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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 7 of 17 — ← Prev page 1 … 5 6 [7] 8 9 … 17 Next page →
| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2020-07-15 20:38 -0600 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <reoeid$jm$1@dont-email.me> |
| In reply to | #21688 |
On 2020-07-15 18:18, olcott wrote:
> On 7/15/2020 12:24 PM, André G. Isaak wrote:
>> On 2020-07-15 09:38, olcott wrote:
>>> On 7/14/2020 10:30 AM, André G. Isaak wrote:
>>>> On 2020-07-14 09:22, olcott wrote:
>>>>> On 7/14/2020 9:57 AM, André G. Isaak wrote:
>>>>>> On 2020-07-14 08:43, olcott wrote:
>>>>>>> On 7/13/2020 9:11 PM, Keith Thompson wrote:
>>>>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>>>> [...]
>>>>>>>>> If a sentence is undecidable in Q then this is merely another
>>>>>>>>> way of
>>>>>>>>> saying that it is neither true nor false in Q.
>>>>>>>>
>>>>>>>> Because you *assume* that all true statements are provable and all
>>>>>>>> provable statements are true.
>>>>>>>
>>>>>>> There is no way in the universe to show that a sentence is true
>>>>>>> besides its provability because its provability is its only
>>>>>>> mapping to Boolean values.
>>>>>>
>>>>>> 'Mapping' means a number of different things in logic and
>>>>>> mathematics. But none of those things have anything to do with
>>>>>> whether something is provable, nor does provability have anything
>>>>>> to do with mapping. You now seem to be equating not only truth but
>>>>>> also mapping with provability.
>>>>>>
>>>>>
>>>>> Do you understand that a formal proof is a special kind of mapping
>>>>> between finite strings?
>>>>
>>>>> It is the mapping from the premises to the consequence through
>>>>> axioms and rule-of-inference.
>>>>
>>>> That isn't how the word 'mapping' is normally used. Even if we were
>>>> to extend the term mapping to include this usage, a proof would
>>>> 'map' things to {theorem}, not to {true}.
>>>>
>>>> André
>>>>
>>>
>>> Yes that is great it seems that we are getting very close to a mutual
>>> understanding, I agree a proof would map things to a {theorem} and
>>> not to {true}.
>>>
>>> We have to add an interpretation to satisfy a formula:
>>>
>>> Satisfiability
>>> A formula is satisfiable if it is possible to find an
>>> interpretation
>>> (model) that makes the formula true.
>>> https://en.wikipedia.org/wiki/Satisfiability
>>>
>>> Interpretation (logic)
>>> An interpretation is an assignment of meaning to the
>>> [non-logical] symbols of a formal language.
>>> https://en.wikipedia.org/wiki/Interpretation_(logic)
>>>
>>> Model theory
>>> A model of a theory is a structure (e.g. an interpretation)
>>> that satisfies the sentences of that theory.
>>> https://en.wikipedia.org/wiki/Model_theory
>>>
>>> Yet there is no way to satisfy a formula that does not require a
>>> chain-of-inference, AKA a formal proof.
>>
>> Of course there is. (A ∧ B) is a well-formed formula of propositional
>> logic. It is most definitely satisfiable. But it cannot be proven
>> because it isn't a theorem.
>>
>
> WRONG !!!
>
> Logical conjunction
> p q p ∧ q
> T T T
> T F F
> F T F
> F F F
You're going to have to supply some actual reasoning here. How exactly
does that truth table make me wrong? It seems to confirm what I said.
You've given the four possible interpretations of (A ∧ B) (though you've
decided to rename them for some reason). One of those interpretations
(the top one) is true, therefore (A ∧ B) is satisfiable.
If (A ∧ B) were a theorem, then (A ∧ B) would be true on *all possible*
interpretations. The above clearly indicates that this is not the case,
since (A ∧ B) is *only* true on the first interpretation given above.
André
--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.
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| From | olcott <NoOne@NoWhere.com> |
|---|---|
| Date | 2020-07-16 16:16 -0500 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <BbWdnQxR6c8EXo3CnZ2dnUU7-bnNnZ2d@giganews.com> |
| In reply to | #21697 |
On 7/15/2020 9:38 PM, André G. Isaak wrote:
> On 2020-07-15 18:18, olcott wrote:
>> On 7/15/2020 12:24 PM, André G. Isaak wrote:
>>> On 2020-07-15 09:38, olcott wrote:
>>>> On 7/14/2020 10:30 AM, André G. Isaak wrote:
>>>>> On 2020-07-14 09:22, olcott wrote:
>>>>>> On 7/14/2020 9:57 AM, André G. Isaak wrote:
>>>>>>> On 2020-07-14 08:43, olcott wrote:
>>>>>>>> On 7/13/2020 9:11 PM, Keith Thompson wrote:
>>>>>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>>>>> [...]
>>>>>>>>>> If a sentence is undecidable in Q then this is merely another
>>>>>>>>>> way of
>>>>>>>>>> saying that it is neither true nor false in Q.
>>>>>>>>>
>>>>>>>>> Because you *assume* that all true statements are provable and all
>>>>>>>>> provable statements are true.
>>>>>>>>
>>>>>>>> There is no way in the universe to show that a sentence is true
>>>>>>>> besides its provability because its provability is its only
>>>>>>>> mapping to Boolean values.
>>>>>>>
>>>>>>> 'Mapping' means a number of different things in logic and
>>>>>>> mathematics. But none of those things have anything to do with
>>>>>>> whether something is provable, nor does provability have anything
>>>>>>> to do with mapping. You now seem to be equating not only truth
>>>>>>> but also mapping with provability.
>>>>>>>
>>>>>>
>>>>>> Do you understand that a formal proof is a special kind of mapping
>>>>>> between finite strings?
>>>>>
>>>>>> It is the mapping from the premises to the consequence through
>>>>>> axioms and rule-of-inference.
>>>>>
>>>>> That isn't how the word 'mapping' is normally used. Even if we were
>>>>> to extend the term mapping to include this usage, a proof would
>>>>> 'map' things to {theorem}, not to {true}.
>>>>>
>>>>> André
>>>>>
>>>>
>>>> Yes that is great it seems that we are getting very close to a
>>>> mutual understanding, I agree a proof would map things to a
>>>> {theorem} and not to {true}.
>>>>
>>>> We have to add an interpretation to satisfy a formula:
>>>>
>>>> Satisfiability
>>>> A formula is satisfiable if it is possible to find an
>>>> interpretation
>>>> (model) that makes the formula true.
>>>> https://en.wikipedia.org/wiki/Satisfiability
>>>>
>>>> Interpretation (logic)
>>>> An interpretation is an assignment of meaning to the
>>>> [non-logical] symbols of a formal language.
>>>> https://en.wikipedia.org/wiki/Interpretation_(logic)
>>>>
>>>> Model theory
>>>> A model of a theory is a structure (e.g. an interpretation)
>>>> that satisfies the sentences of that theory.
>>>> https://en.wikipedia.org/wiki/Model_theory
>>>>
>>>> Yet there is no way to satisfy a formula that does not require a
>>>> chain-of-inference, AKA a formal proof.
>>>
>>> Of course there is. (A ∧ B) is a well-formed formula of propositional
>>> logic. It is most definitely satisfiable. But it cannot be proven
>>> because it isn't a theorem.
>>>
>>
>> WRONG !!!
>>
>> Logical conjunction
>> p q p ∧ q
>> T T T
>> T F F
>> F T F
>> F F F
>
> You're going to have to supply some actual reasoning here. How exactly
> does that truth table make me wrong? It seems to confirm what I said.
>
The above truth table provides the stipulated Boolean values for every
instance of p ∧ q, so it is proven in the same way that an axiom is
proven, it is stipulated to be true.
> You've given the four possible interpretations of (A ∧ B) (though you've
> decided to rename them for some reason).
cut-and-paste from Wikipedia.
> One of those interpretations
> (the top one) is true, therefore (A ∧ B) is satisfiable.
>
> If (A ∧ B) were a theorem, then (A ∧ B) would be true on *all possible*
> interpretations. The above clearly indicates that this is not the case,
> since (A ∧ B) is *only* true on the first interpretation given above.
>
> André
>
If it is only an interpretation (one way of looking at it) then why does
it always work this way all the time?
--
Copyright 2020 Pete Olcott
[toc] | [prev] | [next] | [standalone]
| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2020-07-16 16:01 -0600 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <reqioi$f6r$1@dont-email.me> |
| In reply to | #21713 |
On 2020-07-16 15:16, olcott wrote:
> On 7/15/2020 9:38 PM, André G. Isaak wrote:
>> On 2020-07-15 18:18, olcott wrote:
>>> On 7/15/2020 12:24 PM, André G. Isaak wrote:
>>>> On 2020-07-15 09:38, olcott wrote:
>>>>> On 7/14/2020 10:30 AM, André G. Isaak wrote:
>>>>>> On 2020-07-14 09:22, olcott wrote:
>>>>>>> On 7/14/2020 9:57 AM, André G. Isaak wrote:
>>>>>>>> On 2020-07-14 08:43, olcott wrote:
>>>>>>>>> On 7/13/2020 9:11 PM, Keith Thompson wrote:
>>>>>>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>>>>>> [...]
>>>>>>>>>>> If a sentence is undecidable in Q then this is merely another
>>>>>>>>>>> way of
>>>>>>>>>>> saying that it is neither true nor false in Q.
>>>>>>>>>>
>>>>>>>>>> Because you *assume* that all true statements are provable and
>>>>>>>>>> all
>>>>>>>>>> provable statements are true.
>>>>>>>>>
>>>>>>>>> There is no way in the universe to show that a sentence is true
>>>>>>>>> besides its provability because its provability is its only
>>>>>>>>> mapping to Boolean values.
>>>>>>>>
>>>>>>>> 'Mapping' means a number of different things in logic and
>>>>>>>> mathematics. But none of those things have anything to do with
>>>>>>>> whether something is provable, nor does provability have
>>>>>>>> anything to do with mapping. You now seem to be equating not
>>>>>>>> only truth but also mapping with provability.
>>>>>>>>
>>>>>>>
>>>>>>> Do you understand that a formal proof is a special kind of
>>>>>>> mapping between finite strings?
>>>>>>
>>>>>>> It is the mapping from the premises to the consequence through
>>>>>>> axioms and rule-of-inference.
>>>>>>
>>>>>> That isn't how the word 'mapping' is normally used. Even if we
>>>>>> were to extend the term mapping to include this usage, a proof
>>>>>> would 'map' things to {theorem}, not to {true}.
>>>>>>
>>>>>> André
>>>>>>
>>>>>
>>>>> Yes that is great it seems that we are getting very close to a
>>>>> mutual understanding, I agree a proof would map things to a
>>>>> {theorem} and not to {true}.
>>>>>
>>>>> We have to add an interpretation to satisfy a formula:
>>>>>
>>>>> Satisfiability
>>>>> A formula is satisfiable if it is possible to find an
>>>>> interpretation
>>>>> (model) that makes the formula true.
>>>>> https://en.wikipedia.org/wiki/Satisfiability
>>>>>
>>>>> Interpretation (logic)
>>>>> An interpretation is an assignment of meaning to the
>>>>> [non-logical] symbols of a formal language.
>>>>> https://en.wikipedia.org/wiki/Interpretation_(logic)
>>>>>
>>>>> Model theory
>>>>> A model of a theory is a structure (e.g. an interpretation)
>>>>> that satisfies the sentences of that theory.
