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Groups > comp.theory > #21791 > unrolled thread
| Started by | olcott <NoOne@NoWhere.com> |
|---|---|
| First post | 2020-07-18 15:31 -0500 |
| Last post | 2020-07-24 19:10 -0500 |
| Articles | 20 on this page of 247 — 10 participants |
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Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-18 15:31 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-19 03:35 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-19 11:37 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) André G. Isaak <agisaak@gm.invalid> - 2020-07-19 11:25 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-19 12:46 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) André G. Isaak <agisaak@gm.invalid> - 2020-07-19 11:56 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-19 13:50 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) André G. Isaak <agisaak@gm.invalid> - 2020-07-19 14:06 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-19 13:39 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-19 17:21 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-19 15:57 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-20 01:55 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-19 23:52 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) André G. Isaak <agisaak@gm.invalid> - 2020-07-19 23:15 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-20 12:56 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) André G. Isaak <agisaak@gm.invalid> - 2020-07-20 12:48 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Jeff Barnett <jbb@notatt.com> - 2020-07-20 14:49 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-20 21:42 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) André G. Isaak <agisaak@gm.invalid> - 2020-07-21 10:51 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-21 12:02 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) André G. Isaak <agisaak@gm.invalid> - 2020-07-21 12:12 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-21 20:09 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) André G. Isaak <agisaak@gm.invalid> - 2020-07-21 19:35 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-22 09:47 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) André G. Isaak <agisaak@gm.invalid> - 2020-07-22 09:23 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) André G. Isaak <agisaak@gm.invalid> - 2020-07-22 10:21 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) André G. Isaak <agisaak@gm.invalid> - 2020-07-22 13:22 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-20 18:20 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) André G. Isaak <agisaak@gm.invalid> - 2020-07-20 18:28 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-20 21:41 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-20 11:59 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-20 21:44 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-20 20:03 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-21 10:49 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) André G. Isaak <agisaak@gm.invalid> - 2020-07-21 10:47 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-21 11:54 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) André G. Isaak <agisaak@gm.invalid> - 2020-07-21 11:01 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-21 19:54 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) André G. Isaak <agisaak@gm.invalid> - 2020-07-21 19:23 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-22 09:23 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) André G. Isaak <agisaak@gm.invalid> - 2020-07-22 09:11 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-22 17:02 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-21 11:53 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-21 11:31 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-21 20:22 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-21 00:12 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-20 19:18 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-22 03:32 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Kaz Kylheku <793-849-0957@kylheku.com> - 2020-07-22 05:39 +0000
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-22 10:16 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-23 00:54 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-23 21:12 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Kaz Kylheku <793-849-0957@kylheku.com> - 2020-07-24 16:31 +0000
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-24 12:40 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Jeff Barnett <jbb@notatt.com> - 2020-07-24 12:06 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-24 13:26 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-24 11:38 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-24 14:05 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-24 12:33 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-24 14:42 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Kaz Kylheku <793-849-0957@kylheku.com> - 2020-07-26 15:05 +0000
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-24 19:57 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-24 14:16 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Jeff Barnett <jbb@notatt.com> - 2020-07-24 14:43 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-24 16:00 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) David Kleinecke <dkleinecke@gmail.com> - 2020-07-24 15:38 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-24 17:45 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) David Kleinecke <dkleinecke@gmail.com> - 2020-07-24 15:57 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-24 16:11 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-24 15:52 -0700
Re: Simply defining G"odel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) R Kym Horsell <kym@kymhorsell.com> - 2020-07-24 19:02 +0000
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-26 10:27 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-26 10:36 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-26 10:39 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-24 20:20 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-24 14:37 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-25 00:13 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-24 19:25 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-24 12:02 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-24 23:49 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-24 18:49 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-25 02:28 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Jeff Barnett <jbb@notatt.com> - 2020-07-24 22:18 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) David Kleinecke <dkleinecke@gmail.com> - 2020-07-24 21:58 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Jeff Barnett <jbb@notatt.com> - 2020-07-25 01:46 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) David Kleinecke <dkleinecke@gmail.com> - 2020-07-25 10:31 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-25 12:04 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Jeff Barnett <jbb@notatt.com> - 2020-07-25 14:45 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-26 00:51 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Jeff Barnett <jbb@notatt.com> - 2020-07-25 22:46 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-26 12:10 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-25 22:53 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-26 20:46 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-26 16:42 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-26 15:46 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-26 22:46 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-27 00:28 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-26 17:05 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-27 02:52 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-27 00:30 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-27 00:33 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-26 22:39 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-27 14:12 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-27 12:23 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-27 23:41 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) André G. Isaak <agisaak@gm.invalid> - 2020-07-27 17:37 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-27 19:08 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) André G. Isaak <agisaak@gm.invalid> - 2020-07-27 19:57 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-27 21:14 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) André G. Isaak <agisaak@gm.invalid> - 2020-07-27 20:45 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-27 21:55 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) André G. Isaak <agisaak@gm.invalid> - 2020-07-27 22:58 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-28 00:07 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) André G. Isaak <agisaak@gm.invalid> - 2020-07-27 23:16 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-28 00:38 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) André G. Isaak <agisaak@gm.invalid> - 2020-07-28 00:05 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-28 10:00 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) André G. Isaak <agisaak@gm.invalid> - 2020-07-28 21:32 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-28 22:44 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-28 02:01 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-27 20:45 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-28 03:11 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-27 21:16 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-28 03:24 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-27 21:44 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-28 17:33 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-28 11:40 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-28 11:46 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-28 22:58 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-28 17:13 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-29 00:51 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-28 21:05 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-29 12:04 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-29 12:42 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-29 20:24 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-29 15:38 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-30 01:31 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-30 10:29 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-30 17:45 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-31 10:55 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-28 17:50 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-28 11:57 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-28 20:30 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-28 14:38 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-28 22:52 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-28 17:09 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-29 01:00 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-28 21:29 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2020-07-29 15:47 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-29 13:20 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-29 12:13 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-29 15:16 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-29 14:43 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-29 22:37 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-29 18:34 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-30 02:01 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-31 10:34 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-08-01 01:48 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-08-01 10:58 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-08-01 21:24 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-08-01 15:33 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-08-02 00:24 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-08-02 09:51 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-08-02 17:29 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-08-03 09:41 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-08-03 17:39 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-08-02 18:04 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) André G. Isaak <agisaak@gm.invalid> - 2020-08-01 23:19 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-29 16:33 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-29 13:31 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) David Kleinecke <dkleinecke@gmail.com> - 2020-07-29 12:02 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-29 14:57 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-29 22:47 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) David Kleinecke <dkleinecke@gmail.com> - 2020-07-29 15:33 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-29 20:50 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-29 18:51 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-30 02:38 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-31 10:36 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-08-01 00:47 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-08-01 01:30 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-08-01 12:55 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-08-01 10:45 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-08-01 20:28 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-08-01 15:10 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-08-02 00:11 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-08-02 09:43 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-08-02 17:20 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-08-03 09:39 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-08-03 17:46 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-30 00:28 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-28 23:31 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-29 16:43 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-29 14:02 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-30 00:36 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-29 21:44 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) André G. Isaak <agisaak@gm.invalid> - 2020-07-29 21:50 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) André G. Isaak <agisaak@gm.invalid> - 2020-07-29 21:55 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-31 10:48 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-31 12:45 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-31 14:55 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-31 13:13 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-31 16:16 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) David Kleinecke <dkleinecke@gmail.com> - 2020-07-31 14:42 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-31 18:16 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-31 14:44 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Jeff Barnett <jbb@notatt.com> - 2020-07-31 14:02 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) André G. Isaak <agisaak@gm.invalid> - 2020-07-31 17:58 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-31 22:33 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-31 10:43 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-30 13:39 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-31 10:51 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-08-01 00:58 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-08-01 10:52 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-08-01 21:02 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-08-02 09:28 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-08-02 17:45 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-08-03 09:46 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-08-03 17:46 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-08-03 13:22 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Andy Walker <anw@cuboid.co.uk> - 2020-07-30 20:35 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) David Kleinecke <dkleinecke@gmail.com> - 2020-07-30 12:57 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-30 17:20 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Andy Walker <anw@cuboid.co.uk> - 2020-07-30 23:24 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) David Kleinecke <dkleinecke@gmail.com> - 2020-07-30 16:30 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Andy Walker <anw@cuboid.co.uk> - 2020-07-31 01:31 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) David Kleinecke <dkleinecke@gmail.com> - 2020-07-30 20:36 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-31 00:12 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-30 20:41 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-30 17:10 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) David Kleinecke <dkleinecke@gmail.com> - 2020-07-30 16:31 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-30 20:50 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) David Kleinecke <dkleinecke@gmail.com> - 2020-07-30 20:40 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-31 01:34 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-28 10:41 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-28 12:51 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Kaz Kylheku <793-849-0957@kylheku.com> - 2020-07-28 08:23 +0000
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-28 09:20 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-26 13:00 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-26 22:45 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-27 16:17 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-27 23:51 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-26 22:46 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) olcott <NoOne@NoWhere.com> - 2020-07-27 17:14 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-28 00:22 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-24 15:57 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-07-25 00:03 +0100
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V31 (Semantically Incorrect Defined) olcott <NoOne@NoWhere.com> - 2020-07-24 19:10 -0500
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| From | olcott <NoOne@NoWhere.com> |
|---|---|
| Date | 2020-07-24 18:49 -0500 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <Zd6dnViKxvw77obCnZ2dnUU7-fnNnZ2d@giganews.com> |
| In reply to | #21908 |
On 7/24/2020 5:49 PM, Ben Bacarisse wrote:
> olcott <NoOne@NoWhere.com> writes:
>
>> On 7/22/2020 6:54 PM, Ben Bacarisse wrote:
>>> olcott <NoOne@NoWhere.com> writes:
>>>
>>>>> Try reading page 57 of Mendelson first. I'll answer any reasonable
>>>>> questions you might have about that page.
>>>>
>>>> I have a hard copy of edition 4 and and PDF copy of edition 6. I
>>>> assume you mean edition 6.
>>>
>>> 4th I think. Anyway, the page where the concept of an interpretation
>>> is introduced.
>> I think that I totally understand this:
>> Page 58 fourth edition Mendelson
>> The super script ^n in a name means its number of arguments
>> The subscript _i in a name makes this name unique
>>
>> Definition
>> 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.
>>
>> My understanding: [The set of elements (integers, humans ...) that can
>> be assigned to the free variables]
>
> Not just the free variables. Lets take D = ℕ and use the usual infix
> notation with variables that don't have subscripts. The formula
>
> x < y
>
> has two free variables and is satisfied by some assignment and not for
> others. The sentence
>
> ∀x∀y x < y
>
> has no free variables but is false (in this interpretation) because
> (informally) x < y is not the case for every assignment that could be
> made. Mendelson starts with this rather informal, intuitive notion but
> then makes it technically watertight by explaining how you deal with
> sentences like this on the pages that follow.
>
>> b. For each predicate letter A^n_j of L, an assignment of an n-place
>> relation (A^n_j)^M in D.
>>
>> My understanding: [selects the subset of n-tuples of D that satisfy
>> this relation]
>
> If that wording help you, fine. Technically (A^n_j)^M /is/ a set of
> n-tuples. If D is ℕ and we use the usual interpretation of <, then < is
> just a set of pairs: {(0, 1), (0, 2), ... (1, 2), (1, 3), ... } with a
> funny-looking name.
>
Yet a subset of the set of every n-tuple in ℕ.
(1,0) is an n-tuple in the domain that is excluded.
>> My understanding: [predicates are functions that evaluate to Boolean]
>
> Take care here. That's programmer language. The interpretation of a
> predicate is a set of pairs. There's not really any evaluation and
> there is definitely no need for a predicate to be computable. See the
> next comment for why the distinction actually helps.
Oh I see the predicate Father_of(x, y) normally evaluates to Boolean for
a specific ordered pair of individual people. As a relation on a domain
it defines the whole set of ordered pairs that satisfy that relation.
constant h1 ∈ human beings
constant h2 ∈ human beings
(h1,h2) ∈ Father_of(x, y) would assert that h1 is the father of h2.
It would seem that a predicate that is satisfied would be computable,
because an answer of Yes guarantees that an answer exists.
>> c. For each function letter f^n_j of L, an assignment of an n-place
>> operation (f^n_j)^M in D (that is, a function from D^n into D).
>>
>> My understanding: [functions evaluate to non _Boolean]
>
> More to be wary of here. If the interpretation domain is {true, false}
> or includes {true, false} then functions can "evaluate" to something you
> might call _Boolean. (I don't know exactly what you mean by _Boolean.)
> And that's fine. There is no confusion with predicates because they are
> not functions and you can't mix functions and predicates in arbitrary
> ways.
>
It seems that predicates always [end up with] a possibly empty set of
n-tuples. What is the correct term for [end up with]?
> And again, there is no need for functions to be computable.
>
>> My understanding: [father_of(x) would evaluate to a unique constant
>> element of the set of humans]
>
> I would use use "evaluate". And there is no need for unique -- the
> definition of a function does not permit f(x) to be anything by one
> thing. And the term "constant" is part of the syntax of the language.
> It has no meaning here. If the domain is the set H of humans (not very
> well-defined but let's let that slip) then father_of(x) is an element of
> H.
That sure was simple.
>
> It's unfortunate you picked an example function with the same name as
> Mendelson's example predicate (see below). I hope your example does not
> confuse you.
>
>> My estimate: [functions must always evaluate to elements of D]
>
> Yes, that's explicitly in the text.
>
Great. Any probably also sets of elements of D. GrandfathersOf(x).
>> d. For each individual constant a_i of L, an assignment of some fixed
>> element (a_i)^M of D. According to Wikipedia 0-ary functions are also
>> considered to be constants.
>
> It's the other way round. Some authors don't bother with constants but
> instead permit 0-ary functions to serve the same purpose.
OK that makes things simple.
>> My estimate: [constants are unique elements 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 D^n, 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.
>>
>> My understanding: [if there are n human beings then there are n^2 2-tuples]
>>
>> This is where my understanding drops off. I can't see how his
>> notation translates from the subset of n-tuples of D that satisfy
>> n-ary relation_i.
>
> First, are you ok with this informal notation of whether a formula is
> can be satisfied, and of whether a sentence is true or false, in some
> interpretation? I ask because the technical definition is harder to
> understand than the idea. You may no need to go though every technical
> details provided you grasp the intuitive idea.
