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Groups > comp.theory > #141687 > unrolled thread
| Started by | olcott <polcott333@gmail.com> |
|---|---|
| First post | 2026-06-17 16:14 -0500 |
| Last post | 2026-06-23 09:55 -0500 |
| Articles | 20 on this page of 357 — 11 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 dart200 <user7160@newsgrouper.org.invalid> - 2026-06-19 13:49 -0700
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 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 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 André G. Isaak <agisaak@gm.invalid> - 2026-06-20 11:40 -0600
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 dbush <dbush.mobile@gmail.com> - 2026-06-21 09:14 -0400
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-22 09:16 +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> - 2026-06-23 09:29 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-24 11:23 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-24 15:19 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-25 10:09 +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 Mikko <mikko.levanto@iki.fi> - 2026-06-20 10:54 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Alan Mackenzie <acm@muc.de> - 2026-06-20 10:26 +0000
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 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) foundations of mathematics André G. Isaak <agisaak@gm.invalid> - 2026-06-21 14:18 -0600
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Alan Mackenzie <acm@muc.de> - 2026-06-21 20:44 +0000
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> - 2026-06-23 09:40 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-24 12:45 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-24 15:23 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-25 10:14 +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 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 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> - 2026-06-23 09:47 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-24 12:52 +0300
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-24 15:25 -0500
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics Mikko <mikko.levanto@iki.fi> - 2026-06-25 10:18 +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> - 2026-07-03 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> - 2026-06-28 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> - 2026-06-23 09:52 -0500
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-23 08:54 -0700
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-06-23 09:06 -0700
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction olcott <polcott333@gmail.com> - 2026-06-23 11:56 -0500
Re: Readings in (some of the) foundations of mathematics --- analytic/synthetic distinction Mikko <mikko.levanto@iki.fi> - 2026-06-24 13:06 +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 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) 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
Re: Ross A. Finlayson, readings in (some of the) foundations of mathematics olcott <polcott333@gmail.com> - 2026-06-23 09:55 -0500
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| From | Mikko <mikko.levanto@iki.fi> |
|---|---|
| Date | 2026-06-27 11:05 +0300 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111o07k$149pv$1@dont-email.me> |
| In reply to | #142000 |
On 27/06/2026 01:01, olcott wrote: > On 6/26/2026 2:55 PM, dbush wrote: >> On 6/26/2026 3:38 PM, olcott wrote: >>> On 6/26/2026 2:17 PM, dbush wrote: >>>> On 6/26/2026 3:07 PM, olcott wrote: >>>>> On 6/26/2026 1:51 PM, dbush wrote: >>>>>> On 6/26/2026 2:48 PM, olcott wrote: >>>>>>> On 6/26/2026 1:14 PM, André G. Isaak wrote: >>>>>>>> On 2026-06-26 11:22, olcott wrote: >>>>>>>>> On 6/26/2026 11:08 AM, dbush wrote: >>>>>>>> >>>>>>>>>> By your logic, "no number is equal to its successor" has no >>>>>>>>>> meaning in Robinson arithmetic. >>>>>>>>> >>>>>>>>> In Robinson Arithmetic (often denoted as Q), >>>>>>>>> the statement "no number is equal to its >>>>>>>>> successor" is not provable.While this statement >>>>>>>>> is true for the standard natural numbers, Robinson >>>>>>>>> Arithmetic is too weak to prove it universally >>>>>>>>> (∀ x, S(x) ≠ x). >>>>>>>> >>>>>>>> It's not provable but it certainly has meaning. >>>>>>>> >>>>>>>> André >>>>>>>> >>>>>>> >>>>>>> out-of-scope for Q is more accurate as jargon free. >>>>>>> >>>>>>> PTS does hold the view that meaning is only derived >>>>>>> through inference steps. This simple sentence seems >>>>>>> impossibly too difficult for anyone fully indoctrinated >>>>>>> with alternative views. So I will simply say out-of-scope. >>>>>>> >>>>>> >>>>>> So "out-of-scope" is merely a synonym for unprovable. Then to put >>>>>> things in words you can understand: >>>>>> >>>>> >>>>> "I am driving to Walmart to buy a carton of >>>>> Breyer's natural vanilla ice cream." is also unprovable in PA. >>>>> In both cases the semantics in not represented in PA. >>>> >>>> Not applicable, as that is not a sentence in PA. >>>> >>> >>> It is expressed in PA >> >> False. The above is not a sentence of PA. >> >>> to the same degree that G is expressed >>> in PA has a huge natural number. The semantics of it and >>> the semantics of G are neither expressible in PA. >> >> False. G is simply a sentence like ~∃x x>10 v x<5 but much more complex. >> >>> >>>> "No number is equal to its successor" is a sentence in RA, and it is >>>> true but unprovable in RA (or as your would call it, "out-of-scope"). >>>> >>> >>> If its semantics is not expressible in Q (What RA is called) >>> then it is not actually expressible in Q. >> >> "No number is equal to its successor" is a sentence in the language of >> Q. More formally, it is this: >> >> ~∃x x=S(x) >> >> And this sentence is not provable from the axioms of Q (or, in terms >> you would understand, the above is "out-of-scope" of Q). >> > > OK I checked the details so I need to make my > language more precise. > > Within proof theoretic semantics any expression > that cannot be proven in Q is not semantically > grounded in Q. Nevertheless a sentence like ~∃x x=S(x) is in the language of Q and that way in the theory. -- Mikko
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| From | polcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-27 10:47 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111orae$1kcvi$3@solani.org> |
| In reply to | #142021 |