>>>>> https://en.wikipedia.org/wiki/Model_theory
>>>>>
>>>>> Yet there is no way to satisfy a formula that does not require a
>>>>> chain-of-inference, AKA a formal proof.
>>>>
>>>> Of course there is. (A ∧ B) is a well-formed formula of
>>>> propositional logic. It is most definitely satisfiable. But it
>>>> cannot be proven because it isn't a theorem.
>>>>
>>>
>>> WRONG !!!
>>>
>>> Logical conjunction
>>> p q p ∧ q
>>> T T T
>>> T F F
>>> F T F
>>> F F F
>>
>> You're going to have to supply some actual reasoning here. How exactly
>> does that truth table make me wrong? It seems to confirm what I said.
>>
>
> The above truth table provides the stipulated Boolean values for every
> instance of p ∧ q, so it is proven in the same way that an axiom is
> proven, it is stipulated to be true.
I have absolutely no idea what it is that you are trying to claim.
According to the above (A ∧ B) is false under three of the four possible
interpretations. How can the above possibly constitute a proof?
>> You've given the four possible interpretations of (A ∧ B) (though
>> you've decided to rename them for some reason).
>
> cut-and-paste from Wikipedia.
>
>> One of those interpretations (the top one) is true, therefore (A ∧ B)
>> is satisfiable.
>>
>> If (A ∧ B) were a theorem, then (A ∧ B) would be true on *all
>> possible* interpretations. The above clearly indicates that this is
>> not the case, since (A ∧ B) is *only* true on the first interpretation
>> given above.
>>
>> André
>>
>
> If it is only an interpretation (one way of looking at it) then why does
> it always work this way all the time?
Why don't you learn what 'interpretation' really means?
André
--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.
[toc] | [prev] | [next] | [standalone]
| From | olcott <NoOne@NoWhere.com> |
|---|---|
| Date | 2020-07-16 19:11 -0500 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <equdnWqLB4IhcY3CnZ2dnUU7-d_NnZ2d@giganews.com> |
| In reply to | #21717 |
On 7/16/2020 5:01 PM, André G. Isaak wrote:
> On 2020-07-16 15:16, olcott wrote:
>> On 7/15/2020 9:38 PM, André G. Isaak wrote:
>>> On 2020-07-15 18:18, olcott wrote:
>>>> On 7/15/2020 12:24 PM, André G. Isaak wrote:
>>>>> On 2020-07-15 09:38, olcott wrote:
>>>>>> On 7/14/2020 10:30 AM, André G. Isaak wrote:
>>>>>>> On 2020-07-14 09:22, olcott wrote:
>>>>>>>> On 7/14/2020 9:57 AM, André G. Isaak wrote:
>>>>>>>>> On 2020-07-14 08:43, olcott wrote:
>>>>>>>>>> On 7/13/2020 9:11 PM, Keith Thompson wrote:
>>>>>>>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>>>>>>> [...]
>>>>>>>>>>>> If a sentence is undecidable in Q then this is merely
>>>>>>>>>>>> another way of
>>>>>>>>>>>> saying that it is neither true nor false in Q.
>>>>>>>>>>>
>>>>>>>>>>> Because you *assume* that all true statements are provable
>>>>>>>>>>> and all
>>>>>>>>>>> provable statements are true.
>>>>>>>>>>
>>>>>>>>>> There is no way in the universe to show that a sentence is
>>>>>>>>>> true besides its provability because its provability is its
>>>>>>>>>> only mapping to Boolean values.
>>>>>>>>>
>>>>>>>>> 'Mapping' means a number of different things in logic and
>>>>>>>>> mathematics. But none of those things have anything to do with
>>>>>>>>> whether something is provable, nor does provability have
>>>>>>>>> anything to do with mapping. You now seem to be equating not
>>>>>>>>> only truth but also mapping with provability.
>>>>>>>>>
>>>>>>>>
>>>>>>>> Do you understand that a formal proof is a special kind of
>>>>>>>> mapping between finite strings?
>>>>>>>
>>>>>>>> It is the mapping from the premises to the consequence through
>>>>>>>> axioms and rule-of-inference.
>>>>>>>
>>>>>>> That isn't how the word 'mapping' is normally used. Even if we
>>>>>>> were to extend the term mapping to include this usage, a proof
>>>>>>> would 'map' things to {theorem}, not to {true}.
>>>>>>>
>>>>>>> André
>>>>>>>
>>>>>>
>>>>>> Yes that is great it seems that we are getting very close to a
>>>>>> mutual understanding, I agree a proof would map things to a
>>>>>> {theorem} and not to {true}.
>>>>>>
>>>>>> We have to add an interpretation to satisfy a formula:
>>>>>>
>>>>>> Satisfiability
>>>>>> A formula is satisfiable if it is possible to find an
>>>>>> interpretation
>>>>>> (model) that makes the formula true.
>>>>>> https://en.wikipedia.org/wiki/Satisfiability
>>>>>>
>>>>>> Interpretation (logic)
>>>>>> An interpretation is an assignment of meaning to the
>>>>>> [non-logical] symbols of a formal language.
>>>>>> https://en.wikipedia.org/wiki/Interpretation_(logic)
>>>>>>
>>>>>> Model theory
>>>>>> A model of a theory is a structure (e.g. an interpretation)
>>>>>> that satisfies the sentences of that theory.
>>>>>> https://en.wikipedia.org/wiki/Model_theory
>>>>>>
>>>>>> Yet there is no way to satisfy a formula that does not require a
>>>>>> chain-of-inference, AKA a formal proof.
>>>>>
>>>>> Of course there is. (A ∧ B) is a well-formed formula of
>>>>> propositional logic. It is most definitely satisfiable. But it
>>>>> cannot be proven because it isn't a theorem.
>>>>>
>>>>
>>>> WRONG !!!
>>>>
>>>> Logical conjunction
>>>> p q p ∧ q
>>>> T T T
>>>> T F F
>>>> F T F
>>>> F F F
>>>
>>> You're going to have to supply some actual reasoning here. How
>>> exactly does that truth table make me wrong? It seems to confirm what
>>> I said.
>>>
>>
>> The above truth table provides the stipulated Boolean values for every
>> instance of p ∧ q, so it is proven in the same way that an axiom is
>> proven, it is stipulated to be true.
>
> I have absolutely no idea what it is that you are trying to claim.
> According to the above (A ∧ B) is false under three of the four possible
> interpretations. How can the above possibly constitute a proof?
>
It is much more than a proof it is a stipulated truth.
It provides the algorithm of the meaning of: "∧".
>>> You've given the four possible interpretations of (A ∧ B) (though
>>> you've decided to rename them for some reason).
>>
>> cut-and-paste from Wikipedia.
>>
>>> One of those interpretations (the top one) is true, therefore (A ∧ B)
>>> is satisfiable.
>>>
>>> If (A ∧ B) were a theorem, then (A ∧ B) would be true on *all
>>> possible* interpretations. The above clearly indicates that this is
>>> not the case, since (A ∧ B) is *only* true on the first
>>> interpretation given above.
>>>
>>> André
>>>
>>
>> If it is only an interpretation (one way of looking at it) then why
>> does it always work this way all the time?
>
> Why don't you learn what 'interpretation' really means?
>
> André
>
2.2 First-Order Languages and Their Interpretations:
Satisfiability and Truth: Models
Let L be a first-order language. An interpretation M of L consists of
the following ingredients.
a. A nonempty set D, called the domain of the interpretation.
b. For each predicate letter Anj of L, an assignment of an n-place
relation (Anj)M in D.
c. For each function letter fnj of L, an assignment of an n-place
operation (fnj)M in D (that is, a function from Dn into D).
d. For each individual constant ai of L, an assignment of some fixed
element (ai)M of D.
Given such an interpretation, variables are thought of as ranging over
the set D, and ¬, ⇒ and quantifiers are given their usual meaning.
Remember that an n-place relation in D can be thought of as a subset of
Dn, the set of all n-tuples of elements of D. For example, if D is the
set of human beings, then the relation “father of” can be identified
with the set of all ordered pairs 〈x, y〉 such that x is the father of y.
For a given interpretation of a language L, a wf of L without free
variables (called a closed wf or a sentence) represents a proposition
that is true or false, whereas a wf with free variables may be satisfied
(i.e., true) for some values in the domain and not satisfied (i.e.,
false) for the others. (Mendelson 2015: 57-58)
Mendelson, Elliott 2015. Introduction To Mathematical Logic. Boca Raton:
CRC Press Taylor & Francis Group.
In his example we would not need any separate interpretation where all
predicates have the same name of "A" and are only distinguished by their
two subscripts. We can merely specify ALL of we mean right in the
formula leaving nothing left to be separately interpreted.
∀y ∈ ℤ+ (x ≤ y)
∃x ∈ ℤ+ ∀y ∈ ℤ+ (x ≤ y)
--
Copyright 2020 Pete Olcott
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| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2020-07-16 18:40 -0600 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <reqs2c$qlu$1@dont-email.me> |
| In reply to | #21719 |
On 2020-07-16 18:11, olcott wrote:
> On 7/16/2020 5:01 PM, André G. Isaak wrote:
>> On 2020-07-16 15:16, olcott wrote:
>>> On 7/15/2020 9:38 PM, André G. Isaak wrote:
>>>> On 2020-07-15 18:18, olcott wrote:
>>>>> On 7/15/2020 12:24 PM, André G. Isaak wrote:
>>>>>> On 2020-07-15 09:38, olcott wrote:
>>>>>>> On 7/14/2020 10:30 AM, André G. Isaak wrote:
>>>>>>>> On 2020-07-14 09:22, olcott wrote:
>>>>>>>>> On 7/14/2020 9:57 AM, André G. Isaak wrote:
>>>>>>>>>> On 2020-07-14 08:43, olcott wrote:
>>>>>>>>>>> On 7/13/2020 9:11 PM, Keith Thompson wrote:
>>>>>>>>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>>>>>>>> [...]
>>>>>>>>>>>>> If a sentence is undecidable in Q then this is merely
>>>>>>>>>>>>> another way of
>>>>>>>>>>>>> saying that it is neither true nor false in Q.
>>>>>>>>>>>>
>>>>>>>>>>>> Because you *assume* that all true statements are provable
>>>>>>>>>>>> and all
>>>>>>>>>>>> provable statements are true.
>>>>>>>>>>>
>>>>>>>>>>> There is no way in the universe to show that a sentence is
>>>>>>>>>>> true besides its provability because its provability is its
>>>>>>>>>>> only mapping to Boolean values.
>>>>>>>>>>
>>>>>>>>>> 'Mapping' means a number of different things in logic and
>>>>>>>>>> mathematics. But none of those things have anything to do with
>>>>>>>>>> whether something is provable, nor does provability have
>>>>>>>>>> anything to do with mapping. You now seem to be equating not
>>>>>>>>>> only truth but also mapping with provability.
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> Do you understand that a formal proof is a special kind of
>>>>>>>>> mapping between finite strings?
>>>>>>>>
>>>>>>>>> It is the mapping from the premises to the consequence through
>>>>>>>>> axioms and rule-of-inference.
>>>>>>>>
>>>>>>>> That isn't how the word 'mapping' is normally used. Even if we
>>>>>>>> were to extend the term mapping to include this usage, a proof
>>>>>>>> would 'map' things to {theorem}, not to {true}.
>>>>>>>>
>>>>>>>> André
>>>>>>>>
>>>>>>>
>>>>>>> Yes that is great it seems that we are getting very close to a
>>>>>>> mutual understanding, I agree a proof would map things to a
>>>>>>> {theorem} and not to {true}.
