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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 | dbush <dbush.mobile@gmail.com> |
|---|---|
| Date | 2026-06-26 14:02 -0400 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111mesc$lde0$5@dont-email.me> |
| In reply to | #141991 |
On 6/26/2026 1:53 PM, olcott wrote: > On 6/26/2026 12:42 PM, dbush wrote: >> On 6/26/2026 1:39 PM, olcott wrote: >>> On 6/26/2026 12:25 PM, dbush wrote: >>>> On 6/26/2026 1:22 PM, olcott wrote: >>>>> On 6/26/2026 11:08 AM, 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. >>>>>> >>>>>> 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). >>>> >>>> So you agree that Robinson arithmetic is incomplete. >>>> >>> >>> It is as complete as it was designed to be. >>> >> >> There is no "designed to be". There are sentences in the language of >> Robinson arithmetic that are true but not provable, > > To make is simpler to understand. > In proof theoretic semantics: > unprovable in Q means out-of-scope of Q. In your own words, what does "out-of-scope" mean in this context? > >> therefore making the system incomplete, as you have just agreed, >> meaning that you agree that incompleteness exists. > >
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| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2026-06-26 12:14 -0600 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111mfi9$ns29$1@dont-email.me> |
| In reply to | #141987 |
On 2026-06-26 11:22, olcott wrote: > On 6/26/2026 11:08 AM, dbush wrote: >> By your logic, "no number is equal to its successor" has no meaning in >> Robinson arithmetic. > > In Robinson Arithmetic (often denoted as Q), > the statement "no number is equal to its > successor" is not provable.While this statement > is true for the standard natural numbers, Robinson > Arithmetic is too weak to prove it universally > (∀ x, S(x) ≠ x). It's not provable but it certainly has meaning. André -- To email remove 'invalid' & replace 'gm' with well known Google mail service.
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| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-26 13:48 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111mhh6$oha7$1@dont-email.me> |
| In reply to | #141993 |
On 6/26/2026 1:14 PM, André G. Isaak wrote: > On 2026-06-26 11:22, olcott wrote: >> On 6/26/2026 11:08 AM, dbush wrote: > >>> By your logic, "no number is equal to its successor" has no meaning >>> in Robinson arithmetic. >> >> In Robinson Arithmetic (often denoted as Q), >> the statement "no number is equal to its >> successor" is not provable.While this statement >> is true for the standard natural numbers, Robinson >> Arithmetic is too weak to prove it universally >> (∀ x, S(x) ≠ x). > > It's not provable but it certainly has meaning. > > André > out-of-scope for Q is more accurate as jargon free. PTS does hold the view that meaning is only derived through inference steps. This simple sentence seems impossibly too difficult for anyone fully indoctrinated with alternative views. So I will simply say out-of-scope. -- Copyright 2026 Olcott My 28 year goal has been to make "true on the basis of meaning expressed in language" reliably computable for the entire body of knowledge. The complete structure of this system is now defined. The entire body of knowledge expressed in language is comprised of two types of relations between finite strings: (a) *Axioms* Expressions of language that are stipulated to be true. My system bridges the analytic/synthetic distinction by expressly encoding all empirical "atomic facts" in a formal language such as CycL of the Cyc project. (b) *Inference Rules* Expressions of language that are semantically entailed syntactically from (a) and/or (b).
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| From | dbush <dbush.mobile@gmail.com> |
|---|---|
| Date | 2026-06-26 14:51 -0400 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111mho7$lde0$6@dont-email.me> |
| In reply to | #141994 |
On 6/26/2026 2:48 PM, olcott wrote: > On 6/26/2026 1:14 PM, André G. Isaak wrote: >> On 2026-06-26 11:22, olcott wrote: >>> On 6/26/2026 11:08 AM, dbush wrote: >> >>>> By your logic, "no number is equal to its successor" has no meaning >>>> in Robinson arithmetic. >>> >>> In Robinson Arithmetic (often denoted as Q), >>> the statement "no number is equal to its >>> successor" is not provable.While this statement >>> is true for the standard natural numbers, Robinson >>> Arithmetic is too weak to prove it universally >>> (∀ x, S(x) ≠ x). >> >> It's not provable but it certainly has meaning. >> >> André >> > > out-of-scope for Q is more accurate as jargon free. > > PTS does hold the view that meaning is only derived > through inference steps. This simple sentence seems > impossibly too difficult for anyone fully indoctrinated > with alternative views. So I will simply say out-of-scope. > So "out-of-scope" is merely a synonym for unprovable. Then to put things in words you can understand: Godel proved that any axiomatic system of arithmetic contains out-of-scope statements.