On 6/27/2026 3:05 AM, Mikko wrote: > On 27/06/2026 01:01, olcott wrote: >> On 6/26/2026 2:55 PM, dbush wrote: >>> On 6/26/2026 3:38 PM, olcott wrote: >>>> On 6/26/2026 2:17 PM, dbush wrote: >>>>> On 6/26/2026 3:07 PM, olcott wrote: >>>>>> On 6/26/2026 1:51 PM, dbush wrote: >>>>>>> On 6/26/2026 2:48 PM, olcott wrote: >>>>>>>> On 6/26/2026 1:14 PM, André G. Isaak wrote: >>>>>>>>> On 2026-06-26 11:22, olcott wrote: >>>>>>>>>> On 6/26/2026 11:08 AM, dbush wrote: >>>>>>>>> >>>>>>>>>>> By your logic, "no number is equal to its successor" has no >>>>>>>>>>> meaning in Robinson arithmetic. >>>>>>>>>> >>>>>>>>>> In Robinson Arithmetic (often denoted as Q), >>>>>>>>>> the statement "no number is equal to its >>>>>>>>>> successor" is not provable.While this statement >>>>>>>>>> is true for the standard natural numbers, Robinson >>>>>>>>>> Arithmetic is too weak to prove it universally >>>>>>>>>> (∀ x, S(x) ≠ x). >>>>>>>>> >>>>>>>>> It's not provable but it certainly has meaning. >>>>>>>>> >>>>>>>>> André >>>>>>>>> >>>>>>>> >>>>>>>> out-of-scope for Q is more accurate as jargon free. >>>>>>>> >>>>>>>> PTS does hold the view that meaning is only derived >>>>>>>> through inference steps. This simple sentence seems >>>>>>>> impossibly too difficult for anyone fully indoctrinated >>>>>>>> with alternative views. So I will simply say out-of-scope. >>>>>>>> >>>>>>> >>>>>>> So "out-of-scope" is merely a synonym for unprovable. Then to >>>>>>> put things in words you can understand: >>>>>>> >>>>>> >>>>>> "I am driving to Walmart to buy a carton of >>>>>> Breyer's natural vanilla ice cream." is also unprovable in PA. >>>>>> In both cases the semantics in not represented in PA. >>>>> >>>>> Not applicable, as that is not a sentence in PA. >>>>> >>>> >>>> It is expressed in PA >>> >>> False. The above is not a sentence of PA. >>> >>>> to the same degree that G is expressed >>>> in PA has a huge natural number. The semantics of it and >>>> the semantics of G are neither expressible in PA. >>> >>> False. G is simply a sentence like ~∃x x>10 v x<5 but much more >>> complex. >>> >>>> >>>>> "No number is equal to its successor" is a sentence in RA, and it >>>>> is true but unprovable in RA (or as your would call it, "out-of- >>>>> scope"). >>>>> >>>> >>>> If its semantics is not expressible in Q (What RA is called) >>>> then it is not actually expressible in Q. >>> >>> "No number is equal to its successor" is a sentence in the language >>> of Q. More formally, it is this: >>> >>> ~∃x x=S(x) >>> >>> And this sentence is not provable from the axioms of Q (or, in terms >>> you would understand, the above is "out-of-scope" of Q). >>> >> >> OK I checked the details so I need to make my >> language more precise. >> >> Within proof theoretic semantics any expression >> that cannot be proven in Q is not semantically >> grounded in Q. > > Nevertheless a sentence like ~∃x x=S(x) is in the language of Q and > that way in the theory. > Colorless green ideas sleep furiously was composed by Noam Chomsky in his 1957 book Syntactic Structures as an example of a sentence that is grammatically well-formed, but semantically nonsensical. Proving that syntax is not enough. -- 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 | Ross Finlayson <ross.a.finlayson@gmail.com> |
|---|---|
| Date | 2026-06-27 15:37 -0700 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <PWqdnbW7otUlzd33nZ2dnZfqn_idnZ2d@giganews.com> |
| In reply to | #142034 |
On 06/27/2026 08:47 AM, polcott wrote: > On 6/27/2026 3:05 AM, Mikko wrote: >> On 27/06/2026 01:01, olcott wrote: >>> On 6/26/2026 2:55 PM, dbush wrote: >>>> On 6/26/2026 3:38 PM, olcott wrote: >>>>> On 6/26/2026 2:17 PM, dbush wrote: >>>>>> On 6/26/2026 3:07 PM, olcott wrote: >>>>>>> On 6/26/2026 1:51 PM, dbush wrote: >>>>>>>> On 6/26/2026 2:48 PM, olcott wrote: >>>>>>>>> On 6/26/2026 1:14 PM, André G. Isaak wrote: >>>>>>>>>> On 2026-06-26 11:22, olcott wrote: >>>>>>>>>>> On 6/26/2026 11:08 AM, dbush wrote: >>>>>>>>>> >>>>>>>>>>>> By your logic, "no number is equal to its successor" has no >>>>>>>>>>>> meaning in Robinson arithmetic. >>>>>>>>>>> >>>>>>>>>>> In Robinson Arithmetic (often denoted as Q), >>>>>>>>>>> the statement "no number is equal to its >>>>>>>>>>> successor" is not provable.While this statement >>>>>>>>>>> is true for the standard natural numbers, Robinson >>>>>>>>>>> Arithmetic is too weak to prove it universally >>>>>>>>>>> (∀ x, S(x) ≠ x). >>>>>>>>>> >>>>>>>>>> It's not provable but it certainly has meaning. >>>>>>>>>> >>>>>>>>>> André >>>>>>>>>> >>>>>>>>> >>>>>>>>> out-of-scope for Q is more accurate as jargon free. >>>>>>>>> >>>>>>>>> PTS does hold the view that meaning is only derived >>>>>>>>> through inference steps. This simple sentence seems >>>>>>>>> impossibly too difficult for anyone fully indoctrinated >>>>>>>>> with alternative views. So I will simply say out-of-scope. >>>>>>>>> >>>>>>>> >>>>>>>> So "out-of-scope" is merely a synonym for unprovable. Then to >>>>>>>> put things in words you can understand: >>>>>>>> >>>>>>> >>>>>>> "I am driving to Walmart to buy a carton of >>>>>>> Breyer's natural vanilla ice cream." is also unprovable in PA. >>>>>>> In both cases the semantics in not represented in PA. >>>>>> >>>>>> Not applicable, as that is not a sentence in PA. >>>>>> >>>>> >>>>> It is expressed in PA >>>> >>>> False. The above is not a sentence of PA. >>>> >>>>> to the same degree that G is expressed >>>>> in PA has a huge natural number. The semantics of it and >>>>> the semantics of G are neither expressible in PA. >>>> >>>> False. G is simply a sentence like ~∃x x>10 v x<5 but much more >>>> complex. >>>> >>>>> >>>>>> "No number is equal to its successor" is a sentence in RA, and it >>>>>> is true but unprovable in RA (or as your would call it, "out-of- >>>>>> scope"). >>>>>> >>>>> >>>>> If its semantics is not expressible in Q (What RA is called) >>>>> then it is not actually expressible in Q. >>>> >>>> "No number is equal to its successor" is a sentence in the language >>>> of Q. More formally, it is this: >>>> >>>> ~∃x x=S(x) >>>> >>>> And this sentence is not provable from the axioms of Q (or, in terms >>>> you would understand, the above is "out-of-scope" of Q). >>>> >>> >>> OK I checked the details so I need to make my >>> language more precise. >>> >>> Within proof theoretic semantics any expression >>> that cannot be proven in Q is not semantically >>> grounded in Q. >> >> Nevertheless a sentence like ~∃x x=S(x) is in the language of Q and >> that way in the theory. >> > > > Colorless green ideas sleep furiously > was composed by Noam Chomsky in his 1957 book > Syntactic Structures as an example of a sentence > that is grammatically well-formed, but semantically > nonsensical. > > Proving that syntax is not enough. > "Colorless green" is actually two colors since there's a dual-tristimulus colorspace the chromatic and the prismatic, a fact of the science of the theory of light and color, of which you are ignorant, then making for a reasonable reading of the usual apocryphal comment. Then, ideas can sleep however they want.