>>>>>>>
>>>>>>> We have to add an interpretation to satisfy a formula:
>>>>>>>
>>>>>>> Satisfiability
>>>>>>> A formula is satisfiable if it is possible to find an
>>>>>>> interpretation
>>>>>>> (model) that makes the formula true.
>>>>>>> https://en.wikipedia.org/wiki/Satisfiability
>>>>>>>
>>>>>>> Interpretation (logic)
>>>>>>> An interpretation is an assignment of meaning to the
>>>>>>> [non-logical] symbols of a formal language.
>>>>>>> https://en.wikipedia.org/wiki/Interpretation_(logic)
>>>>>>>
>>>>>>> Model theory
>>>>>>> A model of a theory is a structure (e.g. an interpretation)
>>>>>>> that satisfies the sentences of that theory.
>>>>>>> https://en.wikipedia.org/wiki/Model_theory
>>>>>>>
>>>>>>> Yet there is no way to satisfy a formula that does not require a
>>>>>>> chain-of-inference, AKA a formal proof.
>>>>>>
>>>>>> Of course there is. (A ∧ B) is a well-formed formula of
>>>>>> propositional logic. It is most definitely satisfiable. But it
>>>>>> cannot be proven because it isn't a theorem.
>>>>>>
>>>>>
>>>>> WRONG !!!
>>>>>
>>>>> Logical conjunction
>>>>> p q p ∧ q
>>>>> T T T
>>>>> T F F
>>>>> F T F
>>>>> F F F
>>>>
>>>> You're going to have to supply some actual reasoning here. How
>>>> exactly does that truth table make me wrong? It seems to confirm
>>>> what I said.
>>>>
>>>
>>> The above truth table provides the stipulated Boolean values for
>>> every instance of p ∧ q, so it is proven in the same way that an
>>> axiom is proven, it is stipulated to be true.
>>
>> I have absolutely no idea what it is that you are trying to claim.
>> According to the above (A ∧ B) is false under three of the four
>> possible interpretations. How can the above possibly constitute a proof?
>>
>
> It is much more than a proof it is a stipulated truth.
> It provides the algorithm of the meaning of: "∧".
That is the truth table for ∧. It is not an algorithm.
My original post made two claims, and it is not clear which of these you
are disputing.
CLAIM 1: (A ∧ B) is satisfiable.
CLAIM 2: (A ∧ B) is not a theorem.
The truth table you give is simply a list of the four possible
interpretations of (A ∧ B)
A B (A ∧ B)
(1) T T T
(2) T F F
(3) F T F
(4) F F F
Each of (1)-(4) is an interpretation. Under interpretation (1), (A ∧ B)
is true. Under interpretation (2)-(4) (A ∧ B) is false. That means that
(A ∧ B) is satisfiable because there is one interpretation, namely
interpretation (1), under which (A ∧ B) is true.
For (A ∧ B) to be a theorem, it would need to be true under *every*
interpretation. It isn't. It is false under interpretations (2)-(4).
>>>> You've given the four possible interpretations of (A ∧ B) (though
>>>> you've decided to rename them for some reason).
>>>
>>> cut-and-paste from Wikipedia.
>>>
>>>> One of those interpretations (the top one) is true, therefore (A ∧
>>>> B) is satisfiable.
>>>>
>>>> If (A ∧ B) were a theorem, then (A ∧ B) would be true on *all
>>>> possible* interpretations. The above clearly indicates that this is
>>>> not the case, since (A ∧ B) is *only* true on the first
>>>> interpretation given above.
>>>>
>>>> André
>>>>
>>>
>>> If it is only an interpretation (one way of looking at it) then why
>>> does it always work this way all the time?
>>
>> Why don't you learn what 'interpretation' really means?
>>
>> André
>>
>
> 2.2 First-Order Languages and Their Interpretations:
> Satisfiability and Truth: Models
Quoting someone else's definitions doesn't provide any evidence that you
actually understand what those definitions means.
> Let L be a first-order language. An interpretation M of L consists of
> the following ingredients.
>
> a. A nonempty set D, called the domain of the interpretation.
>
> b. For each predicate letter Anj of L, an assignment of an n-place
> relation (Anj)M in D.
>
> c. For each function letter fnj of L, an assignment of an n-place
> operation (fnj)M in D (that is, a function from Dn into D).
>
> d. For each individual constant ai of L, an assignment of some fixed
> element (ai)M of D.
>
> Given such an interpretation, variables are thought of as ranging over
> the set D, and ¬, ⇒ and quantifiers are given their usual meaning.
> Remember that an n-place relation in D can be thought of as a subset of
> Dn, the set of all n-tuples of elements of D. For example, if D is the
> set of human beings, then the relation “father of” can be identified
> with the set of all ordered pairs 〈x, y〉 such that x is the father of y.
>
> For a given interpretation of a language L, a wf of L without free
> variables (called a closed wf or a sentence) represents a proposition
> that is true or false, whereas a wf with free variables may be satisfied
> (i.e., true) for some values in the domain and not satisfied (i.e.,
> false) for the others. (Mendelson 2015: 57-58)
>
> Mendelson, Elliott 2015. Introduction To Mathematical Logic. Boca Raton:
> CRC Press Taylor & Francis Group.
>
> In his example we would not need any separate interpretation where all
> predicates have the same name of "A" and are only distinguished by their
> two subscripts. We can merely specify ALL of we mean right in the
> formula leaving nothing left to be separately interpreted.
>
> ∀y ∈ ℤ+ (x ≤ y)
> ∃x ∈ ℤ+ ∀y ∈ ℤ+ (x ≤ y)
And what is it that you think either of those formulae have to do with
(A ∧ B), which was the actual formula under discussion?
André
--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.
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| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2020-07-13 23:48 -0600 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <rejgv8$kj2$1@dont-email.me> |
| In reply to | #21636 |
On 2020-07-13 18:49, olcott wrote: > On 7/13/2020 9:05 AM, André G. Isaak wrote: >> On 2020-07-12 21:47, olcott wrote: >>> On 7/12/2020 9:32 PM, Keith Thompson wrote: >>>> olcott <NoOne@NoWhere.com> writes: >>>>> On 7/12/2020 7:22 PM, Keith Thompson wrote: >>>>>> olcott <NoOne@NoWhere.com> writes: >>>>>>> On 7/12/2020 4:04 PM, Keith Thompson wrote: >>>> [...] >>>>>>>> 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? >>> >>> No. That would be like proving that existence never existed or >>> finding some integer Crazy_Number such that Crazy_Number > 5 and >>> Crazy_Number < 3. >>> >>> No thing of all things can be proved false that has been defined to >>> be true. Defined to be true it the ultimate foundation of all truth. >> >> What exactly is it that you are claiming is 'defined to be true'? >> Certainly not the commutativity of addition in Q. In PA, one can prove >> that addition is commutative, but it certainly isn't defined to be true. > > A Boolean function having a WFF as its argument. I don't understand how that is an answer to the question. You claimed it was possible to create a system based on Q in which addition is provably commutative by adding an axiom. You also claimed it was not possible to create a system based on Q in which addition is *not* commutative because nothing can be proven to be false which is defined to be true. But commutativity is certainly not defined to be true in Q, so what is it that is "defined to be true" which would preclude creating a system based on Q in which addition was not commutative? >> >> If it is not possible to add some axiom to Q which makes addition >> non-commutative then that would certainly support what everyone other >> than you is claiming: that addition IS commutative in Q despite the >> fact that this cannot be proven in Q. >> >> André >> > > φ = ∀x ∈ ℕ ∀y ∈ ℕ (y > x) > Boolean Is_Commutativity_of_Addition(φ) > > This function does not exist in Q because Q doesn't know about the > commutativity of addition. That function also does not exist in PA, but commutativity is provable in PA (whether that means PA 'knows' about commutativity is another matter -- I have no idea what it means for a system to 'know' about something). > I am taking the fact that that the commutativity of addition is not > provable in Q to mean that φ is undecidable in Q because φ <is> the > commutativity of addition. > > If a sentence is undecidable in Q then this is merely another way of > saying that it is neither true nor false in Q. Sigh. No it is not. It is a way of saying that Q can neither prove nor disprove that sentence. True and false have nothing to do with it. And Q, being based on standard logic, does not allow propositions which are neither true nor false. André -- To email remove 'invalid' & replace 'gm' with well known Google mail service.
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| From | olcott <NoOne@NoWhere.com> |
|---|---|
| Date | 2020-07-14 10:11 -0500 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <ro6dnRSc2KKXVpDCnZ2dnUU7-RXNnZ2d@giganews.com> |
| In reply to | #21648 |
On 7/14/2020 12:48 AM, André G. Isaak wrote:
> On 2020-07-13 18:49, olcott wrote:
>> On 7/13/2020 9:05 AM, André G. Isaak wrote:
>>> On 2020-07-12 21:47, olcott wrote:
>>>> On 7/12/2020 9:32 PM, Keith Thompson wrote:
>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>> On 7/12/2020 7:22 PM, Keith Thompson wrote:
>>>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>>>> On 7/12/2020 4:04 PM, Keith Thompson wrote:
>>>>> [...]
>>>>>>>>> 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?
>>>>
>>>> No. That would be like proving that existence never existed or
>>>> finding some integer Crazy_Number such that Crazy_Number > 5 and
>>>> Crazy_Number < 3.
>>>>
>>>> No thing of all things can be proved false that has been defined to
>>>> be true. Defined to be true it the ultimate foundation of all truth.
>>>
>>> What exactly is it that you are claiming is 'defined to be true'?
When so ever a Boolean function evaluates a WFF to be true.
Predicates are defined to be Boolean functions.
Boolean True("2+3=5")
Boolean Theorem("2+3=5")
Boolean Provable("2+3=5")
>>> Certainly not the commutativity of addition in Q. In PA, one can
>>> prove that addition is commutative, but it certainly isn't defined to
>>> be true.
>>
>> A Boolean function having a WFF as its argument.
>
> I don't understand how that is an answer to the question.
>
> You claimed it was possible to create a system based on Q in which
> addition is provably commutative by adding an axiom.
>
> You also claimed it was not possible to create a system based on Q in
> which addition is *not* commutative because nothing can be proven to be
> false which is defined to be true. But commutativity is certainly not
> defined to be true in Q, so what is it that is "defined to be true"
> which would preclude creating a system based on Q in which addition was
> not commutative?
>
In the set of human knowledge there is a set of interrelationships
between finite strings that define an algorithm that specifies the
meaning of the commutativity of addition, thus commutativity is defined
to be true on the basis of its meaning. The commutativity of addition is
merely the name of this defined algorithm.
>>>
>>> If it is not possible to add some axiom to Q which makes addition
>>> non-commutative then that would certainly support what everyone other
>>> than you is claiming: that addition IS commutative in Q despite the
>>> fact that this cannot be proven in Q.
>>>
>>> André
>>>
>>
>> φ = ∀x ∈ ℕ ∀y ∈ ℕ (y > x)
>> Boolean Is_Commutativity_of_Addition(φ)
>>
>> This function does not exist in Q because Q doesn't know about the
>> commutativity of addition.
>
> That function also does not exist in PA, but commutativity is provable
> in PA (whether that means PA 'knows' about commutativity is another
> matter -- I have no idea what it means for a system to 'know' about
> something).
Since everyone here is indoctrinated into believing that Gödel is
correct I have to use different terms for provability so that people
will carefully analyze my reasoning and not simply dismiss it
out-of-hand on the basis of their indoctrination.
