>>>>> understand >>>>>>>>>>>>>>>>>>>>>>>>>>> what: "grounded in the atomic base" means is >>>>>>>>>>>>>>>>>>>>>>>>>>> less >>>>>>>>>>>>>>>>>>>>>>>>>>> than no rebuttal at all. >>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>> "grounded in the atomic base of PA" is an >>>>>>>>>>>>>>>>>>>>>>>>>> expression used only by you, and it is one >>>>>>>>>>>>>>>>>>>>>>>>>> which you have never explicitly defined, so >>>>>>>>>>>>>>>>>>>>>>>>>> the fault here certainly doesn't lie with >>>>>>>>>>>>>>>>>>>>>>>>>> Alan. It's certainly not a 'verified fact' >>>>>>>>>>>>>>>>>>>>>>>>>> when you haven't even adequately explained >>>>>>>>>>>>>>>>>>>>>>>>>> what it is that you mean. >>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>> All of knowledge expressed in language is >>>>>>>>>>>>>>>>>>>>>>>>> structured as a tree of semantic relations >>>>>>>>>>>>>>>>>>>>>>>>> specified syntactically between finite strings. >>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>> What makes you believe semantic relations that >>>>>>>>>>>>>>>>>>>>>>>> can be structured as >>>>>>>>>>>>>>>>>>>>>>>> a tree are sufficient to contain all knowledge >>>>>>>>>>>>>>>>>>>>>>>> that is exressed in >>>>>>>>>>>>>>>>>>>>>>>> some language? >>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>> The CycL language and the Cyc Project. >>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>> They use a tree structure for concepts. But why >>>>>>>>>>>>>>>>>>>>>> would one try to >>>>>>>>>>>>>>>>>>>>>> put knowledge in a tree structure? >>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>> It must at least be a directed acyclic graph or >>>>>>>>>>>>>>>>>>>>> the proof gets stuck in an infinite loop and never >>>>>>>>>>>>>>>>>>>>> completes. >>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>> How can any ordering of knowledge prevent getting >>>>>>>>>>>>>>>>>>>> stuck in a loop >>>>>>>>>>>>>>>>>>>> when looking for a proof? >>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>> By looking upward in a type hierarchy. >>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>> If you mean not looking elsewhere that may indeed >>>>>>>>>>>>>>>>>> prevent loops. >>>>>>>>>>>>>>>>>> In most cases that also prevents finding the proof. >>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>> Truth Conditional Semantics (TCS) <is> incoherent >>>>>>>>>>>>>>>>> compared to Proof Theoretic Semantics (PTS). Essentially >>>>>>>>>>>>>>>>> PTS just coherently connects the semantic meanings >>>>>>>>>>>>>>>>> expressed in language together into one coherent body >>>>>>>>>>>>>>>>> of general knowledge. It does this without undecidability >>>>>>>>>>>>>>>>> or mathematical incompleteness. >>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> Looking for a proof does not need any semantics so it is >>>>>>>>>>>>>>>> not obvious >>>>>>>>>>>>>>>> how switching to another semantics could improve it. >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> In proof theoretic semantics an expression only gains >>>>>>>>>>>>>>> semantic meaning by finding a proof. >>>>>>>>>>>>>> >>>>>>>>>>>>>> It should be obvious that finding a proof does not happen >>>>>>>>>>>>>> before >>>>>>>>>>>>>> looking for a proof. >>>>>>>>>>>>>> >>>>>>>>>>>>> >>>>>>>>>>>>> If there is no sequence of inference steps in Q from >>>>>>>>>>>>> ~∃x x=S(x) to the axioms of Q >>>>>>>>>>>> >>>>>>>>>>>> There are, but that sequence is infinite >>>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> If there is no FINITE sequence of inference steps >>>>>>>>>>> in Q from ~∃x x=S(x) to the axioms of Q then ~∃x x=S(x) >>>>>>>>>>> is ungrounded in the PTS atomic base of Q. >>>>>>>>>> >>>>>>>>>> i.e., ~∃x x=S(x) is unprovable is Q, as is commonly known. >>>>>>>>>> >>>>>>>>> Is it commonly known that ~∃x x=S(x) >>>>>>>> >>>>>>>> Which has the semantic meaning "no number is equal to its >>>>>>>> successor" as per the definition of Q. >>>>>>>> >>>>>>> >>>>>>> Since there are no steps in Q that affirm ~∃x x=S(x) >>>>>>> in Q it is an open question in Q and not a confirmed >>>>>>> statement in Q. >>>>>> >>>>>> In other words, unproven as is commonly known. >>>>>> >>>>> Yet never gets to undecidable or in any sense of incomplete. >>>>> >>>> >>>> False, as by definition, Q is incomplete because ~∃x x=S(x) is >>>> unprovable / out-of-scope / not semantically grounded in Q. >>> >>> Proof theoretic semantics DOES NOT DO IT THAT WAY !!! >> >> Irrelevant. The statement that both ∃x x=S(x) and ~∃x x=S(x) are >> sentences of Q but neither is a rheorem or Q does not depend on >> any semantics. > > The entire body of knowledge expressed in language > can be represented as a semantic tautology in an > acyclic directed graph. That knowledge is a DAG was > my very thought on this subject more than 30 years ago. > This single idea gets rid of all undecidability > within the entire body of knowledge. No, it cannot. The usual meaning of knoledge excludes tautologies because the determination of their truth does not need any knowledge beyond a method to determine whether a string is a tautology in the relevant language. When the word "knowledge" is used it usually means knowing about the real world something that cannot be determined without observation. -- Mikko