>
I think that I have a very good gist of all of the basis ideas. The big
correction is that a relation [end up with] a possibly empty set of
n-tuples instead of a Boolean.
(3,7) ∈ Less_Than(x, y) where the domain is ℕ seems to be the way that
you assert: 3 < 7.
> To be sure, I would insist that a student do exercises 2.10 and 2.11.
> Can you do them?
I will look at them now that you made sure my basic understading seems OK.
>
>> Page 59 fourth edition Mendelson
>> 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.
>>
>> Satisfiability will be the fundamental notion, on the basis of which
>> the notion of truth will be defined. Moreover, instead of talking
>> about the n-tuples of objects that satisfy a wf that has n free
>> variables, it is much more convenient from a technical standpoint to
>> deal uniformly with denumerable sequences. What we have in mind is
>> that a denumerable sequence s = (s1, s2, s3, ...) is to be thought of
>> as satisfying a wf B that has xj1, xj2 ,..., xjn as free variables
>> (where j1 < j2 < ... < jn)
>>
>> if the n-tuple <sj1, sj2 ,..., sjn> .satisfies B in the usual
>> sense. For example, a denumerable sequence (s1, s2, s3, ...) of
>> objects in the domain of an interpretation M will turn out to satisfy
>> the wf A21(x2, x5) if and only if the ordered pair, <s2, s5> is in the
>> relation (A21)M assigned to the predicate letter A21 by the
>> interpretation M.
>>
>> Let M be an interpretation of a language L and let D be the domain of
>> M. Let Σ be the set of all denumerable sequences of elements of D. For
>> a wf B of L, we shall define what it means for a sequence s = (s1, s2,
>> ...) in Σ to satisfy B in M. As a preliminary step, for a given s in Σ
>> we shall define a function s* that assigns to each term t of L an
>> element s*(t) in D.
>
> This is just quoting with no questions. I guess your question is what
> it all means, but that is much better discussed after you grasp the
> informal notion because it is all simply detailed text to explain the
> simple idea.
>
> But to get you started, the basic idea is that a sequence (an infinite
> sequence) from the domain is used to provide values for the variables in
> a formula. So (1, 0, 0, 2, 34, 7, 89, ...) assigns 1 to x_1, 0 to x_2
> and so on. For any one formula there will be infinitely many irrelevant
> values, but it turns out to be simpler to define it like this.
>
>> instead of talking
>> about the n-tuples of objects that satisfy a wf that has n free
>> variables,
This is where I get totally lost in the details of the encoding.
>> it is much more convenient from a technical standpoint to
>> deal uniformly with denumerable sequences.
[the n-tuples of objects that satisfy a wf that has n free variables],
are syntactically converted into other notation that seems to make no
sense.
--
Copyright 2020 Pete Olcott
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2020-07-25 02:28 +0100 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <877dusjsda.fsf@bsb.me.uk> |
| In reply to | #21918 |
olcott <NoOne@NoWhere.com> writes:
> On 7/24/2020 5:49 PM, Ben Bacarisse wrote:
>> olcott <NoOne@NoWhere.com> writes:
>>
>>> On 7/22/2020 6:54 PM, Ben Bacarisse wrote:
>>>> olcott <NoOne@NoWhere.com> writes:
>>>>
>>>>>> Try reading page 57 of Mendelson first. I'll answer any reasonable
>>>>>> questions you might have about that page.
>>>>>
>>>>> I have a hard copy of edition 4 and and PDF copy of edition 6. I
>>>>> assume you mean edition 6.
>>>>
>>>> 4th I think. Anyway, the page where the concept of an interpretation
>>>> is introduced.
>>> I think that I totally understand this:
>>> Page 58 fourth edition Mendelson
>>> The super script ^n in a name means its number of arguments
>>> The subscript _i in a name makes this name unique
>>>
>>> Definition
>>> 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.
>>>
>>> My understanding: [The set of elements (integers, humans ...) that can
>>> be assigned to the free variables]
>>
>> Not just the free variables. Lets take D = ℕ and use the usual infix
>> notation with variables that don't have subscripts. The formula
>>
>> x < y
>>
>> has two free variables and is satisfied by some assignment and not for
>> others. The sentence
>>
>> ∀x∀y x < y
>>
>> has no free variables but is false (in this interpretation) because
>> (informally) x < y is not the case for every assignment that could be
>> made. Mendelson starts with this rather informal, intuitive notion but
>> then makes it technically watertight by explaining how you deal with
>> sentences like this on the pages that follow.
>>
>>> b. For each predicate letter A^n_j of L, an assignment of an n-place
>>> relation (A^n_j)^M in D.
>>>
>>> My understanding: [selects the subset of n-tuples of D that satisfy
>>> this relation]
>>
>> If that wording help you, fine. Technically (A^n_j)^M /is/ a set of
>> n-tuples. If D is ℕ and we use the usual interpretation of <, then < is
>> just a set of pairs: {(0, 1), (0, 2), ... (1, 2), (1, 3), ... } with a
>> funny-looking name.
>
> Yet a subset of the set of every n-tuple in ℕ.
> (1,0) is an n-tuple in the domain that is excluded.
I don't understand. I suspect you are using domain wrongly. (1, 0) is
not in the set of pairs that defines the < predicate.
>>> My understanding: [predicates are functions that evaluate to Boolean]
>>
>> Take care here. That's programmer language. The interpretation of a
>> predicate is a set of pairs. There's not really any evaluation and
>> there is definitely no need for a predicate to be computable. See the
>> next comment for why the distinction actually helps.
>
> Oh I see the predicate Father_of(x, y) normally evaluates to Boolean
> for a specific ordered pair of individual people. As a relation on a
> domain it defines the whole set of ordered pairs that satisfy that
> relation.
It /is/ the set of pairs that are in the relation. It does not
"evaluate to Boolean". Nothing is evaluated. For a relation R the
notation R(x, y) is just another way of writing (x, y) ∈ R.
> constant h1 ∈ human beings
> constant h2 ∈ human beings
> (h1,h2) ∈ Father_of(x, y) would assert that h1 is the father of h2.
Please, for the moment, no poems. None of that is written correctly.
At some point you will have to learn the rules for writing actual
mathematics rather than metaphorical hints, but for the moment try to
copy the syntax used by Mendelson.
> It would seem that a predicate that is satisfied would be computable,
> because an answer of Yes guarantees that an answer exists.
Could you refrain from injecting your own ideas for a while? I will get
side-tracked by explaining why things like this are wrong. For the
moment, a predicate in an interpretation is just a set of pairs.
>>> c. For each function letter f^n_j of L, an assignment of an n-place
>>> operation (f^n_j)^M in D (that is, a function from D^n into D).
>>>
>>> My understanding: [functions evaluate to non _Boolean]
>>
>> More to be wary of here. If the interpretation domain is {true, false}
>> or includes {true, false} then functions can "evaluate" to something you
>> might call _Boolean. (I don't know exactly what you mean by _Boolean.)
>> And that's fine. There is no confusion with predicates because they are
>> not functions and you can't mix functions and predicates in arbitrary
>> ways.
>
> It seems that predicates always [end up with] a possibly empty set of
> n-tuples. What is the correct term for [end up with]?
A unary is a subset of D. A binary predicate is a subset of DxD. An
n-ary predicate is a subset of D^n. That's the best I can do. I don't
know what else you are trying to say.
>> And again, there is no need for functions to be computable.
>>
>>> My understanding: [father_of(x) would evaluate to a unique constant
>>> element of the set of humans]
>>
>> I would use use "evaluate". And there is no need for unique -- the
Argh! I meant to say: 'I would /not/ use "evaluate"'.
>> definition of a function does not permit f(x) to be anything by one
>> thing. And the term "constant" is part of the syntax of the language.
>> It has no meaning here. If the domain is the set H of humans (not very
>> well-defined but let's let that slip) then father_of(x) is an element of
>> H.
>
> That sure was simple.
>
>> It's unfortunate you picked an example function with the same name as
>> Mendelson's example predicate (see below). I hope your example does not
>> confuse you.
>>
>>> My estimate: [functions must always evaluate to elements of D]
>>
>> Yes, that's explicitly in the text.
>
> Great. Any probably also sets of elements of D. GrandfathersOf(x).
No. An n-ary function always maps an n-tuple to an element of D, never
to a set of elements of D. Of course, if the domain of the
interpretation includes both people and sets of people, then there is no
problem finding a function like GrandfathersOf(x).
>>> d. For each individual constant a_i of L, an assignment of some fixed
>>> element (a_i)^M of D. According to Wikipedia 0-ary functions are also
>>> considered to be constants.
>>
>> It's the other way round. Some authors don't bother with constants but
>> instead permit 0-ary functions to serve the same purpose.
>
> OK that makes things simple.
>
>>> My estimate: [constants are unique elements 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 D^n, 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.
>>>
>>> My understanding: [if there are n human beings then there are n^2 2-tuples]
>>>
>>> This is where my understanding drops off. I can't see how his
>>> notation translates from the subset of n-tuples of D that satisfy
>>> n-ary relation_i.
>>
>> First, are you ok with this informal notation of whether a formula is
>> can be satisfied, and of whether a sentence is true or false, in some
>> interpretation? I ask because the technical definition is harder to
>> understand than the idea. You may no need to go though every technical
>> details provided you grasp the intuitive idea.
>
> I think that I have a very good gist of all of the basis ideas. The
> big correction is that a relation [end up with] a possibly empty set
> of n-tuples instead of a Boolean.
>
> (3,7) ∈ Less_Than(x, y) where the domain is ℕ seems to be the way that
> you assert: 3 < 7.
I don't know what you mean by Less_Than(x, y).
Here's the problem. You've skipped over what a language is, and I am
not sure you really know, so I can't interpret what you write. In the
"language of arithmetic" 3 < 7, <(3, 7) and Less_Than(3, 7) are all just
syntax for some relation. It means nothing. It is the interpretation
of the language that lets us know what this relation really is. The
trouble is, people don't bother to distinguish because 99.9% of the time
we are talking about the "usual interpretation" -- the domain is N and <
is, in this interpretation, the usual order on N. It's only when
considering the deeper possibilities that we need to keep these
separate.
Hence I advice against using meaningful names in the language because
that will bind you to the possibility that there might be less obvious
interpretations.
Here's an example. The usual axioms of arithmetic are entirely
consistent with an interpretation where the domain is the negative
integers. To make the interpretation work, the interpretation of the
successor function is S(x) = x-1 and the < relation (in the formal
language) will turn out to be > on the negative integers.
>> To be sure, I would insist that a student do exercises 2.10 and 2.11.
>> Can you do them?
>
> I will look at them now that you made sure my basic understading seems OK.
<cut>
> This is where I get totally lost in the details of the encoding.
Do the exercises first. That way I can be sure you really have a sound
intuitive understanding of what the formal stuff is intended to pin
down. It's much easier to understand the next two pages if you are 100%
sure about the informal notion of satisfiable.
--
Ben.
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| From | Jeff Barnett <jbb@notatt.com> |
|---|---|
| Date | 2020-07-24 22:18 -0600 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <rfgbrb$96s$1@dont-email.me> |
| In reply to | #21925 |
On 7/24/2020 7:28 PM, Ben Bacarisse wrote: > No. An n-ary function always maps an n-tuple to an element of D, never > to a set of elements of D. I'm not aware of such a restriction in general set theory or any logic built in or around it. Where was such a restriction introduced? Typically the model/domain of a theory includes all the structures that are needed to instantiate a language. I mean b structures all the things that are "referenced" by language pieces. For example, the elements of a normal set theory are sets. If the set theory has urelements so does your D, etc. And unless you `replace' predicates with functions, logical true and false are not in D; and if you do, they are still not logical true and false. -- Jeff Barnett
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| From | David Kleinecke <dkleinecke@gmail.com> |
|---|---|
| Date | 2020-07-24 21:58 -0700 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <dad34ea1-b1a2-4b62-9d58-5bfce2297421o@googlegroups.com> |
| In reply to | #21929 |
On Friday, July 24, 2020 at 9:18:54 PM UTC-7, Jeff Barnett wrote: > On 7/24/2020 7:28 PM, Ben Bacarisse wrote: > > No. An n-ary function always maps an n-tuple to an element of D, never > > to a set of elements of D. > > I'm not aware of such a restriction in general set theory or any logic > built in or around it. Where was such a restriction introduced? > Typically the model/domain of a theory includes all the structures that > are needed to instantiate a language. I mean b structures all the things > that are "referenced" by language pieces. For example, the elements of a > normal set theory are sets. If the set theory has urelements so does > your D, etc. And unless you `replace' predicates with functions, logical > true and false are not in D; and if you do, they are still not logical > true and false. D was the assumed range - the range being the set of things in function value position. So unless D is a member of itself it can't be in the range. It is usually assumed that everything is a set unless, and this rarely or never happens, it is specifically identified as not a set. Singletons - sets with exactly one member are very common. And that one member is usually a set. IMO the empty set is only one whose existence is obvious. It may be the only example of PO's analytic knowledge.