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| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-26 14:07 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111mild$osp8$1@dont-email.me> |
| In reply to | #141995 |
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. > Godel proved that any axiomatic system of arithmetic contains out-of- > scope statements. Sure, PA also has no idea that driving means operating a motor vehicle. -- 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 | 2026-06-26 15:17 -0400 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111mj8p$lde0$7@dont-email.me> |
| In reply to | #141996 |
On 6/26/2026 3:07 PM, olcott wrote: > On 6/26/2026 1:51 PM, dbush wrote: >> On 6/26/2026 2:48 PM, olcott wrote: >>> On 6/26/2026 1:14 PM, André G. Isaak wrote: >>>> On 2026-06-26 11:22, olcott wrote: >>>>> On 6/26/2026 11:08 AM, dbush wrote: >>>> >>>>>> By your logic, "no number is equal to its successor" has no >>>>>> meaning in Robinson arithmetic. >>>>> >>>>> In Robinson Arithmetic (often denoted as Q), >>>>> the statement "no number is equal to its >>>>> successor" is not provable.While this statement >>>>> is true for the standard natural numbers, Robinson >>>>> Arithmetic is too weak to prove it universally >>>>> (∀ x, S(x) ≠ x). >>>> >>>> It's not provable but it certainly has meaning. >>>> >>>> André >>>> >>> >>> out-of-scope for Q is more accurate as jargon free. >>> >>> PTS does hold the view that meaning is only derived >>> through inference steps. This simple sentence seems >>> impossibly too difficult for anyone fully indoctrinated >>> with alternative views. So I will simply say out-of-scope. >>> >> >> So "out-of-scope" is merely a synonym for unprovable. Then to put >> things in words you can understand: >> > > "I am driving to Walmart to buy a carton of > Breyer's natural vanilla ice cream." is also unprovable in PA. > In both cases the semantics in not represented in PA. Not applicable, as that is not a sentence in PA. "No number is equal to its successor" is a sentence in RA, and it is true but unprovable in RA (or as your would call it, "out-of-scope"). > >> Godel proved that any axiomatic system of arithmetic contains out-of- >> scope statements. > > Sure, PA also has no idea that driving means operating a motor vehicle. > Not applicable.
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| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-26 14:38 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111mkfl$pefu$1@dont-email.me> |
| In reply to | #141997 |
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 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. > "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. >> >>> Godel proved that any axiomatic system of arithmetic contains out-of- >>> scope statements. >> >> Sure, PA also has no idea that driving means operating a motor vehicle. >> > > Not applicable. -- 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 | 2026-06-26 15:55 -0400 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111mlfh$lde0$8@dont-email.me> |
| In reply to | #141998 |
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). > >>> >>>> Godel proved that any axiomatic system of arithmetic contains out- >>>> of- scope statements. >>> >>> Sure, PA also has no idea that driving means operating a motor vehicle. >>> >> >> Not applicable. > >
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| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-26 17:01 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111msrp$rtlv$1@dont-email.me> |
| In reply to | #141999 |
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. -- 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 | 2026-06-26 18:08 -0400 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111mt9p$lde0$9@dont-email.me> |
| In reply to | #142000 |
On 6/26/2026 6:01 PM, 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. > > In your own words, what does it mean for a statement to be "semantically grounded" in a formal system?
