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Groups > comp.theory > #21465 > unrolled thread
| Started by | olcott <NoOne@NoWhere.com> |
|---|---|
| First post | 2020-07-05 22:52 -0500 |
| Last post | 2020-07-08 19:04 -0500 |
| Articles | 20 on this page of 335 — 10 participants |
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Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (axiomatic basis of truth) olcott <NoOne@NoWhere.com> - 2020-07-05 22:52 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (axiomatic basis of truth) André G. Isaak <agisaak@gm.invalid> - 2020-07-05 22:06 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (axiomatic basis of truth) olcott <NoOne@NoWhere.com> - 2020-07-05 23:33 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (axiomatic basis of truth) André G. Isaak <agisaak@gm.invalid> - 2020-07-05 22:58 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (axiomatic basis of truth) olcott <NoOne@NoWhere.com> - 2020-07-06 00:41 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (axiomatic basis of truth) André G. Isaak <agisaak@gm.invalid> - 2020-07-05 23:59 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (axiomatic basis of truth) olcott <NoOne@NoWhere.com> - 2020-07-06 11:20 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (axiomatic basis of truth) André G. Isaak <agisaak@gm.invalid> - 2020-07-06 11:18 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (axiomatic basis of truth) olcott <NoOne@NoWhere.com> - 2020-07-07 13:13 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) olcott <NoOne@NoWhere.com> - 2020-07-07 15:00 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) André G. Isaak <agisaak@gm.invalid> - 2020-07-07 14:17 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) olcott <NoOne@NoWhere.com> - 2020-07-07 15:25 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) André G. Isaak <agisaak@gm.invalid> - 2020-07-07 14:50 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) olcott <NoOne@NoWhere.com> - 2020-07-07 17:12 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) André G. Isaak <agisaak@gm.invalid> - 2020-07-07 18:27 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) olcott <NoOne@NoWhere.com> - 2020-07-07 19:43 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) Jeff Barnett <jbb@notatt.com> - 2020-07-07 19:28 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) olcott <NoOne@NoWhere.com> - 2020-07-07 21:31 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) Jeff Barnett <jbb@notatt.com> - 2020-07-07 21:29 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) olcott <NoOne@NoWhere.com> - 2020-07-07 22:57 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) Jeff Barnett <jbb@notatt.com> - 2020-07-08 12:27 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) olcott <NoOne@NoWhere.com> - 2020-07-08 14:19 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2020-07-10 10:39 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) olcott <NoOne@NoWhere.com> - 2020-07-10 08:41 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) André G. Isaak <agisaak@gm.invalid> - 2020-07-10 08:03 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) olcott <NoOne@NoWhere.com> - 2020-07-10 09:17 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-10 12:41 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) olcott <NoOne@NoWhere.com> - 2020-07-10 09:26 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) André G. Isaak <agisaak@gm.invalid> - 2020-07-07 21:52 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) olcott <NoOne@NoWhere.com> - 2020-07-07 23:00 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 ∃φ (φ ↔ T ⊬ φ) André G. Isaak <agisaak@gm.invalid> - 2020-07-07 22:43 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-08 00:16 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-07 23:39 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-08 00:54 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-08 00:14 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-08 10:11 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-08 09:50 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-08 11:09 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-08 11:29 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-08 11:49 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-09 06:56 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-09 11:02 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-09 11:33 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-09 23:23 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Jeff Barnett <jbb@notatt.com> - 2020-07-10 12:13 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-09 23:50 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-08 12:11 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-09 07:40 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-09 11:14 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-09 12:14 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-09 23:28 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-10 11:54 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Jeff Barnett <jbb@notatt.com> - 2020-07-10 14:46 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-10 16:16 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Jeff Barnett <jbb@notatt.com> - 2020-07-10 17:20 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-10 16:11 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-10 09:12 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-10 09:29 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-10 09:42 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-10 10:54 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-10 10:55 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-10 11:02 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-10 12:16 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-10 11:27 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-10 13:04 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-10 12:12 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-10 15:11 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-10 14:27 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-10 15:42 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-10 15:00 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-10 16:36 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-10 20:19 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-11 04:20 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-11 19:24 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-11 18:57 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (TRUTH BEARER DEFINED) olcott <NoOne@NoWhere.com> - 2020-07-11 22:58 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (TRUTH BEARER DEFINED) André G. Isaak <agisaak@gm.invalid> - 2020-07-12 00:37 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) olcott <NoOne@NoWhere.com> - 2020-07-12 11:43 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) André G. Isaak <agisaak@gm.invalid> - 2020-07-12 12:07 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) olcott <NoOne@NoWhere.com> - 2020-07-12 13:51 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) André G. Isaak <agisaak@gm.invalid> - 2020-07-12 13:36 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) olcott <NoOne@NoWhere.com> - 2020-07-12 15:31 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) olcott <NoOne@NoWhere.com> - 2020-07-12 16:24 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) André G. Isaak <agisaak@gm.invalid> - 2020-07-12 15:37 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) olcott <NoOne@NoWhere.com> - 2020-07-12 18:04 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) André G. Isaak <agisaak@gm.invalid> - 2020-07-12 17:21 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) olcott <NoOne@NoWhere.com> - 2020-07-12 18:53 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) André G. Isaak <agisaak@gm.invalid> - 2020-07-12 18:07 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) olcott <NoOne@NoWhere.com> - 2020-07-12 19:44 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) André G. Isaak <agisaak@gm.invalid> - 2020-07-12 18:58 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) olcott <NoOne@NoWhere.com> - 2020-07-12 23:06 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) André G. Isaak <agisaak@gm.invalid> - 2020-07-13 07:01 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) olcott <NoOne@NoWhere.com> - 2020-07-13 09:32 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) André G. Isaak <agisaak@gm.invalid> - 2020-07-13 08:47 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) olcott <NoOne@NoWhere.com> - 2020-07-13 19:52 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) olcott <NoOne@NoWhere.com> - 2020-07-13 09:07 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) David Kleinecke <dkleinecke@gmail.com> - 2020-07-12 17:28 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (NATURE OF TRUTH ITSELF) olcott <NoOne@NoWhere.com> - 2020-07-12 19:47 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2020-07-10 19:21 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-10 13:35 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2020-07-11 12:25 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-11 19:05 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2020-07-12 14:10 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-12 13:24 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-12 14:04 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-12 18:48 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-12 17:22 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-12 19:52 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-12 19:32 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-12 22:47 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-13 08:05 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-13 19:49 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-13 19:11 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-14 09:43 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-14 08:57 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-14 10:22 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-14 09:30 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-15 10:38 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-15 11:24 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-15 19:18 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-15 20:38 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-16 16:16 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-16 16:01 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-16 19:11 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-16 18:40 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-13 23:48 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-14 10:11 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-14 09:20 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-14 10:26 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-14 09:36 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-15 10:41 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-14 11:25 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-15 10:52 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-15 11:04 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-15 19:07 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-15 18:42 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-16 12:10 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-16 11:46 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-16 16:35 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-16 15:19 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-16 23:19 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-16 22:49 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-17 00:34 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-17 01:04 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-17 17:20 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-17 16:16 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-17 18:59 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-18 03:13 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-17 22:01 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-18 17:17 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-18 12:43 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-18 15:08 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-18 20:28 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-19 03:45 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-19 11:46 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-19 11:05 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-19 12:12 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-19 11:30 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-19 12:36 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-19 20:51 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-19 15:28 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-20 02:44 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-20 12:40 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-21 01:52 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-20 21:35 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-20 19:59 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-21 10:44 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-21 10:00 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-21 19:50 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-21 17:57 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-22 09:07 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-22 02:03 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-22 09:03 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-23 00:30 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-22 09:06 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Jeff Barnett <jbb@notatt.com> - 2020-07-19 22:23 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-20 10:33 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Jeff Barnett <jbb@notatt.com> - 2020-07-20 10:50 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2020-07-17 12:16 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-17 17:04 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-17 17:09 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2020-07-18 00:22 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2020-07-13 13:05 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-13 10:07 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2020-07-13 20:01 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-13 12:24 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 [--Obvious Yet?--] olcott <NoOne@NoWhere.com> - 2020-07-13 14:58 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-13 18:33 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-13 17:46 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-14 09:36 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-14 09:53 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-15 10:49 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-13 23:42 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-13 18:45 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-14 01:26 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-13 22:06 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-14 17:00 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-14 18:15 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-15 02:56 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-14 21:55 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) David Kleinecke <dkleinecke@gmail.com> - 2020-07-14 20:30 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-14 23:13 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Mapping to Boolean) olcott <NoOne@NoWhere.com> - 2020-07-15 09:57 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-15 16:48 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-15 11:46 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-15 11:32 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-15 19:13 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-16 01:37 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-15 22:12 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-16 16:05 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) olcott <NoOne@NoWhere.com> - 2020-07-16 14:18 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-16 13:32 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) olcott <NoOne@NoWhere.com> - 2020-07-16 22:39 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-16 21:00 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-17 02:17 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) olcott <NoOne@NoWhere.com> - 2020-07-16 21:01 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-17 03:54 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) olcott <NoOne@NoWhere.com> - 2020-07-16 23:27 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-17 11:36 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) olcott <NoOne@NoWhere.com> - 2020-07-17 11:10 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) David Kleinecke <dkleinecke@gmail.com> - 2020-07-17 11:11 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) olcott <NoOne@NoWhere.com> - 2020-07-17 14:24 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) Kaz Kylheku <793-849-0957@kylheku.com> - 2020-07-17 20:28 +0000
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) olcott <NoOne@NoWhere.com> - 2020-07-17 16:47 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) Kaz Kylheku <793-849-0957@kylheku.com> - 2020-07-17 20:26 +0000