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| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-27 17:47 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111pjt5$38pef$2@dont-email.me> |
| In reply to | #142076 |
On 6/27/2026 5:37 PM, Ross Finlayson wrote: > On 06/27/2026 08:47 AM, polcott wrote: >> On 6/27/2026 3:05 AM, Mikko wrote: >>> On 27/06/2026 01:01, olcott wrote: >>>> On 6/26/2026 2:55 PM, dbush wrote: >>>>> On 6/26/2026 3:38 PM, olcott wrote: >>>>>> On 6/26/2026 2:17 PM, dbush wrote: >>>>>>> On 6/26/2026 3:07 PM, olcott wrote: >>>>>>>> On 6/26/2026 1:51 PM, dbush wrote: >>>>>>>>> On 6/26/2026 2:48 PM, olcott wrote: >>>>>>>>>> On 6/26/2026 1:14 PM, André G. Isaak wrote: >>>>>>>>>>> On 2026-06-26 11:22, olcott wrote: >>>>>>>>>>>> On 6/26/2026 11:08 AM, dbush wrote: >>>>>>>>>>> >>>>>>>>>>>>> By your logic, "no number is equal to its successor" has no >>>>>>>>>>>>> meaning in Robinson arithmetic. >>>>>>>>>>>> >>>>>>>>>>>> In Robinson Arithmetic (often denoted as Q), >>>>>>>>>>>> the statement "no number is equal to its >>>>>>>>>>>> successor" is not provable.While this statement >>>>>>>>>>>> is true for the standard natural numbers, Robinson >>>>>>>>>>>> Arithmetic is too weak to prove it universally >>>>>>>>>>>> (∀ x, S(x) ≠ x). >>>>>>>>>>> >>>>>>>>>>> It's not provable but it certainly has meaning. >>>>>>>>>>> >>>>>>>>>>> André >>>>>>>>>>> >>>>>>>>>> >>>>>>>>>> out-of-scope for Q is more accurate as jargon free. >>>>>>>>>> >>>>>>>>>> PTS does hold the view that meaning is only derived >>>>>>>>>> through inference steps. This simple sentence seems >>>>>>>>>> impossibly too difficult for anyone fully indoctrinated >>>>>>>>>> with alternative views. So I will simply say out-of-scope. >>>>>>>>>> >>>>>>>>> >>>>>>>>> So "out-of-scope" is merely a synonym for unprovable. Then to >>>>>>>>> put things in words you can understand: >>>>>>>>> >>>>>>>> >>>>>>>> "I am driving to Walmart to buy a carton of >>>>>>>> Breyer's natural vanilla ice cream." is also unprovable in PA. >>>>>>>> In both cases the semantics in not represented in PA. >>>>>>> >>>>>>> Not applicable, as that is not a sentence in PA. >>>>>>> >>>>>> >>>>>> It is expressed in PA >>>>> >>>>> False. The above is not a sentence of PA. >>>>> >>>>>> to the same degree that G is expressed >>>>>> in PA has a huge natural number. The semantics of it and >>>>>> the semantics of G are neither expressible in PA. >>>>> >>>>> False. G is simply a sentence like ~∃x x>10 v x<5 but much more >>>>> complex. >>>>> >>>>>> >>>>>>> "No number is equal to its successor" is a sentence in RA, and it >>>>>>> is true but unprovable in RA (or as your would call it, "out-of- >>>>>>> scope"). >>>>>>> >>>>>> >>>>>> If its semantics is not expressible in Q (What RA is called) >>>>>> then it is not actually expressible in Q. >>>>> >>>>> "No number is equal to its successor" is a sentence in the language >>>>> of Q. More formally, it is this: >>>>> >>>>> ~∃x x=S(x) >>>>> >>>>> And this sentence is not provable from the axioms of Q (or, in terms >>>>> you would understand, the above is "out-of-scope" of Q). >>>>> >>>> >>>> OK I checked the details so I need to make my >>>> language more precise. >>>> >>>> Within proof theoretic semantics any expression >>>> that cannot be proven in Q is not semantically >>>> grounded in Q. >>> >>> Nevertheless a sentence like ~∃x x=S(x) is in the language of Q and >>> that way in the theory. >>> >> >> >> Colorless green ideas sleep furiously >> was composed by Noam Chomsky in his 1957 book >> Syntactic Structures as an example of a sentence >> that is grammatically well-formed, but semantically >> nonsensical. >> >> Proving that syntax is not enough. >> > > "Colorless green" is actually two colors > since there's a dual-tristimulus colorspace > the chromatic and the prismatic, > a fact of the science of the theory of light and color, > of which you are ignorant, then making for a reasonable > reading of the usual apocryphal comment. > > Then, ideas can sleep however they want. > > The point is that syntactically correct expressions can be semantically incoherent. Math always makes sure to ignore this. The gibberish cannot be proven counts as undecidability in math. -- 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 | Ross Finlayson <ross.a.finlayson@gmail.com> |
|---|---|
| Date | 2026-06-27 19:24 -0700 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <z5Ccnfj-1fJ9GN33nZ2dnZfqnPidnZ2d@giganews.com> |
| In reply to | #142079 |
On 06/27/2026 03:47 PM, olcott wrote: > On 6/27/2026 5:37 PM, Ross Finlayson wrote: >> On 06/27/2026 08:47 AM, polcott wrote: >>> On 6/27/2026 3:05 AM, Mikko wrote: >>>> On 27/06/2026 01:01, olcott wrote: >>>>> On 6/26/2026 2:55 PM, dbush wrote: >>>>>> On 6/26/2026 3:38 PM, olcott wrote: >>>>>>> On 6/26/2026 2:17 PM, dbush wrote: >>>>>>>> On 6/26/2026 3:07 PM, olcott wrote: >>>>>>>>> On 6/26/2026 1:51 PM, dbush wrote: >>>>>>>>>> On 6/26/2026 2:48 PM, olcott wrote: >>>>>>>>>>> On 6/26/2026 1:14 PM, André G. Isaak wrote: >>>>>>>>>>>> On 2026-06-26 11:22, olcott wrote: >>>>>>>>>>>>> On 6/26/2026 11:08 AM, dbush wrote: >>>>>>>>>>>> >>>>>>>>>>>>>> By your logic, "no number is equal to its successor" has no >>>>>>>>>>>>>> meaning in Robinson arithmetic. >>>>>>>>>>>>> >>>>>>>>>>>>> In Robinson Arithmetic (often denoted as Q), >>>>>>>>>>>>> the statement "no number is equal to its >>>>>>>>>>>>> successor" is not provable.While this statement >>>>>>>>>>>>> is true for the standard natural numbers, Robinson >>>>>>>>>>>>> Arithmetic is too weak to prove it universally >>>>>>>>>>>>> (∀ x, S(x) ≠ x). >>>>>>>>>>>> >>>>>>>>>>>> It's not provable but it certainly has meaning. >>>>>>>>>>>> >>>>>>>>>>>> André >>>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> out-of-scope for Q is more accurate as jargon free. >>>>>>>>>>> >>>>>>>>>>> PTS does hold the view that meaning is only derived >>>>>>>>>>> through inference steps. This simple sentence seems >>>>>>>>>>> impossibly too difficult for anyone fully indoctrinated >>>>>>>>>>> with alternative views. So I will simply say out-of-scope. >>>>>>>>>>> >>>>>>>>>> >>>>>>>>>> So "out-of-scope" is merely a synonym for unprovable. Then to >>>>>>>>>> put things in words you can understand: >>>>>>>>>> >>>>>>>>> >>>>>>>>> "I am driving to Walmart to buy a carton of >>>>>>>>> Breyer's natural vanilla ice cream." is also unprovable in PA. >>>>>>>>> In both cases the semantics in not represented in PA. >>>>>>>> >>>>>>>> Not applicable, as that is not a sentence in PA. >>>>>>>> >>>>>>> >>>>>>> It is expressed in PA >>>>>> >>>>>> False. The above is not a sentence of PA. >>>>>> >>>>>>> to the same degree that G is expressed >>>>>>> in PA has a huge natural number. The semantics of it and >>>>>>> the semantics of G are neither expressible in PA. >>>>>> >>>>>> False. G is simply a sentence like ~∃x x>10 v x<5 but much more >>>>>> complex. >>>>>> >>>>>>> >>>>>>>> "No number is equal to its successor" is a sentence in RA, and it >>>>>>>> is true but unprovable in RA (or as your would call it, "out-of- >>>>>>>> scope"). >>>>>>>> >>>>>>> >>>>>>> If its semantics is not expressible in Q (What RA is called) >>>>>>> then