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| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-26 17:58 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111n070$spqj$1@dont-email.me> |
| In reply to | #142001 |
On 6/26/2026 5:08 PM, dbush wrote: > On 6/26/2026 6:01 PM, 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. >> >> > > In your own words, what does it mean for a statement to be "semantically > grounded" in a formal system? > > I always do back-chained inference because the typical math way of doing forward chained inference may take an infeasibly long time. A finite set of back-chained inference steps from x to the axioms of Q. I appreciate that you stopped playing head games. -- 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 | 2026-06-26 19:18 -0400 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111n1ch$lde0$10@dont-email.me> |
| In reply to | #142002 |
On 6/26/2026 6:58 PM, olcott wrote: > On 6/26/2026 5:08 PM, dbush wrote: >> On 6/26/2026 6:01 PM, 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. >>> >>> >> >> In your own words, what does it mean for a statement to be >> "semantically grounded" in a formal system? >> >> > > I always do back-chained inference because the typical > math way of doing forward chained inference may take > an infeasibly long time. A finite set of back-chained > inference steps from x to the axioms of Q. Back or forward chained doesn't matter, it's essentially the same steps in a different direction. But in any case, you're saying "semantically grounded" is just another synonym for unprovable. So to again put things in a way you'll understand, Godel proved that any axiom system of arithmetic contains statements that are not semantically grounded. That also means that, using your terminology, it has been proven that the statement ~∃x x=S(x), i.e. "No number is equal to its successor", is not semantically grounded in Q. So you agree with what everyone else is saying, but using different words to say it. > > I appreciate that you stopped playing head games. > I never played head games. I just asked questions that made you realize you were wrong. And you never did answer the question of whether the condition "At least one of the following statements is true" is satisfied in the following natural language statement: -------------------------------------- At least one of the following statements is true: - Earth is the third planet from the sun. - There is a Walmart bag at the deepest point of the Mariana Trench. --------------------------------------
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| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-26 19:05 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111n459$trck$1@dont-email.me> |
| In reply to | #142003 |
On 6/26/2026 6:18 PM, dbush wrote: > On 6/26/2026 6:58 PM, olcott wrote: >> On 6/26/2026 5:08 PM, dbush wrote: >>> On 6/26/2026 6:01 PM, 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. >>>> >>>> >>> >>> In your own words, what does it mean for a statement to be >>> "semantically grounded" in a formal system? >>> >>> >> >> I always do back-chained inference because the typical >> math way of doing forward chained inference may take >> an infeasibly long time. A finite set of back-chained >> inference steps from x to the axioms of Q. > > Back or forward chained doesn't matter, it's essentially the same steps > in a different direction. But in any case, you're saying "semantically > grounded" is just another synonym for unprovable. > > So to again put things in a way you'll understand, Godel proved that any > axiom system of arithmetic contains statements that are not semantically > grounded. > Not quite. G is not semantically grounded in PA yet G is semantically grounded in metamathematics. When an expression in PA only derives semantic meaning in PA when grounded in PA then G has no meaning in PA. > That also means that, using your terminology, it has been proven that > the statement ~∃x x=S(x), i.e. "No number is equal to its successor", is > not semantically grounded in Q. > Thus is meaningless in Q and out-of-scope in Q. 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. -- 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 | 2026-06-26 20:23 -0400 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111n567$u0m6$1@dont-email.me> |
| In reply to | #142004 |
On 6/26/2026 8:05 PM, olcott wrote: > On 6/26/2026 6:18 PM, dbush wrote: >> On 6/26/2026 6:58 PM, olcott wrote: >>> On 6/26/2026 5:08 PM, dbush wrote: >>>> On 6/26/2026 6:01 PM, 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. >>>>> >>>>> >>>> >>>> In your own words, what does it mean for a statement to be >>>> "semantically grounded" in a formal system? >>>> >>>> >>> >>> I always do back-chained inference because the typical >>> math way of doing forward chained inference may take >>> an infeasibly long time. A finite set of back-chained >>> inference steps from x to the axioms of Q. >> >> Back or forward chained doesn't matter, it's essentially the same >> steps in a different direction. But in any case, you're saying >> "semantically grounded" is just another synonym for unprovable. >> >> So to again put things in a way you'll understand, Godel proved that >> any axiom system of arithmetic contains statements that are not >> semantically grounded. >> > > Not quite. G is not semantically grounded i.e. unprovable > in PA > yet G is semantically grounded i.e. provable > in metamathematics. Which is exactly what Godel proved. > When an expression in PA only derives semantic > meaning in PA when grounded in PA > then G has no > meaning in PA. i.e. if a statement is unprovable in PA then it's unprovable in PA. In other words, a meaningless tautology. > >> That also means that, using your terminology, it has been proven that >> the statement ~∃x x=S(x), i.e. "No number is equal to its successor", >> is not semantically grounded in Q. >> > > Thus is meaningless in Q and out-of-scope in Q. Which means the semantically valid statement in Q "No number is equal to its successor" is deemed invalid by PTS, therefore PTS must be discarded as useless.
