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Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V21

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On 7/5/2020 8:17 AM, André G. Isaak wrote:
> 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
> 

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  ∃G_F ∈ WFF(F) ↔ ¬Prov_F(⌈G_F⌉)


-- 
Copyright 2020 Pete Olcott

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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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