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| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-29 08:38 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111tshh$cm19$1@dont-email.me> |
| In reply to | #645833 |
On 6/29/2026 1:23 AM, Mikko wrote: > On 29/06/2026 06:12, olcott wrote: >> On 6/28/2026 4:31 AM, Mikko wrote: >>> On 27/06/2026 22:40, olcott wrote: >>>> On 6/27/2026 2:23 PM, dbush wrote: >>>>> On 6/27/2026 3:16 PM, olcott wrote: >>>>>> On 6/27/2026 2:04 PM, dbush wrote: >>>>>>> On 6/27/2026 3:01 PM, olcott wrote: >>>>>>>> On 6/27/2026 1:39 PM, dbush wrote: >>>>>>>>> On 6/27/2026 2:38 PM, olcott wrote: >>>>>>>>>> On 6/27/2026 1:29 PM, dbush wrote: >>>>>>>>>>> On 6/27/2026 2:27 PM, olcott wrote: >>>>>>>>>>>> On 6/27/2026 1:01 PM, dbush wrote: >>>>>>>>>>>>> On 6/27/2026 11:43 AM, polcott wrote: >>>>>>>>>>>>>> On 6/27/2026 2:35 AM, Mikko wrote: >>>>>>>>>>>>>>> On 26/06/2026 16:10, olcott wrote: >>>>>>>>>>>>>>>> On 6/26/2026 1:39 AM, Mikko wrote: >>>>>>>>>>>>>>>>> On 25/06/2026 19:14, olcott wrote: >>>>>>>>>>>>>>>>>> On 6/25/2026 2:21 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>> On 24/06/2026 23:26, olcott wrote: >>>>>>>>>>>>>>>>>>>> On 6/24/2026 5:00 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>> On 23/06/2026 17:48, olcott wrote: >>>>>>>>>>>>>>>>>>>>>> On 6/23/2026 1:06 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 15:10, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 6/22/2026 1:49 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 02:02, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 4:08 PM, André G. Isaak wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-21 14:42, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 3:04 PM, Alan Mackenzie wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> [ Followup-To: set ] >>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott >>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 6:26 AM, Alan Mackenzie wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I just found the term: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounding in a proof theoretic atomic >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> base" yesterday. >>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You can find any number of terms. That >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> doesn't mean you're capable of >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding them. >>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The above is the key reason why under PTS >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel 1931 incompleteness >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fails. >>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>> I don't believe you. You have no respect >>>>>>>>>>>>>>>>>>>>>>>>>>>>> for or understanding of the >>>>>>>>>>>>>>>>>>>>>>>>>>>>> truth. If you really want to persuade >>>>>>>>>>>>>>>>>>>>>>>>>>>>> anybody that PTS somehow causes >>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's theorem not to hold, then cite an >>>>>>>>>>>>>>>>>>>>>>>>>>>>> academic expert who'll have >>>>>>>>>>>>>>>>>>>>>>>>>>>>> some credibility. >>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If they are mere gibberish words to you >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> then you will not understand. >>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>> You don't understand Proof-theoritic >>>>>>>>>>>>>>>>>>>>>>>>>>>>> Semantics, and you certainly don't >>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand Gödel's Theorem, neither the >>>>>>>>>>>>>>>>>>>>>>>>>>>>> theorem itself nor any proof of >>>>>>>>>>>>>>>>>>>>>>>>>>>>> it. >>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>> It is a verified fact that Gödel's G is >>>>>>>>>>>>>>>>>>>>>>>>>>>> ungrounded >>>>>>>>>>>>>>>>>>>>>>>>>>>> in the atomic base of PA. That you do not >>>>>>>>>>>>>>>>>>>>>>>>>>>> understand >>>>>>>>>>>>>>>>>>>>>>>>>>>> what: "grounded in the atomic base" means is >>>>>>>>>>>>>>>>>>>>>>>>>>>> less >>>>>>>>>>>>>>>>>>>>>>>>>>>> than no rebuttal at all. >>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>> "grounded in the atomic base of PA" is an >>>>>>>>>>>>>>>>>>>>>>>>>>> expression used only by you, and it is one >>>>>>>>>>>>>>>>>>>>>>>>>>> which you have never explicitly defined, so >>>>>>>>>>>>>>>>>>>>>>>>>>> the fault here certainly doesn't lie with >>>>>>>>>>>>>>>>>>>>>>>>>>> Alan. It's certainly not a 'verified fact' >>>>>>>>>>>>>>>>>>>>>>>>>>> when you haven't even adequately explained >>>>>>>>>>>>>>>>>>>>>>>>>>> what it is that you mean. >>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>> All of knowledge expressed in language is >>>>>>>>>>>>>>>>>>>>>>>>>> structured as a tree of semantic relations >>>>>>>>>>>>>>>>>>>>>>>>>> specified syntactically between finite strings. >>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>> What makes you believe semantic relations that >>>>>>>>>>>>>>>>>>>>>>>>> can be structured as >>>>>>>>>>>>>>>>>>>>>>>>> a tree are sufficient to contain all knowledge >>>>>>>>>>>>>>>>>>>>>>>>> that is exressed in >>>>>>>>>>>>>>>>>>>>>>>>> some language? >>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>> The CycL language and the Cyc Project. >>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>> They use