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) olcott <NoOne@NoWhere.com> - 2020-07-17 17:39 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-17 16:06 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) olcott <NoOne@NoWhere.com> - 2020-07-17 18:40 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) André G. Isaak <agisaak@gm.invalid> - 2020-07-17 17:47 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-17 18:01 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) olcott <NoOne@NoWhere.com> - 2020-07-17 22:24 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) André G. Isaak <agisaak@gm.invalid> - 2020-07-17 21:34 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) olcott <NoOne@NoWhere.com> - 2020-07-17 22:44 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) André G. Isaak <agisaak@gm.invalid> - 2020-07-17 22:01 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) olcott <NoOne@NoWhere.com> - 2020-07-18 13:34 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-17 21:09 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) olcott <NoOne@NoWhere.com> - 2020-07-18 10:14 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-18 15:05 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) André G. Isaak <agisaak@gm.invalid> - 2020-07-17 17:23 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) olcott <NoOne@NoWhere.com> - 2020-07-17 18:52 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) André G. Isaak <agisaak@gm.invalid> - 2020-07-17 18:01 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) olcott <NoOne@NoWhere.com> - 2020-07-17 22:35 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) André G. Isaak <agisaak@gm.invalid> - 2020-07-17 21:55 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) olcott <NoOne@NoWhere.com> - 2020-07-18 13:49 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) David Kleinecke <dkleinecke@gmail.com> - 2020-07-17 22:12 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) olcott <NoOne@NoWhere.com> - 2020-07-17 14:20 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-18 02:17 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Membership algorithm) olcott <NoOne@NoWhere.com> - 2020-07-17 21:53 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-15 18:23 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-16 11:51 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-16 11:21 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-16 13:41 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-16 13:10 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-16 22:36 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-16 21:04 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2020-07-17 12:10 +0100
Re: Simply defining G"odel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) R Kym Horsell <kym@kymhorsell.com> - 2020-07-17 11:50 +0000
Re: Simply defining G"odel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-17 17:00 -0500
Re: Simply defining G"odel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-17 17:40 -0600
Re: Simply defining G"odel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-17 17:46 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-17 17:07 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2020-07-18 00:30 +0100
Re: Simply defining G"odel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) R Kym Horsell <kym@kymhorsell.com> - 2020-07-18 02:21 +0000
Re: Simply defining G"odel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2020-07-18 16:19 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-17 22:03 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2020-07-18 16:12 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-18 11:11 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-15 20:25 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-16 16:11 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-16 14:31 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-16 22:45 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-16 21:10 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-16 15:58 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-16 22:47 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-16 21:18 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-16 22:38 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-16 00:35 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-15 18:44 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-16 01:16 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-15 19:28 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) David Kleinecke <dkleinecke@gmail.com> - 2020-07-15 17:44 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-15 20:44 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-16 02:19 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-15 22:20 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-16 16:08 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-16 14:20 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-16 13:12 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-16 22:37 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-13 17:52 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-13 21:12 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-13 20:11 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-13 22:48 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) David Kleinecke <dkleinecke@gmail.com> - 2020-07-13 21:38 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-14 00:03 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) David Kleinecke <dkleinecke@gmail.com> - 2020-07-13 22:26 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-14 00:32 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2020-07-14 14:41 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-14 10:14 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Chris Buckley <alan@sabir.com> - 2020-07-14 18:24 +0000
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-14 17:44 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Chris Buckley <alan@sabir.com> - 2020-07-15 18:08 +0000
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-15 18:47 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) David Kleinecke <dkleinecke@gmail.com> - 2020-07-12 17:30 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-12 19:50 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-12 18:53 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Jeff Barnett <jbb@notatt.com> - 2020-07-12 23:48 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-13 00:58 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Jeff Barnett <jbb@notatt.com> - 2020-07-13 13:07 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-13 14:12 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Jeff Barnett <jbb@notatt.com> - 2020-07-13 15:32 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-13 15:06 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Jeff Barnett <jbb@notatt.com> - 2020-07-14 00:56 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-13 23:26 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-13 16:10 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-13 09:57 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Jeff Barnett <jbb@notatt.com> - 2020-07-13 13:12 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-10 12:53 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-10 16:25 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-10 15:06 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-10 17:21 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) David Kleinecke <dkleinecke@gmail.com> - 2020-07-10 15:58 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-10 18:01 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-11 04:10 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-11 19:13 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Jeff Barnett <jbb@notatt.com> - 2020-07-08 12:39 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-08 23:37 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Jeff Barnett <jbb@notatt.com> - 2020-07-09 00:40 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-09 09:38 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-09 09:18 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-09 12:15 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) olcott <NoOne@NoWhere.com> - 2020-07-09 15:10 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V27 (Simple enough yet?) olcott <NoOne@NoWhere.com> - 2020-07-08 16:25 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V27 (Simple enough yet?) André G. Isaak <agisaak@gm.invalid> - 2020-07-09 07:02 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V27 (Simple enough yet?) olcott <NoOne@NoWhere.com> - 2020-07-09 11:11 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Shell game) olcott <NoOne@NoWhere.com> - 2020-07-08 19:04 -0500
Page 16 of 17 — ← Prev page 1 … 14 15 [16] 17 Next page →
| From | Chris Buckley <alan@sabir.com> |
|---|---|
| Date | 2020-07-15 18:08 +0000 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <slrnrguhgf.g1a.alan@video.sabir.com> |
| In reply to | #21665 |
On 2020-07-14, olcott <NoOne@NoWhere.com> wrote: > On 7/14/2020 1:24 PM, Chris Buckley wrote: >> On 2020-07-14, olcott <NoOne@NoWhere.com> wrote: >>> >>> The only way to show that any expression of the language of any formal >>> system is true in that formal system is to show a mapping in that formal >>> system from this expression in this formal system to a Boolean value. >>> >>> If you understand all of the technical terms you that I just used you >>> will understand that what I just said is a tautology. >>> >>> If the only maps that you know about show streets in a city then you >>> must learn about mathematical maps before you will be able to begin >>> understand me. >> >> No. >> >> Please give your definition of "formal system". You have never done so. >> >> In the classic definition of a formal system, there is no notion at all >> of truth within the formal system. Truth comes in at a layer above >> "formal system", when you start talking about interpretations of the >> formula of the formal system. >> > > OK great. I am referring to Gödel's work being characterized as true and > unprovable. > > https://scholar.google.com/scholar?hl=en&as_sdt=0%2C28&q=%22true+and+unprovable%22+godel&btnG=&oq= > > Simplifying the multiple levels of conventional terminology to a > slightly higher level of abstraction the formal notion of truth is > ultimately the mathematical mapping of a finite string to Boolean values. > > When we talk about the generic notion of truth it is any mapping from an > expression of natural or formal language to a Boolean value. > > When we restrict ourselves to formal systems then the mapping must > remain within the formal system and be specified in a the language of > this system. > >> Nobody understands "all of the technical terms you that I just used" >> since you define them in non-standard ways, and refuse to tell us >> what those definitions are! >> >> Chris >> > > I am referring to the conventional process at a slightly higher level of > abstraction. Ultimately True(T, φ) is a mathematical mapping in T from φ > to a Boolean value. When-so-ever no such mapping exists φ is simply not > a truth bearer in T. > > In classical logic a sentence in a language is true or false under (and > only under) an interpretation and is therefore a truth-bearer. > https://en.wikipedia.org/wiki/Truth > bearer#Sentences_in_languages_of_classical_logic > > If φ and its negation are unsatisfiable in T then φ is not a truth > bearer on T. Again, you are introducing your very own, private, "slightly higher level of abstraction". And then hand-waving a single property of the differences without ever defining them. "Provability" is a property of formulas within a formal system. "Truth" is not. There is nothing in the classical definition of formal system that refers to truth. Godel operated entirely within a formal system for most of his Incompleteness Theorem proof. He was able, *within the formal system*, to come up with a self-referential formula stating that no proof of that formula exists within the formal system. This self-referentiality is not a language trick, like the liar's paradox is. It is derivable in any formal system of sufficient power (and a couple of other restrictions like enumerable axioms.) If you want a layman's idea of how it is accomplished, I just ran across https://www.quantamagazine.org/how-godels-incompleteness-theorems-work-20200714/ It is certainly not a full explanation (there are reasons why a full proof of Godel's Incompleteness Theorem take most of a semester of a graduate level mathematical logic course!), but it does a reasonable job at getting at the gist of the self-referentiality (the comments address at least one major hole.) Once you have that such a formula exists, then you can start considering truth, in whatever model you want. In any consistent model that formula has to be true. Thus we have a true formula in the model which does not have a proof in the formal system. Chris
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| From | olcott <NoOne@NoWhere.com> |
|---|---|
| Date | 2020-07-15 18:47 -0500 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <ssednWuYj5QyCJLCnZ2dnUU7-dnNnZ2d@giganews.com> |
| In reply to | #21681 |
On 7/15/2020 1:08 PM, Chris Buckley wrote:
> On 2020-07-14, olcott <NoOne@NoWhere.com> wrote:
>> On 7/14/2020 1:24 PM, Chris Buckley wrote:
>>> On 2020-07-14, olcott <NoOne@NoWhere.com> wrote:
>>>>
>>>> The only way to show that any expression of the language of any formal
>>>> system is true in that formal system is to show a mapping in that formal
>>>> system from this expression in this formal system to a Boolean value.
>>>>
>>>> If you understand all of the technical terms you that I just used you
>>>> will understand that what I just said is a tautology.
>>>>
>>>> If the only maps that you know about show streets in a city then you
>>>> must learn about mathematical maps before you will be able to begin
>>>> understand me.
>>>
>>> No.
>>>
>>> Please give your definition of "formal system". You have never done so.
>>>
>>> In the classic definition of a formal system, there is no notion at all
>>> of truth within the formal system. Truth comes in at a layer above
>>> "formal system", when you start talking about interpretations of the
>>> formula of the formal system.
>>>
>>
>> OK great. I am referring to Gödel's work being characterized as true and
>> unprovable.
>>
>> https://scholar.google.com/scholar?hl=en&as_sdt=0%2C28&q=%22true+and+unprovable%22+godel&btnG=&oq=
>>
>> Simplifying the multiple levels of conventional terminology to a
>> slightly higher level of abstraction the formal notion of truth is
>> ultimately the mathematical mapping of a finite string to Boolean values.
>>
>> When we talk about the generic notion of truth it is any mapping from an
>> expression of natural or formal language to a Boolean value.
>>
>> When we restrict ourselves to formal systems then the mapping must
>> remain within the formal system and be specified in a the language of
>> this system.
>>
>>> Nobody understands "all of the technical terms you that I just used"
>>> since you define them in non-standard ways, and refuse to tell us
>>> what those definitions are!
>>>
>>> Chris
>>>
>>
>> I am referring to the conventional process at a slightly higher level of
>> abstraction. Ultimately True(T, φ) is a mathematical mapping in T from φ
>> to a Boolean value. When-so-ever no such mapping exists φ is simply not
>> a truth bearer in T.
>>
>> In classical logic a sentence in a language is true or false under (and
>> only under) an interpretation and is therefore a truth-bearer.
>> https://en.wikipedia.org/wiki/Truth
>> bearer#Sentences_in_languages_of_classical_logic
>>
>> If φ and its negation are unsatisfiable in T then φ is not a truth
>> bearer on T.
>
> Again, you are introducing your very own, private, "slightly higher
> level of abstraction". And then hand-waving a single property of the
> differences without ever defining them.
>
Succinct way to show that true and unprovable cannot possibly coexist:
There is no way to show that formula φ of theory T is satisfied in T
that does not require a mathematical mapping from φ through the axioms
and rules-of-inference of T to an element of the set of {true, false}.
> "Provability" is a property of formulas within a formal system. "Truth"
> is not. There is nothing in the classical definition of formal system
> that refers to truth.
>
> Godel operated entirely within a formal system for most of his Incompleteness
> Theorem proof. He was able, *within the formal system*, to come up with
> a self-referential formula stating that no proof of that formula exists
> within the formal system.
>
> This self-referentiality is not a language trick, like the liar's paradox is.
> It is derivable in any formal system of sufficient power (and a couple of
> other restrictions like enumerable axioms.)
>
> If you want a layman's idea of how it is accomplished, I just ran across
> https://www.quantamagazine.org/how-godels-incompleteness-theorems-work-20200714/
>
> It is certainly not a full explanation (there are reasons why a full proof
> of Godel's Incompleteness Theorem take most of a semester of a graduate
> level mathematical logic course!), but it does a reasonable job at getting
> at the gist of the self-referentiality (the comments address at least one
> major hole.)
>
> Once you have that such a formula exists, then you can start
> considering truth, in whatever model you want. In any consistent model
> that formula has to be true. Thus we have a true formula in the model
> which does not have a proof in the formal system.
