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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic,comp.theory
Subject: Re: Undecidability based on epistemological antinomies V2
 --Mendelson--
Date: Sun, 21 Apr 2024 13:05:22 -0400
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On 4/21/24 10:34 AM, olcott wrote:
> On 4/21/2024 2:50 AM, Mikko wrote:
>> On 2024-04-20 16:37:27 +0000, olcott said:
>>
>>> On 4/20/2024 2:41 AM, Mikko wrote:
>>>> On 2024-04-19 02:25:48 +0000, olcott said:
>>>>
>>>>> On 4/18/2024 8:58 PM, Richard Damon wrote:
>>>>
>>>>>> Godel's proof you are quoting from had NOTHING to do with 
>>>>>> undecidability,
>>>>>
>>>>> *Mendelson (and everyone that knows these things) disagrees*
>>>>>
>>>>> https://sistemas.fciencias.unam.mx/~lokylog/images/Notas/la_aldea_de_la_logica/Libros_notas_varios/L_02_MENDELSON,%20E%20-%20Introduction%20to%20Mathematical%20Logic,%206th%20Ed%20-%20CRC%20Press%20(2015).pdf
>>>>
>>>> On questions whether Gödel said something or not the sumpreme authority
>>>> is not Mendelson but Gödel.
>>>>
>>>
>>> When some authors affirm that undecidability and incompleteness
>>> are the exact same thing then whenever Gödel uses the term
>>> incompleteness then he is also referring to the term undecidability.
>>
>> That does not follow. Besides, a reference to the term "undecidability"
>> is not a reference to the concept 'undecidability'.
>>
> 
> In other words you deny the identity principle thus X=X is false.
> 
> An undecidable sentence of a theory K is a closed wf ℬ of K such that
> neither ℬ nor ¬ℬ is a theorem of K, that is, such that not-⊢K ℬ and
> not-⊢K ¬ℬ. (Mendelson: 2015:208)
> 
> Incomplete(F) ≡ ∃x ∈ L ((L ⊬  x) ∧ (L ⊬ ¬x))
> 

Except incompleteness and undecidablity are attributes of different 
sorts of things.

Yes, they are related, as Incompleteness of a Theory K, is a statement 
that in K there exists a theory that is accepted as a valid statement in 
K (and thus, for well defined Theory K that statement has a truth value) 
but that statement is not provable or refutable in the Theory K, which 
in the later terminology says that statement is undecidable.

Note the different, The SYSTEM (K) is incomplete, but it is the 
STATEMENT that is undecidable.

Godel focues on the SYSTEM, and shows that he can construct the 
unprovable but true statement. He doesn't focus on the statement itself, 
as no one is particularlly interested in that particular statement, so 
it being unprovable, by itself, doesn't mean much, except that, by 
existing, he shows that the full system must be incomplete, and thus 
there may well exist MANY such statements that are true but unprovable. 
THAT is an important fact.

Note, this is different then the Halting Problem, where even if Turing 
Computation was shown to be "incomplete" in that there existed some 
mappings that were not computable, being able to compute the Halting 
Mapping would have extreme value, so the focus isn't on the "system" as 
a whole, but that one particular problem, the computing of the Halting 
Function given a description of the machine.

Thus, "Uncomputable" and "Incompleteness" while highly related and 
intertwined are NOT "the same".

Your confusing the two just shows your fundamental lack of understanding 
of what you are talking about.