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| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-26 19:48 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111n6lr$udqn$1@dont-email.me> |
| In reply to | #142005 |
On 6/26/2026 7:23 PM, dbush wrote: > On 6/26/2026 8:05 PM, olcott wrote: >> On 6/26/2026 6:18 PM, dbush wrote: >>> On 6/26/2026 6:58 PM, olcott wrote: >>>> On 6/26/2026 5:08 PM, dbush wrote: >>>>> On 6/26/2026 6:01 PM, 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. >>>>>> >>>>>> >>>>> >>>>> In your own words, what does it mean for a statement to be >>>>> "semantically grounded" in a formal system? >>>>> >>>>> >>>> >>>> I always do back-chained inference because the typical >>>> math way of doing forward chained inference may take >>>> an infeasibly long time. A finite set of back-chained >>>> inference steps from x to the axioms of Q. >>> >>> Back or forward chained doesn't matter, it's essentially the same >>> steps in a different direction. But in any case, you're saying >>> "semantically grounded" is just another synonym for unprovable. >>> >>> So to again put things in a way you'll understand, Godel proved that >>> any axiom system of arithmetic contains statements that are not >>> semantically grounded. >>> >> >> Not quite. G is not semantically grounded > > i.e. unprovable > >> in PA >> yet G is semantically grounded > > i.e. provable > >> in metamathematics. > > Which is exactly what Godel proved. > >> When an expression in PA only derives semantic >> meaning in PA when grounded in PA then G has no >> meaning in PA. > > i.e. if a statement is unprovable in PA then it's unprovable in PA. > > In other words, a meaningless tautology. > >> >>> That also means that, using your terminology, it has been proven that >>> the statement ~∃x x=S(x), i.e. "No number is equal to its successor", >>> is not semantically grounded in Q. >>> >> >> Thus is meaningless in Q and out-of-scope in Q. > > Which means the semantically valid statement in Q does not include ~∃x x=S(x) because to be semantically valid in Q it must connect to the axioms of Q through inference steps in Q. The entire body of knowledge expressed in language is anchored stipulated relations between finite strings. A proof of a finite string in a system L merely involves verifying relations between finite strings reach the axioms of L through inference in finite steps. cats are animals // axiom animals are living things // axiom ∴ cats are living things If we try this in Chinese using a formal system in English this is the same as ~∃x x=S(x) in Q versus ~∃x x=S(x) in PA -- 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 | 2026-06-26 21:11 -0400 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111n80h$u0m6$2@dont-email.me> |
| In reply to | #142006 |
On 6/26/2026 8:48 PM, olcott wrote: > On 6/26/2026 7:23 PM, dbush wrote: >> On 6/26/2026 8:05 PM, olcott wrote: >>> On 6/26/2026 6:18 PM, dbush wrote: >>>> On 6/26/2026 6:58 PM, olcott wrote: >>>>> On 6/26/2026 5:08 PM, dbush wrote: >>>>>> On 6/26/2026 6:01 PM, 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. >>>>>>> >>>>>>> >>>>>> >>>>>> In your own words, what does it mean for a statement to be >>>>>> "semantically grounded" in a formal system? >>>>>> >>>>>> >>>>> >>>>> I always do back-chained inference because the typical >>>>> math way of doing forward chained inference may take >>>>> an infeasibly long time. A finite set of back-chained >>>>> inference steps from x to the axioms of Q. >>>> >>>> Back or forward chained doesn't matter, it's essentially the same >>>> steps in a different direction. But in any case, you're saying >>>> "semantically grounded" is just another synonym for unprovable. >>>> >>>> So to again put things in a way you'll understand, Godel proved that >>>> any axiom system of arithmetic contains statements that are not >>>> semantically grounded. >>>> >>> >>> Not quite. G is not semantically grounded >> >> i.e. unprovable >> >>> in PA >>> yet G is semantically grounded >> >> i.e. provable >> >>> in metamathematics. >> >> Which is exactly what Godel proved. Your lack of response indicates that you agree with Godel, but used different words to do so. >> >>> When an expression in PA only derives semantic >>> meaning in PA when grounded in PA then G has no >>> meaning in PA. >> >> i.e. if a statement is unprovable in PA then it's unprovable in PA. >> >> In other words, a meaningless tautology. >> >>> >>>> That also means that, using your terminology, it has been proven >>>> that the statement ~∃x x=S(x), i.e. "No number is equal to its >>>> successor", is not semantically grounded in Q. >>>> >>> >>> Thus is meaningless in Q and out-of-scope in Q. >> >> Which means the semantically valid statement in Q > > does not include ~∃x x=S(x) False, as it means "no number is equal to its successor", and the concept of a successor and equality have semantic meaning in Q, as does the concept of "all", "none", and "exists". That makes the statement semantically valid, so any alternate system that concludes otherwise is necessarily faulty.