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| From | Jeff Barnett <jbb@notatt.com> |
|---|---|
| Date | 2020-07-25 01:46 -0600 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <rfgo1h$t9o$1@dont-email.me> |
| In reply to | #21930 |
On 7/24/2020 10:58 PM, David Kleinecke wrote: > On Friday, July 24, 2020 at 9:18:54 PM UTC-7, Jeff Barnett wrote: >> On 7/24/2020 7:28 PM, Ben Bacarisse wrote: >>> No. An n-ary function always maps an n-tuple to an element of D, never >>> to a set of elements of D. >> >> I'm not aware of such a restriction in general set theory or any logic >> built in or around it. Where was such a restriction introduced? >> Typically the model/domain of a theory includes all the structures that >> are needed to instantiate a language. I mean b structures all the things >> that are "referenced" by language pieces. For example, the elements of a >> normal set theory are sets. If the set theory has urelements so does >> your D, etc. And unless you `replace' predicates with functions, logical >> true and false are not in D; and if you do, they are still not logical >> true and false. > > D was the assumed range - the range being the set of things in > function value position. So unless D is a member of itself it > can't be in the range. Not in set theory. Typically you end up talking about the class of all sets. N.B. A class is not a set by definition. In any event, no paradox. I did err above in the following sense: D does contain all base elements and "derived" elements so, in a sense, any function argument is in D. But any object in the model can be a functional argument. Assume for a moment that we want to define a function element-of. We will make the values T and F, two distinct constants established by axioms. The function is the set of ordered pairs [x, y] such that y is a non nil set and there is an ordered pair for each element of y (as defined by the "e" predicate of the axioms). Clearly, every set in the model must be a potential function argument. > It is usually assumed that everything is a set unless, and this > rarely or never happens, it is specifically identified as not a > set. Singletons - sets with exactly one member are very common. > And that one member is usually a set. > > IMO the empty set is only one whose existence is obvious. It may > be the only example of PO's analytic knowledge. The world doesn't work that way. In standard set theory, everything - I repeat everything is a set including nil, the empty set. Any thing which isn't a set is called a ur-element. Set theories that have ur-elements have different versions of equivalences of pieces of the axiom of choice hierarchies, etc. I repeat that standard set theory has only sets. Period. For example, if you want to represent integers you might posit nil as your zero and if the set n is an integer, then (n) is its successor. Of course, to not make a pig of everything, you have to do a version of "strong" typing in your axioms and proofs but that is on you and not the theory per se. I will also add that theories with ur-elements are mostly invented and investigated to see what old therms still hold and how to modify some others. -- Jeff Barnett
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| From | David Kleinecke <dkleinecke@gmail.com> |
|---|---|
| Date | 2020-07-25 10:31 -0700 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <7328052a-cf1f-442b-ad0a-ed41d7c01393o@googlegroups.com> |
| In reply to | #21931 |
On Saturday, July 25, 2020 at 12:47:00 AM UTC-7, Jeff Barnett wrote: > On 7/24/2020 10:58 PM, David Kleinecke wrote: > > On Friday, July 24, 2020 at 9:18:54 PM UTC-7, Jeff Barnett wrote: > >> On 7/24/2020 7:28 PM, Ben Bacarisse wrote: > >>> No. An n-ary function always maps an n-tuple to an element of D, never > >>> to a set of elements of D. > >> > >> I'm not aware of such a restriction in general set theory or any logic > >> built in or around it. Where was such a restriction introduced? > >> Typically the model/domain of a theory includes all the structures that > >> are needed to instantiate a language. I mean b structures all the things > >> that are "referenced" by language pieces. For example, the elements of a > >> normal set theory are sets. If the set theory has urelements so does > >> your D, etc. And unless you `replace' predicates with functions, logical > >> true and false are not in D; and if you do, they are still not logical > >> true and false. > > > > D was the assumed range - the range being the set of things in > > function value position. So unless D is a member of itself it > > can't be in the range. > > Not in set theory. Typically you end up talking about the class of all > sets. N.B. A class is not a set by definition. In any event, no paradox. > I did err above in the following sense: D does contain all base > elements and "derived" elements so, in a sense, any function argument is > in D. > > But any object in the model can be a functional argument. Assume for a > moment that we want to define a function element-of. We will make the > values T and F, two distinct constants established by axioms. The > function is the set of ordered pairs [x, y] such that y is a non nil set > and there is an ordered pair for each element of y (as defined by the > "e" predicate of the axioms). Clearly, every set in the model must be a > potential function argument. > > It is usually assumed that everything is a set unless, and this > > rarely or never happens, it is specifically identified as not a > > set. Singletons - sets with exactly one member are very common. > > And that one member is usually a set. > > > > IMO the empty set is only one whose existence is obvious. It may > > be the only example of PO's analytic knowledge. > > The world doesn't work that way. In standard set theory, everything - I > repeat everything is a set including nil, the empty set. Any thing which > isn't a set is called a ur-element. Set theories that have ur-elements > have different versions of equivalences of pieces of the axiom of choice > hierarchies, etc. I repeat that standard set theory has only sets. > Period. For example, if you want to represent integers you might posit > nil as your zero and if the set n is an integer, then (n) is its > successor. Of course, to not make a pig of everything, you have to do a > version of "strong" typing in your axioms and proofs but that is on you > and not the theory per se. > > I will also add that theories with ur-elements are mostly invented and > investigated to see what old therms still hold and how to modify some > others. We have slightly different set theories in mind. My version is pretty crude. But we agree on all points that matter.
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2020-07-25 12:04 +0100 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <87v9ibj1oy.fsf@bsb.me.uk> |
| In reply to | #21929 |
Jeff Barnett <jbb@notatt.com> writes: > On 7/24/2020 7:28 PM, Ben Bacarisse wrote: >> No. An n-ary function always maps an n-tuple to an element of D, never >> to a set of elements of D. > > I'm not aware of such a restriction in general set theory or any logic > built in or around it. Where was such a restriction introduced? In the book PO is using. And in all the other books on the subject I've seen (but that's not many as this was never a subject I taught or researched). More to the point, where is it /not/ introduced? I'd like to see how the details are done because the effect would seem to be that D is no longer the "domain of discourse" but something more like D ∪ 2^D. I don't see any advantage over simply making D be the required set. Do you have a reference to a text that defines an interpretation (AKA a structure) in this way? > Typically the model/domain of a theory includes all the structures > that are needed to instantiate a language. I mean b structures all the > things that are "referenced" by language pieces. For example, the > elements of a normal set theory are sets. If the set theory has > urelements so does your D, etc. And unless you `replace' predicates > with functions, logical true and false are not in D; and if you do, > they are still not logical true and false. I don't understand what you are saying here. -- Ben.
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| From | Jeff Barnett <jbb@notatt.com> |
|---|---|
| Date | 2020-07-25 14:45 -0600 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <rfi5kg$2f3$1@dont-email.me> |
| In reply to | #21932 |
On 7/25/2020 5:04 AM, Ben Bacarisse wrote: > Jeff Barnett <jbb@notatt.com> writes: > >> On 7/24/2020 7:28 PM, Ben Bacarisse wrote: >>> No. An n-ary function always maps an n-tuple to an element of >>> D, never to a set of elements of D. >> >> I'm not aware of such a restriction in general set theory or any >> logic built in or around it. Where was such a restriction >> introduced? > > In the book PO is using. And in all the other books on the subject > I've seen (but that's not many as this was never a subject I taught > or researched). > > More to the point, where is it /not/ introduced? I'd like to see how > the details are done because the effect would seem to be that D is no > longer the "domain of discourse" but something more like D ∪ 2^D. I > don't see any advantage over simply making D be the required set. Do > you have a reference to a text that defines an interpretation (AKA a > structure) in this way? > >> Typically the model/domain of a theory includes all the structures >> that are needed to instantiate a language. I mean b structures >> all the things that are "referenced" by language pieces. For >> example, the elements of a normal set theory are sets. If the set >> theory has urelements so does your D, etc. And unless you >> `replace' predicates with functions, logical true and false are not >> in D; and if you do, they are still not logical true and false. > > I don't understand what you are saying here. Grab a text book on real analysis, point-set theory, or the axiom of choice and browse for conversations on range of formulations. It's been almost 60 years since I mucked about. BTW, in another post close to the above, I rescinded most of what I said: D is indeed the domain of discourse so functions, etc. do map to elements of D, but note, D is considered a class, not a set, so you don't have to screw around with the set of all sets. An absolutely delightful book is "Zermelo's Axiom of Choice" by Gregory H. Moore - $18 including Prime shipping at the US Amazon.com. It was unavailable for decades except in hardback at hundreds of dollars a pop. It's a great recap of the history of foundations, the long debate in the mathematical and philosophical communities, formulations of set theory, hierarchies of axioms and potential axioms, equivalences and relative strengths of various axioms, theorems, and wishes. It also discusses urelements and their weird effects on all this stuff. -- Jeff Barnett
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2020-07-26 00:51 +0100 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <87ft9fi26c.fsf@bsb.me.uk> |
| In reply to | #21934 |
Jeff Barnett <jbb@notatt.com> writes: > On 7/25/2020 5:04 AM, Ben Bacarisse wrote: >> Jeff Barnett <jbb@notatt.com> writes: >> >>> On 7/24/2020 7:28 PM, Ben Bacarisse wrote: >>>> No. An n-ary function always maps an n-tuple to an element of >>>> D, never to a set of elements of D. >>> >>> I'm not aware of such a restriction in general set theory or any >>> logic built in or around it. Where was such a restriction >>> introduced? >> >> In the book PO is using. And in all the other books on the subject >> I've seen (but that's not many as this was never a subject I taught >> or researched). >> >> More to the point, where is it /not/ introduced? I'd like to see how >> the details are done because the effect would seem to be that D is no >> longer the "domain of discourse" but something more like D ∪ 2^D. I >> don't see any advantage over simply making D be the required set. Do >> you have a reference to a text that defines an interpretation (AKA a >> structure) in this way? So who does it the way you say? >>> Typically the model/domain of a theory includes all the structures >>> that are needed to instantiate a language. I mean b structures >>> all the things that are "referenced" by language pieces. For >>> example, the elements of a normal set theory are sets. If the set >>> theory has urelements so does your D, etc. And unless you >>> `replace' predicates with functions, logical true and false are not >>> in D; and if you do, they are still not logical true and false. >> >> I don't understand what you are saying here. > > Grab a text book on real analysis, point-set theory, or the axiom of > choice and browse for conversations on range of formulations. I don't see how re-reading some textbooks will help me understand what you meant by that last paragraph. But maybe it does not matter. Do you really mind if I don't understand you? > An absolutely delightful book is "Zermelo's Axiom of Choice" by > Gregory H. Moore - $18 including Prime shipping at the US > Amazon.com. It was unavailable for decades except in hardback at > hundreds of dollars a pop. It's a great recap of the history of > foundations, the long debate in the mathematical and philosophical > communities, formulations of set theory, hierarchies of axioms and > potential axioms, equivalences and relative strengths of various > axioms, theorems, and wishes. It also discusses urelements and their > weird effects on all this stuff. These are interesting topics but nothing seems directly relevant from your summary. I'm interested in a citation to back up your claim about interpretations/structures. -- Ben.
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| From | Jeff Barnett <jbb@notatt.com> |
|---|---|
| Date | 2020-07-25 22:46 -0600 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <rfj1ru$aal$1@dont-email.me> |
| In reply to | #21935 |
On 7/25/2020 5:51 PM, Ben Bacarisse wrote: > Jeff Barnett <jbb@notatt.com> writes: > >> On 7/25/2020 5:04 AM, Ben Bacarisse wrote: >>> Jeff Barnett <jbb@notatt.com> writes: >>> >>>> On 7/24/2020 7:28 PM, Ben Bacarisse wrote: >>>>> No. An n-ary function always maps an n-tuple to an element of >>>>> D, never to a set of elements of D. >>>> >>>> I'm not aware of such a restriction in general set theory or any >>>> logic built in or around it. Where was such a restriction >>>> introduced? >>> >>> In the book PO is using. And in all the other books on the subject >>> I've seen (but that's not many as this was never a subject I taught >>> or researched). >>> >>> More to the point, where is it /not/ introduced? I'd like to see how >>> the details are done because the effect would seem to be that D is no >>> longer the "domain of discourse" but something more like D ∪ 2^D. I >>> don't see any advantage over simply making D be the required set. Do >>> you have a reference to a text that defines an interpretation (AKA a >>> structure) in this way? > > So who does it the way you say? > >>>> Typically the model/domain of a theory includes all the structures >>>> that are needed to instantiate a language. I mean b structures >>>> all the things that are "referenced" by language pieces. For >>>> example, the elements of a normal set theory are sets. If the set >>>> theory has urelements so does your D, etc. And unless you >>>> `replace' predicates with functions, logical true and false are not >>>> in D; and if you do, they are still not logical true and false. >>> >>> I don't understand what you are saying here. >> >> Grab a text book on real analysis, point-set theory, or the axiom of >> choice and browse for conversations on range of formulations. > > I don't see how re-reading some textbooks will help me understand what > you meant by that last paragraph. But maybe it does not matter. Do you > really mind if I don't understand you? Did you read Moore's book mentioned in the next paragraph? And no, I have no burning desire to be understood. >> An absolutely delightful book is "Zermelo's Axiom of Choice" by >> Gregory H. Moore - $18 including Prime shipping at the US >> Amazon.com. It was unavailable for decades except in hardback at >> hundreds of dollars a pop. It's a great recap of the history of >> foundations, the long debate in the mathematical and philosophical >> communities, formulations of set theory, hierarchies of axioms and >> potential axioms, equivalences and relative strengths of various >> axioms, theorems, and wishes. It also discusses urelements and their >> weird effects on all this stuff. > > These are interesting topics but nothing seems directly relevant from > your summary. I'm interested in a citation to back up your claim about > interpretations/structures. They are relevant. For example the independence of choice and continuum hypotheses are shown with models with and without. The method of forcing to discovery some of those models, etc. There are, also, examples where such problems disappear when a higher order logic is brought to bare. I believe this is the "most fundamental" place to look at such results. -- Jeff Barnett
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2020-07-26 12:10 +0100 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <875zaailac.fsf@bsb.me.uk> |
| In reply to | #21939 |
Jeff Barnett <jbb@notatt.com> writes: > On 7/25/2020 5:51 PM, Ben Bacarisse wrote: >> Jeff Barnett <jbb@notatt.com> writes: >> >>> On 7/25/2020 5:04 AM, Ben Bacarisse wrote: >>>> Jeff Barnett <jbb@notatt.com> writes: >>>> >>>>> On 7/24/2020 7:28 PM, Ben Bacarisse wrote: >>>>>> No. An n-ary function always maps an n-tuple to an element of >>>>>> D, never to a set of elements of D. >>>>> >>>>> I'm not aware of such a restriction in general set theory or any >>>>> logic built in or around it. Where was such a restriction >>>>> introduced? >>>> >>>> In the book PO is using. And in all the other books on the subject >>>> I've seen (but that's not many as this was never a subject I taught >>>> or researched). >>>> >>>> More to the point, where is it /not/ introduced? I'd like to see how >>>> the details are done because the effect would seem to be that D is no >>>> longer the "domain of discourse" but something more like D ∪ 2^D. I >>>> don't see any advantage over simply making D be the required set. Do >>>> you have a reference to a text that defines an interpretation (AKA a >>>> structure) in this way? >> >> So who does it the way you say? I'm guessing no one does. >>>>> Typically the model/domain of a theory includes all the structures >>>>> that are needed to instantiate a language. I mean b structures >>>>> all the things that are "referenced" by language pieces. For >>>>> example, the elements of a normal set theory are sets. If the set >>>>> theory has urelements so does your D, etc. And unless you >>>>> `replace' predicates with functions, logical true and false are not >>>>> in D; and if you do, they are still not logical true and false. >>>> >>>> I don't understand what you are saying here. >>> >>> Grab a text book on real analysis, point-set theory, or the axiom of >>> choice and browse for conversations on range of formulations. >> >> I don't see how re-reading some textbooks will help me understand what >> you meant by that last paragraph. But maybe it does not matter. Do you >> really mind if I don't understand you? > > Did you read Moore's book mentioned in the next paragraph? And no, I > have no burning desire to be understood. OK, I'm good with that too. >>> An absolutely delightful book is "Zermelo's Axiom of Choice" by >>> Gregory H. Moore - $18 including Prime shipping at the US >>> Amazon.com. It was unavailable for decades except in hardback at >>> hundreds of dollars a pop. It's a great recap of the history of >>> foundations, the long debate in the mathematical and philosophical >>> communities, formulations of set theory, hierarchies of axioms and >>> potential axioms, equivalences and relative strengths of various >>> axioms, theorems, and wishes. It also discusses urelements and their >>> weird effects on all this stuff. >> >> These are interesting topics but nothing seems directly relevant from >> your summary. I'm interested in a citation to back up your claim about >> interpretations/structures. > > They are relevant. Which seems to say that the book is not a reference for your claim about how interpretations/structures are defined. I feel you would say it was if that topic was covered in the way you suggested. -- Ben.
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| From | olcott <NoOne@NoWhere.com> |
|---|---|
| Date | 2020-07-25 22:53 -0500 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <KICdnZQ7S4KjY4HCnZ2dnUU7-e3NnZ2d@giganews.com> |
| In reply to | #21925 |
On 7/24/2020 8:28 PM, Ben Bacarisse wrote:
> olcott <NoOne@NoWhere.com> writes:
>
>> On 7/24/2020 5:49 PM, Ben Bacarisse wrote:
>>> olcott <NoOne@NoWhere.com> writes:
>>>
>>>> On 7/22/2020 6:54 PM, Ben Bacarisse wrote:
>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>
>>>>>>> Try reading page 57 of Mendelson first. I'll answer any reasonable
>>>>>>> questions you might have about that page.
>>>>>>
>>>>>> I have a hard copy of edition 4 and and PDF copy of edition 6. I
>>>>>> assume you mean edition 6.
>>>>>
>>>>> 4th I think. Anyway, the page where the concept of an interpretation
>>>>> is introduced.
>>>> I think that I totally understand this:
>>>> Page 58 fourth edition Mendelson
>>>> The super script ^n in a name means its number of arguments
>>>> The subscript _i in a name makes this name unique
>>>>
>>>> Definition
>>>> 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.
>>>>
>>>> My understanding: [The set of elements (integers, humans ...) that can
>>>> be assigned to the free variables]
>>>
>>> Not just the free variables. Lets take D = ℕ and use the usual infix
>>> notation with variables that don't have subscripts. The formula
>>>
>>> x < y
>>>
>>> has two free variables and is satisfied by some assignment and not for
>>> others. The sentence
>>>
>>> ∀x∀y x < y
>>>
>>> has no free variables but is false (in this interpretation) because
>>> (informally) x < y is not the case for every assignment that could be
>>> made. Mendelson starts with this rather informal, intuitive notion but
>>> then makes it technically watertight by explaining how you deal with
>>> sentences like this on the pages that follow.
>>>
>>>> b. For each predicate letter A^n_j of L, an assignment of an n-place
>>>> relation (A^n_j)^M in D.
>>>>
>>>> My understanding: [selects the subset of n-tuples of D that satisfy
>>>> this relation]
>>>
>>> If that wording help you, fine. Technically (A^n_j)^M /is/ a set of
>>> n-tuples. If D is ℕ and we use the usual interpretation of <, then < is
>>> just a set of pairs: {(0, 1), (0, 2), ... (1, 2), (1, 3), ... } with a
>>> funny-looking name.
>>
>> Yet a subset of the set of every n-tuple in ℕ.
>> (1,0) is an n-tuple in the domain that is excluded.
>
> I don't understand. I suspect you are using domain wrongly. (1, 0) is
> not in the set of pairs that defines the < predicate.
>
If the domain is ℕ then some or the ordered pairs satisfy < and some do
not. The ones that do satisfy < are a subset of all of the ordered pairs
of ℕ.
>>>> My understanding: [predicates are functions that evaluate to Boolean]
>>>
>>> Take care here. That's programmer language. The interpretation of a
>>> predicate is a set of pairs. There's not really any evaluation and
>>> there is definitely no need for a predicate to be computable. See the
>>> next comment for why the distinction actually helps.
>>
>> Oh I see the predicate Father_of(x, y) normally evaluates to Boolean
>> for a specific ordered pair of individual people. As a relation on a
>> domain it defines the whole set of ordered pairs that satisfy that
>> relation.
>
> It /is/ the set of pairs that are in the relation. It does not
> "evaluate to Boolean". Nothing is evaluated. For a relation R the
> notation R(x, y) is just another way of writing (x, y) ∈ R.
>
Closed WFF are true or false, Open WFF specify sets.
>> constant h1 ∈ human beings
>> constant h2 ∈ human beings
>> (h1,h2) ∈ Father_of(x, y) would assert that h1 is the father of h2.
>
> Please, for the moment, no poems. None of that is written correctly.
To use the Mendelson notation (a26, a87) ∈ Father_of(x, y)
would be true or false.
> At some point you will have to learn the rules for writing actual
> mathematics rather than metaphorical hints, but for the moment try to
> copy the syntax used by Mendelson.
Yes I just did that.
>> It would seem that a predicate that is satisfied would be computable,
>> because an answer of Yes guarantees that an answer exists.
>
> Could you refrain from injecting your own ideas for a while? I will get
> side-tracked by explaining why things like this are wrong. For the
> moment, a predicate in an interpretation is just a set of pairs.
(a26, a87) ∈ Father_of(x, y)
>
>>>> c. For each function letter f^n_j of L, an assignment of an n-place
>>>> operation (f^n_j)^M in D (that is, a function from D^n into D).
>>>>
>>>> My understanding: [functions evaluate to non _Boolean]
>>>
>>> More to be wary of here. If the interpretation domain is {true, false}
>>> or includes {true, false} then functions can "evaluate" to something you
>>> might call _Boolean. (I don't know exactly what you mean by _Boolean.)
>>> And that's fine. There is no confusion with predicates because they are
>>> not functions and you can't mix functions and predicates in arbitrary
>>> ways.
>>
>> It seems that predicates always [end up with] a possibly empty set of
>> n-tuples. What is the correct term for [end up with]?
>
> A unary is a subset of D. A binary predicate is a subset of DxD. An
> n-ary predicate is a subset of D^n. That's the best I can do. I don't
> know what else you are trying to say.
That works.
>>> And again, there is no need for functions to be computable.
>>>
>>>> My understanding: [father_of(x) would evaluate to a unique constant
>>>> element of the set of humans]
>>>
>>> I would use use "evaluate". And there is no need for unique -- the
>
> Argh! I meant to say: 'I would /not/ use "evaluate"'.
Father_of(x) ∈ Humans = y
>
>>> definition of a function does not permit f(x) to be anything by one
>>> thing. And the term "constant" is part of the syntax of the language.
>>> It has no meaning here. If the domain is the set H of humans (not very
>>> well-defined but let's let that slip) then father_of(x) is an element of
>>> H.
>>
>> That sure was simple.
>>
>>> It's unfortunate you picked an example function with the same name as
>>> Mendelson's example predicate (see below). I hope your example does not
>>> confuse you.
>>>
>>>> My estimate: [functions must always evaluate to elements of D]
>>>
>>> Yes, that's explicitly in the text.
>>
>> Great. Any probably also sets of elements of D. GrandfathersOf(x).
>
> No. An n-ary function always maps an n-tuple to an element of D, never
> to a set of elements of D. Of course, if the domain of the
> interpretation includes both people and sets of people, then there is no
> problem finding a function like GrandfathersOf(x).
An n-ary predicate always maps to a set of n-tuples?
>>>> d. For each individual constant a_i of L, an assignment of some fixed
>>>> element (a_i)^M of D. According to Wikipedia 0-ary functions are also
>>>> considered to be constants.
>>>
>>> It's the other way round. Some authors don't bother with constants but
>>> instead permit 0-ary functions to serve the same purpose.
>>
>> OK that makes things simple.
>>
>>>> My estimate: [constants are unique elements 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 D^n, 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.
>>>>
>>>> My understanding: [if there are n human beings then there are n^2 2-tuples]
>>>>
>>>> This is where my understanding drops off. I can't see how his
>>>> notation translates from the subset of n-tuples of D that satisfy
>>>> n-ary relation_i.
>>>
>>> First, are you ok with this informal notation of whether a formula is
>>> can be satisfied, and of whether a sentence is true or false, in some
>>> interpretation? I ask because the technical definition is harder to
>>> understand than the idea. You may no need to go though every technical
>>> details provided you grasp the intuitive idea.
>>
>> I think that I have a very good gist of all of the basis ideas. The
>> big correction is that a relation [end up with] a possibly empty set
>> of n-tuples instead of a Boolean.
>>
>> (3,7) ∈ Less_Than(x, y) where the domain is ℕ seems to be the way that
>> you assert: 3 < 7.
>
> I don't know what you mean by Less_Than(x, y).
FOL does not have a "<" but it does have predicate names.
> Here's the problem. You've skipped over what a language is, and I am
> not sure you really know, so I can't interpret what you write. In the
> "language of arithmetic" 3 < 7, <(3, 7) and Less_Than(3, 7) are all just
> syntax for some relation. It means nothing. It is the interpretation
> of the language that lets us know what this relation really is. The
> trouble is, people don't bother to distinguish because 99.9% of the time
> we are talking about the "usual interpretation" -- the domain is N and <
> is, in this interpretation, the usual order on N. It's only when
> considering the deeper possibilities that we need to keep these
> separate.
>
Yes like the first time that they examine the Geometric axioms without
assuming geometric objects.
> Hence I advice against using meaningful names in the language because
> that will bind you to the possibility that there might be less obvious
> interpretations.
>
It is better to say exactly what you mean in an unequivocal way.
An algorithm will do that to.
> Here's an example. The usual axioms of arithmetic are entirely
> consistent with an interpretation where the domain is the negative
> integers. To make the interpretation work, the interpretation of the
> successor function is S(x) = x-1 and the < relation (in the formal
> language) will turn out to be > on the negative integers.
>
>>> To be sure, I would insist that a student do exercises 2.10 and 2.11.
>>> Can you do them?
>>
>> I will look at them now that you made sure my basic understading seems OK.
> <cut>
>
>> This is where I get totally lost in the details of the encoding.
>
> Do the exercises first. That way I can be sure you really have a sound
> intuitive understanding of what the formal stuff is intended to pin
> down. It's much easier to understand the next two pages if you are 100%
> sure about the informal notion of satisfiable.
>
Yes I worked on them today. I agree that it is best that I prove my
understanding by those two exercises.
2.10 (a)(i) The set of the product of two positive integers >= 2
(1...,2...) and (2...,1...)
2.10 (b)(i) The set of the sum of two integers = 0
(0,-1,-2...)(0,1,2...)
2.10 (c)(i) The domain is the set of all sets of integers, A^2_1(y,z) is
y ⊆ z, f^1_1(y, z) is y ∩ z, and a1 is the empty set ∅.
⊆(y ∩ z, ∅) I would think that the empty set would have no subsets.
The set of the intersection of y and z is a subset of the empty set.
// I am not sure what this: "⇒" means in this context
ii. A^2_1(x1, x2) ⇒ A^2_1(x2, x1)
iii. (∀x1)(∀x2)(∀x3) (A^2_1(x,x2) ∧ A^2_1(x2,x3) ⇒ A^2_1(x1,x3))
--
Copyright 2020 Pete Olcott
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| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2020-07-26 20:46 +0100 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <87zh7mgiux.fsf@bsb.me.uk> |
| In reply to | #21936 |
olcott <NoOne@NoWhere.com> writes:
> On 7/24/2020 8:28 PM, Ben Bacarisse wrote:
>> olcott <NoOne@NoWhere.com> writes:
>>
>>> On 7/24/2020 5:49 PM, Ben Bacarisse wrote:
>>>> olcott <NoOne@NoWhere.com> writes:
>>>>> On 7/22/2020 6:54 PM, Ben Bacarisse wrote:
>>>>>> olcott <NoOne@NoWhere.com> writes:
<cut>
>>>>> My understanding: [predicates are functions that evaluate to Boolean]
>>>>
>>>> Take care here. That's programmer language. The interpretation of a
>>>> predicate is a set of pairs. There's not really any evaluation and
>>>> there is definitely no need for a predicate to be computable. See the
>>>> next comment for why the distinction actually helps.
>>>
>>> Oh I see the predicate Father_of(x, y) normally evaluates to Boolean
>>> for a specific ordered pair of individual people. As a relation on a
>>> domain it defines the whole set of ordered pairs that satisfy that
>>> relation.
>>
>> It /is/ the set of pairs that are in the relation. It does not
>> "evaluate to Boolean". Nothing is evaluated. For a relation R the
>> notation R(x, y) is just another way of writing (x, y) ∈ R.
>
> Closed WFF are true or false, Open WFF specify sets.
True! But I don't know why you say it here.
>>> constant h1 ∈ human beings
>>> constant h2 ∈ human beings
>>> (h1,h2) ∈ Father_of(x, y) would assert that h1 is the father of h2.
>>
>> Please, for the moment, no poems. None of that is written correctly.
>
> To use the Mendelson notation (a26, a87) ∈ Father_of(x, y)
> would be true or false.
No.
Unfortunately, I don't have time to teach you how to write mathematics,
but the key is precision. I can take a guess at what you mean, but
that's not what the symbols are for. They should remove the need for
guessing.
(My guess: You have a language with a binary relation symbol Father_of
and you are talking about an interpretation that maps that relation
symbol to an actual set of pairs you call Father_of. The domain in
question has (at least) two elements that you call a26 and a87. If so,
you should write (a26, a87) ∈ Father_of, or more conventionally,
Father_of(a26, a87). If this guess is wrong, then I really have no idea
what your symbols are supposed to mean.)