a tree structure for concepts. But why >>>>>>>>>>>>>>>>>>>>>>> would one try to >>>>>>>>>>>>>>>>>>>>>>> put knowledge in a tree structure? >>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>> It must at least be a directed acyclic graph or >>>>>>>>>>>>>>>>>>>>>> the proof gets stuck in an infinite loop and never >>>>>>>>>>>>>>>>>>>>>> completes. >>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>> How can any ordering of knowledge prevent getting >>>>>>>>>>>>>>>>>>>>> stuck in a loop >>>>>>>>>>>>>>>>>>>>> when looking for a proof? >>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>> By looking upward in a type hierarchy. >>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>> If you mean not looking elsewhere that may indeed >>>>>>>>>>>>>>>>>>> prevent loops. >>>>>>>>>>>>>>>>>>> In most cases that also prevents finding the proof. >>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>> Truth Conditional Semantics (TCS) <is> incoherent >>>>>>>>>>>>>>>>>> compared to Proof Theoretic Semantics (PTS). Essentially >>>>>>>>>>>>>>>>>> PTS just coherently connects the semantic meanings >>>>>>>>>>>>>>>>>> expressed in language together into one coherent body >>>>>>>>>>>>>>>>>> of general knowledge. It does this without undecidability >>>>>>>>>>>>>>>>>> or mathematical incompleteness. >>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>> Looking for a proof does not need any semantics so it >>>>>>>>>>>>>>>>> is not obvious >>>>>>>>>>>>>>>>> how switching to another semantics could improve it. >>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> In proof theoretic semantics an expression only gains >>>>>>>>>>>>>>>> semantic meaning by finding a proof. >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> It should be obvious that finding a proof does not happen >>>>>>>>>>>>>>> before >>>>>>>>>>>>>>> looking for a proof. >>>>>>>>>>>>>>> >>>>>>>>>>>>>> >>>>>>>>>>>>>> If there is no sequence of inference steps in Q from >>>>>>>>>>>>>> ~∃x x=S(x) to the axioms of Q >>>>>>>>>>>>> >>>>>>>>>>>>> There are, but that sequence is infinite >>>>>>>>>>>>> >>>>>>>>>>>> >>>>>>>>>>>> If there is no FINITE sequence of inference steps >>>>>>>>>>>> in Q from ~∃x x=S(x) to the axioms of Q then ~∃x x=S(x) >>>>>>>>>>>> is ungrounded in the PTS atomic base of Q. >>>>>>>>>>> >>>>>>>>>>> i.e., ~∃x x=S(x) is unprovable is Q, as is commonly known. >>>>>>>>>>> >>>>>>>>>> Is it commonly known that ~∃x x=S(x) >>>>>>>>> >>>>>>>>> Which has the semantic meaning "no number is equal to its >>>>>>>>> successor" as per the definition of Q. >>>>>>>>> >>>>>>>> >>>>>>>> Since there are no steps in Q that affirm ~∃x x=S(x) >>>>>>>> in Q it is an open question in Q and not a confirmed >>>>>>>> statement in Q. >>>>>>> >>>>>>> In other words, unproven as is commonly known. >>>>>>> >>>>>> Yet never gets to undecidable or in any sense of incomplete. >>>>>> >>>>> >>>>> False, as by definition, Q is incomplete because ~∃x x=S(x) is >>>>> unprovable / out-of-scope / not semantically grounded in Q. >>>> >>>> Proof theoretic semantics DOES NOT DO IT THAT WAY !!! >>> >>> Irrelevant. The statement that both ∃x x=S(x) and ~∃x x=S(x) are >>> sentences of Q but neither is a rheorem or Q does not depend on >>> any semantics. >> >> The entire body of knowledge expressed in language >> can be represented as a semantic tautology in an >> acyclic directed graph. That knowledge is a DAG was >> my very thought on this subject more than 30 years ago. >> This single idea gets rid of all undecidability >> within the entire body of knowledge. > > No, it cannot. The usual meaning of knoledge excludes tautologies You are not paying close enough attention. I did not say logical tautology. I said semantic tautology. That cats are defined to be animals is a semantic tautology. That cats are defined to be cats is a logical tautology. Here is a definition of that fits my definition of semantic tautology. tautology, in logic, a statement so framed that it cannot be denied without inconsistency. Thus, “All humans are mammals” is held to assert with regard to anything whatsoever that either it is not a human or it is a mammal. https://www.britannica.com/topic/tautology https://en.wikipedia.org/wiki/Tautology_(logic) > because the determination of their truth does not need any knowledge > beyond a method to determine whether a string is a tautology in the > relevant language. > > When the word "knowledge" is used it usually means knowing about the > real world something that cannot be determined without observation. > One can know that "cats are animals" when this is stipulated as an axiom. -- Copyright 2026 Olcott My 28 year goal has been to make "true on the basis of meaning expressed in language" reliably computable for the entire body of knowledge. The complete structure of this system is now defined. The entire body of knowledge expressed in language is comprised of two types of relations between finite strings: (a) *Axioms* Expressions of language that are stipulated to be true. My system bridges the analytic/synthetic distinction by expressly encoding all empirical "atomic facts" in a formal language such as CycL of the Cyc project. (b) *Inference Rules* Expressions of language that are semantically entailed syntactically from (a) and/or (b).