>
> Chris
>
--
Copyright 2020 Pete Olcott
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| From | David Kleinecke <dkleinecke@gmail.com> |
|---|---|
| Date | 2020-07-12 17:30 -0700 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <ceddaf04-2956-468f-8ee0-9976a96b48bbo@googlegroups.com> |
| In reply to | #21596 |
On Sunday, July 12, 2020 at 4:48:36 PM UTC-7, olcott wrote: > On 7/12/2020 4:04 PM, Keith Thompson wrote: > > olcott <NoOne@NoWhere.com> writes: > >> On 7/12/2020 8:10 AM, Alan Smaill wrote: > >>> olcott <NoOne@NoWhere.com> writes: > >>>> On 7/11/2020 6:25 AM, Alan Smaill wrote: > >>>>> olcott <NoOne@NoWhere.com> writes: > >>> [...] > >>>>> You miss the point of my comment -- > >>>>> try thinking before you give your stock response. > >>>>> > >>>>> Let's go back to the start and try again: > >>>>> > >>>>>>>> Q ⊬ φ // This is true in Q > >>>>> > >>>>> Agreed. > >>>>> > >>>>>>>> ∴ φ ↔ Q ⊬ φ is not true in Q > >>>>> > >>>>> What makes you think this? > >>>> > >>>> The conventional definition of incompleteness: > >>>> A theory T is incomplete if and only if there is some sentence φ such > >>>> that (T ⊬ φ) and (T ⊬ ¬φ). > >>>> > >>>> Q is incomplete relative to the commutativity of addition: > >>>> φ = (∀x ∀y (x + y = y + x)) > >>>> (Q ⊬ φ ∧ Q ⊬ ¬φ) > >>>> > >>>> https://math.stackexchange.com/questions/998359/robinson-arithmetic-and-its-incompleteness > >>>> > >>>> Nothing can actually be incomplete unless something is missing. In the > >>>> case of Q the commutativity of addition is missing. > > > > You insist on interpreting the word "incomplete" to mean "something > > is missing". That's just not what "incomplete" means. > > You know that is not true. You know that any use of the term: > "incomplete" that does not mean that something is missing is a misnomer. Humpty-Dumpty wants to have a word with you, young man. > > A system > > is by definition "incomplete" if and only there is at least one > > statement such that neither the statement nor its negation can be > > proven. Nothing you say about incompleteness that doesn't refer > > to that definition (or a more correct and rigorous version of it) > > is relevant. > > > >>>> Theories can be called "incomplete" yet if nothing is missing then > >>>> this use is a misnomer. > >>> > >>> Are you saying you accept that Q is incomplete or not? > >>> I can't tell. > >> > >> Q is incomplete relative to the commutativity of addition. > > > > Incompleteness is not "relative to" something. > > Any use by anyone of the term "incomplete" without anything missing is > a misnomer. We could say that Peano arithmetic is incomplete on the > basis the Peano arithmetic cannot prove or disprove that ice cream is a > dairy product. > > We could say that mathematics is incomplete because it cannot prove or > disprove that a cat is biologically an animal because mathematics does > not know about biology. > > > A statement about > > the commutativity of addition can be used to demonstrate that Q is > > incomplete, but that incompleteness is an unqualified characteristic > > of Q. > > > > Do you accept that Q is incomplete? Please start your answer with > > either "Yes." or "No.". Feel free to add whatever discussion you > > like after that. > > The use of the term of the art "incomplete" is incongruous with its > common meaning thus incorrect on that basis. > > >>> You quoted a definition of incompleteness above. > >>> Can you make clear if you are happy with that definition? > >> > >> I am unhappy with that definition because it would decide that a > >> formal system is incomplete (in some cases) based on the fact that the > >> formal system can neither prove nor disprove self-contradictory > >> expressions of its own language. > > > > The definition of incompleteness says nothing about self-contradictory > > statements. In Q, the statement > > ∀x ∀y (x + y = y + x) > > is not self-contradictory. > > > > [...] > > > >>> You accept that Q ⊬ φ. > >>> > >>> What about Q ⊬ ¬φ) ? > >> > >> Is the sentence: "cows are animals" true in mathematics, or is it > >> neither true nor false in mathematics and true in biology? > > > > That's non-responsive. "cow are animals" refers to things outside the > > scope of mathematics. "∀x ∀y (x + y = y + x)" refers to things entirely > > within the scope of Q. > > NO IT DOES NOT. > > Within the scope of the syntax does not entail within the scope of the > semantics. > > (a) ∀x(Ix → Dx) (where D = dairy product and I = ice cream) is true. > (b) ∀x(Ix → Dx) (where D and I are undefined) is neither true nor false. > > > [SNIP] > > > > It's occurred to me that this raises a (perhaps) interesting question > > -- or maybe I'm missing something obvious. > > > > In a system that includes Euclid's first four postulate but not the > > fifth (the parallel postulate), neither the parallel postulate nor > > its negation can be proven. We can construct a consistent system > > with the parallel postulate as an axiom. We can *also* construct > > a consistent system with the negation of the parallel postulate as > > an axiom. > > > > Robinson Arithmetic cannot prove or disprove commutativity > > of addition. We can construct a consistent system based on > > Robinson Arithmetic in which addition is provably commutative. > > Sure just add an axiom: ∀x ∈ ℕ ∃y ∈ ℕ (x + y = y + x) > > > Can we construct a consistent system based on Robinson Arithmetic > > in which addition is provably *not* commutative? > > > > Not within the conventional semantics of the meaning of those terms. > > -- > Copyright 2020 Pete Olcott
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| From | olcott <NoOne@NoWhere.com> |
|---|---|
| Date | 2020-07-12 19:50 -0500 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <5Lmdnelh4P5aMpbCnZ2dnUU7-LfNnZ2d@giganews.com> |
| In reply to | #21601 |
On 7/12/2020 7:30 PM, David Kleinecke wrote: > On Sunday, July 12, 2020 at 4:48:36 PM UTC-7, olcott wrote: >> On 7/12/2020 4:04 PM, Keith Thompson wrote: >>> olcott <NoOne@NoWhere.com> writes: >>>> On 7/12/2020 8:10 AM, Alan Smaill wrote: >>>>> olcott <NoOne@NoWhere.com> writes: >>>>>> On 7/11/2020 6:25 AM, Alan Smaill wrote: >>>>>>> olcott <NoOne@NoWhere.com> writes: >>>>> [...] >>>>>>> You miss the point of my comment -- >>>>>>> try thinking before you give your stock response. >>>>>>> >>>>>>> Let's go back to the start and try again: >>>>>>> >>>>>>>>>> Q ⊬ φ // This is true in Q >>>>>>> >>>>>>> Agreed. >>>>>>> >>>>>>>>>> ∴ φ ↔ Q ⊬ φ is not true in Q >>>>>>> >>>>>>> What makes you think this? >>>>>> >>>>>> The conventional definition of incompleteness: >>>>>> A theory T is incomplete if and only if there is some sentence φ such >>>>>> that (T ⊬ φ) and (T ⊬ ¬φ). >>>>>> >>>>>> Q is incomplete relative to the commutativity of addition: >>>>>> φ = (∀x ∀y (x + y = y + x)) >>>>>> (Q ⊬ φ ∧ Q ⊬ ¬φ) >>>>>> >>>>>> https://math.stackexchange.com/questions/998359/robinson-arithmetic-and-its-incompleteness >>>>>> >>>>>> Nothing can actually be incomplete unless something is missing. In the >>>>>> case of Q the commutativity of addition is missing. >>> >>> You insist on interpreting the word "incomplete" to mean "something >>> is missing". That's just not what "incomplete" means. >> >> You know that is not true. You know that any use of the term: >> "incomplete" that does not mean that something is missing is a misnomer. > > Humpty-Dumpty wants to have a word with you, young man. On my last birthday (I was 65) and celebrated my first day of being old. On my prior birthday I played this song to celebrate: https://www.youtube.com/watch?v=HCTunqv1Xt4 -- Copyright 2020 Pete Olcott
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| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2020-07-12 18:53 -0600 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <regbap$phr$1@dont-email.me> |
| In reply to | #21596 |
On 2020-07-12 17:48, olcott wrote: > On 7/12/2020 4:04 PM, Keith Thompson wrote: >> olcott <NoOne@NoWhere.com> writes: >>> On 7/12/2020 8:10 AM, Alan Smaill wrote: >>>> olcott <NoOne@NoWhere.com> writes: >>>>> Nothing can actually be incomplete unless something is missing. In the >>>>> case of Q the commutativity of addition is missing. >> >> You insist on interpreting the word "incomplete" to mean "something >> is missing". That's just not what "incomplete" means. > > You know that is not true. You know that any use of the term: > "incomplete" that does not mean that something is missing is a misnomer. No, he doesn't know that, because it is rubbish. To an astronomer, nitrogen is a metal. Is that wrong because astronomical usage differs from colloquial usage? Of course not. To a mathematician, a ring is a type algebraic structure which has two operators with certain properties. That has absolutely nothing to do with the colloquial meaning. Does that make it a misnomer? of course not. When physicists refer to quarks as strange or charm, or red or blue or green, these terms obviously have no relation to their colloquial meanings. So what? Not all carnivores are carnivores. Nor are all carnivores carnivores. That's because the word means two entirely different things. Is that a problem? No. André -- To email remove 'invalid' & replace 'gm' with well known Google mail service.
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| From | Jeff Barnett <jbb@notatt.com> |
|---|---|
| Date | 2020-07-12 23:48 -0600 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <regsjh$3ct$1@dont-email.me> |
| In reply to | #21591 |
On 7/12/2020 3:04 PM, Keith Thompson wrote:
> It's occurred to me that this raises a (perhaps) interesting question
> -- or maybe I'm missing something obvious.
>
> In a system that includes Euclid's first four postulate but not the
> fifth (the parallel postulate), neither the parallel postulate nor
> its negation can be proven. We can construct a consistent system
> with the parallel postulate as an axiom. We can *also* construct
> a consistent system with the negation of the parallel postulate as
> an axiom.
>
> Robinson Arithmetic cannot prove or disprove commutativity
> of addition. We can construct a consistent system based on
> Robinson Arithmetic in which addition is provably commutative.
> Can we construct a consistent system based on Robinson Arithmetic
> in which addition is provably *not* commutative?
It's been awhile but my memory might be working. Let's first look at
what a "fixed" use of commutativity might look like: 1+2 = 2+1 expressed
as S0 + SS0 = SS0 + S0, where "s" is the successor function and "0" is a
constant given by the axioms. This can be proven in Robinson (R) and so
can ANY other fixed example. What you can not prove is the combined version:
forall x in I forall y in I x+y = y+x
Since you can prove each instance, within R, there will be no model
where commutativity does not hold, i.e, where
there exist x in I there exist y in I x+y ~= y+x
can be demonstrated. If we endow R with an appropriate induction axiom,
we can prove commutativity within R. There is no extension to R that can
make commutativity false and the extension consistent. There are many
properties of number like systems where all individual examples can be
proven but the combined property cannot.
The above example is very instructive: a formal system besides syntax
rules must have proof rules as we all keep saying. Here we can extend R
to be commutative in at least two ways: 1) make it an axiom of R or 2)
adopt an appropriate induction rule a proof rule.
Now let's make an observation: R is incomplete because it can't prove
something that is true in it. Of course you can't prove its false
either. In this example, you can prove commutativity is true by using
the power of the meta logic which will have some sort of induction
equivalent available.
Let's look at another case floating through these discussions or
corrective lectures to a poor student. Consider Euclid with the standard
four postulates P1, P2, P3. P4 plus a bunch of definitions and the
axioms of measure - sort of the real numbers. We know that P5, the
parallel postulate, can be added and so could its negation. In other
words we know that there can be zero, one, or more lines through a point
(not on another line) that do not intersect that other line. In this
case we don't bother to say that P1, P2, P3, P4 are not complete; rather
we make the much more powerful statement that P5 is INDEPENDENT of basic
Euclid. This is in the same way that the axiom of choice is independent
of the rest of basic set theory.
--
Jeff Barnett
[toc] | [prev] | [next] | [standalone]
| From | Keith Thompson <Keith.S.Thompson+u@gmail.com> |
|---|---|
| Date | 2020-07-13 00:58 -0700 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <874kqb6e3j.fsf@nosuchdomain.example.com> |
| In reply to | #21611 |
Jeff Barnett <jbb@notatt.com> writes:
> On 7/12/2020 3:04 PM, Keith Thompson wrote:
>> It's occurred to me that this raises a (perhaps) interesting question
>> -- or maybe I'm missing something obvious.
>>
>> In a system that includes Euclid's first four postulate but not the
>> fifth (the parallel postulate), neither the parallel postulate nor
>> its negation can be proven. We can construct a consistent system
>> with the parallel postulate as an axiom. We can *also* construct
>> a consistent system with the negation of the parallel postulate as
>> an axiom.
>>
>> Robinson Arithmetic cannot prove or disprove commutativity
>> of addition. We can construct a consistent system based on
>> Robinson Arithmetic in which addition is provably commutative.
>> Can we construct a consistent system based on Robinson Arithmetic
>> in which addition is provably *not* commutative?