>> At some point you will have to learn the rules for writing actual
>> mathematics rather than metaphorical hints, but for the moment try to
>> copy the syntax used by Mendelson.
>
> Yes I just did that.
No, that part is going to take time.
>>> It would seem that a predicate that is satisfied would be computable,
>>> because an answer of Yes guarantees that an answer exists.
>>
>> Could you refrain from injecting your own ideas for a while? I will get
>> side-tracked by explaining why things like this are wrong. For the
>> moment, a predicate in an interpretation is just a set of pairs.
>
> (a26, a87) ∈ Father_of(x, y)
See above. Can't you stick to words for the moment? It will avoid a
lot of side problems.
>>>>> c. For each function letter f^n_j of L, an assignment of an n-place
>>>>> operation (f^n_j)^M in D (that is, a function from D^n into D).
>>>>>
>>>>> My understanding: [functions evaluate to non _Boolean]
>>>>
>>>> More to be wary of here. If the interpretation domain is {true, false}
>>>> or includes {true, false} then functions can "evaluate" to something you
>>>> might call _Boolean. (I don't know exactly what you mean by _Boolean.)
>>>> And that's fine. There is no confusion with predicates because they are
>>>> not functions and you can't mix functions and predicates in arbitrary
>>>> ways.
>>>
>>> It seems that predicates always [end up with] a possibly empty set of
>>> n-tuples. What is the correct term for [end up with]?
>>
>> A unary is a subset of D. A binary predicate is a subset of DxD. An
>> n-ary predicate is a subset of D^n. That's the best I can do. I don't
>> know what else you are trying to say.
>
> That works.
>
>>>> And again, there is no need for functions to be computable.
>>>>
>>>>> My understanding: [father_of(x) would evaluate to a unique constant
>>>>> element of the set of humans]
>>>>
>>>> I would use use "evaluate". And there is no need for unique -- the
>>
>> Argh! I meant to say: 'I would /not/ use "evaluate"'.
>
> Father_of(x) ∈ Humans = y
That just hints at something. I have no idea what you are really
saying. The (modern) way to say that a function maps Humans to Humans
is
Father_of: Humans -> Humans
>>>>> My estimate: [functions must always evaluate to elements of D]
>>>>
>>>> Yes, that's explicitly in the text.
>>>
>>> Great. Any probably also sets of elements of D. GrandfathersOf(x).
>>
>> No. An n-ary function always maps an n-tuple to an element of D, never
>> to a set of elements of D. Of course, if the domain of the
>> interpretation includes both people and sets of people, then there is no
>> problem finding a function like GrandfathersOf(x).
>
> An n-ary predicate always maps to a set of n-tuples?
No. An n-ary predicate is a set of n-tuples. But why are you now
talking about predicates when I was trying explain a misunderstanding
about functions?
>>>>> d. For each individual constant a_i of L, an assignment of some fixed
>>>>> element (a_i)^M of D. According to Wikipedia 0-ary functions are also
>>>>> considered to be constants.
>>>>
>>>> It's the other way round. Some authors don't bother with constants but
>>>> instead permit 0-ary functions to serve the same purpose.
>>>
>>> OK that makes things simple.
>>>
>>>>> My estimate: [constants are unique elements 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 D^n, 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.
>>>>>
>>>>> My understanding: [if there are n human beings then there are n^2 2-tuples]
>>>>>
>>>>> This is where my understanding drops off. I can't see how his
>>>>> notation translates from the subset of n-tuples of D that satisfy
>>>>> n-ary relation_i.
>>>>
>>>> First, are you ok with this informal notation of whether a formula is
>>>> can be satisfied, and of whether a sentence is true or false, in some
>>>> interpretation? I ask because the technical definition is harder to
>>>> understand than the idea. You may no need to go though every technical
>>>> details provided you grasp the intuitive idea.
>>>
>>> I think that I have a very good gist of all of the basis ideas. The
>>> big correction is that a relation [end up with] a possibly empty set
>>> of n-tuples instead of a Boolean.
>>>
>>> (3,7) ∈ Less_Than(x, y) where the domain is ℕ seems to be the way that
>>> you assert: 3 < 7.
>>
>> I don't know what you mean by Less_Than(x, y).
>
> FOL does not have a "<" but it does have predicate names.
My problem was with the (x, y) part, not the name. This is the same
problem as above. If the predicate is called Less_Than, then you say
either (3, 7) ∈ Less_Than or (more conventionally) Less_Than(3, 7).
Your "(3,7) ∈ Less_Than(x, y)" is some mysterious formula with two free
variables. And it suggests that Less_Than is a function symbol not a
predicate symbol.
I think you need to go back a step and learn how Mendelson defines a
language for a first-order theory. Oh, I see I said that last time:
>> Here's the problem. You've skipped over what a language is, and I am
>> not sure you really know, so I can't interpret what you write. In the
>> "language of arithmetic" 3 < 7, <(3, 7) and Less_Than(3, 7) are all just
>> syntax for some relation. It means nothing. It is the interpretation
>> of the language that lets us know what this relation really is. The
>> trouble is, people don't bother to distinguish because 99.9% of the time
>> we are talking about the "usual interpretation" -- the domain is N and <
>> is, in this interpretation, the usual order on N. It's only when
>> considering the deeper possibilities that we need to keep these
>> separate.
>
> Yes like the first time that they examine the Geometric axioms without
> assuming geometric objects.
>
>> Hence I advice against using meaningful names in the language because
>> that will bind you to the possibility that there might be less obvious
(I meant to write "blind you")
>> interpretations.
>
> It is better to say exactly what you mean in an unequivocal way.
> An algorithm will do that to.
No. You are confusing levels again. And then injecting your own as yet
unverified opinions.
Here's what I ask a student to do now to clear this up: Construct an
interpretation of PA (or Q which is simpler) such that the predicate <
(as usually defined) is interpreted to be >.
If you do nothing else in reply to the post, do that.
>> Do the exercises first. That way I can be sure you really have a sound
>> intuitive understanding of what the formal stuff is intended to pin
>> down. It's much easier to understand the next two pages if you are 100%
>> sure about the informal notion of satisfiable.
>>
>
> Yes I worked on them today. I agree that it is best that I prove my
> understanding by those two exercises.
>
> 2.10 (a)(i) The set of the product of two positive integers >= 2
> (1...,2...) and (2...,1...)
I think you know the answer, but you can't write it.
"..." usually means "and so on". You've written only two pairs
(Mendelson uses <a,b> but I assume you are using (a,b) for a pair), both
which appear to have infinite "things" (I can;t tell what) in each part
of the pait.
Unfortunately I can't understand your English either. The phrase "the
set of" is usually followed by a plural not a singular.
So I'm lost. I suggest you use {} for sets and <> for pairs and
consider starting the English with "(i) is satisfied by the set of pairs
such that...". Often is helps to name the members of the pair: "the set
of pairs <a,b> such that...".
> 2.10 (b)(i) The set of the sum of two integers = 0
> (0,-1,-2...)(0,1,2...)
I see no set. I see no pairs. Again, I think you know the answer, but
you can't express it. Try with the form of words above.
> 2.10 (c)(i) The domain is the set of all sets of integers, A^2_1(y,z) is
> y ⊆ z, f^1_1(y, z) is y ∩ z, and a1 is the empty set ∅.
>
> ⊆(y ∩ z, ∅)
Yes, though it's much more common to write ⊆ using infix notation.
> I would think that the empty set would have no subsets.
Except the empty set itself (S ⊆ S for all sets), so the condition is
that (a ∩ b) = ∅, since only when (a ∩ b) = ∅ can (a ∩ b) be a subset of
∅.
> The set of the intersection of y and z is a subset of the empty set.
That's the gist of is. Mendelson is expecting you to know that only the
empty set can be a subset of the empty set, and he probably expects you
to know the technical term for sets with no common elements: they are
called disjoint sets. Can you write the answer succinctly now?
> // I am not sure what this: "⇒" means in this context
> ii. A^2_1(x1, x2) ⇒ A^2_1(x2, x1)
> iii. (∀x1)(∀x2)(∀x3) (A^2_1(x,x2) ∧ A^2_1(x2,x3) ⇒ A^2_1(x1,x3))
It's the logical connective with truth table
T T T
T F F
F T T
F F T
By the way, having everything with a subscript (and many things with a
superscript) makes the formal definition that comes later much simpler
but it's a pain for this kind of thing. I re-wrote then exercises in
the simpler form:
(i) A(f(x, y), a)
(ii) A(x, y) -> A(y, x)
(iii) ∀x∀y∀z[ A(x, y) ∧ A(y, z) -> A(x, z) ]
with these interpretations:
(a) D = Z+, A is >=, f is multiplication and a is 2.
(b) D = Z, A is =, f is addition and a is 0.
(c) D = 2^Z, A is ⊆, f is intersection and a is {}.
--
Ben.
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| From | olcott <NoOne@NoWhere.com> |
|---|---|
| Date | 2020-07-26 16:42 -0500 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <vY-dnW0ku6B8ZYDCnZ2dnUU7-XXNnZ2d@giganews.com> |
| In reply to | #21948 |
On 7/26/2020 2:46 PM, Ben Bacarisse wrote:
> olcott <NoOne@NoWhere.com> writes:
>
>> On 7/24/2020 8:28 PM, Ben Bacarisse wrote:
>>> olcott <NoOne@NoWhere.com> writes:
>>>
>>>> On 7/24/2020 5:49 PM, Ben Bacarisse wrote:
>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>> On 7/22/2020 6:54 PM, Ben Bacarisse wrote:
>>>>>>> olcott <NoOne@NoWhere.com> writes:
> <cut>
>>>>>> My understanding: [predicates are functions that evaluate to Boolean]
>>>>>
>>>>> Take care here. That's programmer language. The interpretation of a
>>>>> predicate is a set of pairs. There's not really any evaluation and
>>>>> there is definitely no need for a predicate to be computable. See the
>>>>> next comment for why the distinction actually helps.
>>>>
>>>> Oh I see the predicate Father_of(x, y) normally evaluates to Boolean
>>>> for a specific ordered pair of individual people. As a relation on a
>>>> domain it defines the whole set of ordered pairs that satisfy that
>>>> relation.
>>>
>>> It /is/ the set of pairs that are in the relation. It does not
>>> "evaluate to Boolean". Nothing is evaluated. For a relation R the
>>> notation R(x, y) is just another way of writing (x, y) ∈ R.
>>
>> Closed WFF are true or false, Open WFF specify sets.
>
> True! But I don't know why you say it here.
Making sure that my understanding is correct.
>
>>>> constant h1 ∈ human beings
>>>> constant h2 ∈ human beings
>>>> (h1,h2) ∈ Father_of(x, y) would assert that h1 is the father of h2.
>>>
>>> Please, for the moment, no poems. None of that is written correctly.
>>
>> To use the Mendelson notation (a26, a87) ∈ Father_of(x, y)
>> would be true or false.
>
> No.
>
> Unfortunately, I don't have time to teach you how to write mathematics,
> but the key is precision. I can take a guess at what you mean, but
> that's not what the symbols are for. They should remove the need for
> guessing.
>
It is crucially important that I understand how to encode constants in
WFF. The ordered pair of two particular individual humans: (a26, a87)
a26 is either the father of a87 or not.
> (My guess: You have a language with a binary relation symbol Father_of
> and you are talking about an interpretation that maps that relation
> symbol to an actual set of pairs you call Father_of. The domain in
> question has (at least) two elements that you call a26 and a87. If so,
> you should write (a26, a87) ∈ Father_of, or more conventionally,
> Father_of(a26, a87). If this guess is wrong, then I really have no idea
> what your symbols are supposed to mean.)
Father_of(x, y) with x and y unbound specifies a set.
Father_of(a26, a87) is a closed WFF that is true or false.
>
>>> At some point you will have to learn the rules for writing actual
>>> mathematics rather than metaphorical hints, but for the moment try to
>>> copy the syntax used by Mendelson.
>>
>> Yes I just did that.
>
> No, that part is going to take time.
Previously I had no experience with open WFF.
>>>> It would seem that a predicate that is satisfied would be computable,
>>>> because an answer of Yes guarantees that an answer exists.
>>>
>>> Could you refrain from injecting your own ideas for a while? I will get
>>> side-tracked by explaining why things like this are wrong. For the
>>> moment, a predicate in an interpretation is just a set of pairs.
>>
>> (a26, a87) ∈ Father_of(x, y)
>
> See above. Can't you stick to words for the moment? It will avoid a
> lot of side problems.
I really need to learn the notation.
>>>>>> c. For each function letter f^n_j of L, an assignment of an n-place
>>>>>> operation (f^n_j)^M in D (that is, a function from D^n into D).
>>>>>>
>>>>>> My understanding: [functions evaluate to non _Boolean]
>>>>>
>>>>> More to be wary of here. If the interpretation domain is {true, false}
>>>>> or includes {true, false} then functions can "evaluate" to something you
>>>>> might call _Boolean. (I don't know exactly what you mean by _Boolean.)
>>>>> And that's fine. There is no confusion with predicates because they are
>>>>> not functions and you can't mix functions and predicates in arbitrary
>>>>> ways.
>>>>
>>>> It seems that predicates always [end up with] a possibly empty set of
>>>> n-tuples. What is the correct term for [end up with]?
>>>
>>> A unary is a subset of D. A binary predicate is a subset of DxD. An
>>> n-ary predicate is a subset of D^n. That's the best I can do. I don't
>>> know what else you are trying to say.
>>
>> That works.
>>
>>>>> And again, there is no need for functions to be computable.
>>>>>
>>>>>> My understanding: [father_of(x) would evaluate to a unique constant
>>>>>> element of the set of humans]
>>>>>
>>>>> I would use use "evaluate". And there is no need for unique -- the
>>>
>>> Argh! I meant to say: 'I would /not/ use "evaluate"'.
>>
>> Father_of(x) ∈ Humans = y
>
> That just hints at something. I have no idea what you are really
> saying. The (modern) way to say that a function maps Humans to Humans
> is
>
> Father_of: Humans -> Humans
The conditional symbol?