Ultimately provability in a formal system is a mathematical mapping from
an expression of language to a Boolean value. Without this mapping the
expression is not a truth bearer and thus neither true nor false.
Is the question: "What time is it?" true or false?
(or not a truth bearer).
>
>> I am taking the fact that that the commutativity of addition is not
>> provable in Q to mean that φ is undecidable in Q because φ <is> the
>> commutativity of addition.
>>
>> If a sentence is undecidable in Q then this is merely another way of
>> saying that it is neither true nor false in Q.
>
> Sigh. No it is not. It is a way of saying that Q can neither prove nor
> disprove that sentence. True and false have nothing to do with it. And
> Q, being based on standard logic, does not allow propositions which are
> neither true nor false.
>
> André
>
In mathematical logic, a sentence of a predicate logic is a
boolean-valued well-formed formula with no free variables. A sentence
can be viewed as expressing a proposition, something that must be true
or false. https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)
The expression of the language of a formal system THAT DOES HAVE FREE
VARIABLES apparently does exist in the language of formal system and
DOES NOT meet the criteria of a proposition.
One can infer from the above quote that the expressions of the language
of formal systems with free variables are neither sentences nor
propositions thus are not able to be true or false.
Interpretation (logic)
An interpretation is an assignment of meaning to the
[non-logical] symbols of a formal language.
https://en.wikipedia.org/wiki/Interpretation_(logic)
"If I say that X went to the the store to buy some Y" it remains
unevaluatable until X and Y have been defined with values providing an
interpretation of the above expression.
--
Copyright 2020 Pete Olcott
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| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2020-07-14 09:20 -0600 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <rekif2$pi6$1@dont-email.me> |
| In reply to | #21654 |
On 2020-07-14 09:11, olcott wrote:
> On 7/14/2020 12:48 AM, André G. Isaak wrote:
>> On 2020-07-13 18:49, olcott wrote:
>>> On 7/13/2020 9:05 AM, André G. Isaak wrote:
>>>> On 2020-07-12 21:47, olcott wrote:
>>>>> On 7/12/2020 9:32 PM, Keith Thompson wrote:
>>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>>> On 7/12/2020 7:22 PM, Keith Thompson wrote:
>>>>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>>>>> On 7/12/2020 4:04 PM, Keith Thompson wrote:
>>>>>> [...]
>>>>>>>>>> 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?
>>>>>
>>>>> No. That would be like proving that existence never existed or
>>>>> finding some integer Crazy_Number such that Crazy_Number > 5 and
>>>>> Crazy_Number < 3.
>>>>>
>>>>> No thing of all things can be proved false that has been defined to
>>>>> be true. Defined to be true it the ultimate foundation of all truth.
>>>>
>>>> What exactly is it that you are claiming is 'defined to be true'?
>
> When so ever a Boolean function evaluates a WFF to be true.
> Predicates are defined to be Boolean functions.
>
> Boolean True("2+3=5")
> Boolean Theorem("2+3=5")
> Boolean Provable("2+3=5")
I have utterly no idea what those are intended to clarify.
>>>> Certainly not the commutativity of addition in Q. In PA, one can
>>>> prove that addition is commutative, but it certainly isn't defined
>>>> to be true.
>>>
>>> A Boolean function having a WFF as its argument.
>>
>> I don't understand how that is an answer to the question.
>>
>> You claimed it was possible to create a system based on Q in which
>> addition is provably commutative by adding an axiom.
>>
>> You also claimed it was not possible to create a system based on Q in
>> which addition is *not* commutative because nothing can be proven to
>> be false which is defined to be true. But commutativity is certainly
>> not defined to be true in Q, so what is it that is "defined to be
>> true" which would preclude creating a system based on Q in which
>> addition was not commutative?
>>
>
> In the set of human knowledge there is a set of interrelationships
> between finite strings that define an algorithm that specifies the
> meaning of the commutativity of addition, thus commutativity is defined
> to be true on the basis of its meaning. The commutativity of addition is
> merely the name of this defined algorithm.
We weren't talking about "the set of human knowledge". We were talking
about Q. What is it that is defined in Q which somehow makes it
impossible to simply define ∀x ∈ ℕ ∀y ∈ ℕ (x + y = y + x) to be false?
You said we could define it as true using an axiom. You also keep
claiming that the sentence is neither true nor false in Q. If it is
neither true nor false, then why should it be acceptable to introduce an
axiom which defines it one way but not the other?
>>>>
>>>> If it is not possible to add some axiom to Q which makes addition
>>>> non-commutative then that would certainly support what everyone
>>>> other than you is claiming: that addition IS commutative in Q
>>>> despite the fact that this cannot be proven in Q.
>>>>
>>>> André
>>>>
>>>
>>> φ = ∀x ∈ ℕ ∀y ∈ ℕ (y > x)
>>> Boolean Is_Commutativity_of_Addition(φ)
>>>
>>> This function does not exist in Q because Q doesn't know about the
>>> commutativity of addition.
>>
>> That function also does not exist in PA, but commutativity is provable
>> in PA (whether that means PA 'knows' about commutativity is another
>> matter -- I have no idea what it means for a system to 'know' about
>> something).
>
> Since everyone here is indoctrinated into believing that Gödel is
> correct I have to use different terms for provability so that people
> will carefully analyze my reasoning and not simply dismiss it
> out-of-hand on the basis of their indoctrination.
>
> Ultimately provability in a formal system is a mathematical mapping from
> an expression of language to a Boolean value. Without this mapping the
> expression is not a truth bearer and thus neither true nor false.
>
> Is the question: "What time is it?" true or false?
> (or not a truth bearer).
Why do you keep repeating this incredibly silly example? As I have
already stated, truth is a property of propositions. Interrogatives are
not propositions. Nor are entities. Nor are properties. etc. Gödel was
talking about *propositions* which are, by definition, truth bearers.
>>
>>> I am taking the fact that that the commutativity of addition is not
>>> provable in Q to mean that φ is undecidable in Q because φ <is> the
>>> commutativity of addition.
>>>
>>> If a sentence is undecidable in Q then this is merely another way of
>>> saying that it is neither true nor false in Q.
>>
>> Sigh. No it is not. It is a way of saying that Q can neither prove nor
>> disprove that sentence. True and false have nothing to do with it. And
>> Q, being based on standard logic, does not allow propositions which
>> are neither true nor false.
>>
>> André
>>
>
> In mathematical logic, a sentence of a predicate logic is a
> boolean-valued well-formed formula with no free variables. A sentence
> can be viewed as expressing a proposition, something that must be true
> or false. https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)
>
> The expression of the language of a formal system THAT DOES HAVE FREE
> VARIABLES apparently does exist in the language of formal system and
> DOES NOT meet the criteria of a proposition.
Except we were talking about this sentence:
∀x ∈ ℕ ∀y ∈ ℕ (x + y = y + x)
There are no free variables in that sentence.
André
> One can infer from the above quote that the expressions of the language
> of formal systems with free variables are neither sentences nor
> propositions thus are not able to be true or false.
>
> Interpretation (logic)
> An interpretation is an assignment of meaning to the
> [non-logical] symbols of a formal language.
> https://en.wikipedia.org/wiki/Interpretation_(logic)
>
> "If I say that X went to the the store to buy some Y" it remains
> unevaluatable until X and Y have been defined with values providing an
> interpretation of the above expression.
>
--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.
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| From | olcott <NoOne@NoWhere.com> |
|---|---|
| Date | 2020-07-14 10:26 -0500 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <YKqdnQpfM98uU5DCnZ2dnUU7-R3NnZ2d@giganews.com> |
| In reply to | #21656 |
On 7/14/2020 10:20 AM, André G. Isaak wrote:
> On 2020-07-14 09:11, olcott wrote:
>> On 7/14/2020 12:48 AM, André G. Isaak wrote:
>>> On 2020-07-13 18:49, olcott wrote:
>>>> On 7/13/2020 9:05 AM, André G. Isaak wrote:
>>>>> On 2020-07-12 21:47, olcott wrote:
>>>>>> On 7/12/2020 9:32 PM, Keith Thompson wrote:
>>>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>>>> On 7/12/2020 7:22 PM, Keith Thompson wrote:
>>>>>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>>>>>> On 7/12/2020 4:04 PM, Keith Thompson wrote:
>>>>>>> [...]
>>>>>>>>>>> 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?
>>>>>>
>>>>>> No. That would be like proving that existence never existed or
>>>>>> finding some integer Crazy_Number such that Crazy_Number > 5 and
>>>>>> Crazy_Number < 3.
>>>>>>
>>>>>> No thing of all things can be proved false that has been defined
>>>>>> to be true. Defined to be true it the ultimate foundation of all
>>>>>> truth.
>>>>>
>>>>> What exactly is it that you are claiming is 'defined to be true'?
>>
>> When so ever a Boolean function evaluates a WFF to be true.
>> Predicates are defined to be Boolean functions.
>>
>> Boolean True("2+3=5")
>> Boolean Theorem("2+3=5")
>> Boolean Provable("2+3=5")
>
> I have utterly no idea what those are intended to clarify.
>
>>>>> Certainly not the commutativity of addition in Q. In PA, one can
>>>>> prove that addition is commutative, but it certainly isn't defined
>>>>> to be true.
>>>>
>>>> A Boolean function having a WFF as its argument.
>>>
>>> I don't understand how that is an answer to the question.
>>>
>>> You claimed it was possible to create a system based on Q in which
>>> addition is provably commutative by adding an axiom.
>>>
>>> You also claimed it was not possible to create a system based on Q in
>>> which addition is *not* commutative because nothing can be proven to
>>> be false which is defined to be true. But commutativity is certainly
>>> not defined to be true in Q, so what is it that is "defined to be
>>> true" which would preclude creating a system based on Q in which
>>> addition was not commutative?
>>>
>>
>> In the set of human knowledge there is a set of interrelationships
>> between finite strings that define an algorithm that specifies the
>> meaning of the commutativity of addition, thus commutativity is
>> defined to be true on the basis of its meaning. The commutativity of
>> addition is merely the name of this defined algorithm.
>
> We weren't talking about "the set of human knowledge". We were talking
> about Q. What is it that is defined in Q which somehow makes it
> impossible to simply define ∀x ∈ ℕ ∀y ∈ ℕ (x + y = y + x) to be false?
> You said we could define it as true using an axiom. You also keep
> claiming that the sentence is neither true nor false in Q. If it is
> neither true nor false, then why should it be acceptable to introduce an
> axiom which defines it one way but not the other?
>
>>>>>
>>>>> If it is not possible to add some axiom to Q which makes addition
>>>>> non-commutative then that would certainly support what everyone
>>>>> other than you is claiming: that addition IS commutative in Q
>>>>> despite the fact that this cannot be proven in Q.
>>>>>
>>>>> André
>>>>>
>>>>
>>>> φ = ∀x ∈ ℕ ∀y ∈ ℕ (y > x)
>>>> Boolean Is_Commutativity_of_Addition(φ)
>>>>
>>>> This function does not exist in Q because Q doesn't know about the
>>>> commutativity of addition.
>>>
>>> That function also does not exist in PA, but commutativity is
>>> provable in PA (whether that means PA 'knows' about commutativity is
>>> another matter -- I have no idea what it means for a system to 'know'
>>> about something).
>>
>> Since everyone here is indoctrinated into believing that Gödel is
>> correct I have to use different terms for provability so that people
>> will carefully analyze my reasoning and not simply dismiss it
>> out-of-hand on the basis of their indoctrination.
