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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic,comp.theory
Subject: Re: Undecidability based on epistemological antinomies V2
 --Mendelson--
Date: Tue, 23 Apr 2024 09:54:09 -0500
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On 4/22/2024 3:26 AM, Mikko wrote:
> On 2024-04-21 14:34:44 +0000, olcott said:
> 
>> 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.
> 
> It is not a good idea to lie where the truth can be seen.
> 

It is not a good idea to say gibberish nonsense and
expect it to be understood.
 >>> a reference to the term "undecidability"
 >>> is not a reference to the concept 'undecidability'.


>> 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)
> 
> So that is what "undecideble" means in Mendelson: 2015. Elsewhere it may
> mean something else.
> 

It usually means one cannot make up one's mind.
In math it means an epistemological antinomy expression
is not a proposition thus a type mismatch error for every
bivalent system of logic.

not-⊢K ℬ and not-⊢K ¬ℬ. (Mendelson: 2015:208)
K ⊬ ℬ and K ⊬ ¬ℬ. // switching notational conventions

>> Incomplete(F) ≡ ∃x ∈ L ((L ⊬  x) ∧ (L ⊬ ¬x))
> 
> So not the same.
> 

When an expression cannot be proved or refuted is a formal system
this is exactly the same as an expression cannot be proved or refuted
in a formal system.

-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer