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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 09:44:43 -0500
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On 4/22/2024 5:54 PM, Richard Damon wrote:
> On 4/22/24 10:03 AM, olcott wrote:
>> 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.
>>>
>>
>>  >>>"undecidability" is not a reference to the concept 'undecidability'.
>> That is the best that I could make about the above quote. There is no
>> standard practice of using different kind of quotes that I am aware of.
> 
> Except that undeciability and incompleteness are not the EXACT same thing.
> 

So you were paying attention?
He said that undecidability is not the same thing as undecidability.
Somehow he felt that two different kinds of quotes mean something.

> They CAN'T be, because they apply to different class of objects.
> 
> Of course, you are too stupid to understand that, because you logic is 
> based on making category errors.

In this case the issue is that you did not pay attention.
You glanced at a couple of words without even seeing them
and then spouted off a canned rebuttal that does not apply.

>>
>>>> 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 never means anything else.
> 
> LIE.
> 
> It also means (as the ORIGINAL definition) a computation problem for 
> which no computation can be created that always gives the correct answer.
> 

That is the theory of computation way of saying it.
Mendelson translates the same idea into the math way of saying it.

>>
>>>> Incomplete(F) ≡ ∃x ∈ L ((L ⊬  x) ∧ (L ⊬ ¬x))
>>>
>>> So not the same.
>>>
>>
>> Not provable or refutable in a formal system is exactly
>> the same as not provable of refutable in a formal system.
>> I think that you are playing head games.
>>
> 
> But that isn't what the above says, itr says that F HAS a statement that 
> is not provable or refutable, while undecidable (when applied to a 
> statement) says THAT STATEMENT is not provable or refutable.
> 
> SYSTEMS are not STATEMENTS, so you are shows to be just wrong.

When an expression is neither provable or refutable because it is not
a statement/proposition that has a truth value then it must be rejected 
as a type mismatch error for ever bivalent system of logic.

A proposition is a central concept in the philosophy of language,
semantics, logic, and related fields, often characterized as the primary
bearer of truth or falsity.  https://en.wikipedia.org/wiki/Proposition

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