>>
>> Ultimately provability in a formal system is a mathematical mapping
>> from an expression of language to a Boolean value. Without this
>> mapping the expression is not a truth bearer and thus neither true nor
>> false.
>>
>> Is the question: "What time is it?" true or false?
>> (or not a truth bearer).
>
> Why do you keep repeating this incredibly silly example? As I have
> already stated, truth is a property of propositions. Interrogatives are
> not propositions. Nor are entities. Nor are properties. etc. Gödel was
> talking about *propositions* which are, by definition, truth bearers.
>
>>>
>>>> I am taking the fact that that the commutativity of addition is not
>>>> provable in Q to mean that φ is undecidable in Q because φ <is> the
>>>> commutativity of addition.
>>>>
>>>> If a sentence is undecidable in Q then this is merely another way of
>>>> saying that it is neither true nor false in Q.
>>>
>>> Sigh. No it is not. It is a way of saying that Q can neither prove
>>> nor disprove that sentence. True and false have nothing to do with
>>> it. And Q, being based on standard logic, does not allow propositions
>>> which are neither true nor false.
>>>
>>> André
>>>
>>
>> In mathematical logic, a sentence of a predicate logic is a
>> boolean-valued well-formed formula with no free variables. A sentence
>> can be viewed as expressing a proposition, something that must be true
>> or false. https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)
>>
>> The expression of the language of a formal system THAT DOES HAVE FREE
>> VARIABLES apparently does exist in the language of formal system and
>> DOES NOT meet the criteria of a proposition.
>
> Except we were talking about this sentence:
>
> ∀x ∈ ℕ ∀y ∈ ℕ (x + y = y + x)
>
> There are no free variables in that sentence.
I am starting with the basic idea that there are some WFF of a formal
system that are neither true nor false so that you can see that this
basic idea is fulfilled.
After you understand that some WFF really are neither true nor false
because they are not closed WFF then we move on to the next step of
additional elaboration.
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
>
> André
>
>> One can infer from the above quote that the expressions of the
>> language of formal systems with free variables are neither sentences
>> nor propositions thus are not able to be true or false.
>>
>> Interpretation (logic)
>> An interpretation is an assignment of meaning to the
>> [non-logical] symbols of a formal language.
>> https://en.wikipedia.org/wiki/Interpretation_(logic)
>>
>> "If I say that X went to the the store to buy some Y" it remains
>> unevaluatable until X and Y have been defined with values providing an
>> interpretation of the above expression.
>>
>
>
--
Copyright 2020 Pete Olcott
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| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Date | 2020-07-14 09:36 -0600 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <rekjdm$tia$2@dont-email.me> |
| In reply to | #21658 |
On 2020-07-14 09:26, olcott wrote: > On 7/14/2020 10:20 AM, André G. Isaak wrote: >> On 2020-07-14 09:11, olcott wrote: >>> The expression of the language of a formal system THAT DOES HAVE FREE >>> VARIABLES apparently does exist in the language of formal system and >>> DOES NOT meet the criteria of a proposition. >> >> Except we were talking about this sentence: >> >> ∀x ∈ ℕ ∀y ∈ ℕ (x + y = y + x) >> >> There are no free variables in that sentence. > > I am starting with the basic idea that there are some WFF of a formal > system that are neither true nor false so that you can see that this > basic idea is fulfilled. > > After you understand that some WFF really are neither true nor false > because they are not closed WFF then we move on to the next step of > additional elaboration. No one here has been making claims about non-closed WFFs, so that is entirely irrelevant. And there are no non-closed WFFs in Q. In Q, all variables are implicitly bound by a universal quantifier unless an explicit existential quantifier is present. André -- To email remove 'invalid' & replace 'gm' with well known Google mail service.
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| From | olcott <NoOne@NoWhere.com> |
|---|---|
| Date | 2020-07-15 10:41 -0500 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <irCdnc5xxrkSvpLCnZ2dnUU7-aWdnZ2d@giganews.com> |
| In reply to | #21660 |
On 7/14/2020 10:36 AM, André G. Isaak wrote: > On 2020-07-14 09:26, olcott wrote: >> On 7/14/2020 10:20 AM, André G. Isaak wrote: >>> On 2020-07-14 09:11, olcott wrote: > > >>>> The expression of the language of a formal system THAT DOES HAVE >>>> FREE VARIABLES apparently does exist in the language of formal >>>> system and DOES NOT meet the criteria of a proposition. >>> >>> Except we were talking about this sentence: >>> >>> ∀x ∈ ℕ ∀y ∈ ℕ (x + y = y + x) >>> >>> There are no free variables in that sentence. >> >> I am starting with the basic idea that there are some WFF of a formal >> system that are neither true nor false so that you can see that this >> basic idea is fulfilled. >> >> After you understand that some WFF really are neither true nor false >> because they are not closed WFF then we move on to the next step of >> additional elaboration. > > No one here has been making claims about non-closed WFFs, so that is > entirely irrelevant. > It is helpful to see that the notion of truth bearer applies to very common situations before trying to apply it to much more difficult cases. > And there are no non-closed WFFs in Q. In Q, all variables are > implicitly bound by a universal quantifier unless an explicit > existential quantifier is present. > > André -- Copyright 2020 Pete Olcott
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| From | Keith Thompson <Keith.S.Thompson+u@gmail.com> |
|---|---|
| Date | 2020-07-14 11:25 -0700 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <87zh820x98.fsf@nosuchdomain.example.com> |
| In reply to | #21654 |
olcott <NoOne@NoWhere.com> writes:
[...]
> Since everyone here is indoctrinated into believing that Gödel is
> correct I have to use different terms for provability so that people
> will carefully analyze my reasoning and not simply dismiss it
> out-of-hand on the basis of their indoctrination.
It seems to me that the best way to demonstrate that Gödel is
incorrect would be to demonstrate a flaw in what he actually wrote.
I haven't read everything you've written here, but I don't recall
you ever directly quoting Gödel's proof.
You seem to assert that provability and truth must be the same thing
because of course they are, and how could anyone believe otherwise?
I haven't read Gödel's proof myself, and likely wouldn't understand
all of it, but there are plenty of people who have and would.
[...]
--
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 | 2020-07-15 10:52 -0500 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <VLWdnefp55Dfu5LCnZ2dnUU7-KmdnZ2d@giganews.com> |
| In reply to | #21664 |
On 7/14/2020 1:25 PM, Keith Thompson wrote: > olcott <NoOne@NoWhere.com> writes: > [...] >> Since everyone here is indoctrinated into believing that Gödel is >> correct I have to use different terms for provability so that people >> will carefully analyze my reasoning and not simply dismiss it >> out-of-hand on the basis of their indoctrination. > > It seems to me that the best way to demonstrate that Gödel is > incorrect would be to demonstrate a flaw in what he actually wrote. > I haven't read everything you've written here, but I don't recall > you ever directly quoting Gödel's proof. > Not really. When we refute the enormously simplified key result of his claim: true and unprovable can possibly coexist, then the steps that he used to get to this key result are moot. > You seem to assert that provability and truth must be the same thing > because of course they are, and how could anyone believe otherwise? > > I haven't read Gödel's proof myself, and likely wouldn't understand > all of it, but there are plenty of people who have and would. > > [...] > -- Copyright 2020 Pete Olcott
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| From | Keith Thompson <Keith.S.Thompson+u@gmail.com> |
|---|---|
| Date | 2020-07-15 11:04 -0700 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <87imeo1wov.fsf@nosuchdomain.example.com> |
| In reply to | #21676 |
olcott <NoOne@NoWhere.com> writes:
> On 7/14/2020 1:25 PM, Keith Thompson wrote:
>> olcott <NoOne@NoWhere.com> writes:
>> [...]
>>> Since everyone here is indoctrinated into believing that Gödel is
>>> correct I have to use different terms for provability so that people
>>> will carefully analyze my reasoning and not simply dismiss it
>>> out-of-hand on the basis of their indoctrination.
>>
>> It seems to me that the best way to demonstrate that Gödel is
>> incorrect would be to demonstrate a flaw in what he actually wrote.
>> I haven't read everything you've written here, but I don't recall
>> you ever directly quoting Gödel's proof.
>
> Not really. When we refute the enormously simplified key result of his
> claim: true and unprovable can possibly coexist, then the steps that
> he used to get to this key result are moot.
You've been asserting that for years, and nobody believes you.
Do you think that's going to change if you assert it just one
more time? What is your goal here?
Whether it's the best way or not, surely *a* way to demonstrate
that Gödel is incorrect would be to demonstrate a flaw in what he
actually wrote. Not in some summary of his proof, but in his actual
proof as he wrote it. Something like "In step 42, Gödel makes use
of this assumption, but previously in step 23 he showed that that
assumption does not hold in all cases". (That's a hypothetical
example, of course.)
Are you able to cite a specific flaw in Gödel's proof? If you're
actually able to do that, I think people would pay attention.
If you're not able to do that, it just might imply that you're
wrong about all this.
>> You seem to assert that provability and truth must be the same thing
>> because of course they are, and how could anyone believe otherwise?
>>
>> I haven't read Gödel's proof myself, and likely wouldn't understand
>> all of it, but there are plenty of people who have and would.
>>
>> [...]
>>
--
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 | 2020-07-15 19:07 -0500 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <DKmdnUY8tKTHB5LCnZ2dnUU7-fXNnZ2d@giganews.com> |
| In reply to | #21680 |
On 7/15/2020 1:04 PM, Keith Thompson wrote: > olcott <NoOne@NoWhere.com> writes: >> On 7/14/2020 1:25 PM, Keith Thompson wrote: >>> olcott <NoOne@NoWhere.com> writes: >>> [...] >>>> Since everyone here is indoctrinated into believing that Gödel is >>>> correct I have to use different terms for provability so that people >>>> will carefully analyze my reasoning and not simply dismiss it >>>> out-of-hand on the basis of their indoctrination. >>> >>> It seems to me that the best way to demonstrate that Gödel is >>> incorrect would be to demonstrate a flaw in what he actually wrote. >>> I haven't read everything you've written here, but I don't recall >>> you ever directly quoting Gödel's proof. >> >> Not really. When we refute the enormously simplified key result of his >> claim: true and unprovable can possibly coexist, then the steps that >> he used to get to this key result are moot. > > You've been asserting that for years, and nobody believes you https://scholar.google.com/scholar?hl=en&as_sdt=0%2C28&q=%22true+and+unprovable%22+godel&btnG=&oq=%22true+and+unprovable%22 > Do you think that's going to change if you assert it just one > more time? What is your goal here? > > Whether it's the best way or not, surely *a* way to demonstrate > that Gödel is incorrect would be to demonstrate a flaw in what he > actually wrote. Not in some summary of his proof, but in his actual > proof as he wrote it. Something like "In step 42, Gödel makes use > of this assumption, but previously in step 23 he showed that that > assumption does not hold in all cases". (That's a hypothetical > example, of course.) > His mistake can only be seen through a refutation of the essence of his conclusion. There is his own conclusion: The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F. and there are many scholars that interpret what this means: https://scholar.google.com/scholar?hl=en&as_sdt=0%2C28&q=%22true+and+unprovable%22+godel&btnG=&oq=%22true+and+unprovable%22 It is far simpler to refute the scholars interpretation because it can be shown that true and unprovable cannot coexist and people have a rational understanding of the terms "true" and provable On the other hand apparently if "incomplete" was defined as: "painted my pickup truck green" most math people would agree that a formal system is "incomplete" if someone just painted their pickup truck green because that is the way it has been defined. > Are you able to cite a specific flaw in Gödel's proof? If you're > actually able to do that, I think people would pay attention. > If you're not able to do that, it just might imply that you're > wrong about all this. > If a complex proof have an enormously complex proof with a large number of steps concludes that 3 > 7 do we need find exactly which detail of the proof went astray? >>> You seem to assert that provability and truth must be the same thing >>> because of course they are, and how could anyone believe otherwise? >>> >>> I haven't read Gödel's proof myself, and likely wouldn't understand >>> all of it, but there are plenty of people who have and would. >>> >>> [...] >>> > -- Copyright 2020 Pete Olcott
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| From | Keith Thompson <Keith.S.Thompson+u@gmail.com> |
|---|---|
| Date | 2020-07-15 18:42 -0700 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <87a7001bhr.fsf@nosuchdomain.example.com> |
| In reply to | #21685 |
olcott <NoOne@NoWhere.com> writes:
> On 7/15/2020 1:04 PM, Keith Thompson wrote:
>> olcott <NoOne@NoWhere.com> writes:
>>> On 7/14/2020 1:25 PM, Keith Thompson wrote:
>>>> olcott <NoOne@NoWhere.com> writes:
>>>> [...]
