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Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Simple_Arithmetic)

Subject Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Simple_Arithmetic)
Newsgroups comp.theory, comp.ai.philosophy, comp.ai.nat-lang, sci.lang.semantics
References <QYqdne2udKHUrmPDnZ2dnUU7-evNnZ2d@giganews.com> <rdle1p$haf$1@dont-email.me> <MLCdnSmK2a-RoGPDnZ2dnUU7-dfNnZ2d@giganews.com>
From olcott <NoOne@NoWhere.com>
Date 2020-07-02 15:31 -0500
Message-ID <DrKdnSFFdoqZ2WPDnZ2dnUU7-R_NnZ2d@giganews.com> (permalink)

Cross-posted to 4 groups.

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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)

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.


-- 
Copyright 2020 Pete Olcott

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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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