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| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-26 20:39 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111n9jn$v2ar$1@dont-email.me> |
| In reply to | #142007 |
On 6/26/2026 8:11 PM, dbush wrote: > On 6/26/2026 8:48 PM, olcott wrote: >> On 6/26/2026 7:23 PM, dbush wrote: >>> On 6/26/2026 8:05 PM, olcott wrote: >>>> On 6/26/2026 6:18 PM, dbush wrote: >>>>> On 6/26/2026 6:58 PM, olcott wrote: >>>>>> On 6/26/2026 5:08 PM, dbush wrote: >>>>>>> On 6/26/2026 6:01 PM, 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. >>>>>>>> >>>>>>>> >>>>>>> >>>>>>> In your own words, what does it mean for a statement to be >>>>>>> "semantically grounded" in a formal system? >>>>>>> >>>>>>> >>>>>> >>>>>> I always do back-chained inference because the typical >>>>>> math way of doing forward chained inference may take >>>>>> an infeasibly long time. A finite set of back-chained >>>>>> inference steps from x to the axioms of Q. >>>>> >>>>> Back or forward chained doesn't matter, it's essentially the same >>>>> steps in a different direction. But in any case, you're saying >>>>> "semantically grounded" is just another synonym for unprovable. >>>>> >>>>> So to again put things in a way you'll understand, Godel proved >>>>> that any axiom system of arithmetic contains statements that are >>>>> not semantically grounded. >>>>> >>>> >>>> Not quite. G is not semantically grounded >>> >>> i.e. unprovable >>> >>>> in PA >>>> yet G is semantically grounded >>> >>> i.e. provable >>> >>>> in metamathematics. >>> >>> Which is exactly what Godel proved. > > Your lack of response indicates that you agree with Godel, but used > different words to do so. > >>> >>>> When an expression in PA only derives semantic >>>> meaning in PA when grounded in PA then G has no >>>> meaning in PA. >>> >>> i.e. if a statement is unprovable in PA then it's unprovable in PA. >>> >>> In other words, a meaningless tautology. >>> >>>> >>>>> That also means that, using your terminology, it has been proven >>>>> that the statement ~∃x x=S(x), i.e. "No number is equal to its >>>>> successor", is not semantically grounded in Q. >>>>> >>>> >>>> Thus is meaningless in Q and out-of-scope in Q. >>> >>> Which means the semantically valid statement in Q >> >> does not include ~∃x x=S(x) > > False, as it means "no number is equal to its successor", and the > concept of a successor and equality have semantic meaning in Q, as does > the concept of "all", "none", and "exists". > 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. PTS implements the generic model that knowledge expressed in language merely connects ideas to their definitions. A PTS proof verifies that connection, else failure means undefined. > That makes the statement semantically valid, so any alternate system > that concludes otherwise is necessarily faulty. > -- 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 | 2026-06-26 21:51 -0400 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111naal$u0m6$3@dont-email.me> |
| In reply to | #142008 |
On 6/26/2026 9:39 PM, olcott wrote: > On 6/26/2026 8:11 PM, dbush wrote: >> On 6/26/2026 8:48 PM, olcott wrote: >>> On 6/26/2026 7:23 PM, dbush wrote: >>>> On 6/26/2026 8:05 PM, olcott wrote: >>>>> On 6/26/2026 6:18 PM, dbush wrote: >>>>>> On 6/26/2026 6:58 PM, olcott wrote: >>>>>>> On 6/26/2026 5:08 PM, dbush wrote: >>>>>>>> On 6/26/2026 6:01 PM, 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. >>>>>>>>> >>>>>>>>> >>>>>>>> >>>>>>>> In your own words, what does it mean for a statement to be >>>>>>>> "semantically grounded" in a formal system? >>>>>>>> >>>>>>>> >>>>>>> >>>>>>> I always do back-chained inference because the typical >>>>>>> math way of doing forward chained inference may take >>>>>>> an infeasibly long time. A finite set of back-chained >>>>>>> inference steps from x to the axioms of Q. >>>>>> >>>>>> Back or forward chained doesn't matter, it's essentially the same >>>>>> steps in a different direction. But in any case, you're saying >>>>>> "semantically grounded" is just another synonym for unprovable. >>>>>> >>>>>> So