>>>>> Since everyone here is indoctrinated into believing that Gödel is
>>>>> correct I have to use different terms for provability so that people
>>>>> will carefully analyze my reasoning and not simply dismiss it
>>>>> out-of-hand on the basis of their indoctrination.
>>>>
>>>> It seems to me that the best way to demonstrate that Gödel is
>>>> incorrect would be to demonstrate a flaw in what he actually wrote.
>>>> I haven't read everything you've written here, but I don't recall
>>>> you ever directly quoting Gödel's proof.
>>>
>>> Not really. When we refute the enormously simplified key result of his
>>> claim: true and unprovable can possibly coexist, then the steps that
>>> he used to get to this key result are moot.
>>
>> You've been asserting that for years, and nobody believes you
> https://scholar.google.com/scholar?hl=en&as_sdt=0%2C28&q=%22true+and+unprovable%22+godel&btnG=&oq=%22true+and+unprovable%22
125 results. No, I'm not going to read them.
>> Do you think that's going to change if you assert it just one
>> more time? What is your goal here?
>>
>> Whether it's the best way or not, surely *a* way to demonstrate
>> that Gödel is incorrect would be to demonstrate a flaw in what he
>> actually wrote. Not in some summary of his proof, but in his actual
>> proof as he wrote it. Something like "In step 42, Gödel makes use
>> of this assumption, but previously in step 23 he showed that that
>> assumption does not hold in all cases". (That's a hypothetical
>> example, of course.)
>
> His mistake can only be seen through a refutation of the essence of
> his conclusion.
Ah, now that's an interesting assertion. Did you really mean "only"?
So are you saying that you *cannot* demonstrate that Gödel proof is
incorrect by citing a specific error within the proof. It seems to me
that that's equivalent to saying that Gödel's proof is correct. I'm
sure that's not what you meant. Did you mean specifically that *you*
cannot do that? I doubt that that's what you meant either.
Are you saying that it's possible for every step of Gödel's proof
to be valid, but for the proof as a whole to be invalid, yielding a
false conclusion? If so, that's a remarkable assertion from someone
who says that a complex system can be complete and consistent.
Do you believe there is a specific flaw in Gödel's proof?
(This question is not about what that flaw is, just whether you
think there is one.)
> There is his own conclusion:
>
> The first incompleteness theorem states that in any consistent formal
> system F within which a certain amount of arithmetic can be carried
> out, there are statements of the language of F which can neither be
> proved nor disproved in F.
>
> and there are many scholars that interpret what this means:
> https://scholar.google.com/scholar?hl=en&as_sdt=0%2C28&q=%22true+and+unprovable%22+godel&btnG=&oq=%22true+and+unprovable%22
>
> It is far simpler to refute the scholars interpretation because it can
> be shown that true and unprovable cannot coexist and people have a
> rational understanding of the terms "true" and provable
>
> On the other hand apparently if "incomplete" was defined as:
> "painted my pickup truck green" most math people would agree that a
> formal system is "incomplete" if someone just painted their pickup
> truck green because that is the way it has been defined.
>
>> Are you able to cite a specific flaw in Gödel's proof? If you're
>> actually able to do that, I think people would pay attention.
>> If you're not able to do that, it just might imply that you're
>> wrong about all this.
>
> If a complex proof have an enormously complex proof with a large
> number of steps concludes that 3 > 7 do we need find exactly which
> detail of the proof went astray?
Once again, I asked you a yes or no question, and your response did not
included the word "yes" or "no".
Are you able to cite a specific flaw in Gödel's proof?
Obviously you've decided that the best way for you to refute Gödel's
proof is to refute "the essence of his conclusion". But you know by now
that that doesn't convince the other participants in this discussion.
If a mathematician published a complex proof that 3 > 7, I personally
probably wouldn't bother to find a flaw in it -- but surely it would
be possible to find such a flaw. If such a proof had been published
nearly a century ago and generally accepted by the mathematics
community, the first person to demonstrate a flaw would be famous.
If your goal is to repeatedly make the same claims and not convince
anyone, keep doing what you're doing. If your goal is to demonstrate
to the satisfaction of experts in mathematical logic that Gödel
was wrong, I suggest that finding a specific flaw in Gödel's proof
is the best way to do that. (To be honest, I don't expect that to
be possible, because I think Gödel was right and you're wrong, but
I have no rigorous proof of that other than Gödel's proof itself,
which as I've said I don't claim to understand in full.)
>>>> You seem to assert that provability and truth must be the same thing
>>>> because of course they are, and how could anyone believe otherwise?
>>>>
>>>> I haven't read Gödel's proof myself, and likely wouldn't understand
>>>> all of it, but there are plenty of people who have and would.
>>>>
>>>> [...]
--
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 | 2020-07-16 12:10 -0500 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <KdKdnbB7kbtpFI3CnZ2dnUU7-S3NnZ2d@giganews.com> |
| In reply to | #21694 |
On 7/15/2020 8:42 PM, Keith Thompson wrote: > olcott <NoOne@NoWhere.com> writes: >> On 7/15/2020 1:04 PM, Keith Thompson wrote: >>> olcott <NoOne@NoWhere.com> writes: >>>> On 7/14/2020 1:25 PM, Keith Thompson wrote: >>>>> olcott <NoOne@NoWhere.com> writes: >>>>> [...] >>>>>> Since everyone here is indoctrinated into believing that Gödel is >>>>>> correct I have to use different terms for provability so that people >>>>>> will carefully analyze my reasoning and not simply dismiss it >>>>>> out-of-hand on the basis of their indoctrination. >>>>> >>>>> It seems to me that the best way to demonstrate that Gödel is >>>>> incorrect would be to demonstrate a flaw in what he actually wrote. >>>>> I haven't read everything you've written here, but I don't recall >>>>> you ever directly quoting Gödel's proof. >>>> >>>> Not really. When we refute the enormously simplified key result of his >>>> claim: true and unprovable can possibly coexist, then the steps that >>>> he used to get to this key result are moot. >>> >>> You've been asserting that for years, and nobody believes you >> https://scholar.google.com/scholar?hl=en&as_sdt=0%2C28&q=%22true+and+unprovable%22+godel&btnG=&oq=%22true+and+unprovable%22 > > 125 results. No, I'm not going to read them. 125 different people that all believe that Gödel showed that true and unprovable formulas exists, and 125 > 0, thus "nobody believes you" is proven to be false. > >>> Do you think that's going to change if you assert it just one >>> more time? What is your goal here? >>> >>> Whether it's the best way or not, surely *a* way to demonstrate >>> that Gödel is incorrect would be to demonstrate a flaw in what he >>> actually wrote. Not in some summary of his proof, but in his actual >>> proof as he wrote it. Something like "In step 42, Gödel makes use >>> of this assumption, but previously in step 23 he showed that that >>> assumption does not hold in all cases". (That's a hypothetical >>> example, of course.) >> >> His mistake can only be seen through a refutation of the essence of >> his conclusion. > > Ah, now that's an interesting assertion. Did you really mean "only"? "Needle in a hay stack" When you are looking for a particular needle in a humongous stack of needles it is very helpful to move this needle far away from all the other needles or you can't even see it separately. > So are you saying that you *cannot* demonstrate that Gödel proof is > incorrect by citing a specific error within the proof. It seems to me > that that's equivalent to saying that Gödel's proof is correct. I'm > sure that's not what you meant. Did you mean specifically that *you* > cannot do that? I doubt that that's what you meant either. If Gödel's proof is correct except for a single key false assumption then Gödel's proof is incorrect. > Are you saying that it's possible for every step of Gödel's proof > to be valid, but for the proof as a whole to be invalid, yielding a > false conclusion? If so, that's a remarkable assertion from someone > who says that a complex system can be complete and consistent. > A single false premise makes the conclusion unsound. > Do you believe there is a specific flaw in Gödel's proof? > (This question is not about what that flaw is, just whether you > think there is one.) > The definition of incompleteness is its flaw. We could define "incomplete" as a term of the art of mathematics such that every formal system that uses conjunction: "∧", disjunction: "∨", or negation: "¬" is "defined" to be "incomplete". This definition: A theory T is incomplete if and only if there is some sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ) is equally ridiculous when all of its implications are very carefully examined. Because people are so fully indoctrinated into that definition of incompleteness it is much much easier to prove that the next level inference based on that definition: "true and unprovable can coexist" is impossible. -- Copyright 2020 Pete Olcott
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| From | Keith Thompson <Keith.S.Thompson+u@gmail.com> |
|---|---|
| Date | 2020-07-16 11:46 -0700 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <87sgdrz49w.fsf@nosuchdomain.example.com> |
| In reply to | #21703 |
olcott <NoOne@NoWhere.com> writes:
> On 7/15/2020 8:42 PM, Keith Thompson wrote:
>> olcott <NoOne@NoWhere.com> writes:
>>> On 7/15/2020 1:04 PM, Keith Thompson wrote:
>>>> olcott <NoOne@NoWhere.com> writes:
>>>>> On 7/14/2020 1:25 PM, Keith Thompson wrote:
>>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>> [...]
>>>>>>> Since everyone here is indoctrinated into believing that Gödel is
>>>>>>> correct I have to use different terms for provability so that people
>>>>>>> will carefully analyze my reasoning and not simply dismiss it
>>>>>>> out-of-hand on the basis of their indoctrination.
>>>>>>
>>>>>> It seems to me that the best way to demonstrate that Gödel is
>>>>>> incorrect would be to demonstrate a flaw in what he actually wrote.
>>>>>> I haven't read everything you've written here, but I don't recall
>>>>>> you ever directly quoting Gödel's proof.
>>>>>
>>>>> Not really. When we refute the enormously simplified key result of his
>>>>> claim: true and unprovable can possibly coexist, then the steps that
>>>>> he used to get to this key result are moot.
>>>>
>>>> You've been asserting that for years, and nobody believes you
>>> https://scholar.google.com/scholar?hl=en&as_sdt=0%2C28&q=%22true+and+unprovable%22+godel&btnG=&oq=%22true+and+unprovable%22
>>
>> 125 results. No, I'm not going to read them.
>
> 125 different people that all believe that Gödel showed that true and
> unprovable formulas exists, and 125 > 0, thus "nobody believes you" is
> proven to be false.
