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Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V14

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On 2020-06-28 20:21, olcott wrote:
> On 6/28/2020 1:18 PM, André G. Isaak wrote:
>> On 2020-06-28 11:22, olcott wrote:
>>> On 6/28/2020 3:16 AM, André G. Isaak wrote:
>>>> On 2020-06-27 23:24, olcott wrote:
>>>>> 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'?
>>>>
>>>> Again, I ask. How is Euclid's P5 '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?
>>>>
>>>> Again, you fail to answer the above. How can this claim be 
>>>> self-contradictory? [Answer: It isn't, and this is what Gödel relies 
>>>> on in his proof, not the example you keep attributing to him].
>>>>
>>>>>> 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 isn't even coherent. Self-contradictory sentences 
>>>> would not meet the definition of incompleteness as they would be 
>>>> inconsistent which is different from undecidable.
>>>
>>>
>>> // If this was true that would make it false thus self contradictory
>>> (G ↔ ((F ⊬ G)
>> Strictly speaking, that would not be self-contradictory. Just strange.
> 
> --- ∃F ∈ Formal_Systems ∃G ∈ WFF(F) (G ↔ (F ⊬ G))
> What is the Boolean value of the above expression, and why?

You still have not grasped that ⊬ IS NOT a truth functional operator. (G 
↔ ¬G) is obviously false, but (G ↔ ⊬G) means something entirely 
different and cannot be evaluated except with reference to some actual 
SPECIFIED formal system (not just a variable like F) and where the 
actual content of G is fully specified. Your ∃F ∈ Formal_Systems is 
entirely meaningless since Formal_Systems is not a well-defined set. The 
critical point I am making is simply that the above, regardless whether 
it is true or false, is not self-contradictory.

>>> // If this was true that would make it false thus self contradictory
>>> (G ↔ ((F ⊬ ¬G))
>>
>> How would that be self-contradictory? There's nothing remotely 
>> self-contradictory about that statement.
> 
> --- ∃F ∈ Formal_Systems ∃G ∈ WFF(F) (G ↔ (F ⊬ G))
> What is the Boolean value of the above expression, and why?
why are you switching from G ↔ ((F ⊬ ¬G) to G ↔ ((F ⊬ G)? Those are 
entirely different claims.

>>> // If this was true that would make G not Boolean and
>>> // cause F to meet this spec:
>>> //
>>> // A theory T is incomplete if and only if there is some
>>> // sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ).
>>> (G ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))
>>
>> That definition says a theory would be incomplete if (F ⊬ G) ∧ (F ⊬ 
>> ¬G) The biconditional G ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)) which you keep 
>> insisting on bringing up isn't mentioned in that definition, and I 
>> have absolutely no idea where this comes from or why you keep raising 
>> it, apart from your obsession with self-reference (which this is not 
>> an example of anyways).
> 
> You didn't notice that I have quoted that from here fifteen times ???
> https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom

Except you are introducing this in the context which has absolutely 
nothing to do with the argument being laid out in that particular link. 
So even if you have referenced this article in the past, that provides 
no explanation for why you keep introducing "G ↔" at the beginning of 
statements entirely unrelated to that article.

>> I already gave you two examples which you have refused to comment on.
> 
> I am going to always absolutely refuse deflection, I wasted decades on 
> tolerating deflection away for the point.
> 
> X proves Y, Well what about Z, Z is off topic !!!
> 
>> Euclid's P5 cannot be proven from P1-P4. Nor can ¬P5 be proven from 
>> P1-P4. Therefore P5 is undecidable in a system consisting solely of 
>> P1-P4.
> 
> The point is that incompleteness is defined when-so-ever any WFF is 
> neither provable nor refutable.

Which means that unless you can demonstrate that either P5 or ¬P5 can be 
derived from P1-P4, P1-P4 is incomplete.

> We don't have to frigging examine every single damn case of WFF that are 
> neither provable nor refutable.
> We only need to examine the one case that proves that the definition of 
> incompleteness doesn't actually mean any sort of
> {not having all the necessary or appropriate parts}

The above is completely unparsable. I have no idea what "incompleteness 
doesn't actually mean any sort of {not having all the necessary or 
appropriate parts}" is supposed to mean, nor have I ever heard anyone 
claim that.

> If there is one case of this then the definition of incompleness is 
> incorrect

Once again, definitions *cannot* be incorrect. They can be useful or 
non-useful.

André


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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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