>
> It's been awhile but my memory might be working. Let's first look at
> what a "fixed" use of commutativity might look like: 1+2 = 2+1
> expressed as S0 + SS0 = SS0 + S0, where "s" is the successor function
> and "0" is a constant given by the axioms. This can be proven in
> Robinson (R) and so can ANY other fixed example. What you can not
> prove is the combined version:
> forall x in I forall y in I x+y = y+x
> Since you can prove each instance, within R, there will be no model
> where commutativity does not hold, i.e, where
> there exist x in I there exist y in I x+y ~= y+x
> can be demonstrated. If we endow R with an appropriate induction
> axiom, we can prove commutativity within R. There is no extension to R
> that can make commutativity false and the extension consistent. There
> are many properties of number like systems where all individual
> examples can be proven but the combined property cannot.
>
> The above example is very instructive: a formal system besides syntax
> rules must have proof rules as we all keep saying. Here we can extend
> R to be commutative in at least two ways: 1) make it an axiom of R or
> 2) adopt an appropriate induction rule a proof rule.
(Apparently Robinson Arithmetic is usually called Q, not R.)
So Q can't prove that addition is commutative, but Q+induction
can, yes?
I wonder if a system in which induction specifically *doesn't* work,
such as Q+¬induction, so that a statement that would be true in
Q+induction might be provably false, could be made to work.
But I digress.
> Now let's make an observation: R is incomplete because it can't prove
> something that is true in it. Of course you can't prove its false
> either. In this example, you can prove commutativity is true by using
> the power of the meta logic which will have some sort of induction
> equivalent available.
>
> Let's look at another case floating through these discussions or
> corrective lectures to a poor student. Consider Euclid with the
> standard four postulates P1, P2, P3. P4 plus a bunch of definitions
> and the axioms of measure - sort of the real numbers. We know that P5,
> the parallel postulate, can be added and so could its negation. In
> other words we know that there can be zero, one, or more lines through
> a point (not on another line) that do not intersect that other
> line. In this case we don't bother to say that P1, P2, P3, P4 are not
> complete; rather we make the much more powerful statement that P5 is
> INDEPENDENT of basic Euclid. This is in the same way that the axiom of
> choice is independent of the rest of basic set theory.
That makes sense. So the relationship between Q and the
commutativity of addition is fundamentally different from the
relationship between Euclid-P5 (Euclid without the parallel
postulate) and P5. The former is true but unprovable in Q,
but P5 can consistently be made either true or false in a system
built on top of Euclid-P5. (But if we could create a system like
Q+¬induction, they might be more similar. It seems difficult to
think about such as system.)
--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Philips Healthcare
void Void(void) { Void(); } /* The recursive call of the void */
[toc] | [prev] | [next] | [standalone]
| From | Jeff Barnett <jbb@notatt.com> |
|---|---|
| Date | 2020-07-13 13:07 -0600 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <reibd3$p3e$1@dont-email.me> |
| In reply to | #21612 |
On 7/13/2020 1:58 AM, Keith Thompson wrote: > Jeff Barnett <jbb@notatt.com> writes: >> On 7/12/2020 3:04 PM, Keith Thompson wrote: >>> It's occurred to me that this raises a (perhaps) interesting question >>> -- or maybe I'm missing something obvious. >>> >>> In a system that includes Euclid's first four postulate but not the >>> fifth (the parallel postulate), neither the parallel postulate nor >>> its negation can be proven. We can construct a consistent system >>> with the parallel postulate as an axiom. We can *also* construct >>> a consistent system with the negation of the parallel postulate as >>> an axiom. >>> >>> Robinson Arithmetic cannot prove or disprove commutativity >>> of addition. We can construct a consistent system based on >>> Robinson Arithmetic in which addition is provably commutative. >>> Can we construct a consistent system based on Robinson Arithmetic >>> in which addition is provably *not* commutative? >> >> It's been awhile but my memory might be working. Let's first look at >> what a "fixed" use of commutativity might look like: 1+2 = 2+1 >> expressed as S0 + SS0 = SS0 + S0, where "s" is the successor function >> and "0" is a constant given by the axioms. This can be proven in >> Robinson (R) and so can ANY other fixed example. What you can not >> prove is the combined version: >> forall x in I forall y in I x+y = y+x >> Since you can prove each instance, within R, there will be no model >> where commutativity does not hold, i.e, where >> there exist x in I there exist y in I x+y ~= y+x >> can be demonstrated. If we endow R with an appropriate induction >> axiom, we can prove commutativity within R. There is no extension to R >> that can make commutativity false and the extension consistent. There >> are many properties of number like systems where all individual >> examples can be proven but the combined property cannot. >> >> The above example is very instructive: a formal system besides syntax >> rules must have proof rules as we all keep saying. Here we can extend >> R to be commutative in at least two ways: 1) make it an axiom of R or >> 2) adopt an appropriate induction rule a proof rule. > > (Apparently Robinson Arithmetic is usually called Q, not R.) > > So Q can't prove that addition is commutative, but Q+induction > can, yes? > > I wonder if a system in which induction specifically *doesn't* work, > such as Q+¬induction, so that a statement that would be true in > Q+induction might be provably false, could be made to work. > > But I digress. > >> Now let's make an observation: R is incomplete because it can't prove >> something that is true in it. Of course you can't prove its false >> either. In this example, you can prove commutativity is true by using >> the power of the meta logic which will have some sort of induction >> equivalent available. >> >> Let's look at another case floating through these discussions or >> corrective lectures to a poor student. Consider Euclid with the >> standard four postulates P1, P2, P3. P4 plus a bunch of definitions >> and the axioms of measure - sort of the real numbers. We know that P5, >> the parallel postulate, can be added and so could its negation. In >> other words we know that there can be zero, one, or more lines through >> a point (not on another line) that do not intersect that other >> line. In this case we don't bother to say that P1, P2, P3, P4 are not >> complete; rather we make the much more powerful statement that P5 is >> INDEPENDENT of basic Euclid. This is in the same way that the axiom of >> choice is independent of the rest of basic set theory. > > That makes sense. So the relationship between Q and the > commutativity of addition is fundamentally different from the > relationship between Euclid-P5 (Euclid without the parallel > postulate) and P5. The former is true but unprovable in Q, > but P5 can consistently be made either true or false in a system > built on top of Euclid-P5. (But if we could create a system like > Q+¬induction, they might be more similar. It seems difficult to > think about such as system.) Recall, we can't prove commutativity in R|Q period. However, our meta logic can do the proof because it provides additional mechanisms. Let's look at a Peano-like induction rule where I is the integers and S is the successor function: If Z subset-of I, 0 in Z, and whenever x in Z then Sx in Z; then Z = I. How would an anti-induction rule be expressed? What would it mean? And would it lead to inconsistency? In other words would it falsify even one theorem provable in R|Q by the remaining axioms? -- Jeff Barnett
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| From | Keith Thompson <Keith.S.Thompson+u@gmail.com> |
|---|---|
| Date | 2020-07-13 14:12 -0700 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <87eepf3yr5.fsf@nosuchdomain.example.com> |
| In reply to | #21622 |
Jeff Barnett <jbb@notatt.com> writes:
> On 7/13/2020 1:58 AM, Keith Thompson wrote:
>> Jeff Barnett <jbb@notatt.com> writes:
>>> On 7/12/2020 3:04 PM, Keith Thompson wrote:
>>>> It's occurred to me that this raises a (perhaps) interesting question
>>>> -- or maybe I'm missing something obvious.
>>>>
>>>> In a system that includes Euclid's first four postulate but not the
>>>> fifth (the parallel postulate), neither the parallel postulate nor
>>>> its negation can be proven. We can construct a consistent system
>>>> with the parallel postulate as an axiom. We can *also* construct
>>>> a consistent system with the negation of the parallel postulate as
>>>> an axiom.
>>>>
>>>> Robinson Arithmetic cannot prove or disprove commutativity
>>>> of addition. We can construct a consistent system based on
>>>> Robinson Arithmetic in which addition is provably commutative.
>>>> Can we construct a consistent system based on Robinson Arithmetic
>>>> in which addition is provably *not* commutative?
>>>
>>> It's been awhile but my memory might be working. Let's first look at
>>> what a "fixed" use of commutativity might look like: 1+2 = 2+1
>>> expressed as S0 + SS0 = SS0 + S0, where "s" is the successor function
>>> and "0" is a constant given by the axioms. This can be proven in
>>> Robinson (R) and so can ANY other fixed example. What you can not
>>> prove is the combined version:
>>> forall x in I forall y in I x+y = y+x
>>> Since you can prove each instance, within R, there will be no model
>>> where commutativity does not hold, i.e, where
>>> there exist x in I there exist y in I x+y ~= y+x
>>> can be demonstrated. If we endow R with an appropriate induction
>>> axiom, we can prove commutativity within R. There is no extension to R
>>> that can make commutativity false and the extension consistent. There
>>> are many properties of number like systems where all individual
>>> examples can be proven but the combined property cannot.
>>>
>>> The above example is very instructive: a formal system besides syntax
>>> rules must have proof rules as we all keep saying. Here we can extend
>>> R to be commutative in at least two ways: 1) make it an axiom of R or
>>> 2) adopt an appropriate induction rule a proof rule.
>>
>> (Apparently Robinson Arithmetic is usually called Q, not R.)
>>
>> So Q can't prove that addition is commutative, but Q+induction
>> can, yes?
>>
>> I wonder if a system in which induction specifically *doesn't* work,
>> such as Q+¬induction, so that a statement that would be true in
>> Q+induction might be provably false, could be made to work.
>>
>> But I digress.
>>
>>> Now let's make an observation: R is incomplete because it can't prove
>>> something that is true in it. Of course you can't prove its false
>>> either. In this example, you can prove commutativity is true by using
>>> the power of the meta logic which will have some sort of induction
>>> equivalent available.
>>>
>>> Let's look at another case floating through these discussions or
>>> corrective lectures to a poor student. Consider Euclid with the
>>> standard four postulates P1, P2, P3. P4 plus a bunch of definitions
>>> and the axioms of measure - sort of the real numbers. We know that P5,
>>> the parallel postulate, can be added and so could its negation. In
>>> other words we know that there can be zero, one, or more lines through
>>> a point (not on another line) that do not intersect that other
>>> line. In this case we don't bother to say that P1, P2, P3, P4 are not
>>> complete; rather we make the much more powerful statement that P5 is
>>> INDEPENDENT of basic Euclid. This is in the same way that the axiom of
>>> choice is independent of the rest of basic set theory.
>>
>> That makes sense. So the relationship between Q and the
>> commutativity of addition is fundamentally different from the
>> relationship between Euclid-P5 (Euclid without the parallel
>> postulate) and P5. The former is true but unprovable in Q,
>> but P5 can consistently be made either true or false in a system
>> built on top of Euclid-P5. (But if we could create a system like
>> Q+¬induction, they might be more similar. It seems difficult to
>> think about such as system.)
>
> Recall, we can't prove commutativity in R|Q period. However, our meta
> logic can do the proof because it provides additional
> mechanisms. Let's look at a Peano-like induction rule where I is the
> integers and S is the successor function: If Z subset-of I, 0 in Z,
> and whenever x in Z then Sx in Z; then Z = I. How would an
> anti-induction rule be expressed? What would it mean? And would it
> lead to inconsistency? In other words would it falsify even one
> theorem provable in R|Q by the remaining axioms?
The answer to most of that is "I don't know", but I can sketch one
possible partial answer.
https://en.wikipedia.org/wiki/Robinson_arithmetic#Axioms
Consider a set consisting of blue natural numbers (integers >= 0,
colored blue) and red integers (all integers, negative, 0, and positive,
colored red). The successor of any blue number is a blue number, and
the successor of any red number is a red number. (red 0 is zero in the
model; blue 0 is not. Blue 0 is the successor of blue -1.)
