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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 V20 (Simple_Arithmetic) |
| Date | 2020-07-02 23:24 -0600 |
| Organization | Christians and Atheists United Against Creeping Agnosticism |
| Message-ID | <rdmfen$m8a$1@dont-email.me> (permalink) |
| References | (1 earlier) <rdle1p$haf$1@dont-email.me> <MLCdnSmK2a-RoGPDnZ2dnUU7-dfNnZ2d@giganews.com> <DrKdnSFFdoqZ2WPDnZ2dnUU7-R_NnZ2d@giganews.com> <rdlleq$tee$1@dont-email.me> <Xu6dnS2UHtvbwWPDnZ2dnUU7-RudnZ2d@giganews.com> |
Cross-posted to 4 groups.
On 2020-07-02 16:14, olcott wrote: > On 7/2/2020 5:00 PM, André G. Isaak wrote: >> On 2020-07-02 14:31, olcott wrote: >>> On 7/2/2020 3:01 PM, olcott wrote: >>>> On 7/2/2020 2:54 PM, André G. Isaak wrote: >>>>> On 2020-07-02 13:19, olcott wrote: >>>>>> The definition of incompleteness:A theory T is incomplete if and >>>>>> only if there is some sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ). >>>>>> >>>>>> Violates valid/sound deductive inference model: >>>>>> ∀F ∈ Formal_Systems ∀C ∈ WFF(F) ((F ⊢ C) ↔ True(F, C)) >>>>>> ∀F ∈ Formal_Systems ∀C ∈ WFF(F) ((F ⊢ ¬C) ↔ False(F, C)) >>>>>> ∀F ∈ Formal_Systems ∀C ∈ WFF(F) ((F ⊬ C) ↔ NonSequitur(F, C)) >>>>> >>>>> As far as I can tell, that entails that if some φ can be proven to >>>>> be false but cannot be proven to be true, then it is both false and >>>>> a non-sequitur. >>>>> >>>>> So what is the difference between the two? >>>>> >>>>> André >>>>> >>>> >>>> Ah you got me. I did not encode it the way that I usually do: >>>> ∀F ∈ Formal_Systems ∀C ∈ WFF(F) >>>> (((F ⊬ C) ∧ (F ⊬ ¬C)) ↔ NonSequitur(F, C)) >>> >>> Let's try all that using the formal system of Simple_Arithmetic (SA) >> >> Unless you actually define what you mean by Simple_Arithmetic, the >> below is entirely meaningless. >> >> André >> >>> A := "2 + 3 = 5" (SA ⊢ A)(true) (SA ⊢ ¬A)(false) >>> B := "2 + 3 = 9" (SA ⊢ A)(false) (SA ⊢ ¬A)(true) >>> B := "2 # 3 = 9" (SA ⊢ A)(false) (SA ⊢ ¬A)(false) >>> >>> Since Simple_Arithmetic cannot express self-contradictory sentences >>> the only finite strings that cannot be proved or disproved in the >>> system are strings that are not WFF. >>> > > I have a whole C++ program for people that disingenuously claim that > they can't begin to understand that "2 + 3 = 5" is true "2 + 3 = 9" is > false and "2 # 3 = 9" is gibberish. That C++ program is a computer program, not a formal system. It is possible that it instantiates some formal system called SA, but unless you actually provide us with such a formal system we'll never know. If I were to attempt to guess what SA is supposed to be based on this program, all I can come up with is some system which doesn't satisfy the conditions which Gödel sets out in his proof. Gödel's theorem only applies to formal systems in which a certain minimal amount of arithmetic can be performed and, AFAICT, your SA doesn't come close to meeting that minimal threshold. Ergo, it is entirely irrelevant to any discussion of Gödel. <snip pointless program> 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 V20 olcott <NoOne@NoWhere.com> - 2020-07-02 14:19 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 André G. Isaak <agisaak@gm.invalid> - 2020-07-02 13:54 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 olcott <NoOne@NoWhere.com> - 2020-07-02 15:01 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 André G. Isaak <agisaak@gm.invalid> - 2020-07-02 14:28 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Simple_Arithmetic) olcott <NoOne@NoWhere.com> - 2020-07-02 16:01 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 olcott <NoOne@NoWhere.com> - 2020-07-02 16:50 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 olcott <NoOne@NoWhere.com> - 2020-07-02 17:10 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Simple_Arithmetic) olcott <NoOne@NoWhere.com> - 2020-07-02 15:31 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Simple_Arithmetic) olcott <NoOne@NoWhere.com> - 2020-07-02 15:56 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Simple_Arithmetic) André G. Isaak <agisaak@gm.invalid> - 2020-07-02 16:00 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Simple_Arithmetic) olcott <NoOne@NoWhere.com> - 2020-07-02 17:14 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Simple_Arithmetic) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-02 18:41 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Simple_Arithmetic) olcott <NoOne@NoWhere.com> - 2020-07-02 21:03 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Simple_Arithmetic) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-02 21:17 -0700
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Simple_Arithmetic) olcott <NoOne@NoWhere.com> - 2020-07-03 00:33 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Simple_Arithmetic) André G. Isaak <agisaak@gm.invalid> - 2020-07-02 23:10 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Simple_Arithmetic) André G. Isaak <agisaak@gm.invalid> - 2020-07-02 23:24 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Simple_Arithmetic) olcott <NoOne@NoWhere.com> - 2020-07-03 01:00 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Simple_Arithmetic) André G. Isaak <agisaak@gm.invalid> - 2020-07-03 01:02 -0600
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Simple_Arithmetic) olcott <NoOne@NoWhere.com> - 2020-07-03 12:56 -0500
Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Simple_Arithmetic) André G. Isaak <agisaak@gm.invalid> - 2020-07-03 12:17 -0600
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