it is not actually expressible in Q. >>>>>> >>>>>> "No number is equal to its successor" is a sentence in the language >>>>>> of Q. More formally, it is this: >>>>>> >>>>>> ~∃x x=S(x) >>>>>> >>>>>> And this sentence is not provable from the axioms of Q (or, in terms >>>>>> you would understand, the above is "out-of-scope" of Q). >>>>>> >>>>> >>>>> OK I checked the details so I need to make my >>>>> language more precise. >>>>> >>>>> Within proof theoretic semantics any expression >>>>> that cannot be proven in Q is not semantically >>>>> grounded in Q. >>>> >>>> Nevertheless a sentence like ~∃x x=S(x) is in the language of Q and >>>> that way in the theory. >>>> >>> >>> >>> Colorless green ideas sleep furiously >>> was composed by Noam Chomsky in his 1957 book >>> Syntactic Structures as an example of a sentence >>> that is grammatically well-formed, but semantically >>> nonsensical. >>> >>> Proving that syntax is not enough. >>> >> >> "Colorless green" is actually two colors >> since there's a dual-tristimulus colorspace >> the chromatic and the prismatic, >> a fact of the science of the theory of light and color, >> of which you are ignorant, then making for a reasonable >> reading of the usual apocryphal comment. >> >> Then, ideas can sleep however they want. >> >> > > The point is that syntactically correct expressions > can be semantically incoherent. Math always makes > sure to ignore this. The gibberish cannot be proven > counts as undecidability in math. > > That's yanking one's own chain, and doesn't work on others. The "grammar" hierarchy of Chomsky is of a very limited and simple model of computation and a very direct connection to "regular expressions", with regards to formal methods, finite automata, linear, right linear, and regular expressions, and of accounts of the various amounts of look-ahead or memory in scanners what result productions, that then in any account of source models involves linking and dictionaries of symbols, that essentially it's not saying much and isn't much of "grammar". Notions for example of the "railroad diagram" simply equip what are models of languages like "SQL" that are complicated in Chomsky to be simple in alternatives/optionals/loops with regards to the declarations of "grammars". Any sort of usual useful "grammar" involves a "multi-pass parser", with regards to parsing, for example for natural language, which usually has a direct account of nouns and verbs, when really the infinitives are always interrupted by instantiating a verb tense, and nouns are particulars and simple. Then, for natural language, all readers of natural human language using something alike "Tesniere grammars" as of "dependency grammars" that all learned in school with regards to diagramming any well-formed sentence. Aristotle is not a fool - and Aristotle won't be made a fool. That that that that that that that, .... The problem is not that "colorless green ideas sleep furiously" is given a _context_ where it's not simply mimsy as the borogoves, then that besides, all utterances are in a large overall context. So now you don't know grammar, either.
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| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-27 22:21 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111q3v1$3c6ln$1@dont-email.me> |
| In reply to | #142098 |
On 6/27/2026 9:24 PM, Ross Finlayson wrote: > On 06/27/2026 03:47 PM, olcott wrote: >> On 6/27/2026 5:37 PM, Ross Finlayson wrote: >>> On 06/27/2026 08:47 AM, polcott wrote: >>>> On 6/27/2026 3:05 AM, Mikko wrote: >>>>> On 27/06/2026 01:01, olcott wrote: >>>>>> On 6/26/2026 2:55 PM, dbush wrote: >>>>>>> On 6/26/2026 3:38 PM, olcott wrote: >>>>>>>> On 6/26/2026 2:17 PM, dbush wrote: >>>>>>>>> On 6/26/2026 3:07 PM, olcott wrote: >>>>>>>>>> On 6/26/2026 1:51 PM, dbush wrote: >>>>>>>>>>> On 6/26/2026 2:48 PM, olcott wrote: >>>>>>>>>>>> On 6/26/2026 1:14 PM, André G. Isaak wrote: >>>>>>>>>>>>> On 2026-06-26 11:22, olcott wrote: >>>>>>>>>>>>>> On 6/26/2026 11:08 AM, dbush wrote: >>>>>>>>>>>>> >>>>>>>>>>>>>>> By your logic, "no number is equal to its successor" has no >>>>>>>>>>>>>>> meaning in Robinson arithmetic. >>>>>>>>>>>>>> >>>>>>>>>>>>>> In Robinson Arithmetic (often denoted as Q), >>>>>>>>>>>>>> the statement "no number is equal to its >>>>>>>>>>>>>> successor" is not provable.While this statement >>>>>>>>>>>>>> is true for the standard natural numbers, Robinson >>>>>>>>>>>>>> Arithmetic is too weak to prove it universally >>>>>>>>>>>>>> (∀ x, S(x) ≠ x). >>>>>>>>>>>>> >>>>>>>>>>>>> It's not provable but it certainly has meaning. >>>>>>>>>>>>> >>>>>>>>>>>>> André >>>>>>>>>>>>> >>>>>>>>>>>> >>>>>>>>>>>> out-of-scope for Q is more accurate as jargon free. >>>>>>>>>>>> >>>>>>>>>>>> PTS does hold the view that meaning is only derived >>>>>>>>>>>> through inference steps. This simple sentence seems >>>>>>>>>>>> impossibly too difficult for anyone fully indoctrinated >>>>>>>>>>>> with alternative views. So I will simply say out-of-scope. >>>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> So "out-of-scope" is merely a synonym for unprovable. Then to >>>>>>>>>>> put things in words you can understand: >>>>>>>>>>> >>>>>>>>>> >>>>>>>>>> "I am driving to Walmart to buy a carton of >>>>>>>>>> Breyer's natural vanilla ice cream." is also unprovable in PA. >>>>>>>>>> In both cases the semantics in not represented in PA. >>>>>>>>> >>>>>>>>> Not applicable, as that is not a sentence in PA. >>>>>>>>> >>>>>>>> >>>>>>>> It is expressed in PA >>>>>>> >>>>>>> False. The above is not a sentence of PA. >>>>>>> >>>>>>>> to the same degree that G is expressed >>>>>>>> in PA has a huge natural number. The semantics of it and >>>>>>>> the semantics of G are neither expressible in PA. >>>>>>> >>>>>>> False. G is simply a sentence like ~∃x x>10 v x<5 but much more >>>>>>> complex. >>>>>>> >>>>>>>> >>>>>>>>> "No number is equal to its successor" is a sentence in RA, and it >>>>>>>>> is true but unprovable in RA (or as your would call it, "out-of- >>>>>>>>> scope"). >>>>>>>>> >>>>>>>> >>>>>>>> If its semantics is not expressible in Q (What RA is called) >>>>>>>> then it is not actually expressible in Q. >>>>>>> >>>>>>> "No number is equal to its successor" is a sentence in the language >>>>>>> of Q. More formally, it is this: >>>>>>> >>>>>>> ~∃x x=S(x) >>>>>>> >>>>>>> And this sentence is not provable from the axioms of Q (or, in terms >>>>>>> you would understand, the above is "out-of-scope" of Q). >>>>>>> >>>>>> >>>>>> OK I checked the details so I need to make my >>>>>> language more precise. >>>>>> >>>>>> Within proof theoretic semantics any expression >>>>>> that cannot be proven in Q is not semantically >>>>>> grounded in Q. >>>>> >>>>> Nevertheless a sentence like ~∃x x=S(x) is in the language of Q and >>>>> that way in the theory. >>>>> >>>> >>>> >>>> Colorless green ideas sleep furiously >>>> was composed by Noam Chomsky in his 1957 book >>>> Syntactic Structures as an example of a sentence >>>> that is grammatically well-formed, but semantically >>>> nonsensical. >>>> >>>> Proving that syntax is not enough. >>>> >>> >>> "Colorless green" is actually