This satisfies at least the first 3 of the 7 axioms described at the
link above:
1. Sx ≠ 0
2. (Sx = Sy) → x = y
3. y=0 ∨ ∃x (Sx = y)
It breaks induction, since induction would indicate that all numbers
are red.
I suspect that it breaks one or both of the 5th and 7th axioms, the ones
that define addition and multiplication:
4. x + 0 = x
5. x + Sy = S(x + y)
6. x·0 = 0
7. x·Sy = (x·y) + x
You'd have to assign meanings to blue-red, red-blue, and red-red
addition and multiplication, and I haven't taken the time to try to do
so consistently.
But if that doesn't work, what I'm wondering is whether there's
*some* formulation that's consistent with Robinson Arithmetic in
which induction doesn't work. And if there is, that *might* lead
to a system consistent with Q in which addition is not commutative.
Is induction the only way to prove that Q addition is commutative?
--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Philips Healthcare
void Void(void) { Void(); } /* The recursive call of the void */
[toc] | [prev] | [next] | [standalone]
| From | Jeff Barnett <jbb@notatt.com> |
|---|---|
| Date | 2020-07-13 15:32 -0600 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <reijsu$f6o$1@dont-email.me> |
| In reply to | #21626 |
On 7/13/2020 3:12 PM, Keith Thompson wrote: > Jeff Barnett <jbb@notatt.com> writes: >> On 7/13/2020 1:58 AM, Keith Thompson wrote: >>> Jeff Barnett <jbb@notatt.com> writes: >>>> On 7/12/2020 3:04 PM, Keith Thompson wrote: >>>>> It's occurred to me that this raises a (perhaps) interesting question >>>>> -- or maybe I'm missing something obvious. >>>>> >>>>> In a system that includes Euclid's first four postulate but not the >>>>> fifth (the parallel postulate), neither the parallel postulate nor >>>>> its negation can be proven. We can construct a consistent system >>>>> with the parallel postulate as an axiom. We can *also* construct >>>>> a consistent system with the negation of the parallel postulate as >>>>> an axiom. >>>>> >>>>> Robinson Arithmetic cannot prove or disprove commutativity >>>>> of addition. We can construct a consistent system based on >>>>> Robinson Arithmetic in which addition is provably commutative. >>>>> Can we construct a consistent system based on Robinson Arithmetic >>>>> in which addition is provably *not* commutative? >>>> >>>> It's been awhile but my memory might be working. Let's first look at >>>> what a "fixed" use of commutativity might look like: 1+2 = 2+1 >>>> expressed as S0 + SS0 = SS0 + S0, where "s" is the successor function >>>> and "0" is a constant given by the axioms. This can be proven in >>>> Robinson (R) and so can ANY other fixed example. What you can not >>>> prove is the combined version: >>>> forall x in I forall y in I x+y = y+x >>>> Since you can prove each instance, within R, there will be no model >>>> where commutativity does not hold, i.e, where >>>> there exist x in I there exist y in I x+y ~= y+x >>>> can be demonstrated. If we endow R with an appropriate induction >>>> axiom, we can prove commutativity within R. There is no extension to R >>>> that can make commutativity false and the extension consistent. There >>>> are many properties of number like systems where all individual >>>> examples can be proven but the combined property cannot. >>>> >>>> The above example is very instructive: a formal system besides syntax >>>> rules must have proof rules as we all keep saying. Here we can extend >>>> R to be commutative in at least two ways: 1) make it an axiom of R or >>>> 2) adopt an appropriate induction rule a proof rule. >>> >>> (Apparently Robinson Arithmetic is usually called Q, not R.) >>> >>> So Q can't prove that addition is commutative, but Q+induction >>> can, yes? >>> >>> I wonder if a system in which induction specifically *doesn't* work, >>> such as Q+¬induction, so that a statement that would be true in >>> Q+induction might be provably false, could be made to work. >>> >>> But I digress. >>> >>>> Now let's make an observation: R is incomplete because it can't prove >>>> something that is true in it. Of course you can't prove its false >>>> either. In this example, you can prove commutativity is true by using >>>> the power of the meta logic which will have some sort of induction >>>> equivalent available. >>>> >>>> Let's look at another case floating through these discussions or >>>> corrective lectures to a poor student. Consider Euclid with the >>>> standard four postulates P1, P2, P3. P4 plus a bunch of definitions >>>> and the axioms of measure - sort of the real numbers. We know that P5, >>>> the parallel postulate, can be added and so could its negation. In >>>> other words we know that there can be zero, one, or more lines through >>>> a point (not on another line) that do not intersect that other >>>> line. In this case we don't bother to say that P1, P2, P3, P4 are not >>>> complete; rather we make the much more powerful statement that P5 is >>>> INDEPENDENT of basic Euclid. This is in the same way that the axiom of >>>> choice is independent of the rest of basic set theory. >>> >>> That makes sense. So the relationship between Q and the >>> commutativity of addition is fundamentally different from the >>> relationship between Euclid-P5 (Euclid without the parallel >>> postulate) and P5. The former is true but unprovable in Q, >>> but P5 can consistently be made either true or false in a system >>> built on top of Euclid-P5. (But if we could create a system like >>> Q+¬induction, they might be more similar. It seems difficult to >>> think about such as system.) >> >> Recall, we can't prove commutativity in R|Q period. However, our meta >> logic can do the proof because it provides additional >> mechanisms. Let's look at a Peano-like induction rule where I is the >> integers and S is the successor function: If Z subset-of I, 0 in Z, >> and whenever x in Z then Sx in Z; then Z = I. How would an >> anti-induction rule be expressed? What would it mean? And would it >> lead to inconsistency? In other words would it falsify even one >> theorem provable in R|Q by the remaining axioms? > > The answer to most of that is "I don't know", but I can sketch one > possible partial answer. > > https://en.wikipedia.org/wiki/Robinson_arithmetic#Axioms > > Consider a set consisting of blue natural numbers (integers >= 0, > colored blue) and red integers (all integers, negative, 0, and positive, > colored red). The successor of any blue number is a blue number, and > the successor of any red number is a red number. (red 0 is zero in the > model; blue 0 is not. Blue 0 is the successor of blue -1.) If I read this correctly: 1) All non-negative integers are blue. 2) All integers less than, equal to, or greater than zero are read => all integers are read. 3) Therefore, every blue integer is also red. I don't believe that is what you mean so I ask for a clarification before continuing to what follows. I understand that I might be misreading the above paragraph but I keep getting what I just said in my paraphrase. > This satisfies at least the first 3 of the 7 axioms described at the > link above: > 1. Sx ≠ 0 > 2. (Sx = Sy) → x = y > 3. y=0 ∨ ∃x (Sx = y) > > It breaks induction, since induction would indicate that all numbers > are red. > > I suspect that it breaks one or both of the 5th and 7th axioms, the ones > that define addition and multiplication: > 4. x + 0 = x > 5. x + Sy = S(x + y) > 6. x·0 = 0 > 7. x·Sy = (x·y) + x > > You'd have to assign meanings to blue-red, red-blue, and red-red > addition and multiplication, and I haven't taken the time to try to do > so consistently. > > But if that doesn't work, what I'm wondering is whether there's > *some* formulation that's consistent with Robinson Arithmetic in > which induction doesn't work. And if there is, that *might* lead > to a system consistent with Q in which addition is not commutative. > > Is induction the only way to prove that Q addition is commutative? -- Jeff Barnett
[toc] | [prev] | [next] | [standalone]
| From | Keith Thompson <Keith.S.Thompson+u@gmail.com> |
|---|---|
| Date | 2020-07-13 15:06 -0700 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <87a7033w9l.fsf@nosuchdomain.example.com> |
| In reply to | #21627 |
Jeff Barnett <jbb@notatt.com> writes:
> On 7/13/2020 3:12 PM, Keith Thompson wrote:
>> Jeff Barnett <jbb@notatt.com> writes:
>>> On 7/13/2020 1:58 AM, Keith Thompson wrote:
>>>> Jeff Barnett <jbb@notatt.com> writes:
>>>>> On 7/12/2020 3:04 PM, Keith Thompson wrote:
>>>>>> It's occurred to me that this raises a (perhaps) interesting question
>>>>>> -- or maybe I'm missing something obvious.
>>>>>>
>>>>>> In a system that includes Euclid's first four postulate but not the
>>>>>> fifth (the parallel postulate), neither the parallel postulate nor
>>>>>> its negation can be proven. We can construct a consistent system
>>>>>> with the parallel postulate as an axiom. We can *also* construct
>>>>>> a consistent system with the negation of the parallel postulate as
>>>>>> an axiom.
>>>>>>
>>>>>> Robinson Arithmetic cannot prove or disprove commutativity
>>>>>> of addition. We can construct a consistent system based on
>>>>>> Robinson Arithmetic in which addition is provably commutative.
>>>>>> Can we construct a consistent system based on Robinson Arithmetic
>>>>>> in which addition is provably *not* commutative?
>>>>>
>>>>> It's been awhile but my memory might be working. Let's first look at
>>>>> what a "fixed" use of commutativity might look like: 1+2 = 2+1
>>>>> expressed as S0 + SS0 = SS0 + S0, where "s" is the successor function
>>>>> and "0" is a constant given by the axioms. This can be proven in
>>>>> Robinson (R) and so can ANY other fixed example. What you can not
>>>>> prove is the combined version:
>>>>> forall x in I forall y in I x+y = y+x
>>>>> Since you can prove each instance, within R, there will be no model
>>>>> where commutativity does not hold, i.e, where
>>>>> there exist x in I there exist y in I x+y ~= y+x
>>>>> can be demonstrated. If we endow R with an appropriate induction
>>>>> axiom, we can prove commutativity within R. There is no extension to R
>>>>> that can make commutativity false and the extension consistent. There
>>>>> are many properties of number like systems where all individual
>>>>> examples can be proven but the combined property cannot.
>>>>>
>>>>> The above example is very instructive: a formal system besides syntax
>>>>> rules must have proof rules as we all keep saying. Here we can extend
>>>>> R to be commutative in at least two ways: 1) make it an axiom of R or
>>>>> 2) adopt an appropriate induction rule a proof rule.
>>>>
>>>> (Apparently Robinson Arithmetic is usually called Q, not R.)
>>>>
>>>> So Q can't prove that addition is commutative, but Q+induction
>>>> can, yes?
>>>>
>>>> I wonder if a system in which induction specifically *doesn't* work,
>>>> such as Q+¬induction, so that a statement that would be true in
>>>> Q+induction might be provably false, could be made to work.
>>>>
>>>> But I digress.
>>>>
>>>>> Now let's make an observation: R is incomplete because it can't prove
>>>>> something that is true in it. Of course you can't prove its false
>>>>> either. In this example, you can prove commutativity is true by using
>>>>> the power of the meta logic which will have some sort of induction
>>>>> equivalent available.
>>>>>
>>>>> Let's look at another case floating through these discussions or
>>>>> corrective lectures to a poor student. Consider Euclid with the
>>>>> standard four postulates P1, P2, P3. P4 plus a bunch of definitions
>>>>> and the axioms of measure - sort of the real numbers. We know that P5,
>>>>> the parallel postulate, can be added and so could its negation. In
>>>>> other words we know that there can be zero, one, or more lines through
>>>>> a point (not on another line) that do not intersect that other
>>>>> line. In this case we don't bother to say that P1, P2, P3, P4 are not
>>>>> complete; rather we make the much more powerful statement that P5 is
>>>>> INDEPENDENT of basic Euclid. This is in the same way that the axiom of
>>>>> choice is independent of the rest of basic set theory.
>>>>
>>>> That makes sense. So the relationship between Q and the
>>>> commutativity of addition is fundamentally different from the
>>>> relationship between Euclid-P5 (Euclid without the parallel
>>>> postulate) and P5. The former is true but unprovable in Q,
>>>> but P5 can consistently be made either true or false in a system
>>>> built on top of Euclid-P5. (But if we could create a system like
>>>> Q+¬induction, they might be more similar. It seems difficult to
>>>> think about such as system.)