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| From | Mikko <mikko.levanto@iki.fi> |
|---|---|
| Date | 2026-06-30 10:48 +0300 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111vscc$trjp$1@dont-email.me> |
| In reply to | #645841 |
On 29/06/2026 16:38, olcott wrote: > On 6/29/2026 1:23 AM, Mikko wrote: >> On 29/06/2026 06:12, olcott wrote: >>> On 6/28/2026 4:31 AM, Mikko wrote: >>>> On 27/06/2026 22:40, olcott wrote: >>>>> On 6/27/2026 2:23 PM, dbush wrote: >>>>>> On 6/27/2026 3:16 PM, olcott wrote: >>>>>>> On 6/27/2026 2:04 PM, dbush wrote: >>>>>>>> On 6/27/2026 3:01 PM, olcott wrote: >>>>>>>>> On 6/27/2026 1:39 PM, dbush wrote: >>>>>>>>>> On 6/27/2026 2:38 PM, olcott wrote: >>>>>>>>>>> On 6/27/2026 1:29 PM, dbush wrote: >>>>>>>>>>>> On 6/27/2026 2:27 PM, olcott wrote: >>>>>>>>>>>>> On 6/27/2026 1:01 PM, dbush wrote: >>>>>>>>>>>>>> On 6/27/2026 11:43 AM, polcott wrote: >>>>>>>>>>>>>>> On 6/27/2026 2:35 AM, Mikko wrote: >>>>>>>>>>>>>>>> On 26/06/2026 16:10, olcott wrote: >>>>>>>>>>>>>>>>> On 6/26/2026 1:39 AM, Mikko wrote: >>>>>>>>>>>>>>>>>> On 25/06/2026 19:14, olcott wrote: >>>>>>>>>>>>>>>>>>> On 6/25/2026 2:21 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>> On 24/06/2026 23:26, olcott wrote: >>>>>>>>>>>>>>>>>>>>> On 6/24/2026 5:00 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>> On 23/06/2026 17:48, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>> On 6/23/2026 1:06 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 15:10, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 6/22/2026 1:49 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 02:02, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 4:08 PM, André G. Isaak wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-21 14:42, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 3:04 PM, Alan Mackenzie wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> [ Followup-To: set ] >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 6:26 AM, Alan Mackenzie wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I just found the term: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounding in a proof theoretic atomic >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> base" yesterday. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You can find any number of terms. That >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> doesn't mean you're capable of >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding them. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The above is the key reason why under PTS >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel 1931 incompleteness >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fails. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I don't believe you. You have no respect >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> for or understanding of the >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> truth. If you really want to persuade >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> anybody that PTS somehow causes >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's theorem not to hold, then cite an >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> academic expert who'll have >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some credibility. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If they are mere gibberish words to you >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> then you will not understand. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You don't understand Proof-theoritic >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Semantics, and you certainly don't >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand Gödel's Theorem, neither the >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theorem itself nor any proof of >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> it. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>> It is a verified fact that Gödel's G is >>>>>>>>>>>>>>>>>>>>>>>>>>>>> ungrounded >>>>>>>>>>>>>>>>>>>>>>>>>>>>> in the atomic base of PA. That you do not >>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand >>>>>>>>>>>>>>>>>>>>>>>>>>>>> what: "grounded in the atomic base" means >>>>>>>>>>>>>>>>>>>>>>>>>>>>> is less >>>>>>>>>>>>>>>>>>>>>>>>>>>>> than no rebuttal at all. >>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounded in the atomic base of PA" is an >>>>>>>>>>>>>>>>>>>>>>>>>>>> expression used only by you, and it is one >>>>>>>>>>>>>>>>>>>>>>>>>>>> which you have never explicitly defined, so >>>>>>>>>>>>>>>>>>>>>>>>>>>> the fault here certainly doesn't lie with >>>>>>>>>>>>>>>>>>>>>>>>>>>> Alan. It's certainly not a 'verified fact' >>>>>>>>>>>>>>>>>>>>>>>>>>>> when you haven't even adequately explained >>>>>>>>>>>>>>>>>>>>>>>>>>>> what it is that you mean. >>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>> All of knowledge expressed in language is >>>>>>>>>>>>>>>>>>>>>>>>>>> structured as a tree of semantic relations >>>>>>>>>>>>>>>>>>>>>>>>>>> specified syntactically between finite strings. >>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>> What makes you believe semantic relations that >>>>>>>>>>>>>>>>>>>>>>>>>> can be structured as >>>>>>>>>>>>>>>>>>>>>>>>>> a tree are sufficient to contain all knowledge >>>>>>>>>>>>>>>>>>>>>>>>>> that is exressed in >>>>>>>>>>>>>>>>>>>>>>>>>> some language? >>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>> The CycL language and the Cyc Project. >>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>> They use a tree structure for concepts. But why >>>>>>>>>>>>>>>>>>>>>>>> would one try to >>>>>>>>>>>>>>>>>>>>>>>> put knowledge in a tree structure? >>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>> It must at least be a directed acyclic graph or >>>>>>>>>>>>>>>>>>>>>>> the proof gets stuck in an infinite loop and never >>>>>>>>>>>>>>>>>>>>>>> completes. >>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>> How can any ordering of knowledge prevent getting >>>>>>>>>>>>>>>>>>>>>> stuck in a loop >>>>>>>>>>>>>>>>>>>>>> when looking for a proof? >>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>> By looking upward in a type hierarchy. >>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>> If you mean not looking elsewhere that may indeed >>>>>>>>>>>>>>>>>>>> prevent loops. >>>>>>>>>>>>>>>>>>>> In most cases that also prevents finding the proof. >>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>> Truth Conditional Semantics (TCS) <is> incoherent >>>>>>>>>>>>>>>>>>> compared to Proof Theoretic Semantics (PTS). Essentially >>>>>>>>>>>>>>>>>>> PTS just coherently connects the semantic meanings >>>>>>>>>>>>>>>>>>> expressed in language together into one coherent body >>>>>>>>>>>>>>>>>>> of general knowledge. It does this without >>>>>>>>>>>>>>>>>>> undecidability >>>>>>>>>>>>>>>>>>> or mathematical incompleteness. >>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>> Looking for a proof does not need any semantics so it >>>>>>>>>>>>>>>>>> is not obvious >>>>>>>>>>>>>>>>>> how switching to another semantics could improve it. >>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>> In proof theoretic semantics an expression only gains >>>>>>>>>>>>>>>>> semantic meaning by finding a