Wait, what? Is that really what you meant to say? Gödel *did* show
that true and unprovable formulas exist. Did you omit a "not"?
OK, "nobody believes you" was hyperbole. I've seen nobody posting here
who believes that you've successfully refuted Gödel's proof. If you can
cite an exception, I suppose it would be mildly interesting, but not
particularly relevant other than to refute my statement. I'll gladly
revise it to "Hardly anybody believes you".
>>>> Do you think that's going to change if you assert it just one
>>>> more time? What is your goal here?
>>>>
>>>> Whether it's the best way or not, surely *a* way to demonstrate
>>>> that Gödel is incorrect would be to demonstrate a flaw in what he
>>>> actually wrote. Not in some summary of his proof, but in his actual
>>>> proof as he wrote it. Something like "In step 42, Gödel makes use
>>>> of this assumption, but previously in step 23 he showed that that
>>>> assumption does not hold in all cases". (That's a hypothetical
>>>> example, of course.)
>>>
>>> His mistake can only be seen through a refutation of the essence of
>>> his conclusion.
>>
>> Ah, now that's an interesting assertion. Did you really mean "only"?
>
> "Needle in a hay stack"
> When you are looking for a particular needle in a humongous stack of
> needles it is very helpful to move this needle far away from all the
> other needles or you can't even see it separately.
And again, your response to a yes or no question does not include the
word "yes" or "no".
>> So are you saying that you *cannot* demonstrate that Gödel proof is
>> incorrect by citing a specific error within the proof. It seems to me
>> that that's equivalent to saying that Gödel's proof is correct. I'm
>> sure that's not what you meant. Did you mean specifically that *you*
>> cannot do that? I doubt that that's what you meant either.
>
> If Gödel's proof is correct except for a single key false assumption
> then Gödel's proof is incorrect.
And again.
>> Are you saying that it's possible for every step of Gödel's proof
>> to be valid, but for the proof as a whole to be invalid, yielding a
>> false conclusion? If so, that's a remarkable assertion from someone
>> who says that a complex system can be complete and consistent.
>
> A single false premise makes the conclusion unsound.
And again. If I ask you a yes or no question, I will ignore any
response that does not include the word "yes" or "no", or explain
why neither "yes" nor "no" would be meaningful.
>> Do you believe there is a specific flaw in Gödel's proof?
>> (This question is not about what that flaw is, just whether you
>> think there is one.)
>>
> The definition of incompleteness is its flaw.
I'll take that as a yes, but next time I'll ask you to include the
word "yes" in your answer if that's what you mean.
Really? Is that your whole problem with Gödel's proof, that you
don't like the way he defines "incompleteness" (or more likely
"Unvollständigkeit")? (Of course the concept existed before Gödel.)
> We could define "incomplete" as a term of the art of mathematics such
> that every formal system that uses conjunction: "∧", disjunction: "∨",
> or negation: "¬" is "defined" to be "incomplete".
>
> This definition: A theory T is incomplete if and only if there is some
> sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ) is equally ridiculous when
> all of its implications are very carefully examined.
May I presume you have a rigorous definition of "ridiculous"?
> Because people are so fully indoctrinated into that definition of
> incompleteness it is much much easier to prove that the next level
> inference based on that definition: "true and unprovable can coexist"
> is impossible.
"Incompleteness" is just a word. I understand that you don't like the
way it's used.
So let's use different words. Perform the following replacements:
complete --> blurgicious
incomplete --> unblurgicious
completeness --> blurgitude
incompleteness --> unblurgitude
(I've tried to pick words that have no existing baggage, avoiding any
preconceived notions about what they mean, so we can define them
rigorously and without ambiguity.)
Suppose I gave you a copy of (an English translation of) Gödel's
proof with the above substitutions performed on it. Adapting a
definition from Wikipedia, a set of axioms is blurgicious if and
only if, for any statement in the axioms' language, that statement or
its negation is provable from the axioms. "Blurgicious" does *not*
mean "something is missing". Gödel proved that no sufficiently
complex formal system (basically <handwave>one able to represent the
axioms of the natural numbers </handwave>) can be both consistent and
blurgicious. (I'm assuming you don't have any issues with the word
"consistent". If you do, we can invent a replacement for it too.)
Now that we're not using the word "complete" or any form of it,
how do you refute this version of Gödel's proof?
--
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 | 2020-07-16 16:35 -0500 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <F5qdnUiiueGBVY3CnZ2dnUU7-Q-dnZ2d@giganews.com> |
| In reply to | #21706 |
On 7/16/2020 1:46 PM, Keith Thompson wrote: > olcott <NoOne@NoWhere.com> writes: >> On 7/15/2020 8:42 PM, Keith Thompson wrote: >>> olcott <NoOne@NoWhere.com> writes: >>>> On 7/15/2020 1:04 PM, Keith Thompson wrote: >>>>> olcott <NoOne@NoWhere.com> writes: >>>>>> On 7/14/2020 1:25 PM, Keith Thompson wrote: >>>>>>> olcott <NoOne@NoWhere.com> writes: >>>>>>> [...] >>>>>>>> Since everyone here is indoctrinated into believing that Gödel is >>>>>>>> correct I have to use different terms for provability so that people >>>>>>>> will carefully analyze my reasoning and not simply dismiss it >>>>>>>> out-of-hand on the basis of their indoctrination. >>>>>>> >>>>>>> It seems to me that the best way to demonstrate that Gödel is >>>>>>> incorrect would be to demonstrate a flaw in what he actually wrote. >>>>>>> I haven't read everything you've written here, but I don't recall >>>>>>> you ever directly quoting Gödel's proof. >>>>>> >>>>>> Not really. When we refute the enormously simplified key result of his >>>>>> claim: true and unprovable can possibly coexist, then the steps that >>>>>> he used to get to this key result are moot. >>>>> >>>>> You've been asserting that for years, and nobody believes you >>>> https://scholar.google.com/scholar?hl=en&as_sdt=0%2C28&q=%22true+and+unprovable%22+godel&btnG=&oq=%22true+and+unprovable%22 >>> >>> 125 results. No, I'm not going to read them. >> >> 125 different people that all believe that Gödel showed that true and >> unprovable formulas exists, and 125 > 0, thus "nobody believes you" is >> proven to be false. > > Wait, what? Is that really what you meant to say? Gödel *did* show > that true and unprovable formulas exist. Did you omit a "not"? > OK that is even better. I thought that he only concluded that some formulas are neither provable nor disprovable. > OK, "nobody believes you" was hyperbole. I've seen nobody posting here > who believes that you've successfully refuted Gödel's proof. If you can > cite an exception, I suppose it would be mildly interesting, but not > particularly relevant other than to refute my statement. I'll gladly > revise it to "Hardly anybody believes you". > How can you show that Gödel really showed that true and unprovable formulas exist. In other words and this is not merely an interpretation that someone added later on. The best way would be to quote his paper where he would say: "I just proved that true and unprovable formulas exist". Even his use of natural language seems to be about as convoluted as he can possibly make it. If he were to say: "I just proved that true and unprovable formulas exist" it would take him at least fifteen pages. >>>>> Do you think that's going to change if you assert it just one >>>>> more time? What is your goal here? >>>>> >>>>> Whether it's the best way or not, surely *a* way to demonstrate >>>>> that Gödel is incorrect would be to demonstrate a flaw in what he >>>>> actually wrote. Not in some summary of his proof, but in his actual >>>>> proof as he wrote it. Something like "In step 42, Gödel makes use >>>>> of this assumption, but previously in step 23 he showed that that >>>>> assumption does not hold in all cases". (That's a hypothetical >>>>> example, of course.) >>>> >>>> His mistake can only be seen through a refutation of the essence of >>>> his conclusion. >>> >>> Ah, now that's an interesting assertion. Did you really mean "only"? >> Yes >> "Needle in a hay stack" >> When you are looking for a particular needle in a humongous stack of >> needles it is very helpful to move this needle far away from all the >> other needles or you can't even see it separately. > > And again, your response to a yes or no question does not include the > word "yes" or "no". > >>> So are you saying that you *cannot* demonstrate that Gödel proof is >>> incorrect by citing a specific error within the proof. It seems to me >>> that that's equivalent to saying that Gödel's proof is correct. I'm >>> sure that's not what you meant. Did you mean specifically that *you* >>> cannot do that? I doubt that that's what you meant either. >> >> If Gödel's proof is correct except for a single key false assumption >> then Gödel's proof is incorrect. > > And again. > >>> Are you saying that it's possible for every step of Gödel's proof >>> to be valid, but for the proof as a whole to be invalid, yielding a >>> false conclusion? If so, that's a remarkable assertion from someone >>> who says that a complex system can be complete and consistent. >> >> A single false premise makes the conclusion unsound. > > And again. If I ask you a yes or no question, I will ignore any > response that does not include the word "yes" or "no", or explain > why neither "yes" nor "no" would be meaningful. I am fully refuting your general whole point as it can be applied to Gödel or anything else. If it can be proved that a conclusion is incorrect then there must have been some error somewhere in the reasoning that lead to the conclusion. No need to even look at this reasoning as long as its conclsion can be proved to be incorrect. > >>> Do you believe there is a specific flaw in Gödel's proof? >>> (This question is not about what that flaw is, just whether you >>> think there is one.) >>> >> The definition of incompleteness is its flaw. > > I'll take that as a yes, but next time I'll ask you to include the > word "yes" in your answer if that's what you mean. > > Really? Is that your whole problem with Gödel's proof, that you > don't like the way he defines "incompleteness" (or more likely > "Unvollständigkeit")? (Of course the concept existed before Gödel.) If he proved that there are true and unprovable formulas once you understand how True(x) really works you will see that it is the same as if he proved 3 > 7, utterly impossibly correct. > >> We could define "incomplete" as a term of the art of mathematics such >> that every formal system that uses conjunction: "∧", disjunction: "∨", >> or negation: "¬" is "defined" to be "incomplete". >> >> This definition: A theory T is incomplete if and only if there is some >> sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ) is equally ridiculous when >> all of its implications are very carefully examined. > > May I presume you have a rigorous definition of "ridiculous"? Ideas deserving of disparagement? > >> Because people are so fully indoctrinated into that definition of >> incompleteness it is much much easier to prove that the next level >> inference based on that definition: "true and unprovable can coexist" >> is impossible. > > "Incompleteness" is just a word. I understand that you don't like the > way it's used. It it much much easier to refer to "true and unprovable" because these have corresponding algorithms. The fact that math people believe that an X is a Y has no corresponding algorithm. > So let's use different words. Perform the following replacements: > complete --> blurgicious > incomplete --> unblurgicious > completeness --> blurgitude > incompleteness --> unblurgitude > > (I've tried to pick words that have no existing baggage, avoiding any > preconceived notions about what they mean, so we can define them > rigorously and without ambiguity.) > > Suppose I gave you a copy of (an English translation of) Gödel's > proof with the above substitutions performed on it. Adapting a > definition from Wikipedia, a set of axioms is blurgicious if and > only if, for any statement in the axioms' language, that statement or > its negation is provable from the axioms. "Blurgicious" does *not* > mean "something is missing". Gödel proved that no sufficiently > complex formal system (basically <handwave>one able to represent the > axioms of the natural numbers </handwave>) can be both consistent and > blurgicious. (I'm assuming you don't have any issues with the word > "consistent". If you do, we can invent a replacement for it too.) > > Now that we're not using the word "complete" or any form of it, > how do you refute this version of Gödel's proof? > There must be some mathematical mapping from a finite string (or expression of language) "A" to a Boolean value: "B". It can be a mapping through rules-of-inference, axioms, and models, but it still must be a mapping from A to B or it is incorrect. It cannot be a mapping to a short-circuit that leaps from "A" to "B". The problem with mathematics is that key aspects can only be described using natural language. This is the source of its inference gaps. When the same ideas are specified in computer science all the gaps can be closed. A language L on Σ is said to be recursive if there exists a Turing machine M that accepts L and halts on every w in Σ+. In other words, a language is recursive if and only if there exists a membership algorithm for it. (Linz 1990:288). When we look at the concept of analytical truth in its most abstract and general form True(x) is decided by a membership algorithm. Is-a-member(x) Is-a-theorem(x) Is-true(x) are all different names for the exact same thing. Since they are all names for the exact same thing that they are ever mutually exclusive is utterly impossible. -- Copyright 2020 Pete Olcott
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| From | Keith Thompson <Keith.S.Thompson+u@gmail.com> |
|---|---|
| Date | 2020-07-16 15:19 -0700 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V24 (Are we there yet?) |
| Message-ID | <874kq7yug9.fsf@nosuchdomain.example.com> |
| In reply to | #21715 |
olcott <NoOne@NoWhere.com> writes:
> On 7/16/2020 1:46 PM, Keith Thompson wrote:
>> olcott <NoOne@NoWhere.com> writes:
>>> On 7/15/2020 8:42 PM, Keith Thompson wrote:
>>>> olcott <NoOne@NoWhere.com> writes:
>>>>> On 7/15/2020 1:04 PM, Keith Thompson wrote:
>>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>>> On 7/14/2020 1:25 PM, Keith Thompson wrote:
>>>>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>>>> [...]