>>>
>>> Recall, we can't prove commutativity in R|Q period. However, our meta
>>> logic can do the proof because it provides additional
>>> mechanisms. Let's look at a Peano-like induction rule where I is the
>>> integers and S is the successor function: If Z subset-of I, 0 in Z,
>>> and whenever x in Z then Sx in Z; then Z = I. How would an
>>> anti-induction rule be expressed? What would it mean? And would it
>>> lead to inconsistency? In other words would it falsify even one
>>> theorem provable in R|Q by the remaining axioms?
>>
>> The answer to most of that is "I don't know", but I can sketch one
>> possible partial answer.
>>
>> https://en.wikipedia.org/wiki/Robinson_arithmetic#Axioms
>>
>> Consider a set consisting of blue natural numbers (integers >= 0,
>> colored blue) and red integers (all integers, negative, 0, and positive,
>> colored red). The successor of any blue number is a blue number, and
>> the successor of any red number is a red number. (red 0 is zero in the
>> model; blue 0 is not. Blue 0 is the successor of blue -1.)
>
> If I read this correctly: 1) All non-negative integers are blue. 2)
> All integers less than, equal to, or greater than zero are read => all
> integers are read. 3) Therefore, every blue integer is also red. I
> don't believe that is what you mean so I ask for a clarification
> before continuing to what follows. I understand that I might be
> misreading the above paragraph but I keep getting what I just said in
> my paraphrase.
Not quite.
The set consists of blue non-negative integers and red integers. Red
and blue integers with the same value are distinct. So these are all
distinct elements of the set:
blue 0
blue 1
blue 2
...
red -2
red -1
red 0
red 1
red 2
Blue 0 (but not red 0) plays the role of "zero" in the axioms.
It's designed so there's a least element (with no predecessor),
every element has a successor, and every element other than zero has
a predecessor, but (unlike the natural numbers) there are elements
you can never reach by repeatedly applying the successor operation
starting at zero.
[...]
--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Philips Healthcare
void Void(void) { Void(); } /* The recursive call of the void */
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| From | Jeff Barnett <jbb@notatt.com> |
|---|---|
| Date | 2020-07-14 00:56 -0600 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <rejkur$7bs$1@dont-email.me> |
| In reply to | #21628 |
On 7/13/2020 4:06 PM, Keith Thompson wrote: > Jeff Barnett <jbb@notatt.com> writes: >> On 7/13/2020 3:12 PM, Keith Thompson wrote: >>> Jeff Barnett <jbb@notatt.com> writes: >>>> On 7/13/2020 1:58 AM, Keith Thompson wrote: >>>>> Jeff Barnett <jbb@notatt.com> writes: >>>>>> On 7/12/2020 3:04 PM, Keith Thompson wrote: >>>>>>> It's occurred to me that this raises a (perhaps) interesting question >>>>>>> -- or maybe I'm missing something obvious. >>>>>>> >>>>>>> In a system that includes Euclid's first four postulate but not the >>>>>>> fifth (the parallel postulate), neither the parallel postulate nor >>>>>>> its negation can be proven. We can construct a consistent system >>>>>>> with the parallel postulate as an axiom. We can *also* construct >>>>>>> a consistent system with the negation of the parallel postulate as >>>>>>> an axiom. >>>>>>> >>>>>>> Robinson Arithmetic cannot prove or disprove commutativity >>>>>>> of addition. We can construct a consistent system based on >>>>>>> Robinson Arithmetic in which addition is provably commutative. >>>>>>> Can we construct a consistent system based on Robinson Arithmetic >>>>>>> in which addition is provably *not* commutative? >>>>>> >>>>>> It's been awhile but my memory might be working. Let's first look at >>>>>> what a "fixed" use of commutativity might look like: 1+2 = 2+1 >>>>>> expressed as S0 + SS0 = SS0 + S0, where "s" is the successor function >>>>>> and "0" is a constant given by the axioms. This can be proven in >>>>>> Robinson (R) and so can ANY other fixed example. What you can not >>>>>> prove is the combined version: >>>>>> forall x in I forall y in I x+y = y+x >>>>>> Since you can prove each instance, within R, there will be no model >>>>>> where commutativity does not hold, i.e, where >>>>>> there exist x in I there exist y in I x+y ~= y+x >>>>>> can be demonstrated. If we endow R with an appropriate induction >>>>>> axiom, we can prove commutativity within R. There is no extension to R >>>>>> that can make commutativity false and the extension consistent. There >>>>>> are many properties of number like systems where all individual >>>>>> examples can be proven but the combined property cannot. >>>>>> >>>>>> The above example is very instructive: a formal system besides syntax >>>>>> rules must have proof rules as we all keep saying. Here we can extend >>>>>> R to be commutative in at least two ways: 1) make it an axiom of R or >>>>>> 2) adopt an appropriate induction rule a proof rule. >>>>> >>>>> (Apparently Robinson Arithmetic is usually called Q, not R.) >>>>> >>>>> So Q can't prove that addition is commutative, but Q+induction >>>>> can, yes? >>>>> >>>>> I wonder if a system in which induction specifically *doesn't* work, >>>>> such as Q+¬induction, so that a statement that would be true in >>>>> Q+induction might be provably false, could be made to work. >>>>> >>>>> But I digress. >>>>> >>>>>> Now let's make an observation: R is incomplete because it can't prove >>>>>> something that is true in it. Of course you can't prove its false >>>>>> either. In this example, you can prove commutativity is true by using >>>>>> the power of the meta logic which will have some sort of induction >>>>>> equivalent available. >>>>>> >>>>>> Let's look at another case floating through these discussions or >>>>>> corrective lectures to a poor student. Consider Euclid with the >>>>>> standard four postulates P1, P2, P3. P4 plus a bunch of definitions >>>>>> and the axioms of measure - sort of the real numbers. We know that P5, >>>>>> the parallel postulate, can be added and so could its negation. In >>>>>> other words we know that there can be zero, one, or more lines through >>>>>> a point (not on another line) that do not intersect that other >>>>>> line. In this case we don't bother to say that P1, P2, P3, P4 are not >>>>>> complete; rather we make the much more powerful statement that P5 is >>>>>> INDEPENDENT of basic Euclid. This is in the same way that the axiom of >>>>>> choice is independent of the rest of basic set theory. >>>>> >>>>> That makes sense. So the relationship between Q and the >>>>> commutativity of addition is fundamentally different from the >>>>> relationship between Euclid-P5 (Euclid without the parallel >>>>> postulate) and P5. The former is true but unprovable in Q, >>>>> but P5 can consistently be made either true or false in a system >>>>> built on top of Euclid-P5. (But if we could create a system like >>>>> Q+¬induction, they might be more similar. It seems difficult to >>>>> think about such as system.) >>>> >>>> Recall, we can't prove commutativity in R|Q period. However, our meta >>>> logic can do the proof because it provides additional >>>> mechanisms. Let's look at a Peano-like induction rule where I is the >>>> integers and S is the successor function: If Z subset-of I, 0 in Z, >>>> and whenever x in Z then Sx in Z; then Z = I. How would an >>>> anti-induction rule be expressed? What would it mean? And would it >>>> lead to inconsistency? In other words would it falsify even one >>>> theorem provable in R|Q by the remaining axioms? >>> >>> The answer to most of that is "I don't know", but I can sketch one >>> possible partial answer. >>> >>> https://en.wikipedia.org/wiki/Robinson_arithmetic#Axioms >>> >>> Consider a set consisting of blue natural numbers (integers >= 0, >>> colored blue) and red integers (all integers, negative, 0, and positive, >>> colored red). The successor of any blue number is a blue number, and >>> the successor of any red number is a red number. (red 0 is zero in the >>> model; blue 0 is not. Blue 0 is the successor of blue -1.) >> >> If I read this correctly: 1) All non-negative integers are blue. 2) >> All integers less than, equal to, or greater than zero are read => all >> integers are read. 3) Therefore, every blue integer is also red. I >> don't believe that is what you mean so I ask for a clarification >> before continuing to what follows. I understand that I might be >> misreading the above paragraph but I keep getting what I just said in >> my paraphrase. > > Not quite. > > The set consists of blue non-negative integers and red integers. Red > and blue integers with the same value are distinct. So these are all > distinct elements of the set: > > blue 0 > blue 1 > blue 2 > ... > red -2 > red -1 > red 0 > red 1 > red 2 > > Blue 0 (but not red 0) plays the role of "zero" in the axioms. > > It's designed so there's a least element (with no predecessor), > every element has a successor, and every element other than zero has > a predecessor, but (unlike the natural numbers) there are elements > you can never reach by repeatedly applying the successor operation > starting at zero. Got it. By the way, there are standard constructions so that you have arbitrary elements with no successor (and also arbitrary elements with no predecessor) and you do or don't have max and min elements. All of this can be packed into a denumerable set or larger. A major accomplishment of Peano was to eliminate all of this by linearizing the elements; another was the recognition that an induction axiom is necessary. -- Jeff Barnett
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2020-07-13 23:26 +0100 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <878sfnnj9s.fsf@bsb.me.uk> |
| In reply to | #21627 |
Jeff Barnett <jbb@notatt.com> writes:
> On 7/13/2020 3:12 PM, Keith Thompson wrote:
>> Jeff Barnett <jbb@notatt.com> writes:
<cut>
>>> Recall, we can't prove commutativity in R|Q period. However, our meta
>>> logic can do the proof because it provides additional
>>> mechanisms. Let's look at a Peano-like induction rule where I is the
>>> integers and S is the successor function: If Z subset-of I, 0 in Z,
>>> and whenever x in Z then Sx in Z; then Z = I. How would an
>>> anti-induction rule be expressed? What would it mean? And would it
>>> lead to inconsistency? In other words would it falsify even one
>>> theorem provable in R|Q by the remaining axioms?
>>
>> The answer to most of that is "I don't know", but I can sketch one
>> possible partial answer.
>>
>> https://en.wikipedia.org/wiki/Robinson_arithmetic#Axioms
>>
>> Consider a set consisting of blue natural numbers (integers >= 0,
>> colored blue) and red integers (all integers, negative, 0, and positive,
>> colored red). The successor of any blue number is a blue number, and
>> the successor of any red number is a red number. (red 0 is zero in the
>> model; blue 0 is not. Blue 0 is the successor of blue -1.)
>
> If I read this correctly: 1) All non-negative integers are blue. 2)
> All integers less than, equal to, or greater than zero are read => all
> integers are read. 3) Therefore, every blue integer is also red. I
> don't believe that is what you mean so I ask for a clarification
> before continuing to what follows. I understand that I might be
> misreading the above paragraph but I keep getting what I just said in
> my paraphrase.
The model is ({blue} x N) U ({red} x Z). I.e. the elements are pairs:
{ (blue, 0), (blue, 1), (blue, 2), ...
... (red, -2), (red, -1), (red, 0), (red, 1), (red, 2), ... }
The successor is defined as
S(blue, n) = (blue, n+1)
S(red, n) = (red, n+1)
Q's zero is (red, 0).
I've not been following the sub-thread so I won't say if this model
shows what is claimed -- indeed I'm not yet sure what is claimed -- but
since I /was/ pretty sure I know what Keith was suggesting, I thought I'd
say something.
--
Ben.
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| From | Keith Thompson <Keith.S.Thompson+u@gmail.com> |
|---|---|
| Date | 2020-07-13 16:10 -0700 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <875zar3tb3.fsf@nosuchdomain.example.com> |
| In reply to | #21629 |
Ben Bacarisse <ben.usenet@bsb.me.uk> writes:
> Jeff Barnett <jbb@notatt.com> writes:
>> On 7/13/2020 3:12 PM, Keith Thompson wrote:
>>> Jeff Barnett <jbb@notatt.com> writes:
> <cut>
>>>> Recall, we can't prove commutativity in R|Q period. However, our meta
>>>> logic can do the proof because it provides additional
>>>> mechanisms. Let's look at a Peano-like induction rule where I is the
>>>> integers and S is the successor function: If Z subset-of I, 0 in Z,
>>>> and whenever x in Z then Sx in Z; then Z = I. How would an
>>>> anti-induction rule be expressed? What would it mean? And would it
>>>> lead to inconsistency? In other words would it falsify even one
>>>> theorem provable in R|Q by the remaining axioms?