proof. >>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> It should be obvious that finding a proof does not >>>>>>>>>>>>>>>> happen before >>>>>>>>>>>>>>>> looking for a proof. >>>>>>>>>>>>>>>> >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> If there is no sequence of inference steps in Q from >>>>>>>>>>>>>>> ~∃x x=S(x) to the axioms of Q >>>>>>>>>>>>>> >>>>>>>>>>>>>> There are, but that sequence is infinite >>>>>>>>>>>>>> >>>>>>>>>>>>> >>>>>>>>>>>>> If there is no FINITE sequence of inference steps >>>>>>>>>>>>> in Q from ~∃x x=S(x) to the axioms of Q then ~∃x x=S(x) >>>>>>>>>>>>> is ungrounded in the PTS atomic base of Q. >>>>>>>>>>>> >>>>>>>>>>>> i.e., ~∃x x=S(x) is unprovable is Q, as is commonly known. >>>>>>>>>>>> >>>>>>>>>>> Is it commonly known that ~∃x x=S(x) >>>>>>>>>> >>>>>>>>>> Which has the semantic meaning "no number is equal to its >>>>>>>>>> successor" as per the definition of Q. >>>>>>>>>> >>>>>>>>> >>>>>>>>> Since there are no steps in Q that affirm ~∃x x=S(x) >>>>>>>>> in Q it is an open question in Q and not a confirmed >>>>>>>>> statement in Q. >>>>>>>> >>>>>>>> In other words, unproven as is commonly known. >>>>>>>> >>>>>>> Yet never gets to undecidable or in any sense of incomplete. >>>>>>> >>>>>> >>>>>> False, as by definition, Q is incomplete because ~∃x x=S(x) is >>>>>> unprovable / out-of-scope / not semantically grounded in Q. >>>>> >>>>> Proof theoretic semantics DOES NOT DO IT THAT WAY !!! >>>> >>>> Irrelevant. The statement that both ∃x x=S(x) and ~∃x x=S(x) are >>>> sentences of Q but neither is a rheorem or Q does not depend on >>>> any semantics. >>> >>> The entire body of knowledge expressed in language >>> can be represented as a semantic tautology in an >>> acyclic directed graph. That knowledge is a DAG was >>> my very thought on this subject more than 30 years ago. >>> This single idea gets rid of all undecidability >>> within the entire body of knowledge. >> >> No, it cannot. The usual meaning of knoledge excludes tautologies > > You are not paying close enough attention. I did not say logical > tautology. I said semantic tautology. That cats are defined to > be animals is a semantic tautology. That cats are defined to be > cats is a logical tautology. Here is a definition of that fits > my definition of semantic tautology. > > tautology, in logic, a statement so framed that > it cannot be denied without inconsistency. Thus, > “All humans are mammals” is held to assert with > regard to anything whatsoever that either it is > not a human or it is a mammal. > https://www.britannica.com/topic/tautology > > https://en.wikipedia.org/wiki/Tautology_(logic) >> because the determination of their truth does not need any knowledge >> beyond a method to determine whether a string is a tautology in the >> relevant language. >> >> When the word "knowledge" is used it usually means knowing about the >> real world something that cannot be determined without observation. > > One can know that "cats are animals" when this is > stipulated as an axiom. That axiom only realtes the words "cat" and "animal". It does not tell anything about the real world. -- Mikko
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| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-30 08:43 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <1120h6o$1494d$1@dont-email.me> |
| In reply to | #645874 |
On 6/30/2026 2:48 AM, Mikko wrote: > On 29/06/2026 16:38, olcott wrote: >> On 6/29/2026 1:23 AM, Mikko wrote: >>> On 29/06/2026 06:12, olcott wrote: >>>> On 6/28/2026 4:31 AM, Mikko wrote: >>>>> On 27/06/2026 22:40, olcott wrote: >>>>>> On 6/27/2026 2:23 PM, dbush wrote: >>>>>>> On 6/27/2026 3:16 PM, olcott wrote: >>>>>>>> On 6/27/2026 2:04 PM, dbush wrote: >>>>>>>>> On 6/27/2026 3:01 PM, olcott wrote: >>>>>>>>>> On 6/27/2026 1:39 PM, dbush wrote: >>>>>>>>>>> On 6/27/2026 2:38 PM, olcott wrote: >>>>>>>>>>>> On 6/27/2026 1:29 PM, dbush wrote: >>>>>>>>>>>>> On 6/27/2026 2:27 PM, olcott wrote: >>>>>>>>>>>>>> On 6/27/2026 1:01 PM, dbush wrote: >>>>>>>>>>>>>>> On 6/27/2026 11:43 AM, polcott wrote: >>>>>>>>>>>>>>>> On 6/27/2026 2:35 AM, Mikko wrote: >>>>>>>>>>>>>>>>> On 26/06/2026 16:10, olcott wrote: >>>>>>>>>>>>>>>>>> On 6/26/2026 1:39 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>> On 25/06/2026 19:14, olcott wrote: >>>>>>>>>>>>>>>>>>>> On 6/25/2026 2:21 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>> On 24/06/2026 23:26, olcott wrote: >>>>>>>>>>>>>>>>>>>>>> On 6/24/2026 5:00 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 23/06/2026 17:48, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 6/23/2026 1:06 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 15:10, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 6/22/2026 1:49 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 02:02, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 4:08 PM, André G. Isaak wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-21 14:42, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 3:04 PM, Alan Mackenzie wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> [ Followup-To: set ] >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 6:26 AM, Alan Mackenzie wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I just found the term: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounding in a proof theoretic atomic >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> base" yesterday. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You can find any number of terms. That >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> doesn't mean you're capable of >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding them. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The above is the key reason why under >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PTS Gödel 1931 incompleteness >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fails. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I don't believe you. You have no respect >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> for or understanding of the >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> truth. If you really want to persuade >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> anybody that PTS somehow causes >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's theorem not to hold, then cite an >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> academic expert who'll have >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some credibility. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If they are mere gibberish words to you >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> then you will not understand. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You don't understand Proof-theoritic >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Semantics, and you certainly don't >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand Gödel's Theorem, neither the >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theorem itself nor any proof of >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> it. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It is a verified fact that Gödel's G is >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> ungrounded >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in the atomic base of PA. That you do not >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> what: "grounded in the atomic base" means >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> is less >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> than no rebuttal at all. >>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounded in the atomic base of PA" is an >>>>>>>>>>>>>>>>>>>>>>>>>>>>> expression used only by you, and it is one >>>>>>>>>>>>>>>>>>>>>>>>>>>>> which you have never explicitly defined, so >>>>>>>>>>>>>>>>>>>>>>>>>>>>> the fault here certainly doesn't lie with >>>>>>>>>>>>>>>>>>>>>>>>>>>>> Alan. It's certainly not a 'verified fact' >>>>>>>>>>>>>>>>>>>>>>>>>>>>> when you haven't even adequately explained >>>>>>>>>>>>>>>>>>>>>>>>>>>>> what it is that you mean. >>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>> All of knowledge expressed in language is >>>>>>>>>>>>>>>>>>>>>>>>>>>> structured as a tree of semantic relations >>>>>>>>>>>>>>>>>>>>>>>>>>>> specified syntactically between finite strings. >>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>> What makes you believe semantic relations >>>>>>>>>>>>>>>>>>>>>>>>>>> that can be structured as >>>>>>>>>>>>>>>>>>>>>>>>>>> a tree are sufficient to contain all >>>>>>>>>>>>>>>>>>>>>>>>>>> knowledge that is exressed in >>>>>>>>>>>>>>>>>>>>>>>>>>> some language? >>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>> The CycL language and the Cyc Project. >>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>> They use a tree structure for concepts. But why >>>>>>>>>>>>>>>>>>>>>>>>> would one try to >>>>>>>>>>>>>>>>>>>>>>>>> put knowledge in a tree structure? >>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>> It must at least be a directed acyclic graph or >>>>>>>>>>>>>>>>>>>>>>>> the proof gets stuck in an infinite loop and never >>>>>>>>>>>>>>>>>>>>>>>> completes. >>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>> How can any ordering of knowledge prevent getting >>>>>>>>>>>>>>>>>>>>>>> stuck in a loop >>>>>>>>>>>>>>>>>>>>>>> when looking for a proof? >>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>> By looking upward in a type hierarchy. >>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>> If you mean not looking elsewhere that may indeed >>>>>>>>>>>>>>>>>>>>> prevent loops. >>>>>>>>>>>>>>>>>>>>> In most cases that also prevents finding the proof. >>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>> Truth Conditional Semantics (TCS) <is> incoherent >>>>>>>>>>>>>>>>>>>> compared to Proof Theoretic Semantics (PTS). >>>>>>>>>>>>>>>>>>>> Essentially >>>>>>>>>>>>>>>>>>>> PTS just coherently connects the semantic meanings >>>>>>>>>>>>>>>>>>>> expressed in language together into one coherent body >>>>>>>>>>>>>>>>>>>> of general knowledge. It does this without >>>>>>>>>>>>>>>>>>>> undecidability >>>>>>>>>>>>>>>>>>>> or mathematical incompleteness. >>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>> Looking for a proof does not need any semantics so it >>>>>>>>>>>>>>>>>>> is not obvious >>>>>>>>>>>>>>>>>>> how switching to another semantics could improve it. >>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>> In proof theoretic semantics an expression only gains >>>>>>>>>>>>>>>>>> semantic meaning by finding a proof. >>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>> It should be obvious that finding a proof does not >>>>>>>>>>>>>>>>> happen before >>>>>>>>>>>>>>>>> looking for a proof. >>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> If there is no sequence of inference steps in Q from >>>>>>>>>>>>>>>> ~∃x x=S(x) to the axioms of Q >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> There are, but that sequence is infinite >>>>>>>>>>>>>>> >>>>>>>>>>>>>> >>>>>>>>>>>>>> If there is no FINITE sequence of inference steps >>>>>>>>>>>>>> in Q from ~∃x x=S(x) to the axioms of Q then ~∃x x=S(x) >>>>>>>>>>>>>> is ungrounded in the PTS atomic base of Q. >>>>>>>>>>>>> >>>>>>>>>>>>> i.e., ~∃x x=S(x) is unprovable is Q, as is commonly known. >>>>>>>>>>>>> >>>>>>>>>>>> Is it commonly known that ~∃x x=S(x) >>>>>>>>>>> >>>>>>>>>>> Which has the semantic meaning "no number is equal to its >>>>>>>>>>> successor" as per the definition of Q. >>>>>>>>>>> >>>>>>>>>> >>>>>>>>>> Since there are no steps in Q that affirm ~∃x x=S(x) >>>>>>>>>> in Q it is an open question in Q and not a confirmed >>>>>>>>>> statement in Q. >>>>>>>>> >>>>>>>>> In other words, unproven as is commonly known. >>>>>>>>> >>>>>>>> Yet never gets to undecidable or in any sense of incomplete. >>>>>>>> >>>>>>> >>>>>>> False, as by definition, Q is incomplete because ~∃x x=S(x) is >>>>>>> unprovable / out-of-scope / not semantically grounded in Q. >>>>>> >>>>>> Proof theoretic semantics DOES NOT DO IT THAT WAY !!! >>>>> >>>>> Irrelevant. The statement that both ∃x x=S(x) and ~∃x x=S(x) are >>>>> sentences of Q but neither is a rheorem or Q does not depend on >>>>> any semantics. >>>> >>>> The entire body of knowledge expressed in language >>>> can be represented as a semantic tautology in an >>>> acyclic directed graph. That knowledge is a DAG was >>>> my very thought on this subject more than 30 years ago. >>>> This single idea gets rid of all undecidability >>>> within the entire body of knowledge. >>> >>> No, it cannot. The usual meaning of knoledge excludes tautologies >> >> You are not paying close enough attention. I did not say logical >> tautology. I said semantic tautology. That cats are defined to >> be animals is a semantic tautology. That cats are defined to be >> cats is a logical tautology. Here is a definition of that fits >> my definition of semantic tautology. >> >> tautology, in logic, a statement so framed that >> it cannot be denied without inconsistency. Thus, >> “All humans are mammals” is held to assert with >> regard to anything whatsoever that either it is >> not a human or it is a mammal. >> https://www.britannica.com/topic/tautology >> >> https://en.wikipedia.org/wiki/Tautology_(logic) >>> because the determination of their truth does not need any knowledge >>> beyond a method to determine whether a string is a tautology in the >>> relevant language. >>> >>> When the word "knowledge" is used it usually means knowing about the >>> real world something that cannot be determined without observation. >> >> One can know that "cats are animals" when this is >> stipulated as an axiom. > > That axiom only realtes the words "cat" and "animal". It does not tell > anything about the real world. > It tells us exactly one thing about the real world. cats have paws animals are living things The Earth orbits the Sun -- Copyright 2026 Olcott My 28 year goal has been to make "true on the basis of meaning expressed in language" reliably computable for the entire body of knowledge. The complete structure of this system is now defined. The entire body of knowledge expressed in language is comprised of two types of relations between finite strings: (a) *Axioms* Expressions of language that are stipulated to be true. My system bridges the analytic/synthetic distinction by expressly encoding all empirical "atomic facts" in a formal language such as CycL of the Cyc project. (b) *Inference Rules* Expressions of language that are semantically entailed syntactically from (a) and/or (b).