>>>>>>>>> Since everyone here is indoctrinated into believing that Gödel is
>>>>>>>>> correct I have to use different terms for provability so that people
>>>>>>>>> will carefully analyze my reasoning and not simply dismiss it
>>>>>>>>> out-of-hand on the basis of their indoctrination.
>>>>>>>>
>>>>>>>> It seems to me that the best way to demonstrate that Gödel is
>>>>>>>> incorrect would be to demonstrate a flaw in what he actually wrote.
>>>>>>>> I haven't read everything you've written here, but I don't recall
>>>>>>>> you ever directly quoting Gödel's proof.
>>>>>>>
>>>>>>> Not really. When we refute the enormously simplified key result of his
>>>>>>> claim: true and unprovable can possibly coexist, then the steps that
>>>>>>> he used to get to this key result are moot.
>>>>>>
>>>>>> You've been asserting that for years, and nobody believes you
>>>>> https://scholar.google.com/scholar?hl=en&as_sdt=0%2C28&q=%22true+and+unprovable%22+godel&btnG=&oq=%22true+and+unprovable%22
>>>>
>>>> 125 results. No, I'm not going to read them.
>>>
>>> 125 different people that all believe that Gödel showed that true and
>>> unprovable formulas exists, and 125 > 0, thus "nobody believes you" is
>>> proven to be false.
>>
>> Wait, what? Is that really what you meant to say? Gödel *did* show
>> that true and unprovable formulas exist. Did you omit a "not"?
>
> OK that is even better. I thought that he only concluded that some
> formulas are neither provable nor disprovable.
Quoting https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems :
The first incompleteness theorem states that no consistent system
of axioms whose theorems can be listed by an effective procedure
(i.e., an algorithm) is capable of proving all truths about the
arithmetic of natural numbers. For any such consistent formal
system, there will always be statements about natural numbers
that are true, but that are unprovable within the system. The
second incompleteness theorem, an extension of the first,
shows that the system cannot demonstrate its own consistency.
Your "OK that is even better" remark seems to imply that you didn't know
that, which is frankly staggering. You've spent years claiming to have
refuted Gödel's proof, but you don't even know what he proved.
>> OK, "nobody believes you" was hyperbole. I've seen nobody posting here
>> who believes that you've successfully refuted Gödel's proof. If you can
>> cite an exception, I suppose it would be mildly interesting, but not
>> particularly relevant other than to refute my statement. I'll gladly
>> revise it to "Hardly anybody believes you".
>
> How can you show that Gödel really showed that true and unprovable
> formulas exist. In other words and this is not merely an
> interpretation that someone added later on.
By reading and understanding his proof. I'm honestly not prepared
to do that myself. I think that some other participants here have,
and I'll let them comment further if they choose to do so.
Here's another yes or no question: Have you read Gödel's proof?
> The best way would be to quote his paper where he would say:
> "I just proved that true and unprovable formulas exist".
>
> Even his use of natural language seems to be about as convoluted as he
> can possibly make it. If he were to say: "I just proved that true and
> unprovable formulas exist" it would take him at least fifteen pages.
>
>>>>>> Do you think that's going to change if you assert it just one
>>>>>> more time? What is your goal here?
>>>>>>
>>>>>> Whether it's the best way or not, surely *a* way to demonstrate
>>>>>> that Gödel is incorrect would be to demonstrate a flaw in what he
>>>>>> actually wrote. Not in some summary of his proof, but in his actual
>>>>>> proof as he wrote it. Something like "In step 42, Gödel makes use
>>>>>> of this assumption, but previously in step 23 he showed that that
>>>>>> assumption does not hold in all cases". (That's a hypothetical
>>>>>> example, of course.)
>>>>>
>>>>> His mistake can only be seen through a refutation of the essence of
>>>>> his conclusion.
>>>>
>>>> Ah, now that's an interesting assertion. Did you really mean "only"?
>
> Yes
Well, I wasn't expecting that answer.
You seem to be saying that it would *not* be possible to refute Gödel's
proof by finding a specific flaw in it. Is that really what you mean?
>>> "Needle in a hay stack"
>>> When you are looking for a particular needle in a humongous stack of
>>> needles it is very helpful to move this needle far away from all the
>>> other needles or you can't even see it separately.
>>
>> And again, your response to a yes or no question does not include the
>> word "yes" or "no".
>>
>>>> So are you saying that you *cannot* demonstrate that Gödel proof is
>>>> incorrect by citing a specific error within the proof. It seems to me
>>>> that that's equivalent to saying that Gödel's proof is correct. I'm
>>>> sure that's not what you meant. Did you mean specifically that *you*
>>>> cannot do that? I doubt that that's what you meant either.
>>>
>>> If Gödel's proof is correct except for a single key false assumption
>>> then Gödel's proof is incorrect.
>>
>> And again.
>>
>>>> Are you saying that it's possible for every step of Gödel's proof
>>>> to be valid, but for the proof as a whole to be invalid, yielding a
>>>> false conclusion? If so, that's a remarkable assertion from someone
>>>> who says that a complex system can be complete and consistent.
>>>
>>> A single false premise makes the conclusion unsound.
>>
>> And again. If I ask you a yes or no question, I will ignore any
>> response that does not include the word "yes" or "no", or explain
>> why neither "yes" nor "no" would be meaningful.
>
> I am fully refuting your general whole point as it can be applied to
> Gödel or anything else.
>
> If it can be proved that a conclusion is incorrect then there must
> have been some error somewhere in the reasoning that lead to the
> conclusion. No need to even look at this reasoning as long as its
> conclsion can be proved to be incorrect.
You have not answered my question.
Are you saying that it's possible for every step of Gödel's proof to be
valid, but for the proof as a whole to be invalid, yielding a false
conclusion? Yes or no, please.
>>>> Do you believe there is a specific flaw in Gödel's proof?
>>>> (This question is not about what that flaw is, just whether you
>>>> think there is one.)
>>>>
>>> The definition of incompleteness is its flaw.
>>
>> I'll take that as a yes, but next time I'll ask you to include the
>> word "yes" in your answer if that's what you mean.
>>
>> Really? Is that your whole problem with Gödel's proof, that you
>> don't like the way he defines "incompleteness" (or more likely
>> "Unvollständigkeit")? (Of course the concept existed before Gödel.)
>
> If he proved that there are true and unprovable formulas once you
> understand how True(x) really works you will see that it is the same
> as if he proved 3 > 7, utterly impossibly correct.
Yes or no, please.
>>> We could define "incomplete" as a term of the art of mathematics such
>>> that every formal system that uses conjunction: "∧", disjunction: "∨",
>>> or negation: "¬" is "defined" to be "incomplete".
>>>
>>> This definition: A theory T is incomplete if and only if there is some
>>> sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ) is equally ridiculous when
>>> all of its implications are very carefully examined.
>>
>> May I presume you have a rigorous definition of "ridiculous"?
>
> Ideas deserving of disparagement?
I'll take that as a no, you don't have a rigorous definition of
"ridiculous".
In my opinion, concepts like "ridiculous", "deserving", and
"disparagement" have no legitimate place in this discussion.
>>> Because people are so fully indoctrinated into that definition of
>>> incompleteness it is much much easier to prove that the next level
>>> inference based on that definition: "true and unprovable can coexist"
>>> is impossible.
>>
>> "Incompleteness" is just a word. I understand that you don't like the
>> way it's used.
>
> It it much much easier to refer to "true and unprovable" because these
> have corresponding algorithms.
>
> The fact that math people believe that an X is a Y has no
> corresponding algorithm.
>
>> So let's use different words. Perform the following replacements:
>> complete --> blurgicious
>> incomplete --> unblurgicious
>> completeness --> blurgitude
>> incompleteness --> unblurgitude
>>
>> (I've tried to pick words that have no existing baggage, avoiding any
>> preconceived notions about what they mean, so we can define them
>> rigorously and without ambiguity.)
>>
>> Suppose I gave you a copy of (an English translation of) Gödel's
>> proof with the above substitutions performed on it. Adapting a
>> definition from Wikipedia, a set of axioms is blurgicious if and
>> only if, for any statement in the axioms' language, that statement or
>> its negation is provable from the axioms. "Blurgicious" does *not*
>> mean "something is missing". Gödel proved that no sufficiently
>> complex formal system (basically <handwave>one able to represent the
>> axioms of the natural numbers </handwave>) can be both consistent and
>> blurgicious. (I'm assuming you don't have any issues with the word
>> "consistent". If you do, we can invent a replacement for it too.)
>>
>> Now that we're not using the word "complete" or any form of it,
>> how do you refute this version of Gödel's proof?
>
> There must be some mathematical mapping from a finite string (or
> expression of language) "A" to a Boolean value: "B".
>
> It can be a mapping through rules-of-inference, axioms, and models,
> but it still must be a mapping from A to B or it is incorrect.
>
> It cannot be a mapping to a short-circuit that leaps from "A" to "B".
>
> The problem with mathematics is that key aspects can only be described
> using natural language. This is the source of its inference gaps. When
> the same ideas are specified in computer science all the gaps can be
> closed.
>
> A language L on Σ is said to be recursive if there exists a Turing
> machine M that accepts L and halts on every w in Σ+. In other words, a
> language is recursive if and only if there exists a membership
> algorithm for it. (Linz 1990:288).
>
> When we look at the concept of analytical truth in its most abstract
> and general form True(x) is decided by a membership algorithm.
>
> Is-a-member(x) Is-a-theorem(x) Is-true(x) are all different names for
> the exact same thing. Since they are all names for the exact same
> thing that they are ever mutually exclusive is utterly impossible.
I see no meaningful response to my question other than a restatement
of your previous claims.
If Gödel's proof is invalid, then you should be able to demonstrate a
specific flaw in the proof itself. Please do so.
--
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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