>>>
>>> The answer to most of that is "I don't know", but I can sketch one
>>> possible partial answer.
>>>
>>> https://en.wikipedia.org/wiki/Robinson_arithmetic#Axioms
>>>
>>> Consider a set consisting of blue natural numbers (integers >= 0,
>>> colored blue) and red integers (all integers, negative, 0, and positive,
>>> colored red). The successor of any blue number is a blue number, and
>>> the successor of any red number is a red number. (red 0 is zero in the
>>> model; blue 0 is not. Blue 0 is the successor of blue -1.)
CORRECTION: I should have written:
... (blue 0 is zero in the
model; red 0 is not. Red 0 is the successor of red -1.)
>> If I read this correctly: 1) All non-negative integers are blue. 2)
>> All integers less than, equal to, or greater than zero are read => all
>> integers are read. 3) Therefore, every blue integer is also red. I
>> don't believe that is what you mean so I ask for a clarification
>> before continuing to what follows. I understand that I might be
>> misreading the above paragraph but I keep getting what I just said in
>> my paraphrase.
>
> The model is ({blue} x N) U ({red} x Z). I.e. the elements are pairs:
>
> { (blue, 0), (blue, 1), (blue, 2), ...
> ... (red, -2), (red, -1), (red, 0), (red, 1), (red, 2), ... }
>
> The successor is defined as
>
> S(blue, n) = (blue, n+1)
> S(red, n) = (red, n+1)
Yes.
> Q's zero is (red, 0).
No, Q's zero is (blue, 0) (which has no predecessor). I misstated
that above.
> I've not been following the sub-thread so I won't say if this model
> shows what is claimed -- indeed I'm not yet sure what is claimed -- but
> since I /was/ pretty sure I know what Keith was suggesting, I thought I'd
> say something.
--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Philips Healthcare
void Void(void) { Void(); } /* The recursive call of the void */
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| From | olcott <NoOne@NoWhere.com> |
|---|---|
| Date | 2020-07-13 09:57 -0500 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <yMqdnfGlOqX365HCnZ2dnUU7-QvNnZ2d@giganews.com> |
| In reply to | #21611 |
On 7/13/2020 12:48 AM, Jeff Barnett wrote: > On 7/12/2020 3:04 PM, Keith Thompson wrote: > >> It's occurred to me that this raises a (perhaps) interesting question >> -- or maybe I'm missing something obvious. >> >> In a system that includes Euclid's first four postulate but not the >> fifth (the parallel postulate), neither the parallel postulate nor >> its negation can be proven. We can construct a consistent system >> with the parallel postulate as an axiom. We can *also* construct >> a consistent system with the negation of the parallel postulate as >> an axiom. >> >> Robinson Arithmetic cannot prove or disprove commutativity >> of addition. We can construct a consistent system based on >> Robinson Arithmetic in which addition is provably commutative. >> Can we construct a consistent system based on Robinson Arithmetic >> in which addition is provably *not* commutative? > > It's been awhile but my memory might be working. Let's first look at > what a "fixed" use of commutativity might look like: 1+2 = 2+1 expressed > as S0 + SS0 = SS0 + S0, where "s" is the successor function and "0" is a > constant given by the axioms. This can be proven in Robinson (R) and so > can ANY other fixed example. What you can not prove is the combined > version: > forall x in I forall y in I x+y = y+x > Since you can prove each instance, within R, there will be no model > where commutativity does not hold, i.e, where > there exist x in I there exist y in I x+y ~= y+x > can be demonstrated. If we endow R with an appropriate induction axiom, > we can prove commutativity within R. There is no extension to R that can > make commutativity false and the extension consistent. There are many > properties of number like systems where all individual examples can be > proven but the combined property cannot. > > The above example is very instructive: a formal system besides syntax > rules must have proof rules as we all keep saying. Here we can extend R > to be commutative in at least two ways: 1) make it an axiom of R or 2) > adopt an appropriate induction rule a proof rule. > > Now let's make an observation: R is incomplete because it can't prove > something that is true in it. For the same reason that "cats are animals" is true in biology and not true in mathematics there cannot be any expression of the language of any formal system that is true in this formal system yet lacks a mapping in this formal system from this expression to a Boolean value. > Of course you can't prove its false > either. In this example, you can prove commutativity is true by using > the power of the meta logic which will have some sort of induction > equivalent available. > > Let's look at another case floating through these discussions or > corrective lectures to a poor student. Consider Euclid with the standard > four postulates P1, P2, P3. P4 plus a bunch of definitions and the > axioms of measure - sort of the real numbers. We know that P5, the > parallel postulate, can be added and so could its negation. In other > words we know that there can be zero, one, or more lines through a point > (not on another line) that do not intersect that other line. In this > case we don't bother to say that P1, P2, P3, P4 are not complete; rather > we make the much more powerful statement that P5 is INDEPENDENT of basic > Euclid. This is in the same way that the axiom of choice is independent > of the rest of basic set theory. -- Copyright 2020 Pete Olcott
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| From | Jeff Barnett <jbb@notatt.com> |
|---|---|
| Date | 2020-07-13 13:12 -0600 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <reibmp$rej$1@dont-email.me> |
| In reply to | #21619 |
On 7/13/2020 8:57 AM, olcott wrote: > On 7/13/2020 12:48 AM, Jeff Barnett wrote: >> On 7/12/2020 3:04 PM, Keith Thompson wrote: >> >>> It's occurred to me that this raises a (perhaps) interesting question >>> -- or maybe I'm missing something obvious. >>> >>> In a system that includes Euclid's first four postulate but not the >>> fifth (the parallel postulate), neither the parallel postulate nor >>> its negation can be proven. We can construct a consistent system >>> with the parallel postulate as an axiom. We can *also* construct >>> a consistent system with the negation of the parallel postulate as >>> an axiom. >>> >>> Robinson Arithmetic cannot prove or disprove commutativity >>> of addition. We can construct a consistent system based on >>> Robinson Arithmetic in which addition is provably commutative. >>> Can we construct a consistent system based on Robinson Arithmetic >>> in which addition is provably *not* commutative? >> >> It's been awhile but my memory might be working. Let's first look at >> what a "fixed" use of commutativity might look like: 1+2 = 2+1 >> expressed as S0 + SS0 = SS0 + S0, where "s" is the successor function >> and "0" is a constant given by the axioms. This can be proven in >> Robinson (R) and so can ANY other fixed example. What you can not >> prove is the combined version: >> forall x in I forall y in I x+y = y+x >> Since you can prove each instance, within R, there will be no model >> where commutativity does not hold, i.e, where >> there exist x in I there exist y in I x+y ~= y+x >> can be demonstrated. If we endow R with an appropriate induction >> axiom, we can prove commutativity within R. There is no extension to R >> that can make commutativity false and the extension consistent. There >> are many properties of number like systems where all individual >> examples can be proven but the combined property cannot. >> >> The above example is very instructive: a formal system besides syntax >> rules must have proof rules as we all keep saying. Here we can extend >> R to be commutative in at least two ways: 1) make it an axiom of R or >> 2) adopt an appropriate induction rule a proof rule. >> >> Now let's make an observation: R is incomplete because it can't prove >> something that is true in it. > > For the same reason that "cats are animals" is true in biology and not > true in mathematics there cannot be any expression of the language of > any formal system that is true in this formal system yet lacks a mapping > in this formal system from this expression to a Boolean value. Total gibberish as usual. Note, I didn't gave my reply to one of your messages since it was certain that you would misread it and strike out screaming in the wrong direction. I noticed that there are at least a half dozen new versions of your crap that no one has replied to. You need to read starting with the basics. -- Jeff Barnett
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| From | Keith Thompson <Keith.S.Thompson+u@gmail.com> |
|---|---|
| Date | 2020-07-10 12:53 -0700 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <87v9iv6t9z.fsf@nosuchdomain.example.com> |
| In reply to | #21541 |
olcott <NoOne@NoWhere.com> writes:
> On 7/9/2020 2:14 PM, Keith Thompson wrote:
>> olcott <NoOne@NoWhere.com> writes:
>>> On 7/9/2020 8:40 AM, André G. Isaak wrote:
>> [...]
>>>> I've asked you repeatedly about Robinson's arithmetic, in which x +
>>>> y = y + x is not provable. Neither is ¬(x + y = y + x) provable. The
>>>> law of the excluded middle demands that one of those be true, so
>>>> there exists a true statement in Q which is not provable in Q.
>>>>
>>>> And one can prove that x + y = y + x is true in Q. You just can't
>>>> prove it from within Q.
>>>
>>> That is the exactly same key mistake that you, Tarski and presumably
>>> Gödel made. How do we know that it is true IN Q when it is not
>>> provable IN Q (We look outside of Q). THEN IT IS NOT TRUE IN Q, IT IS
>>> ONLY TRUE OUTSIDE OF Q.
>>
>> If it is not true in Q, then there are values x and y in Q such that
>> x + y = y + x is false in Q.
>>
>> In fact there are no such values. (You could refute that if you could
>> provide such values.)
>>
>> I'm assuming that "x + y = y + x is true in Q" and "x + y = y + x is
>> false in Q" are the only possibilities (law of the excluded middle).
>> Do you accept that assumption?
>
> This is my current best guess of the correct use of the term
> satisfiable if the term satisfiable can even be applied to a single
> theory:
>
> ∃φ (Q ⊢ "x + y = y + x") would seem to be unsatisfiable in Q.
> ∃φ ¬(Q ⊢ "x + y = y + x") would also seem to be unsatisfiable in Q.
>
> This would seem to indicate that Q is incomplete relative to commutativity.
>
> I am certain that the ideas are correct. I am uncertain if my use of
> the term unsatisfiable corresponds to its conventional use.
>
> I am certain that my use of the term incomplete correctly augments the
> conventional use of the term such that my use is more correct than the
> conventional use.
And this is an example of why trying to have a conversation with you is
so frustrating.
I asked what I thought was a straightforward yes or no question, "Do you
accept that assumption?". Your response did not include the word "yes"
or "no", nor did it attempt to demonstrate that neither "yes" nor "no"
would be a meaningful answer. Instead you wrote several paragraphs
about the meaning of "satisfiable".
By all means, write all you like about the meaning of "satisifiable",
but please don't do so in a context that makes it look like you're
trying to answer my question. Perhaps what you wrote has some relevance
to what I asked, but I don't see it.
You have not answered my question. "Yes" or "No" would be an answer.
