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Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14

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On 6/27/2020 11:48 PM, André G. Isaak wrote:
> On 2020-06-27 22:01, olcott wrote:
>> On 6/27/2020 10:03 PM, André G. Isaak wrote:
>>> On 2020-06-27 14:58, olcott wrote:
>>>> On 6/27/2020 2:22 PM, André G. Isaak wrote:
>>>>> On 2020-06-27 12:03, olcott wrote:
>>>>>> Defining Gödel Incompleteness Away
>>>>>>
>>>>>> We can simply define Gödel 1931 Incompleteness away by redefining 
>>>>>> the meaning of the standard definition of Incompleteness: A theory 
>>>>>> T is incomplete if and only if there is some sentence φ such that 
>>>>>> (T ⊬ φ) and (T ⊬ ¬φ) to cease defining formal systems as 
>>>>>> incomplete on the basis that its self contradictory expressions 
>>>>>> can be neither proven nor disproven.
>>>>>
>>>>> You keep introducing the notion of 'self-contradictory expressions' 
>>>>> into your discussions of incompleteness, but I have no idea why. 
>>>>> The definition you cite above states that a system is incomplete if 
>>>>> there is some φ such that neither φ nor ¬φ can be derived as 
>>>>> theorems of the system. There's nothing in that definition which 
>>>>> claims that φ is self-contradictory, so trying to figure out how to 
>>>>> exclude such propositions is going to have no bearing on 
>>>>> incompleteness.
>>>>>
>>>>> As a concrete example, consider a system of geometry which consists 
>>>>> of Euclid's first four postulates, but not the fifth postulate. P5 
>>>>> cannot be proven from P1 through P4. Nor can it be disproven from 
>>>>> P1 through P4. But there is absolutely nothing self-contradictory 
>>>>> about the fifth postulate.
> 
> You continuously skip over this. In what sense can P5 be said to be 
> 'self-contradictory'?
> 
>>>>> André
>>>>>
>>>>
>>>> You have to actually read all of what I say and not merely glance at 
>>>> a few words before forming your rebuttal.
>>>>
>>>> If you read the rest of what I said you will see that the simplified 
>>>> essence of the Gödel sentence is only unprovable because it is 
>>>> self-contradictory.
>>>
>>> You have made this claim many times, but it is simply wrong. Your 
>>> 'simplified essence' of the Gödel sentence bears no resemblance to 
>>> what Gödel actually wrote.
> 
> You conveniently snipped this line:
> 
> Gödel's sentence asserts that X = 0 (where X is a complex polynomial) 
> has a solution. How can such an assertion possibly be self-contradictory?
> 
> Please answer the question.
> 
>> Here is the cite of my source.
>> Raatikainen, Panu, "Gödel’s Incompleteness Theorems", The Stanford 
>> Encyclopedia of Philosophy (Summer 2020 Edition), Edward N. Zalta 
>> (ed.), URL = 
>> <https://plato.stanford.edu/archives/sum2020/entries/goedel-incompleteness/>. 
>>
>>
>> The conventional definition of incompleteness:
>> A theory T is incomplete if and only if there is some sentence φ such 
>> that (T ⊬ φ) and (T ⊬ ¬φ).
>>
>> The way that we can definitely tell that this minimal essence of the 
>> Gödel sentence sufficiently matches the essence of the original 
>> sentence is that this sentence meets the above definition of 
>> incompleteness thus proving incompleteness:
>>
>> ∃F ∈ Formal_Systems ∃G ∈ WFF(F) (G ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))
> 
> That sentence is not the same as the definition given above and says 
> nothing whatsoever about Gödel.

-    14 Every epistemological antinomy can likewise be used for
-    a similar undecidability proof.(Godel 1931:40)

The fact that all self-contradictory sentences meet the definition
of incompleteness unequivocally proves that the definition of 
incompleteness is erroneous.

The above sentence is an epistemological antinomy that proves 
incompleteness thus sufficiently meeting Gödel's own measure of 
sufficiency and the definition of incompleteness itself.



-- 
Copyright 2020 Pete Olcott

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Simply defining Gödel Incompleteness and Tarski Undefinability away V14 olcott <NoOne@NoWhere.com> - 2020-06-27 13:03 -0500
  Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14 André G. Isaak <agisaak@gm.invalid> - 2020-06-27 13:22 -0600
    Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14 olcott <NoOne@NoWhere.com> - 2020-06-27 15:58 -0500
      Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14 André G. Isaak <agisaak@gm.invalid> - 2020-06-27 21:03 -0600
        Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14 olcott <NoOne@NoWhere.com> - 2020-06-27 23:01 -0500
          Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14 André G. Isaak <agisaak@gm.invalid> - 2020-06-27 22:48 -0600
            Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14 olcott <NoOne@NoWhere.com> - 2020-06-28 00:24 -0500
              Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14 olcott <NoOne@NoWhere.com> - 2020-06-28 00:51 -0500
              Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14 André G. Isaak <agisaak@gm.invalid> - 2020-06-28 02:16 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14 olcott <NoOne@NoWhere.com> - 2020-06-28 12:22 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14 André G. Isaak <agisaak@gm.invalid> - 2020-06-28 12:18 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14 olcott <NoOne@NoWhere.com> - 2020-06-28 21:21 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14 André G. Isaak <agisaak@gm.invalid> - 2020-06-28 21:23 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14 olcott <NoOne@NoWhere.com> - 2020-06-29 14:23 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14 André G. Isaak <agisaak@gm.invalid> - 2020-06-29 13:47 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14 olcott <NoOne@NoWhere.com> - 2020-06-29 14:55 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14 André G. Isaak <agisaak@gm.invalid> - 2020-06-29 14:10 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14 olcott <NoOne@NoWhere.com> - 2020-06-29 15:14 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14 David Kleinecke <dkleinecke@gmail.com> - 2020-06-29 14:05 -0700
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14 André G. Isaak <agisaak@gm.invalid> - 2020-06-29 15:19 -0600

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