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| From | André G. Isaak <agisaak@gm.invalid> |
|---|---|
| Newsgroups | comp.theory, comp.ai.philosophy, comp.ai.nat-lang, sci.lang.semantics |
| Subject | Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 |
| Date | 2020-07-05 07:17 -0600 |
| Organization | Christians and Atheists United Against Creeping Agnosticism |
| Message-ID | <rdsjtm$tv0$1@dont-email.me> (permalink) |
| References | <TKydnUzgKLe6LZ3CnZ2dnUU7-S_NnZ2d@giganews.com> <rdqhbj$fnp$1@dont-email.me> <_MGdnVMFZeIeaZ3CnZ2dnUU7-WHNnZ2d@giganews.com> |
Cross-posted to 4 groups.
On 2020-07-04 15:28, olcott wrote: > On 7/4/2020 1:21 PM, André G. Isaak wrote: >> On 2020-07-04 10:36, olcott wrote: >>> OVERVIEW: >>> The sentence used in the SEP article to show the essential gist of >>> the 1931 Gödel incompleteness sentence >>> >>> https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom >>> (G) F ⊢ G_F ↔ ¬Prov_F(⌈G_F⌉) >>> >>> has been shown to not meet the standard definition of incompleteness: >> >> Umm. Of course this doesn't meet the definition of incompleteness. >> Incompleteness is a property of *systems*. What you've given above is >> a *statement*, not a formal system. >> >>> A theory T is incomplete if and only if there is some sentence φ such >>> that (T ⊬ φ) and (T ⊬ ¬φ). Because its negation is provable in F. >>> >>> This is not understood to be any failing of the simplified essence to >>> sufficiently correspond to the gist of the orginal Gödel sentence. It >>> is understood to mean that the Gödel incompleteness sentence does not >>> actually prove incompleteness at all. >> >> nor has it been claimed to prove incompleteness. >> >> The significance of >> >> F ⊢ G_F ↔ ¬Prov_F(⌈G_F⌉) >> >> Is that this statement can only be true if EITHER F is inconsistent OR >> if F is incomplete. And since Gödel provides a mechanical procedure >> for generating a proposition G_F which satisfies the above, ONE of >> these two things must be true. >> >> This only proves that F is incomplete once we add the stipulation that >> F is consistent. Thus, this only proves that F is incomplete once we >> recall Gödel claims his proof only holds true for CONSISTENT formal >> systems in which some minimal amount of arithmetic can be performed. > > We are doing way too many steps at once we will never get resolution at > the current rate because we always slip-slid into extraneous side issues. The problem is we seriously disagree on what count as side issues. > Discussing this one step at a time until that step is 100% resolved. > > Can you see how this can be existentially quantified: > F ⊢ G_F ↔ ¬Prov_F(⌈G_F⌉) such as this: ∃G_F ∈ WFF(F) ¬Prov_F(⌈G_F⌉) Why are you eliminating the biconditional here? Is there some justification for that? André -- To email remove 'invalid' & replace 'gm' with well known Google mail service.
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Simply defining Gödel Incompleteness and Tarski Undefinability away V21 olcott <NoOne@NoWhere.com> - 2020-07-04 11:36 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 André G. Isaak <agisaak@gm.invalid> - 2020-07-04 12:21 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 olcott <NoOne@NoWhere.com> - 2020-07-04 16:28 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 André G. Isaak <agisaak@gm.invalid> - 2020-07-05 07:17 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 olcott <NoOne@NoWhere.com> - 2020-07-05 10:31 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 André G. Isaak <agisaak@gm.invalid> - 2020-07-05 10:28 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 olcott <NoOne@NoWhere.com> - 2020-07-05 11:42 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 André G. Isaak <agisaak@gm.invalid> - 2020-07-05 10:53 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 olcott <NoOne@NoWhere.com> - 2020-07-05 13:38 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 André G. Isaak <agisaak@gm.invalid> - 2020-07-05 12:44 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 olcott <NoOne@NoWhere.com> - 2020-07-05 13:56 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 André G. Isaak <agisaak@gm.invalid> - 2020-07-05 13:16 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 olcott <NoOne@NoWhere.com> - 2020-07-05 15:25 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 André G. Isaak <agisaak@gm.invalid> - 2020-07-05 14:46 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 olcott <NoOne@NoWhere.com> - 2020-07-05 16:08 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 David Kleinecke <dkleinecke@gmail.com> - 2020-07-05 15:28 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 (axiomatic basis of truth) olcott <NoOne@NoWhere.com> - 2020-07-05 17:50 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 (axiomatic basis of truth) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-05 17:13 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 (axiomatic basis of truth) olcott <NoOne@NoWhere.com> - 2020-07-05 20:37 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 (axiomatic basis of truth) André G. Isaak <agisaak@gm.invalid> - 2020-07-05 20:46 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 olcott <NoOne@NoWhere.com> - 2020-07-04 17:39 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21 olcott <NoOne@NoWhere.com> - 2020-07-05 13:15 -0500
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