two colors >>> since there's a dual-tristimulus colorspace >>> the chromatic and the prismatic, >>> a fact of the science of the theory of light and color, >>> of which you are ignorant, then making for a reasonable >>> reading of the usual apocryphal comment. >>> >>> Then, ideas can sleep however they want. >>> >>> >> >> The point is that syntactically correct expressions >> can be semantically incoherent. Math always makes >> sure to ignore this. The gibberish cannot be proven >> counts as undecidability in math. >> >> > > That's yanking one's own chain, and doesn't work on others. > > The "grammar" hierarchy of Chomsky is of a very limited and simple model > of computation and a very direct connection to "regular expressions", > with regards to formal methods, finite automata, > linear, right linear, and regular expressions, and of accounts > of the various amounts of look-ahead or memory in scanners what > result productions, that then in any account of source models > involves linking and dictionaries of symbols, that essentially > it's not saying much and isn't much of "grammar". > > Notions for example of the "railroad diagram" simply equip what > are models of languages like "SQL" that are complicated in Chomsky > to be simple in alternatives/optionals/loops with regards to the > declarations of "grammars". > > Any sort of usual useful "grammar" involves a "multi-pass parser", > with regards to parsing, for example for natural language, which > usually has a direct account of nouns and verbs, when really the > infinitives are always interrupted by instantiating a verb tense, > and nouns are particulars and simple. > > Then, for natural language, all readers of natural human language > using something alike "Tesniere grammars" as of "dependency grammars" > that all learned in school with regards to diagramming any well-formed > sentence. > > > Aristotle is not a fool - and Aristotle won't be made a fool. > > > > > That that that that that that that, .... > > > > The problem is not that "colorless green ideas sleep furiously" > is given a _context_ where it's not simply mimsy as the borogoves, > then that besides, all utterances are in a large overall context. > > > > So now you don't know grammar, either. > > In Robinson Arithmetic (often denoted as Q), the statement "no number is equal to its successor" is not provable. While this statement is true for the standard natural numbers, Robinson Arithmetic is too weak to prove it universally (∀ x, S(x) ≠ x). -- 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 | Ross Finlayson <ross.a.finlayson@gmail.com> |
|---|---|
| Date | 2026-06-27 19:25 -0700 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <z5Ccnfv-1fKrG933nZ2dnZfqnPhj4p2d@giganews.com> |
| In reply to | #142079 |
On 06/27/2026 03:47 PM, olcott wrote: > On 6/27/2026 5:37 PM, Ross Finlayson wrote: >> On 06/27/2026 08:47 AM, polcott wrote: >>> On 6/27/2026 3:05 AM, Mikko wrote: >>>> On 27/06/2026 01:01, olcott wrote: >>>>> On 6/26/2026 2:55 PM, dbush wrote: >>>>>> On 6/26/2026 3:38 PM, olcott wrote: >>>>>>> On 6/26/2026 2:17 PM, dbush wrote: >>>>>>>> On 6/26/2026 3:07 PM, olcott wrote: >>>>>>>>> On 6/26/2026 1:51 PM, dbush wrote: >>>>>>>>>> On 6/26/2026 2:48 PM, olcott wrote: >>>>>>>>>>> On 6/26/2026 1:14 PM, André G. Isaak wrote: >>>>>>>>>>>> On 2026-06-26 11:22, olcott wrote: >>>>>>>>>>>>> On 6/26/2026 11:08 AM, dbush wrote: >>>>>>>>>>>> >>>>>>>>>>>>>> By your logic, "no number is equal to its successor" has no >>>>>>>>>>>>>> meaning in Robinson arithmetic. >>>>>>>>>>>>> >>>>>>>>>>>>> In Robinson Arithmetic (often denoted as Q), >>>>>>>>>>>>> the statement "no number is equal to its >>>>>>>>>>>>> successor" is not provable.While this statement >>>>>>>>>>>>> is true for the standard natural numbers, Robinson >>>>>>>>>>>>> Arithmetic is too weak to prove it universally >>>>>>>>>>>>> (∀ x, S(x) ≠ x). >>>>>>>>>>>> >>>>>>>>>>>> It's not provable but it certainly has meaning. >>>>>>>>>>>> >>>>>>>>>>>> André >>>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> out-of-scope for Q is more accurate as jargon free. >>>>>>>>>>> >>>>>>>>>>> PTS does hold the view that meaning is only derived >>>>>>>>>>> through inference steps. This simple sentence seems >>>>>>>>>>> impossibly too difficult for anyone fully indoctrinated >>>>>>>>>>> with alternative views. So I will simply say out-of-scope. >>>>>>>>>>> >>>>>>>>>> >>>>>>>>>> So "out-of-scope" is merely a synonym for unprovable. Then to >>>>>>>>>> put things in words you can understand: >>>>>>>>>> >>>>>>>>> >>>>>>>>> "I am driving to Walmart to buy a carton of >>>>>>>>> Breyer's natural vanilla ice cream." is also unprovable in PA. >>>>>>>>> In both cases the semantics in not represented in PA. >>>>>>>> >>>>>>>> Not applicable, as that is not a sentence in PA. >>>>>>>> >>>>>>> >>>>>>> It is expressed in PA >>>>>> >>>>>> False. The above is not a sentence of PA. >>>>>> >>>>>>> to the same degree that G is expressed >>>>>>> in PA has a huge natural number. The semantics of it and >>>>>>> the semantics of G are neither expressible in PA. >>>>>> >>>>>> False. G is simply a sentence like ~∃x x>10 v x<5 but much more >>>>>> complex. >>>>>> >>>>>>> >>>>>>>> "No number is equal to its successor" is a sentence in RA, and it >>>>>>>> is true but unprovable in RA (or as your would call it, "out-of- >>>>>>>> scope"). >>>>>>>> >>>>>>> >>>>>>> If its semantics is not expressible in Q (What RA is called) >>>>>>> then it is not actually expressible in Q. >>>>>> >>>>>> "No number is equal to its successor" is a sentence in the language >>>>>> of Q. More formally, it is this: >>>>>> >>>>>> ~∃x x=S(x) >>>>>> >>>>>> And this sentence is not provable from the axioms of Q (or, in terms >>>>>> you would understand, the above is "out-of-scope" of Q). >>>>>> >>>>> >>>>> OK I checked the details so I need to make my >>>>> language more precise. >>>>> >>>>> Within proof theoretic semantics any expression >>>>> that cannot be proven in Q is not semantically >>>>> grounded in Q. >>>> >>>> Nevertheless a sentence like ~∃x x=S(x) is in the language of Q and >>>> that way in the theory. >>>> >>> >>> >>> Colorless green ideas sleep furiously >>> was composed by Noam Chomsky in his 1957 book >>> Syntactic Structures as an example of a sentence >>> that is grammatically well-formed, but semantically >>> nonsensical. >>> >>> Proving that syntax is not enough. >>> >> >> "Colorless green" is actually two colors >> since there's a dual-tristimulus colorspace >> the chromatic and the prismatic, >> a fact of the science of the theory of light and color, >> of which you are ignorant, then making for a reasonable >> reading of the usual apocryphal comment. >> >> Then, ideas can sleep however they want. >> >> > > The point is that syntactically correct expressions > can be semantically incoherent. Math always makes > sure to ignore this. The gibberish cannot be proven > counts as undecidability in math. > > That's yanking one's own chain, and doesn't work on others. The "grammar" hierarchy of Chomsky is of a very limited and simple model of computation and a very direct connection to "regular expressions", with regards