--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Philips Healthcare
void Void(void) { Void(); } /* The recursive call of the void */
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| From | olcott <NoOne@NoWhere.com> |
|---|---|
| Date | 2020-07-10 16:25 -0500 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <lYmdnV6N6OlUQZXCnZ2dnUU7-b_NnZ2d@giganews.com> |
| In reply to | #21558 |
On 7/10/2020 2:53 PM, Keith Thompson wrote: > olcott <NoOne@NoWhere.com> writes: >> On 7/9/2020 2:14 PM, Keith Thompson wrote: >>> olcott <NoOne@NoWhere.com> writes: >>>> On 7/9/2020 8:40 AM, André G. Isaak wrote: >>> [...] >>>>> I've asked you repeatedly about Robinson's arithmetic, in which x + >>>>> y = y + x is not provable. Neither is ¬(x + y = y + x) provable. The >>>>> law of the excluded middle demands that one of those be true, so >>>>> there exists a true statement in Q which is not provable in Q. >>>>> >>>>> And one can prove that x + y = y + x is true in Q. You just can't >>>>> prove it from within Q. >>>> >>>> That is the exactly same key mistake that you, Tarski and presumably >>>> Gödel made. How do we know that it is true IN Q when it is not >>>> provable IN Q (We look outside of Q). THEN IT IS NOT TRUE IN Q, IT IS >>>> ONLY TRUE OUTSIDE OF Q. >>> >>> If it is not true in Q, then there are values x and y in Q such that >>> x + y = y + x is false in Q. >>> >>> In fact there are no such values. (You could refute that if you could >>> provide such values.) >>> >>> I'm assuming that "x + y = y + x is true in Q" and "x + y = y + x is >>> false in Q" are the only possibilities (law of the excluded middle). >>> Do you accept that assumption? No I do not accept that assumption. Q does not know about the commutative property of addition so it is neither true nor false in Q. >> >> This is my current best guess of the correct use of the term >> satisfiable if the term satisfiable can even be applied to a single >> theory: >> >> ∃φ (Q ⊢ "x + y = y + x") would seem to be unsatisfiable in Q. >> ∃φ ¬(Q ⊢ "x + y = y + x") would also seem to be unsatisfiable in Q. >> >> This would seem to indicate that Q is incomplete relative to commutativity. >> >> I am certain that the ideas are correct. I am uncertain if my use of >> the term unsatisfiable corresponds to its conventional use. >> >> I am certain that my use of the term incomplete correctly augments the >> conventional use of the term such that my use is more correct than the >> conventional use. > > And this is an example of why trying to have a conversation with you is > so frustrating. > > I asked what I thought was a straightforward yes or no question, I answered with all of the reasoning behind the correct answer. It is like you asked me are their any five million pound giant humans? I answer that there is no animal that weighs more than 330,000 lbs. Then you said I did not answer your question. > "Do you > accept that assumption?". Your response did not include the word "yes" > or "no", nor did it attempt to demonstrate that neither "yes" nor "no" > would be a meaningful answer. Instead you wrote several paragraphs > about the meaning of "satisfiable". > > By all means, write all you like about the meaning of "satisifiable", > but please don't do so in a context that makes it look like you're > trying to answer my question. Perhaps what you wrote has some relevance > to what I asked, but I don't see it. > > You have not answered my question. "Yes" or "No" would be an answer. > -- Copyright 2020 Pete Olcott
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| From | Keith Thompson <Keith.S.Thompson+u@gmail.com> |
|---|---|
| Date | 2020-07-10 15:06 -0700 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <878sfr6n54.fsf@nosuchdomain.example.com> |
| In reply to | #21566 |
olcott <NoOne@NoWhere.com> writes:
> On 7/10/2020 2:53 PM, Keith Thompson wrote:
>> olcott <NoOne@NoWhere.com> writes:
>>> On 7/9/2020 2:14 PM, Keith Thompson wrote:
>>>> olcott <NoOne@NoWhere.com> writes:
>>>>> On 7/9/2020 8:40 AM, André G. Isaak wrote:
>>>> [...]
>>>>>> I've asked you repeatedly about Robinson's arithmetic, in which x +
>>>>>> y = y + x is not provable. Neither is ¬(x + y = y + x) provable. The
>>>>>> law of the excluded middle demands that one of those be true, so
>>>>>> there exists a true statement in Q which is not provable in Q.
>>>>>>
>>>>>> And one can prove that x + y = y + x is true in Q. You just can't
>>>>>> prove it from within Q.
>>>>>
>>>>> That is the exactly same key mistake that you, Tarski and presumably
>>>>> Gödel made. How do we know that it is true IN Q when it is not
>>>>> provable IN Q (We look outside of Q). THEN IT IS NOT TRUE IN Q, IT IS
>>>>> ONLY TRUE OUTSIDE OF Q.
>>>>
>>>> If it is not true in Q, then there are values x and y in Q such that
>>>> x + y = y + x is false in Q.
>>>>
>>>> In fact there are no such values. (You could refute that if you could
>>>> provide such values.)
>>>>
>>>> I'm assuming that "x + y = y + x is true in Q" and "x + y = y + x is
>>>> false in Q" are the only possibilities (law of the excluded middle).
>>>> Do you accept that assumption?
>
> No I do not accept that assumption. Q does not know about the
> commutative property of addition so it is neither true nor false in Q.
>
>>>
>>> This is my current best guess of the correct use of the term
>>> satisfiable if the term satisfiable can even be applied to a single
>>> theory:
>>>
>>> ∃φ (Q ⊢ "x + y = y + x") would seem to be unsatisfiable in Q.
>>> ∃φ ¬(Q ⊢ "x + y = y + x") would also seem to be unsatisfiable in Q.
>>>
>>> This would seem to indicate that Q is incomplete relative to commutativity.
>>>
>>> I am certain that the ideas are correct. I am uncertain if my use of
>>> the term unsatisfiable corresponds to its conventional use.
>>>
>>> I am certain that my use of the term incomplete correctly augments the
>>> conventional use of the term such that my use is more correct than the
>>> conventional use.
>>
>> And this is an example of why trying to have a conversation with you is
>> so frustrating.
>>
>> I asked what I thought was a straightforward yes or no question,
>
> I answered with all of the reasoning behind the correct answer.
> It is like you asked me are their any five million pound giant humans?
> I answer that there is no animal that weighs more than 330,000 lbs.
> Then you said I did not answer your question.
That kind of answer is not useful if the person asking the question does
not accept understand the reasoning behind the answer. In your example,
for that to be an answer I'd have to accept that humans are animals and
that five million pounds is more than 333,000 pounds. Of course I do
accept both of those, but that's not the case with your statements about
mathematical logic, and you should stop assuming that it is.
I won't address (at least not in this post) whether I don't understand
your reasoning because I'm too stupid to understand your brilliance or
because you're wrong.
You seem to be frustrated that the rest of us either don't understand or
don't accept your claims. I get that, and I believe you're sincere.
But if your intent is to communicate, you must at least *accept the
observed fact* that the rest of us either don't understand or don't
accept your claims.
If I ask you a yes or no question (which I've spent time carefully
constructing for the purpose of eliciting information), it would be
helpful if your answer would include the word "yes" or "no", preferably
at the beginning -- or an assertion, or ideally an explanation, that
neither "yes" nor "no" would be meaningful. Whatever else you want to
say after that is fine.
For example:
Q: Are their any five million pound giant humans?
A: No, because there is no animal that weighs more than 330,000 lbs.
Q: OK, but how does your answer follow from your following statement?
What are you assuming?
... and so on.
>> "Do you
>> accept that assumption?". Your response did not include the word "yes"
>> or "no", nor did it attempt to demonstrate that neither "yes" nor "no"
>> would be a meaningful answer. Instead you wrote several paragraphs
>> about the meaning of "satisfiable".
>>
>> By all means, write all you like about the meaning of "satisifiable",
>> but please don't do so in a context that makes it look like you're
>> trying to answer my question. Perhaps what you wrote has some relevance
>> to what I asked, but I don't see it.
>>
>> You have not answered my question. "Yes" or "No" would be an answer.
--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Philips Healthcare
void Void(void) { Void(); } /* The recursive call of the void */
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| From | olcott <NoOne@NoWhere.com> |
|---|---|
| Date | 2020-07-10 17:21 -0500 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <9LWdne-PZcBhdJXCnZ2dnUU7-LPNnZ2d@giganews.com> |
| In reply to | #21568 |
On 7/10/2020 5:06 PM, Keith Thompson wrote: > olcott <NoOne@NoWhere.com> writes: >> On 7/10/2020 2:53 PM, Keith Thompson wrote: >>> olcott <NoOne@NoWhere.com> writes: >>>> On 7/9/2020 2:14 PM, Keith Thompson wrote: >>>>> olcott <NoOne@NoWhere.com> writes: >>>>>> On 7/9/2020 8:40 AM, André G. Isaak wrote: >>>>> [...] >>>>>>> I've asked you repeatedly about Robinson's arithmetic, in which x + >>>>>>> y = y + x is not provable. Neither is ¬(x + y = y + x) provable. The >>>>>>> law of the excluded middle demands that one of those be true, so >>>>>>> there exists a true statement in Q which is not provable in Q. >>>>>>> >>>>>>> And one can prove that x + y = y + x is true in Q. You just can't >>>>>>> prove it from within Q. >>>>>> >>>>>> That is the exactly same key mistake that you, Tarski and presumably >>>>>> Gödel made. How do we know that it is true IN Q when it is not >>>>>> provable IN Q (We look outside of Q). THEN IT IS NOT TRUE IN Q, IT IS >>>>>> ONLY TRUE OUTSIDE OF Q. >>>>> >>>>> If it is not true in Q, then there are values x and y in Q such that >>>>> x + y = y + x is false in Q. >>>>> >>>>> In fact there are no such values. (You could refute that if you could >>>>> provide such values.) >>>>> >>>>> I'm assuming that "x + y = y + x is true in Q" and "x + y = y + x is >>>>> false in Q" are the only possibilities (law of the excluded middle). >>>>> Do you accept that assumption? >> >> No I do not accept that assumption. Q does not know about the >> commutative property of addition so it is neither true nor false in Q. >> >>>> >>>> This is my current best guess of the correct use of the term >>>> satisfiable if the term satisfiable can even be applied to a single >>>> theory: >>>> >>>> ∃φ (Q ⊢ "x + y = y + x") would seem to be unsatisfiable in Q. >>>> ∃φ ¬(Q ⊢ "x + y = y + x") would also seem to be unsatisfiable in Q. >>>> >>>> This would seem to indicate that Q is incomplete relative to commutativity. >>>> >>>> I am certain that the ideas are correct. I am uncertain if my use of >>>> the term unsatisfiable corresponds to its conventional use. >>>> >>>> I am certain that my use of the term incomplete correctly augments the >>>> conventional use of the term such that my use is more correct than the >>>> conventional use. >>> >>> And this is an example of why trying to have a conversation with you is >>> so frustrating. >>> >>> I asked what I thought was a straightforward yes or no question, >> >> I answered with all of the reasoning behind the correct answer. >> It is like you asked me are their any five million pound giant humans? >> I answer that there is no animal that weighs more than 330,000 lbs. >> Then you said I did not answer your question. > > That kind of answer is not useful if the person asking the question does > not accept understand the reasoning behind the answer. OK. I will strive to do better on this with you. > In your example, > for that to be an answer I'd have to accept that humans are animals and > that five million pounds is more than 333,000 pounds. Of course I do > accept both of those, but that's not the case with your statements about > mathematical logic, and you should stop assuming that it is. > > I won't address (at least not in this post) whether I don't understand > your reasoning because I'm too stupid to understand your brilliance or > because you're wrong. I am looking at these things at a much higher level of philosophical abstraction related to the fundemental nature of true itself. > You seem to be frustrated that the rest of us either don't understand or > don't accept your claims. I get that, and I believe you're sincere. > But if your intent is to communicate, you must at least *accept the > observed fact* that the rest of us either don't understand or don't > accept your claims. If I was understood what I said would have to be accepted because it is a tautology. > If I ask you a yes or no question (which I've spent time carefully > constructing for the purpose of eliciting information), it would be > helpful if your answer would include the word "yes" or "no", preferably OK withe you I will do that. With everyone else I say yes and they simply say that they disagree. I provide the reasoning behind what I say so that any disagreement is cut-off before it begins. > at the beginning -- or an assertion, or ideally an explanation, that > neither "yes" nor "no" would be meaningful. Whatever else you want to > say after that is fine. OK that is reasonable. It is almost always the case that whenever I say that an expression of language is neither true nor false everyone's mind short-circuits. They cannot (even after many hundreds of dialogues) begin to understand that syntactically correct expressions can be semantically wrong. So far after at least 12,000 hours at this over twenty-two years not one person ever acknowledged that any WFF could be semantically incorrect. -- Copyright 2020 Pete Olcott
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