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| From | Mikko <mikko.levanto@iki.fi> |
|---|---|
| Date | 2026-07-01 10:01 +0300 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <1122e12$1mdml$1@dont-email.me> |
| In reply to | #645882 |
On 30/06/2026 16:43, olcott wrote: > On 6/30/2026 2:48 AM, Mikko wrote: >> On 29/06/2026 16:38, olcott wrote: >>> On 6/29/2026 1:23 AM, Mikko wrote: >>>> On 29/06/2026 06:12, olcott wrote: >>>>> On 6/28/2026 4:31 AM, Mikko wrote: >>>>>> On 27/06/2026 22:40, olcott wrote: >>>>>>> On 6/27/2026 2:23 PM, dbush wrote: >>>>>>>> On 6/27/2026 3:16 PM, olcott wrote: >>>>>>>>> On 6/27/2026 2:04 PM, dbush wrote: >>>>>>>>>> On 6/27/2026 3:01 PM, olcott wrote: >>>>>>>>>>> On 6/27/2026 1:39 PM, dbush wrote: >>>>>>>>>>>> On 6/27/2026 2:38 PM, olcott wrote: >>>>>>>>>>>>> On 6/27/2026 1:29 PM, dbush wrote: >>>>>>>>>>>>>> On 6/27/2026 2:27 PM, olcott wrote: >>>>>>>>>>>>>>> On 6/27/2026 1:01 PM, dbush wrote: >>>>>>>>>>>>>>>> On 6/27/2026 11:43 AM, polcott wrote: >>>>>>>>>>>>>>>>> On 6/27/2026 2:35 AM, Mikko wrote: >>>>>>>>>>>>>>>>>> On 26/06/2026 16:10, olcott wrote: >>>>>>>>>>>>>>>>>>> On 6/26/2026 1:39 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>> On 25/06/2026 19:14, olcott wrote: >>>>>>>>>>>>>>>>>>>>> On 6/25/2026 2:21 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>> On 24/06/2026 23:26, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>> On 6/24/2026 5:00 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 23/06/2026 17:48, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 6/23/2026 1:06 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 15:10, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/22/2026 1:49 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> On 22/06/2026 02:02, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 4:08 PM, André G. Isaak wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-21 14:42, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 3:04 PM, Alan Mackenzie wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> [ Followup-To: set ] >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/21/2026 6:26 AM, Alan Mackenzie >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In comp.theory olcott >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> <polcott333@gmail.com> wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I just found the term: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounding in a proof theoretic >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> atomic base" yesterday. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You can find any number of terms. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> That doesn't mean you're capable of >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understanding them. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The above is the key reason why under >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PTS Gödel 1931 incompleteness >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fails. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I don't believe you. You have no >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> respect for or understanding of the >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> truth. If you really want to persuade >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> anybody that PTS somehow causes >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Gödel's theorem not to hold, then cite >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> an academic expert who'll have >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> some credibility. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> If they are mere gibberish words to you >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> then you will not understand. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You don't understand Proof-theoritic >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Semantics, and you certainly don't >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand Gödel's Theorem, neither the >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> theorem itself nor any proof of >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> it. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> It is a verified fact that Gödel's G is >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> ungrounded >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in the atomic base of PA. That you do not >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> what: "grounded in the atomic base" means >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> is less >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> than no rebuttal at all. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> "grounded in the atomic base of PA" is an >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> expression used only by you, and it is one >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> which you have never explicitly defined, >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> so the fault here certainly doesn't lie >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> with Alan. It's certainly not a 'verified >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> fact' when you haven't even adequately >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> explained what it is that you mean. >>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>> All of knowledge expressed in language is >>>>>>>>>>>>>>>>>>>>>>>>>>>>> structured as a tree of semantic relations >>>>>>>>>>>>>>>>>>>>>>>>>>>>> specified syntactically between finite >>>>>>>>>>>>>>>>>>>>>>>>>>>>> strings. >>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>> What makes you believe semantic relations >>>>>>>>>>>>>>>>>>>>>>>>>>>> that can be structured