to formal methods, finite automata, linear, right linear, and regular expressions, and of accounts of the various amounts of look-ahead or memory in scanners what result productions, that then in any account of source models involves linking and dictionaries of symbols, that essentially it's not saying much and isn't much of "grammar". Notions for example of the "railroad diagram" simply equip what are models of languages like "SQL" that are complicated in Chomsky to be simple in alternatives/optionals/loops with regards to the declarations of "grammars". Any sort of usual useful "grammar" involves a "multi-pass parser", with regards to parsing, for example for natural language, which usually has a direct account of nouns and verbs, when really the infinitives are always interrupted by instantiating a verb tense, and nouns are particulars and simple. Then, for natural language, all readers of natural human language using something alike "Tesniere grammars" as of "dependency grammars" that all learned in school with regards to diagramming any well-formed sentence. Aristotle is not a fool - and Aristotle won't be made a fool. That that that that that that that, .... The problem is not that "colorless green ideas sleep furiously" is given a _context_ where it's not simply mimsy as the borogoves, then that besides, all utterances are in a large overall context. So now you don't know grammar, either.
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| From | Mikko <mikko.levanto@iki.fi> |
|---|---|
| Date | 2026-06-28 11:22 +0300 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111qlkc$3g61e$1@dont-email.me> |
| In reply to | #142076 |
On 28/06/2026 01:37, Ross Finlayson wrote: > On 06/27/2026 08:47 AM, polcott wrote: >> On 6/27/2026 3:05 AM, Mikko wrote: >>> On 27/06/2026 01:01, olcott wrote: >>>> On 6/26/2026 2:55 PM, dbush wrote: >>>>> On 6/26/2026 3:38 PM, olcott wrote: >>>>>> On 6/26/2026 2:17 PM, dbush wrote: >>>>>>> On 6/26/2026 3:07 PM, olcott wrote: >>>>>>>> On 6/26/2026 1:51 PM, dbush wrote: >>>>>>>>> On 6/26/2026 2:48 PM, olcott wrote: >>>>>>>>>> On 6/26/2026 1:14 PM, André G. Isaak wrote: >>>>>>>>>>> On 2026-06-26 11:22, olcott wrote: >>>>>>>>>>>> On 6/26/2026 11:08 AM, dbush wrote: >>>>>>>>>>> >>>>>>>>>>>>> By your logic, "no number is equal to its successor" has no >>>>>>>>>>>>> meaning in Robinson arithmetic. >>>>>>>>>>>> >>>>>>>>>>>> In Robinson Arithmetic (often denoted as Q), >>>>>>>>>>>> the statement "no number is equal to its >>>>>>>>>>>> successor" is not provable.While this statement >>>>>>>>>>>> is true for the standard natural numbers, Robinson >>>>>>>>>>>> Arithmetic is too weak to prove it universally >>>>>>>>>>>> (∀ x, S(x) ≠ x). >>>>>>>>>>> >>>>>>>>>>> It's not provable but it certainly has meaning. >>>>>>>>>>> >>>>>>>>>>> André >>>>>>>>>>> >>>>>>>>>> >>>>>>>>>> out-of-scope for Q is more accurate as jargon free. >>>>>>>>>> >>>>>>>>>> PTS does hold the view that meaning is only derived >>>>>>>>>> through inference steps. This simple sentence seems >>>>>>>>>> impossibly too difficult for anyone fully indoctrinated >>>>>>>>>> with alternative views. So I will simply say out-of-scope. >>>>>>>>>> >>>>>>>>> >>>>>>>>> So "out-of-scope" is merely a synonym for unprovable. Then to >>>>>>>>> put things in words you can understand: >>>>>>>>> >>>>>>>> >>>>>>>> "I am driving to Walmart to buy a carton of >>>>>>>> Breyer's natural vanilla ice cream." is also unprovable in PA. >>>>>>>> In both cases the semantics in not represented in PA. >>>>>>> >>>>>>> Not applicable, as that is not a sentence in PA. >>>>>>> >>>>>> >>>>>> It is expressed in PA >>>>> >>>>> False. The above is not a sentence of PA. >>>>> >>>>>> to the same degree that G is expressed >>>>>> in PA has a huge natural number. The semantics of it and >>>>>> the semantics of G are neither expressible in PA. >>>>> >>>>> False. G is simply a sentence like ~∃x x>10 v x<5 but much more >>>>> complex. >>>>> >>>>>> >>>>>>> "No number is equal to its successor" is a sentence in RA, and it >>>>>>> is true but unprovable in RA (or as your would call it, "out-of- >>>>>>> scope"). >>>>>>> >>>>>> >>>>>> If its semantics is not expressible in Q (What RA is called) >>>>>> then it is not actually expressible in Q. >>>>> >>>>> "No number is equal to its successor" is a sentence in the language >>>>> of Q. More formally, it is this: >>>>> >>>>> ~∃x x=S(x) >>>>> >>>>> And this sentence is not provable from the axioms of Q (or, in terms >>>>> you would understand, the above is "out-of-scope" of Q). >>>>> >>>> >>>> OK I checked the details so I need to make my >>>> language more precise. >>>> >>>> Within proof theoretic semantics any expression >>>> that cannot be proven in Q is not semantically >>>> grounded in Q. >>> >>> Nevertheless a sentence like ~∃x x=S(x) is in the language of Q and >>> that way in the theory. >>> >> Colorless green ideas sleep furiously >> was composed by Noam Chomsky in his 1957 book >> Syntactic Structures as an example of a sentence >> that is grammatically well-formed, but semantically >> nonsensical. >> >> Proving that syntax is not enough. > > "Colorless green" is actually two colors > since there's a dual-tristimulus colorspace > the chromatic and the prismatic, > a fact of the science of the theory of light and color, > of which you are ignorant, then making for a reasonable > reading of the usual apocryphal comment. A problem with color theories is that some people have a different color system from the usual ones. There is even more variation among other animals. -- Mikko
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| From | Mikko <mikko.levanto@iki.fi> |
|---|---|
| Date | 2026-06-28 11:17 +0300 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111qlbn$3g407$1@dont-email.me> |
| In reply to | #142034 |
On 27/06/2026 18:47, polcott wrote: > On 6/27/2026 3:05 AM, Mikko wrote: >> On 27/06/2026 01:01, olcott wrote: >>> On 6/26/2026 2:55 PM, dbush wrote: >>>> On 6/26/2026 3:38 PM, olcott wrote: >>>>> On 6/26/2026 2:17 PM, dbush wrote: >>>>>> On 6/26/2026 3:07 PM, olcott wrote: >>>>>>> On 6/26/2026 1:51 PM, dbush wrote: >>>>>>>> On 6/26/2026 2:48 PM, olcott wrote: >>>>>>>>> On 6/26/2026 1:14 PM, André G. Isaak wrote: >>>>>>>>>> On 2026-06-26 11:22, olcott wrote: >>>>>>>>>>> On 6/26/2026 11:08 AM, dbush wrote: >>>>>>>>>> >>>>>>>>>>>> By your logic, "no number is equal to its successor" has no >>>>>>>>>>>> meaning in Robinson arithmetic. >>>>>>>>>>> >>>>>>>>>>> In Robinson Arithmetic (often denoted as Q), >>>>>>>>>>> the statement "no number is equal to its >>>>>>>>>>> successor" is not provable.While this statement >>>>>>>>>>> is true for the standard natural numbers, Robinson >>>>>>>>>>> Arithmetic is too weak to prove it universally >>>>>>>>>>> (∀ x, S(x) ≠ x). >>>>>>>>>> >>>>>>>>>> It's not provable but it certainly has meaning. >>>>>>>>>> >>>>>>>>>> André >>>>>>>>>> >>>>>>>>> >>>>>>>>> out-of-scope for Q is more accurate as jargon free. >>>>>>>>> >>>>>>>>> PTS does hold the view that meaning is only derived >>>>>>>>> through inference steps. This simple sentence seems >>>>>>>>> impossibly too difficult for anyone fully indoctrinated >>>>>>>>> with alternative views. So I will simply say out-of-scope. >>>>>>>>> >>>>>>>> >>>>>>>> So "out-of-scope" is merely a synonym for unprovable. Then to >>>>>>>> put things in words you can understand: >>>>>>>> >>>>>>> >>>>>>> "I am driving to Walmart to buy a carton of >>>>>>> Breyer's natural vanilla ice cream." is also unprovable in PA. >>>>>>> In both cases the semantics in not represented in PA. >>>>>> >>>>>> Not applicable, as that is not a sentence in PA. >>>>>> >>>>> >>>>> It is expressed in PA >>>> >>>> False. The above is not a sentence of PA. >>>> >>>>> to the same degree that G is expressed >>>>> in PA has a huge natural number. The semantics of it and >>>>> the semantics of G are neither expressible in PA. >>>> >>>> False. G is simply a sentence like ~∃x x>10 v x<5 but much more >>>> complex. >>>> >>>>> >>>>>> "No number is equal to its successor" is a sentence in RA, and it >>>>>> is true but unprovable in RA (or as your would call it, "out-of- >>>>>> scope"). >>>>>> >>>>> >>>>> If its semantics is not expressible in Q (What RA is called) >>>>> then it is not actually expressible in Q. >>>> >>>> "No number is equal to its successor" is a sentence in the language >>>> of Q. More formally, it is this: >>>> >>>> ~∃x x=S(x) >>>> >>>> And this sentence is not provable from the axioms of Q (or, in terms >>>> you would understand, the above is "out-of-scope" of Q). >>>> >>> >>> OK I checked the details so I need to make my >>> language more precise. >>> >>> Within proof theoretic semantics any expression >>> that cannot be proven in Q is not semantically >>> grounded in Q. >> >> Nevertheless a sentence like ~∃x x=S(x) is in the language of Q and >> that way in the theory. > > Colorless green ideas sleep furiously > was composed by Noam Chomsky in his 1957 book > Syntactic Structures as an example of a sentence > that is grammatically well-formed, but semantically > nonsensical. It is not obvious that it is grammatically correct. The actual grammar may be different from what Chomsky assumed. For example, there may be constraints involving sorts of words or something like that. It is also possible that there are less common meanings of the words that allow a sensible interpretation. > Proving that syntax is not enough. A syntax meay be insufficient or even irrelevant for a description of a natural language. Formal languages are deffined with a syntax, which therefore is both necessary and sufficient for language membership. Though one could accept as formal any language for which there is a Turing decider that accepts all strings in the language and rejects all other strings. Whether that Turing decider is called a syntax is a matter of definition. -- Mikko
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| From | Mikko <mikko.levanto@iki.fi> |
|---|---|
| Date | 2026-06-27 10:48 +0300 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111nv8k$142mq$1@dont-email.me> |
| In reply to | #141985 |
On 26/06/2026 19:08, dbush wrote: > On 6/26/2026 10:24 AM, olcott wrote: >> On 6/26/2026 8:57 AM, dbush wrote: >>> On 6/26/2026 9:45 AM, olcott wrote: >>>> On 6/26/2026 8:20 AM, dbush wrote: >>>>> On 6/26/2026 9:10 AM, 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. >>>>> >>>>> In other words, you're saying that the sentence "no number is equal >>>>> to its successor" has no meaning in Robinson Arithmetic. >>>>> >>>>> >>>> >>>> In proof theoretic semantics an expression only gains >>>> semantic meaning by finding a proof from within a >>>> stipulated atomic base of its own axioms like the one >>>> that you provided. >>>> >>>> >>> >>> Then you agree that the above natural language sentence that is >>> semantically required to be either true or false has no meaning? >>> >> >> Your sentence would be what it always has been >> a stipulated true sentence axiom. > > False, as that statement is not one of the axioms of Robinson > arithmetic, but it is a statement in its language, and one that has > *only* an infinite connection to the axioms of that system. What infinite connection? The statement is false in natural numbers, which is one model of Robinson Arithmetic but not the only one. In another model there may be a number that is its successor. There may even be more than one such number. > By your logic, "no number is equal to its successor" has no meaning in > Robinson arithmetic. -- Mikko
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| From | polcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-27 10:45 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111or6i$1kcvi$2@solani.org> |
| In reply to | #142020 |
On 6/27/2026 2:48 AM, Mikko wrote: > On 26/06/2026 19:08, dbush wrote: >> On 6/26/2026 10:24 AM, olcott wrote: >>> On 6/26/2026 8:57 AM, dbush wrote: >>>> On 6/26/2026 9:45 AM, olcott wrote: >>>>> On 6/26/2026 8:20 AM, dbush wrote: >>>>>> On 6/26/2026 9:10 AM, 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. >>>>>> >>>>>> In other words, you're saying that the sentence "no number is >>>>>> equal to its successor" has no meaning in Robinson Arithmetic. >>>>>> >>>>>> >>>>> >>>>> In proof theoretic semantics an expression only gains >>>>> semantic meaning by finding a proof from within a >>>>> stipulated atomic base of its own axioms like the one >>>>> that you provided. >>>>> >>>>> >>>> >>>> Then you agree that the above natural language sentence that is >>>> semantically required to be either true or false has no meaning? >>>> >>> >>> Your sentence would be what it always has been >>> a stipulated true sentence axiom. >> >> False, as that statement is not one of the axioms of Robinson >> arithmetic, but it is a statement in its language, and one that has >> *only* an infinite connection to the axioms of that system. > > What infinite connection? The statement is false in natural numbers, > which is one model of Robinson Arithmetic but not the only one. > In another model there may be a number that is its successor. There > may even be more than one such number. > It cannot be proved in Q and can be proved in PA. Thus its semantic meaning is out-of-scope in Q. >> By your logic, "no number is equal to its successor" has no meaning in >> Robinson arithmetic. > -- 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 11:38 +0300 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111qmil$3gd0r$2@dont-email.me> |
| In reply to | #142033 |