Father_of: Humans ⇒ Humans
∀y ∈ Humans ∃x ∈ Humans Father_Of(x, y)
D = Humans ∀y ∃x Father_Of(x, y)
>>>>>> My estimate: [functions must always evaluate to elements of D]
>>>>>
>>>>> Yes, that's explicitly in the text.
>>>>
>>>> Great. Any probably also sets of elements of D. GrandfathersOf(x).
>>>
>>> No. An n-ary function always maps an n-tuple to an element of D, never
>>> to a set of elements of D. Of course, if the domain of the
>>> interpretation includes both people and sets of people, then there is no
>>> problem finding a function like GrandfathersOf(x).
>>
>> An n-ary predicate always maps to a set of n-tuples?
>
> No. An n-ary predicate is a set of n-tuples. But why are you now
> talking about predicates when I was trying explain a misunderstanding
> about functions?
<quote Mendelson>
For each predicate letter A^n_j of L, an assignment of an n-place
relation (A^n_j)^M in D.
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.
</quote Mendelson>
It can't possibly be the case that an n-ary predicate <is> a set of
n-tuples because predicates have arguments and n-tuples do no have
arguments.
Are these correct:
An open n-ary predicate specifies a set of n-tuples.
An closed n-ary predicate specifies a Boolean value.
A function specifies one element of the domain.
>>>>>> d. For each individual constant a_i of L, an assignment of some fixed
>>>>>> element (a_i)^M of D. According to Wikipedia 0-ary functions are also
>>>>>> considered to be constants.
>>>>>
>>>>> It's the other way round. Some authors don't bother with constants but
>>>>> instead permit 0-ary functions to serve the same purpose.
>>>>
>>>> OK that makes things simple.
>>>>
>>>>>> My estimate: [constants are unique elements 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 D^n, 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.
>>>>>>
>>>>>> My understanding: [if there are n human beings then there are n^2 2-tuples]
>>>>>>
>>>>>> This is where my understanding drops off. I can't see how his
>>>>>> notation translates from the subset of n-tuples of D that satisfy
>>>>>> n-ary relation_i.
>>>>>
>>>>> First, are you ok with this informal notation of whether a formula is
>>>>> can be satisfied, and of whether a sentence is true or false, in some
>>>>> interpretation? I ask because the technical definition is harder to
>>>>> understand than the idea. You may no need to go though every technical
>>>>> details provided you grasp the intuitive idea.
>>>>
>>>> I think that I have a very good gist of all of the basis ideas. The
>>>> big correction is that a relation [end up with] a possibly empty set
>>>> of n-tuples instead of a Boolean.
>>>>
>>>> (3,7) ∈ Less_Than(x, y) where the domain is ℕ seems to be the way that
>>>> you assert: 3 < 7.
>>>
>>> I don't know what you mean by Less_Than(x, y).
>>
>> FOL does not have a "<" but it does have predicate names.
>
> My problem was with the (x, y) part, not the name. This is the same
> problem as above. If the predicate is called Less_Than, then you say
> either (3, 7) ∈ Less_Than or (more conventionally) Less_Than(3, 7).
The latter convention makes perfect sense. I got confused because I
never ever dealt with open WFF before.
> Your "(3,7) ∈ Less_Than(x, y)" is some mysterious formula with two free
> variables. And it suggests that Less_Than is a function symbol not a
> predicate symbol.
Now that I have dealt with open WFF, I know that
Less_Than(x, y) and (x < y) specify sets of ordered pairs in ℕ.
Whereas Less_Than(3, 7) and (3 < 7) specify Boolean true.
> I think you need to go back a step and learn how Mendelson defines a
> language for a first-order theory. Oh, I see I said that last time:
>
He leaves a lot of stuff out.
>>> Here's the problem. You've skipped over what a language is, and I am
>>> not sure you really know, so I can't interpret what you write. In the
>>> "language of arithmetic" 3 < 7, <(3, 7) and Less_Than(3, 7) are all just
>>> syntax for some relation. It means nothing. It is the interpretation
>>> of the language that lets us know what this relation really is. The
>>> trouble is, people don't bother to distinguish because 99.9% of the time
>>> we are talking about the "usual interpretation" -- the domain is N and <
>>> is, in this interpretation, the usual order on N. It's only when
>>> considering the deeper possibilities that we need to keep these
>>> separate.
>>
>> Yes like the first time that they examine the Geometric axioms without
>> assuming geometric objects.
>>
>>> Hence I advice against using meaningful names in the language because
>>> that will bind you to the possibility that there might be less obvious
> (I meant to write "blind you")
>>> interpretations.
>>
>> It is better to say exactly what you mean in an unequivocal way.
>> An algorithm will do that to.
>
> No. You are confusing levels again. And then injecting your own as yet
> unverified opinions.
If you specify the entire process as operations on finite strings then
all the semantics is self-defined.
Equal(Sum(2,5), 7) is transformed into the constant: TRUE.
UTF-32 could be the alphabet with some special symbols such as
0xffffffff to stand for Boolean TRUE.
> Here's what I ask a student to do now to clear this up: Construct an
> interpretation of PA (or Q which is simpler) such that the predicate <
> (as usually defined) is interpreted to be >.
>
> If you do nothing else in reply to the post, do that.
∃x (Sy = x) // (x > y)
∃y (Sx = y) // (x < y)
>>> Do the exercises first. That way I can be sure you really have a sound
>>> intuitive understanding of what the formal stuff is intended to pin
>>> down. It's much easier to understand the next two pages if you are 100%
>>> sure about the informal notion of satisfiable.
>>>
>>
>> Yes I worked on them today. I agree that it is best that I prove my
>> understanding by those two exercises.
>>
>> 2.10 (a)(i) The set of the product of two positive integers >= 2
>> (1...,2...) and (2...,1...)
>
> I think you know the answer, but you can't write it.
>
> "..." usually means "and so on". You've written only two pairs
> (Mendelson uses <a,b> but I assume you are using (a,b) for a pair), both
> which appear to have infinite "things" (I can;t tell what) in each part
> of the pair.
>
> Unfortunately I can't understand your English either. The phrase "the
> set of" is usually followed by a plural not a singular.
Every pair of positive integers (x,y) such that x*y >= 2.
> So I'm lost. I suggest you use {} for sets and <> for pairs and
> consider starting the English with "(i) is satisfied by the set of pairs
> such that...". Often is helps to name the members of the pair: "the set
> of pairs <a,b> such that...".
>
>> 2.10 (b)(i) The set of the sum of two integers = 0
>> (0,-1,-2...)(0,1,2...)
> I see no set. I see no pairs. Again, I think you know the answer, but
> you can't express it. Try with the form of words above.
The set of ordered pairs (x,y) such that (x + y = 0)
(a) Every negative integer x and its absolute value
(b) Every positive integer x and x * -1
(c) 0, 0
>> 2.10 (c)(i) The domain is the set of all sets of integers, A^2_1(y,z) is
>> y ⊆ z, f^1_1(y, z) is y ∩ z, and a1 is the empty set ∅.
>>
>> ⊆(y ∩ z, ∅)
>
> Yes, though it's much more common to write ⊆ using infix notation.
I was writing it the way that it was specified.
To use infix notation would require rearranging the order from the order
specified.
>> I would think that the empty set would have no subsets.
>
> Except the empty set itself (S ⊆ S for all sets), so the condition is
> that (a ∩ b) = ∅, since only when (a ∩ b) = ∅ can (a ∩ b) be a subset of
> ∅.
That a set is a subset of itself is incoherent.
Likewise with a set being a member of itself.
>> The set of the intersection of y and z is a subset of the empty set.
>
> That's the gist of is. Mendelson is expecting you to know that only the
> empty set can be a subset of the empty set, and he probably expects you
> to know the technical term for sets with no common elements: they are
> called disjoint sets. Can you write the answer succinctly now?
>
I disagree that the empty set has any subsets.
>> // I am not sure what this: "⇒" means in this context
>> ii. A^2_1(x1, x2) ⇒ A^2_1(x2, x1)
>> iii. (∀x1)(∀x2)(∀x3) (A^2_1(x,x2) ∧ A^2_1(x2,x3) ⇒ A^2_1(x1,x3))
>
> It's the logical connective with truth table
>
> T T T
> T F F
> F T T
> F F T
>
> By the way, having everything with a subscript (and many things with a
> superscript) makes the formal definition that comes later much simpler
> but it's a pain for this kind of thing. I re-wrote then exercises in
> the simpler form:
>
> (i) A(f(x, y), a)
> (ii) A(x, y) -> A(y, x)
> (iii) ∀x∀y∀z[ A(x, y) ∧ A(y, z) -> A(x, z) ]
>
> with these interpretations:
>
> (a) D = Z+, A is >=, f is multiplication and a is 2.
> (b) D = Z, A is =, f is addition and a is 0.
> (c) D = 2^Z, A is ⊆, f is intersection and a is {}.
>
The key question is how do we apply the Boolean relation ⇒ between sets
of ordered pairs?
I already took a good guess about this in another reply. You can answer
this question there.
--
Copyright 2020 Pete Olcott
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| From | Keith Thompson <Keith.S.Thompson+u@gmail.com> |
|---|---|
| Date | 2020-07-26 15:46 -0700 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <87pn8hdhe9.fsf@nosuchdomain.example.com> |
| In reply to | #21949 |
olcott <NoOne@NoWhere.com> writes:
[...]
> That a set is a subset of itself is incoherent.
[...]
> I disagree that the empty set has any subsets.
[...]
Are you familiar with the term "proper subset"?
A is a proper subset of B if A is a subset of B and A is not B.
If a set could not be a subset of itself, there would be no need
for the term "proper subset".
Every set is a subset of itself. The empty set is not an
exception to this. (The word "proper" is not a moral judgement.
Proper subsets and improper subsets are both subsets.)
https://en.wikipedia.org/wiki/Subset
--
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 | 2020-07-26 22:46 -0500 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <YPSdnclzy4eG04PCnZ2dnUU7-dGdnZ2d@giganews.com> |
| In reply to | #21952 |
On 7/26/2020 5:46 PM, Keith Thompson wrote: > olcott <NoOne@NoWhere.com> writes: > [...] >> That a set is a subset of itself is incoherent. > [...] >> I disagree that the empty set has any subsets. > [...] > > Are you familiar with the term "proper subset"? I momentarily forgot about that. > > A is a proper subset of B if A is a subset of B and A is not B. > If a set could not be a subset of itself, there would be no need > for the term "proper subset". > > Every set is a subset of itself. The empty set is not an > exception to this. (The word "proper" is not a moral judgement. > Proper subsets and improper subsets are both subsets.) > > https://en.wikipedia.org/wiki/Subset > -- Copyright 2020 Pete Olcott
[toc] | [prev] | [next] | [standalone]
| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2020-07-27 00:28 +0100 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <87ime9hn5h.fsf@bsb.me.uk> |
| In reply to | #21949 |
olcott <NoOne@NoWhere.com> writes:
> On 7/26/2020 2:46 PM, Ben Bacarisse wrote:
>> olcott <NoOne@NoWhere.com> writes:
>>
>>> On 7/24/2020 8:28 PM, Ben Bacarisse wrote:
>>>> olcott <NoOne@NoWhere.com> writes:
>>>>
>>>>> On 7/24/2020 5:49 PM, Ben Bacarisse wrote:
>>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>>> On 7/22/2020 6:54 PM, Ben Bacarisse wrote:
>>>>>>>> olcott <NoOne@NoWhere.com> writes:
>> <cut>
>>>>>>> My understanding: [predicates are functions that evaluate to Boolean]
>>>>>>
>>>>>> Take care here. That's programmer language. The interpretation of a
>>>>>> predicate is a set of pairs. There's not really any evaluation and
>>>>>> there is definitely no need for a predicate to be computable. See the
>>>>>> next comment for why the distinction actually helps.
>>>>>
>>>>> Oh I see the predicate Father_of(x, y) normally evaluates to Boolean
>>>>> for a specific ordered pair of individual people. As a relation on a
>>>>> domain it defines the whole set of ordered pairs that satisfy that
>>>>> relation.
>>>>
>>>> It /is/ the set of pairs that are in the relation. It does not
>>>> "evaluate to Boolean". Nothing is evaluated. For a relation R the
>>>> notation R(x, y) is just another way of writing (x, y) ∈ R.
>>>
>>> Closed WFF are true or false, Open WFF specify sets.
>>
>> True! But I don't know why you say it here.
>
> Making sure that my understanding is correct.
I asked why /here/. The fact that you say it here where it is
irrelevant makes me suspect you don't understand the concepts, despite
having said a true thing. Was it in reply to what I said or a random
true thing you thought you'd say?
>>>>> constant h1 ∈ human beings
>>>>> constant h2 ∈ human beings
>>>>> (h1,h2) ∈ Father_of(x, y) would assert that h1 is the father of h2.
>>>>
>>>> Please, for the moment, no poems. None of that is written correctly.
>>>
>>> To use the Mendelson notation (a26, a87) ∈ Father_of(x, y)
>>> would be true or false.
>>
>> No.
>>
>> Unfortunately, I don't have time to teach you how to write mathematics,
>> but the key is precision. I can take a guess at what you mean, but
>> that's not what the symbols are for. They should remove the need for
>> guessing.
>
> It is crucially important that I understand how to encode constants in
> WFF.
Then you are on your own because I don't know what you mean by encode.
I will concentrate on trying to help you understand languages and
interpretations.
> The ordered pair of two particular individual humans: (a26, a87) a26
> is either the father of a87 or not.
There are always two kinds of formula involved here. There are those of
the formal language, and those we use to talk about an interpretation.
If you don't make it clear which you are talking about there is no way I
can understand you.
But I will say that the usual terminology is to talk about constants (or
0-ary functions) in the formal language, and individuals in the
interpretation.
>> (My guess: You have a language with a binary relation symbol Father_of
>> and you are talking about an interpretation that maps that relation
>> symbol to an actual set of pairs you call Father_of. The domain in
>> question has (at least) two elements that you call a26 and a87. If so,
>> you should write (a26, a87) ∈ Father_of, or more conventionally,
>> Father_of(a26, a87). If this guess is wrong, then I really have no idea
>> what your symbols are supposed to mean.)
>
> Father_of(x, y) with x and y unbound specifies a set.
It could specify a banana. You have not said what formal language you
are using, nor what interpretation you are applying. And even then I
might not know which you are talking about.