to again put things in a way you'll understand, Godel proved >>>>>> that any axiom system of arithmetic contains statements that are >>>>>> not semantically grounded. >>>>>> >>>>> >>>>> Not quite. G is not semantically grounded >>>> >>>> i.e. unprovable >>>> >>>>> in PA >>>>> yet G is semantically grounded >>>> >>>> i.e. provable >>>> >>>>> in metamathematics. >>>> >>>> Which is exactly what Godel proved. >> >> Your lack of response indicates that you agree with Godel, but used >> different words to do so. >> >>>> >>>>> When an expression in PA only derives semantic >>>>> meaning in PA when grounded in PA then G has no >>>>> meaning in PA. >>>> >>>> i.e. if a statement is unprovable in PA then it's unprovable in PA. >>>> >>>> In other words, a meaningless tautology. >>>> >>>>> >>>>>> That also means that, using your terminology, it has been proven >>>>>> that the statement ~∃x x=S(x), i.e. "No number is equal to its >>>>>> successor", is not semantically grounded in Q. >>>>>> >>>>> >>>>> Thus is meaningless in Q and out-of-scope in Q. >>>> >>>> Which means the semantically valid statement in Q >>> >>> does not include ~∃x x=S(x) >> >> False, as it means "no number is equal to its successor", and the >> concept of a successor and equality have semantic meaning in Q, as >> does the concept of "all", "none", and "exists". >> > > 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. In other words, ~∃x x=S(x) is unprovable in Q, as is commonly known. So once again, you agree with everyone else, but are using different words to say so.
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| From | olcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-26 21:00 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111nard$vb4e$1@dont-email.me> |
| In reply to | #142009 |
On 6/26/2026 8:51 PM, dbush wrote: > On 6/26/2026 9:39 PM, olcott wrote: >> On 6/26/2026 8:11 PM, dbush wrote: >>> On 6/26/2026 8:48 PM, olcott wrote: >>>> On 6/26/2026 7:23 PM, dbush wrote: >>>>> On 6/26/2026 8:05 PM, olcott wrote: >>>>>> On 6/26/2026 6:18 PM, dbush wrote: >>>>>>> On 6/26/2026 6:58 PM, olcott wrote: >>>>>>>> On 6/26/2026 5:08 PM, dbush wrote: >>>>>>>>> On 6/26/2026 6:01 PM, 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. >>>>>>>>>> >>>>>>>>>> >>>>>>>>> >>>>>>>>> In your own words, what does it mean for a statement to be >>>>>>>>> "semantically grounded" in a formal system? >>>>>>>>> >>>>>>>>> >>>>>>>> >>>>>>>> I always do back-chained inference because the typical >>>>>>>> math way of doing forward chained inference may take >>>>>>>> an infeasibly long time. A finite set of back-chained >>>>>>>> inference steps from x to the axioms of Q. >>>>>>> >>>>>>> Back or forward chained doesn't matter, it's essentially the same >>>>>>> steps in a different direction. But in any case, you're saying >>>>>>> "semantically grounded" is just another synonym for unprovable. >>>>>>> >>>>>>> So to again put things in a way you'll understand, Godel proved >>>>>>> that any axiom system of arithmetic contains statements that are >>>>>>> not semantically grounded. >>>>>>> >>>>>> >>>>>> Not quite. G is not semantically grounded >>>>> >>>>> i.e. unprovable >>>>> >>>>>> in PA >>>>>> yet G is semantically grounded >>>>> >>>>> i.e. provable >>>>> >>>>>> in metamathematics. >>>>> >>>>> Which is exactly what Godel proved. >>> >>> Your lack of response indicates that you agree with Godel, but used >>> different words to do so. >>> >>>>> >>>>>> When an expression in PA only derives semantic >>>>>> meaning in PA when grounded in PA then G has no >>>>>> meaning in PA. >>>>> >>>>> i.e. if a statement is unprovable in PA then it's unprovable in PA. >>>>> >>>>> In other words, a meaningless tautology. >>>>> >>>>>> >>>>>>> That also means that, using your terminology, it has been proven >>>>>>> that the statement ~∃x x=S(x), i.e. "No number is equal to its >>>>>>> successor", is not semantically grounded in Q. >>>>>>> >>>>>> >>>>>> Thus is meaningless in Q and out-of-scope in Q. >>>>> >>>>> Which means the semantically valid statement in Q >>>> >>>> does not include ~∃x