as >>>>>>>>>>>>>>>>>>>>>>>>>>>> a tree are sufficient to contain all >>>>>>>>>>>>>>>>>>>>>>>>>>>> knowledge that is exressed in >>>>>>>>>>>>>>>>>>>>>>>>>>>> some language? >>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>> The CycL language and the Cyc Project. >>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>> They use a tree structure for concepts. But >>>>>>>>>>>>>>>>>>>>>>>>>> why would one try to >>>>>>>>>>>>>>>>>>>>>>>>>> put knowledge in a tree structure? >>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>> It must at least be a directed acyclic graph or >>>>>>>>>>>>>>>>>>>>>>>>> the proof gets stuck in an infinite loop and never >>>>>>>>>>>>>>>>>>>>>>>>> completes. >>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>> How can any ordering of knowledge prevent >>>>>>>>>>>>>>>>>>>>>>>> getting stuck in a loop >>>>>>>>>>>>>>>>>>>>>>>> when looking for a proof? >>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>> By looking upward in a type hierarchy. >>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>> If you mean not looking elsewhere that may indeed >>>>>>>>>>>>>>>>>>>>>> prevent loops. >>>>>>>>>>>>>>>>>>>>>> In most cases that also prevents finding the proof. >>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>> Truth Conditional Semantics (TCS) <is> incoherent >>>>>>>>>>>>>>>>>>>>> compared to Proof Theoretic Semantics (PTS). >>>>>>>>>>>>>>>>>>>>> Essentially >>>>>>>>>>>>>>>>>>>>> PTS just coherently connects the semantic meanings >>>>>>>>>>>>>>>>>>>>> expressed in language together into one coherent body >>>>>>>>>>>>>>>>>>>>> of general knowledge. It does this without >>>>>>>>>>>>>>>>>>>>> undecidability >>>>>>>>>>>>>>>>>>>>> or mathematical incompleteness. >>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>> Looking for a proof does not need any semantics so >>>>>>>>>>>>>>>>>>>> it is not obvious >>>>>>>>>>>>>>>>>>>> how switching to another semantics could improve it. >>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>> In proof theoretic semantics an expression only gains >>>>>>>>>>>>>>>>>>> semantic meaning by finding a proof. >>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>> It should be obvious that finding a proof does not >>>>>>>>>>>>>>>>>> happen before >>>>>>>>>>>>>>>>>> looking for a proof. >>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>> If there is no sequence of inference steps in Q from >>>>>>>>>>>>>>>>> ~∃x x=S(x) to the axioms of Q >>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> There are, but that sequence is infinite >>>>>>>>>>>>>>>> >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> If there is no FINITE sequence of inference steps >>>>>>>>>>>>>>> in Q from ~∃x x=S(x) to the axioms of Q then ~∃x x=S(x) >>>>>>>>>>>>>>> is ungrounded in the PTS atomic base of Q. >>>>>>>>>>>>>> >>>>>>>>>>>>>> i.e., ~∃x x=S(x) is unprovable is Q, as is commonly known. >>>>>>>>>>>>>> >>>>>>>>>>>>> Is it commonly known that ~∃x x=S(x) >>>>>>>>>>>> >>>>>>>>>>>> Which has the semantic meaning "no number is equal to its >>>>>>>>>>>> successor" as per the definition of Q. >>>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> Since there are no steps in Q that affirm ~∃x x=S(x) >>>>>>>>>>> in Q it is an open question in Q and not a confirmed >>>>>>>>>>> statement in Q. >>>>>>>>>> >>>>>>>>>> In other words, unproven as is commonly known. >>>>>>>>>> >>>>>>>>> Yet never gets to undecidable or in any sense of incomplete. >>>>>>>>> >>>>>>>> >>>>>>>> False, as by definition, Q is incomplete because ~∃x x=S(x) is >>>>>>>> unprovable / out-of-scope / not semantically grounded in Q. >>>>>>> >>>>>>> Proof theoretic semantics DOES NOT DO IT THAT WAY !!! >>>>>> >>>>>> Irrelevant. The statement that both ∃x x=S(x) and ~∃x x=S(x) are >>>>>> sentences of Q but neither is a rheorem or Q does not depend on >>>>>> any semantics. >>>>> >>>>> The entire body of knowledge expressed in language >>>>> can be represented as a semantic tautology in an >>>>> acyclic directed graph. That knowledge is a DAG was >>>>> my very thought on this subject more than 30 years ago. >>>>> This single idea gets rid of all undecidability >>>>> within the entire body of knowledge. >>>> >>>> No, it cannot. The usual meaning of knoledge excludes tautologies >>> >>> You are not paying close enough attention. I did not say logical >>> tautology. I said semantic tautology. That cats are defined to >>> be animals is a semantic tautology. That cats are defined to be >>> cats is a logical tautology. Here is a definition of that fits >>> my definition of semantic tautology. >>> >>> tautology, in logic, a statement so framed that >>> it cannot be denied without inconsistency. Thus, >>> “All humans are mammals” is held to assert with >>> regard to anything whatsoever that either it is >>> not a human or it is a mammal. >>> https://www.britannica.com/topic/tautology >>> >>> https://en.wikipedia.org/wiki/Tautology_(logic) >>>> because the determination of their truth does not need any knowledge >>>> beyond a method to determine whether a string is a tautology in the >>>> relevant language. >>>> >>>> When the word "knowledge" is used it usually means knowing about the >>>> real world something that cannot be determined without observation. >>> >>> One can know that "cats are animals" when this is >>> stipulated as an axiom. >> >> That axiom only realtes the words "cat" and "animal". It does not tell >> anything about the real world. > > It tells us exactly one thing about the real world. Only to those who already know that there are things called "cat" in the real world and know what kind of things the words "cat" and "animal" refer to but don't already know that every thing that the word "cat" refers to is a thing that the word "animal" refers to. -- Mikko
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