On 27/06/2026 18:45, polcott wrote: > On 6/27/2026 2:48 AM, Mikko wrote: >> On 26/06/2026 19:08, dbush wrote: >>> On 6/26/2026 10:24 AM, olcott wrote: >>>> On 6/26/2026 8:57 AM, dbush wrote: >>>>> On 6/26/2026 9:45 AM, olcott wrote: >>>>>> On 6/26/2026 8:20 AM, dbush wrote: >>>>>>> On 6/26/2026 9:10 AM, 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. >>>>>>> >>>>>>> In other words, you're saying that the sentence "no number is >>>>>>> equal to its successor" has no meaning in Robinson Arithmetic. >>>>>>> >>>>>>> >>>>>> >>>>>> In proof theoretic semantics an expression only gains >>>>>> semantic meaning by finding a proof from within a >>>>>> stipulated atomic base of its own axioms like the one >>>>>> that you provided. >>>>>> >>>>>> >>>>> >>>>> Then you agree that the above natural language sentence that is >>>>> semantically required to be either true or false has no meaning? >>>>> >>>> >>>> Your sentence would be what it always has been >>>> a stipulated true sentence axiom. >>> >>> False, as that statement is not one of the axioms of Robinson >>> arithmetic, but it is a statement in its language, and one that has >>> *only* an infinite connection to the axioms of that system. >> >> What infinite connection? The statement is false in natural numbers, >> which is one model of Robinson Arithmetic but not the only one. >> In another model there may be a number that is its successor. There >> may even be more than one such number. >> > > It cannot be proved in Q and can be proved in PA. > Thus its semantic meaning is out-of-scope in Q. But its unprovability is a fact about Q. And the expression has some truth value in every model of Q. >>> By your logic, "no number is equal to its successor" has no meaning >>> in Robinson arithmetic. -- Mikko
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| From | Mikko <mikko.levanto@iki.fi> |
|---|---|
| Date | 2026-06-27 10:35 +0300 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111nufn$13s5b$1@dont-email.me> |
| In reply to | #141975 |
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. > This is the same sort of thing as finding the defined > meaning of a word. If you cannot find its recursively > defined meaning then it never gains any meaning. That does not follow. Words have meanings even without definitions. You can't present the first definition unless you already have meaningful words. Typically the presentation of a formal theory begins with the introduction of undefined symbols. But the symbols are not fully meaningless. They get some amount of meaning from being introduces as symbols of a particular syntactic category and more from being used in the postulates of the theory. -- Mikko
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| From | polcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-27 10:43 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111or2v$1kcvi$1@solani.org> |
| In reply to | #142019 |
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 then ~∃x x=S(x) is ungrounded in the PTS atomic base of Q. This does not mean undecidable or incomplete it means that ~∃x x=S(x) is out-of-scope for Q. >> This is the same sort of thing as finding the defined >> meaning of a word. If you cannot find its recursively >> defined meaning then it never gains any meaning. > That does not follow. Words have meanings even without definitions. > You can't present the first definition unless you already have > meaningful words. > A particular new word can only be defined in terms of other existing words that already have definitions. PTS works in a similar way. If ~∃x x=S(x) cannot connect to its meanings in Q the it remains undefined in Q. > Typically the presentation of a formal theory begins with the > introduction of undefined symbols. But the symbols are not > fully meaningless. They get some amount of meaning from being > introduces as symbols of a particular syntactic category and > more from being used in the postulates of the theory. > The body of knowledge expressed in language starts with an atomic basis of expressions of language that are stipulated to be true. -- 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 14:01 -0400 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111p35m$34avj$2@dont-email.me> |
| In reply to | #142032 |
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 > then ~∃x x=S(x) is > ungrounded in the PTS atomic base of Q. i.e. it is unproven in Q > This does not mean undecidable or incomplete false, see below > it means that ~∃x x=S(x) is out-of-scope for Q. i.e. ~∃x x=S(x) is unprovable in Q, therefore making Q incomplete. A formal system is incomplete if it contains statements that are unprovable / out-of-scope / not semantically grounded. > >>> This is the same sort of thing as finding the defined >>> meaning of a word. If you cannot find its recursively >>> defined meaning then it never gains any meaning. > >> That does not follow. Words have meanings even without definitions. >> You can't present the first definition unless you already have >> meaningful words. >> > > A particular new word can only be defined in terms > of other existing words that already have definitions. > PTS works in a similar way. If ~∃x x=S(x) i.e. "No number is equal to its successor" > cannot connect > to its meanings in Q the it remains undefined in Q. It is connected, as "∃", "S", "~", and "=" are defined giving it the semantic meaning above. So if PTS claims it has no semantic meaning then PTS must be discarded as useless.
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| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-27 13:27 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111p4mh$3504e$1@dont-email.me> |
| In reply to | #142041 |
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. PTS also says FINITE sequence. I cannot use the convoluted way that PTS says it in all of their different author-by-author terms-of-the-art and still be understood. The above version is very close to the way that one PTS author would say it and does convey the same gist of meanings that other PTS authors accept. -- 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 14:29 -0400 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111p4qo$34asl$2@dont-email.me> |
| In reply to | #142044 |
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. So again, you're agreeing with everyone else but using different words.
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| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-27 13:38 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111p5aa$355in$1@dont-email.me> |
| In reply to | #142046 |
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) is semantic nonsense in Q? All of logic took a psychotic break from reality when they took semantics out of logic and put it in a separate model. With mistakes like this one can conclude that liars always tell the whole truth and nothing but the truth. -- 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 14:39 -0400 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111p5du$34asl$4@dont-email.me> |
| In reply to | #142048 |
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. > is semantic nonsense in Q? False, see above.
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| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-27 14:01 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111p6mk$35hhj$1@dont-email.me> |
| In reply to | #142049 |
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. It is more accurate to say it this way than to say that it is semantically incoherent in Q. It is great that you brought this up: ~∃x x=S(x). We can have much clearer communication about that then we can about Gödel's 1931 Incompleteness. >> is semantic nonsense in Q? > > False, see above. > -- 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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