x=S(x) >>> >>> False, as it means "no number is equal to its successor", and the >>> concept of a successor and equality have semantic meaning in Q, as >>> does the concept of "all", "none", and "exists". >>> >> >> 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. > > In other words, ~∃x x=S(x) is unprovable in Q, as is commonly known. > > So once again, you agree with everyone else, but are using different > words to say so. > The big change is that undecidability is construed ether as semantic incoherence or as in the truth value of the Goldbach conjecture currently unknown. This is the best example that can possibly exist of proof theoretic semantics incoherent semantics. % This sentence is not true. ?- LP = not(true(LP)). LP = not(true(LP)). ?- unify_with_occurs_check(LP, not(true(LP))). false. -- 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 | polcott <polcott333@gmail.com> |
|---|---|
| Date | 2026-06-27 08:34 -0500 |
| Subject | Re: Readings in (some of the) foundations of mathematics --- tree of knowledge |
| Message-ID | <111ojgg$sfoa$2@solani.org> |
| In reply to | #142009 |
On 6/26/2026 8:51 PM, dbush wrote: > On 6/26/2026 9:39 PM, olcott wrote: >> On 6/26/2026 8:11 PM, dbush wrote: >>> On 6/26/2026 8:48 PM, olcott wrote: >>>> On 6/26/2026 7:23 PM, dbush wrote: >>>>> On 6/26/2026 8:05 PM, olcott wrote: >>>>>> On 6/26/2026 6:18 PM, dbush wrote: >>>>>>> On 6/26/2026 6:58 PM, olcott wrote: >>>>>>>> On 6/26/2026 5:08 PM, dbush wrote: >>>>>>>>> On 6/26/2026 6:01 PM, 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. >>>>>>>>>> >>>>>>>>>> >>>>>>>>> >>>>>>>>> In your own words, what does it mean for a statement to be >>>>>>>>> "semantically grounded" in a formal system? >>>>>>>>> >>>>>>>>> >>>>>>>> >>>>>>>> I always do back-chained inference because the typical >>>>>>>> math way of doing forward chained inference may take >>>>>>>> an infeasibly long time. A finite set of back-chained >>>>>>>> inference steps from x to the axioms of Q. >>>>>>> >>>>>>> Back or forward chained doesn't matter, it's essentially the same >>>>>>> steps in a different direction. But in any case, you're saying >>>>>>> "semantically grounded" is just another synonym for unprovable. >>>>>>> >>>>>>> So to again put things in a way you'll understand, Godel proved >>>>>>> that any axiom system of arithmetic contains statements that are >>>>>>> not semantically grounded. >>>>>>> >>>>>> >>>>>> Not quite. G is not semantically grounded >>>>> >>>>> i.e. unprovable >>>>> >>>>>> in PA >>>>>> yet G is semantically grounded >>>>> >>>>> i.e. provable >>>>> >>>>>> in metamathematics. >>>>> >>>>> Which is exactly what Godel proved. >>> >>> Your lack of response indicates that you agree with Godel, but used >>> different words to do so. >>> >>>>> >>>>>> When an expression in PA only derives semantic >>>>>> meaning in PA when grounded in PA then G has no >>>>>> meaning in PA. >>>>> >>>>> i.e. if a statement is unprovable in PA then it's unprovable in PA. >>>>> >>>>> In other words, a meaningless tautology. >>>>> >>>>>> >>>>>>> That also means that, using your terminology, it has been proven >>>>>>> that the statement ~∃x x=S(x), i.e. "No number is equal to its >>>>>>> successor", is not semantically grounded in Q. >>>>>>> >>>>>> >>>>>> Thus is meaningless in Q and out-of-scope in Q. >>>>> >>>>> Which means the semantically valid statement in Q >>>> >>>> does not include ~∃x x=S(x) >>> >>> False, as it means "no number is equal to its successor", and the >>> concept of a successor and equality have semantic meaning in Q, as >>> does the concept of "all", "none", and "exists". >>> >> >> 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. > > In other words, ~∃x x=S(x) is unprovable in Q, as is commonly known. > > So once again, you agree with everyone else, but are using different > words to say so. > Your move to ~∃x x=S(x) and Q was brilliant. It is the exact same situation as G and PA yet a much simpler example with none of the emotional baggage of Gödel. -- 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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