> Father_of(a26, a87) is a closed WFF that is true or false.
WF are true or false in an interpretation. If the domain of discourse
is London train stations, Father_of(Paddington, Mile End) might indeed
be true. It depends what relation Father_of denotes. But equally
Father_of could be a function symbol and Father_of(Paddington, Mile End)
could be Green Park. You are relying on the English meaning of the
symbol to convey something the syntax does not. You really should stop
doing that.
>>> (a26, a87) ∈ Father_of(x, y)
>>
>> See above. Can't you stick to words for the moment? It will avoid a
>> lot of side problems.
>
> I really need to learn the notation.
Have you got a basic textbook on mathematics?
>>>>>>> c. For each function letter f^n_j of L, an assignment of an n-place
>>>>>>> operation (f^n_j)^M in D (that is, a function from D^n into D).
>>>>>>>
>>>>>>> My understanding: [functions evaluate to non _Boolean]
>>>>>>
>>>>>> More to be wary of here. If the interpretation domain is {true, false}
>>>>>> or includes {true, false} then functions can "evaluate" to something you
>>>>>> might call _Boolean. (I don't know exactly what you mean by _Boolean.)
>>>>>> And that's fine. There is no confusion with predicates because they are
>>>>>> not functions and you can't mix functions and predicates in arbitrary
>>>>>> ways.
>>>>>
>>>>> It seems that predicates always [end up with] a possibly empty set of
>>>>> n-tuples. What is the correct term for [end up with]?
>>>>
>>>> A unary is a subset of D. A binary predicate is a subset of DxD. An
>>>> n-ary predicate is a subset of D^n. That's the best I can do. I don't
>>>> know what else you are trying to say.
>>>
>>> That works.
>>>
>>>>>> And again, there is no need for functions to be computable.
>>>>>>
>>>>>>> My understanding: [father_of(x) would evaluate to a unique constant
>>>>>>> element of the set of humans]
>>>>>>
>>>>>> I would use use "evaluate". And there is no need for unique -- the
>>>>
>>>> Argh! I meant to say: 'I would /not/ use "evaluate"'.
>>>
>>> Father_of(x) ∈ Humans = y
>>
>> That just hints at something. I have no idea what you are really
>> saying. The (modern) way to say that a function maps Humans to Humans
>> is
>>
>> Father_of: Humans -> Humans
>
> The conditional symbol?
No. It's an arrow. Sometimes ↦ is used. Some people use an arrow for
implication as well. There are just not enough symbols to go round, but
we can use ⇒ for the logical connective so → should be clear.
> Father_of: Humans ⇒ Humans
No, what I wrote.
>>>>>>> My estimate: [functions must always evaluate to elements of D]
>>>>>>
>>>>>> Yes, that's explicitly in the text.
>>>>>
>>>>> Great. Any probably also sets of elements of D. GrandfathersOf(x).
>>>>
>>>> No. An n-ary function always maps an n-tuple to an element of D, never
>>>> to a set of elements of D. Of course, if the domain of the
>>>> interpretation includes both people and sets of people, then there is no
>>>> problem finding a function like GrandfathersOf(x).
>>>
>>> An n-ary predicate always maps to a set of n-tuples?
>>
>> No. An n-ary predicate is a set of n-tuples. But why are you now
>> talking about predicates when I was trying explain a misunderstanding
>> about functions?
>
> <quote Mendelson>
> For each predicate letter A^n_j of L, an assignment of an n-place
> relation (A^n_j)^M in D.
>
> 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.
> </quote Mendelson>
OK. Given the context where I used "maps to" in relation to functions I
thought you were using it in the same sense. You meant "maps to" in the
sense of Mendelson's "can be identified with". There's nothing wrong
about that, but the context made your words very confusing.
(And my wording was bad. I should have said "an n-ary relation (not
predicate) is a set of n-tuples". Then it would have been clear we were
talking at cross purposes.)
> It can't possibly be the case that an n-ary predicate <is> a set of
> n-tuples because predicates have arguments and n-tuples do no have
> arguments.
Actually I meant relation, not predicate. A predicate is just a symbol
in a formal language. It is nothing more nor less than marks on a page
(so to speak). But we write relations as if they have arguments:
a R b
R(a, b)
(a, b) ∈ R
are all ways to write the same thing: that a and b are on some specified
relationship with each other.
> Are these correct:
> An open n-ary predicate specifies a set of n-tuples.
Hm... not really. It's the relation the predicate is identified with
that matters, and it's a formula, not a predicate that is satisfied.
Both they formulas R(x, x) and R(x, y) may be satisfied in some
interpretation -- i.e. they "specify a set of n-tuples -- but the
predicate on it's own does not. Note the the arity of n-tuples is
determined by the formula, not the predicate.
> An closed n-ary predicate specifies a Boolean value.
Closed is a property of formulas. All closed formulas of a language L
are either true of false under some interpretation of L. I can guess
what you mean by "a closed predicate" but it's not helpful.
> A function specifies one element of the domain.
Kind of. A function symbol, in the formal language, specifies nothing.
In some interpretation, a function is a mapping from D^n to D. Formally
it's a subset of D^n x D with the important function property (that f(x)
is unique). So the function (in an interpretation) specifies the whole
mapping. A particular application of the function specified one element
of the domain.
> Now that I have dealt with open WFF, I know that
> Less_Than(x, y) and (x < y) specify sets of ordered pairs in ℕ.
If the interpretation has domain ℕ and Less_Than is identified with the
usual less than relation, then yes, those formulas are satisfied by some
subset of ℕ x ℕ.
> Whereas Less_Than(3, 7) and (3 < 7) specify Boolean true.
Are these formulas in the formal language? Is 3 just shorthand for
S(S(S(0)))? If so, then the usual interpretation will determine that
they are true.
> If you specify the entire process as operations on finite strings then
> all the semantics is self-defined.
Too vague and I don't want to spend time trying to find out what you
really mean. I know how computable functions of string relate to
provability and truth and I don't think you can add to that
understanding.
> Equal(Sum(2,5), 7) is transformed into the constant: TRUE.
Stop writing formulas without saying what the language is. Stop saying
what is or is not true without saying what interpretation you are
considering.
>> Here's what I ask a student to do now to clear this up: Construct an
>> interpretation of PA (or Q which is simpler) such that the predicate <
>> (as usually defined) is interpreted to be >.
>>
>> If you do nothing else in reply to the post, do that.
>
> ∃x (Sy = x) // (x > y)
> ∃y (Sx = y) // (x < y)
You've just read what an interpretation is in the book. I asked for
one. These are two formulas that seem to be attempts to define < and >.
Can you specify a language for either Q or PA? Once you have the
language you will see what is needed to have an interpretation of the
formulas of that language.
>>>> Do the exercises first. That way I can be sure you really have a sound
>>>> intuitive understanding of what the formal stuff is intended to pin
>>>> down. It's much easier to understand the next two pages if you are 100%
>>>> sure about the informal notion of satisfiable.
>>>>
>>>
>>> Yes I worked on them today. I agree that it is best that I prove my
>>> understanding by those two exercises.
>>>
>>> 2.10 (a)(i) The set of the product of two positive integers >= 2
>>> (1...,2...) and (2...,1...)
>>
>> I think you know the answer, but you can't write it.
>>
>> "..." usually means "and so on". You've written only two pairs
>> (Mendelson uses <a,b> but I assume you are using (a,b) for a pair), both
>> which appear to have infinite "things" (I can;t tell what) in each part
>> of the pair.
>>
>> Unfortunately I can't understand your English either. The phrase "the
>> set of" is usually followed by a plural not a singular.
>
> Every pair of positive integers (x,y) such that x*y >= 2.
Yes!
>> So I'm lost. I suggest you use {} for sets and <> for pairs and
>> consider starting the English with "(i) is satisfied by the set of pairs
>> such that...". Often is helps to name the members of the pair: "the set
>> of pairs <a,b> such that...".
>>
>>> 2.10 (b)(i) The set of the sum of two integers = 0
>>> (0,-1,-2...)(0,1,2...)
>> I see no set. I see no pairs. Again, I think you know the answer, but
>> you can't express it. Try with the form of words above.
>
> The set of ordered pairs (x,y) such that (x + y = 0)
> (a) Every negative integer x and its absolute value
> (b) Every positive integer x and x * -1
> (c) 0, 0
Yes. I'd say the pairs <x, -x>.
>>> 2.10 (c)(i) The domain is the set of all sets of integers, A^2_1(y,z) is
>>> y ⊆ z, f^1_1(y, z) is y ∩ z, and a1 is the empty set ∅.
>>>
>>> ⊆(y ∩ z, ∅)
>>
>> Yes, though it's much more common to write ⊆ using infix notation.
>
> I was writing it the way that it was specified.
Then why not do that for the function and write ⊆(∩(y, z))?
> To use infix notation would require rearranging the order from the
> order specified.
Yes, as you did for the function. It makes things much clearer but I
really don't mind. I can read either.
>>> I would think that the empty set would have no subsets.
>>
>> Except the empty set itself (S ⊆ S for all sets), so the condition is
>> that (a ∩ b) = ∅, since only when (a ∩ b) = ∅ can (a ∩ b) be a subset of
>> ∅.
>
> That a set is a subset of itself is incoherent.
x ≤ x but not(x < x). x ⊆ x but not(x ⊂ x). But it's not important
what you think about the words. There are two set relation that are
just like ≤ and <. One includes equality and the other does not.
> Likewise with a set being a member of itself.
Entirely different. That would lead to all kinds of technical issues.
The fact that ⊆ is defined to be ⊂ or = (just like ≤) is a trivial
simplification to avoid writing (a ⊂ b) ∨ (a = b) all the time. By all
means make the voice in your head say "proper subset of or equal to"
every time you see ⊆. The mathematics is what matters.
>>> The set of the intersection of y and z is a subset of the empty set.
>>
>> That's the gist of is. Mendelson is expecting you to know that only the
>> empty set can be a subset of the empty set, and he probably expects you
>> to know the technical term for sets with no common elements: they are
>> called disjoint sets. Can you write the answer succinctly now?
>
> I disagree that the empty set has any subsets.
No one cares. That boat has sailed. The empty set has no proper
subset, but every set has itself as a subset. If you don't want to use
those words you need make up your own and make it clear you are using
your own.
>>> // I am not sure what this: "⇒" means in this context
>>> ii. A^2_1(x1, x2) ⇒ A^2_1(x2, x1)
>>> iii. (∀x1)(∀x2)(∀x3) (A^2_1(x,x2) ∧ A^2_1(x2,x3) ⇒ A^2_1(x1,x3))
>>
>> It's the logical connective with truth table
>>
>> T T T
>> T F F
>> F T T
>> F F T
>>
>> By the way, having everything with a subscript (and many things with a
>> superscript) makes the formal definition that comes later much simpler
>> but it's a pain for this kind of thing. I re-wrote then exercises in
>> the simpler form:
>>
>> (i) A(f(x, y), a)
>> (ii) A(x, y) -> A(y, x)
>> (iii) ∀x∀y∀z[ A(x, y) ∧ A(y, z) -> A(x, z) ]
>>
>> with these interpretations:
>>
>> (a) D = Z+, A is >=, f is multiplication and a is 2.
>> (b) D = Z, A is =, f is addition and a is 0.
>> (c) D = 2^Z, A is ⊆, f is intersection and a is {}.
>
> The key question is how do we apply the Boolean relation ⇒ between
> sets of ordered pairs?
You don't. It's not a programming language. F1 ⇒ F2 is satisfied by
all assignments except those that satisfy F1 and not F2. (F1 ⇒ F2 is
defined to be ¬F1 ∨ F2. Its negation is therefore F1 ∧ ¬F2.)
> I already took a good guess about this in another reply. You can
> answer this question there.
Yes, I have done. There, you clearly got the idea about satisfying
formulas with connectives like ⇒.
--
Ben.
[toc] | [prev] | [next] | [standalone]
| From | Keith Thompson <Keith.S.Thompson+u@gmail.com> |
|---|---|
| Date | 2020-07-26 17:05 -0700 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <87ft9dddqn.fsf@nosuchdomain.example.com> |
| In reply to | #21953 |
Ben Bacarisse <ben.usenet@bsb.me.uk> writes:
[407 lines deleted]
Ben, you posted the same thing 3 times over about 5 minutes.
The bodies of all three articles are identical (I checked).
FYI.
--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Philips Healthcare
void Void(void) { Void(); } /* The recursive call of the void */
[toc] | [prev] | [next] | [standalone]
| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2020-07-27 02:52 +0100 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <87v9i9g1ww.fsf@bsb.me.uk> |
| In reply to | #21956 |
Keith Thompson <Keith.S.Thompson+u@gmail.com> writes: > Ben Bacarisse <ben.usenet@bsb.me.uk> writes: > [407 lines deleted] > > Ben, you posted the same thing 3 times over about 5 minutes. > The bodies of all three articles are identical (I checked). > FYI. Thanks. The posting appeared to fail (after a very long wait) at this end so I retried. I don't know what was happening. -- Ben.
[toc] | [prev] | [next] | [standalone]
| From | Ben Bacarisse <ben.usenet@bsb.me.uk> |
|---|---|
| Date | 2020-07-27 00:30 +0100 |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) |
| Message-ID | <87d04hhn1h.fsf@bsb.me.uk> |
| In reply to | #21949 |
olcott <NoOne@NoWhere.com> writes:
> On 7/26/2020 2:46 PM, Ben Bacarisse wrote:
>> olcott <NoOne@NoWhere.com> writes:
>>
>>> On 7/24/2020 8:28 PM, Ben Bacarisse wrote:
>>>> olcott <NoOne@NoWhere.com> writes:
>>>>
>>>>> On 7/24/2020 5:49 PM, Ben Bacarisse wrote:
>>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>>> On 7/22/2020 6:54 PM, Ben Bacarisse wrote:
>>>>>>>> olcott <NoOne@NoWhere.com> writes:
>> <cut>
>>>>>>> My understanding: [predicates are functions that evaluate to Boolean]
>>>>>>
>>>>>> Take care here. That's programmer language. The interpretation of a
>>>>>> predicate is a set of pairs. There's not really any evaluation and
>>>>>> there is definitely no need for a predicate to be computable. See the
>>>>>> next comment for why the distinction actually helps.
>>>>>
>>>>> Oh I see the predicate Father_of(x, y) normally evaluates to Boolean
>>>>> for a specific ordered pair of individual people. As a relation on a
>>>>> domain it defines the whole set of ordered pairs that satisfy that
>>>>> relation.
>>>>
>>>> It /is/ the set of pairs that are in the relation. It does not
>>>> "evaluate to Boolean". Nothing is evaluated. For a relation R the
>>>> notation R(x, y) is just another way of writing (x, y) ∈ R.
>>>
>>> Closed WFF are true or false, Open WFF specify sets.
>>
>> True! But I don't know why you say it here.
>
> Making sure that my understanding is correct.
I asked why /here/. The fact that you say it here where it is
irrelevant makes me suspect you don't understand the concepts, despite
having said a true thing. Was it in reply to what I said or a random
true thing you thought you'd say?
>>>>> constant h1 ∈ human beings
>>>>> constant h2 ∈ human beings
>>>>> (h1,h2) ∈ Father_of(x, y) would assert that h1 is the father of h2.
>>>>
>>>> Please, for the moment, no poems. None of that is written correctly.
>>>
>>> To use the Mendelson notation (a26, a87) ∈ Father_of(x, y)
>>> would be true or false.
>>
>> No.
>>
>> Unfortunately, I don't have time to teach you how to write mathematics,
>> but the key is precision. I can take a guess at what you mean, but
>> that's not what the symbols are for. They should remove the need for
>> guessing.
>
> It is crucially important that I understand how to encode constants in
> WFF.
Then you are on your own because I don't know what you mean by encode.
I will concentrate on trying to help you understand languages and
interpretations.
> The ordered pair of two particular individual humans: (a26, a87) a26
> is either the father of a87 or not.
There are always two kinds of formula involved here. There are those of
the formal language, and those we use to talk about an interpretation.
If you don't make it clear which you are talking about there is no way I
can understand you.
But I will say that the usual terminology is to talk about constants (or
0-ary functions) in the formal language, and individuals in the
interpretation.
>> (My guess: You have a language with a binary relation symbol Father_of
>> and you are talking about an interpretation that maps that relation
>> symbol to an actual set of pairs you call Father_of. The domain in
>> question has (at least) two elements that you call a26 and a87. If so,
>> you should write (a26, a87) ∈ Father_of, or more conventionally,
>> Father_of(a26, a87). If this guess is wrong, then I really have no idea
>> what your symbols are supposed to mean.)
>
> Father_of(x, y) with x and y unbound specifies a set.
It could specify a banana. You have not said what formal language you
are using, nor what interpretation you are applying. And even then I
might not know which you are talking about.
> Father_of(a26, a87) is a closed WFF that is true or false.
WF are true or false in an interpretation. If the domain of discourse
is London train stations, Father_of(Paddington, Mile End) might indeed
be true. It depends what relation Father_of denotes. But equally
Father_of could be a function symbol and Father_of(Paddington, Mile End)
could be Green Park. You are relying on the English meaning of the
symbol to convey something the syntax does not. You really should stop
doing that.
>>> (a26, a87) ∈ Father_of(x, y)
>>
>> See above. Can't you stick to words for the moment? It will avoid a
>> lot of side problems.
>
> I really need to learn the notation.
Have you got a basic textbook on mathematics?
>>>>>>> c. For each function letter f^n_j of L, an assignment of an n-place
>>>>>>> operation (f^n_j)^M in D (that is, a function from D^n into D).
>>>>>>>
>>>>>>> My understanding: [functions evaluate to non _Boolean]
>>>>>>
>>>>>> More to be wary of here. If the interpretation domain is {true, false}
>>>>>> or includes {true, false} then functions can "evaluate" to something you
>>>>>> might call _Boolean. (I don't know exactly what you mean by _Boolean.)
>>>>>> And that's fine. There is no confusion with predicates because they are
>>>>>> not functions and you can't mix functions and predicates in arbitrary
>>>>>> ways.
>>>>>
>>>>> It seems that predicates always [end up with] a possibly empty set of
>>>>> n-tuples. What is the correct term for [end up with]?
>>>>
>>>> A unary is a subset of D. A binary predicate is a subset of DxD. An
>>>> n-ary predicate is a subset of D^n. That's the best I can do. I don't
>>>> know what else you are trying to say.
>>>
>>> That works.
>>>
>>>>>> And again, there is no need for functions to be computable.
>>>>>>
>>>>>>> My understanding: [father_of(x) would evaluate to a unique constant
>>>>>>> element of the set of humans]
>>>>>>
>>>>>> I would use use "evaluate". And there is no need for unique -- the
>>>>
>>>> Argh! I meant to say: 'I would /not/ use "evaluate"'.
>>>
>>> Father_of(x) ∈ Humans = y
>>
>> That just hints at something. I have no idea what you are really
>> saying. The (modern) way to say that a function maps Humans to Humans
>> is
>>
>> Father_of: Humans -> Humans
>
> The conditional symbol?
No. It's an arrow. Sometimes ↦ is used. Some people use an arrow for
implication as well. There are just not enough symbols to go round, but
we can use ⇒ for the logical connective so → should be clear.
> Father_of: Humans ⇒ Humans
No, what I wrote.
>>>>>>> My estimate: [functions must always evaluate to elements of D]
>>>>>>
>>>>>> Yes, that's explicitly in the text.
>>>>>
>>>>> Great. Any probably also sets of elements of D. GrandfathersOf(x).
>>>>
>>>> No. An n-ary function always maps an n-tuple to an element of D, never
>>>> to a set of elements of D. Of course, if the domain of the
>>>> interpretation includes both people and sets of people, then there is no
>>>> problem finding a function like GrandfathersOf(x).
>>>
>>> An n-ary predicate always maps to a set of n-tuples?
>>
>> No. An n-ary predicate is a set of n-tuples. But why are you now
>> talking about predicates when I was trying explain a misunderstanding
>> about functions?
>
> <quote Mendelson>
> For each predicate letter A^n_j of L, an assignment of an n-place
> relation (A^n_j)^M in D.
>
> 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.
> </quote Mendelson>
OK. Given the context where I used "maps to" in relation to functions I
thought you were using it in the same sense. You meant "maps to" in the
sense of Mendelson's "can be identified with". There's nothing wrong
about that, but the context made your words very confusing.
(And my wording was bad. I should have said "an n-ary relation (not
predicate) is a set of n-tuples". Then it would have been clear we were
talking at cross purposes.)
> It can't possibly be the case that an n-ary predicate <is> a set of
> n-tuples because predicates have arguments and n-tuples do no have
> arguments.
Actually I meant relation, not predicate. A predicate is just a symbol
in a formal language. It is nothing more nor less than marks on a page
(so to speak). But we write relations as if they have arguments:
a R b
R(a, b)
(a, b) ∈ R
are all ways to write the same thing: that a and b are on some specified
relationship with each other.
> Are these correct:
> An open n-ary predicate specifies a set of n-tuples.
Hm... not really. It's the relation the predicate is identified with
that matters, and it's a formula, not a predicate that is satisfied.
Both they formulas R(x, x) and R(x, y) may be satisfied in some
interpretation -- i.e. they "specify a set of n-tuples -- but the
predicate on it's own does not. Note the the arity of n-tuples is
determined by the formula, not the predicate.
> An closed n-ary predicate specifies a Boolean value.
Closed is a property of formulas. All closed formulas of a language L
are either true of false under some interpretation of L. I can guess
what you mean by "a closed predicate" but it's not helpful.
> A function specifies one element of the domain.
Kind of. A function symbol, in the formal language, specifies nothing.
In some interpretation, a function is a mapping from D^n to D. Formally
it's a subset of D^n x D with the important function property (that f(x)
is unique). So the function (in an interpretation) specifies the whole
mapping. A particular application of the function specified one element
of the domain.
> Now that I have dealt with open WFF, I know that
> Less_Than(x, y) and (x < y) specify sets of ordered pairs in ℕ.
If the interpretation has domain ℕ and Less_Than is identified with the
usual less than relation, then yes, those formulas are satisfied by some
subset of ℕ x ℕ.
> Whereas Less_Than(3, 7) and (3 < 7) specify Boolean true.
Are these formulas in the formal language? Is 3 just shorthand for
S(S(S(0)))? If so, then the usual interpretation will determine that
they are true.
> If you specify the entire process as operations on finite strings then
> all the semantics is self-defined.
Too vague and I don't want to spend time trying to find out what you
really mean. I know how computable functions of string relate to
provability and truth and I don't think you can add to that
understanding.
> Equal(Sum(2,5), 7) is transformed into the constant: TRUE.
Stop writing formulas without saying what the language is. Stop saying
what is or is not true without saying what interpretation you are
considering.
>> Here's what I ask a student to do now to clear this up: Construct an
>> interpretation of PA (or Q which is simpler) such that the predicate <
>> (as usually defined) is interpreted to be >.
>>
>> If you do nothing else in reply to the post, do that.
>
> ∃x (Sy = x) // (x > y)
> ∃y (Sx = y) // (x < y)
You've just read what an interpretation is in the book. I asked for
one. These are two formulas that seem to be attempts to define < and >.
Can you specify a language for either Q or PA? Once you have the
language you will see what is needed to have an interpretation of the
formulas of that language.
>>>> Do the exercises first. That way I can be sure you really have a sound
>>>> intuitive understanding of what the formal stuff is intended to pin
>>>> down. It's much easier to understand the next two pages if you are 100%
>>>> sure about the informal notion of satisfiable.
>>>>
>>>
>>> Yes I worked on them today. I agree that it is best that I prove my
>>> understanding by those two exercises.
>>>
>>> 2.10 (a)(i) The set of the product of two positive integers >= 2
>>> (1...,2...) and (2...,1...)
>>
>> I think you know the answer, but you can't write it.
>>
>> "..." usually means "and so on". You've written only two pairs
>> (Mendelson uses <a,b> but I assume you are using (a,b) for a pair), both
>> which appear to have infinite "things" (I can;t tell what) in each part
>> of the pair.
>>
>> Unfortunately I can't understand your English either. The phrase "the
>> set of" is usually followed by a plural not a singular.
>
> Every pair of positive integers (x,y) such that x*y >= 2.
Yes!
>> So I'm lost. I suggest you use {} for sets and <> for pairs and
>> consider starting the English with "(i) is satisfied by the set of pairs
>> such that...". Often is helps to name the members of the pair: "the set
>> of pairs <a,b> such that...".
>>
>>> 2.10 (b)(i) The set of the sum of two integers = 0
>>> (0,-1,-2...)(0,1,2...)
>> I see no set. I see no pairs. Again, I think you know the answer, but
>> you can't express it. Try with the form of words above.
>
> The set of ordered pairs (x,y) such that (x + y = 0)
> (a) Every negative integer x and its absolute value
> (b) Every positive integer x and x * -1
> (c) 0, 0
Yes. I'd say the pairs <x, -x>.
>>> 2.10 (c)(i) The domain is the set of all sets of integers, A^2_1(y,z) is
>>> y ⊆ z, f^1_1(y, z) is y ∩ z, and a1 is the empty set ∅.
>>>
>>> ⊆(y ∩ z, ∅)
>>
>> Yes, though it's much more common to write ⊆ using infix notation.
>
> I was writing it the way that it was specified.
Then why not do that for the function and write ⊆(∩(y, z))?
> To use infix notation would require rearranging the order from the
> order specified.
Yes, as you did for the function. It makes things much clearer but I
really don't mind. I can read either.
>>> I would think that the empty set would have no subsets.
>>
>> Except the empty set itself (S ⊆ S for all sets), so the condition is
>> that (a ∩ b) = ∅, since only when (a ∩ b) = ∅ can (a ∩ b) be a subset of
>> ∅.
>
> That a set is a subset of itself is incoherent.
x ≤ x but not(x < x). x ⊆ x but not(x ⊂ x). But it's not important
what you think about the words. There are two set relation that are
just like ≤ and <. One includes equality and the other does not.
> Likewise with a set being a member of itself.
Entirely different. That would lead to all kinds of technical issues.
The fact that ⊆ is defined to be ⊂ or = (just like ≤) is a trivial
simplification to avoid writing (a ⊂ b) ∨ (a = b) all the time. By all
means make the voice in your head say "proper subset of or equal to"
every time you see ⊆. The mathematics is what matters.
>>> The set of the intersection of y and z is a subset of the empty set.
>>
>> That's the gist of is. Mendelson is expecting you to know that only the
>> empty set can be a subset of the empty set, and he probably expects you
>> to know the technical term for sets with no common elements: they are
>> called disjoint sets. Can you write the answer succinctly now?
>
> I disagree that the empty set has any subsets.
No one cares. That boat has sailed. The empty set has no proper
subset, but every set has itself as a subset. If you don't want to use
those words you need make up your own and make it clear you are using
your own.
>>> // I am not sure what this: "⇒" means in this context
>>> ii. A^2_1(x1, x2) ⇒ A^2_1(x2, x1)
>>> iii. (∀x1)(∀x2)(∀x3) (A^2_1(x,x2) ∧ A^2_1(x2,x3) ⇒ A^2_1(x1,x3))
>>
>> It's the logical connective with truth table
>>
>> T T T
>> T F F
>> F T T
>> F F T
>>
>> By the way, having everything with a subscript (and many things with a
>> superscript) makes the formal definition that comes later much simpler
>> but it's a pain for this kind of thing. I re-wrote then exercises in
>> the simpler form:
>>
>> (i) A(f(x, y), a)
>> (ii) A(x, y) -> A(y, x)
>> (iii) ∀x∀y∀z[ A(x, y) ∧ A(y, z) -> A(x, z) ]
>>
>> with these interpretations:
>>
>> (a) D = Z+, A is >=, f is multiplication and a is 2.
>> (b) D = Z, A is =, f is addition and a is 0.
>> (c) D = 2^Z, A is ⊆, f is intersection and a is {}.
>
> The key question is how do we apply the Boolean relation ⇒ between
> sets of ordered pairs?
You don't. It's not a programming language. F1 ⇒ F2 is satisfied by
all assignments except those that satisfy F1 and not F2. (F1 ⇒ F2 is
defined to be ¬F1 ∨ F2. Its negation is therefore F1 ∧ ¬F2.)
> I already took a good guess about this in another reply. You can
> answer this question there.
Yes, I have done. There, you clearly got the idea about satisfying
formulas with connectives like ⇒.
--
Ben.
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