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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: Undecidability based on epistemological antinomies V2 --Mendelson--
Date: Wed, 24 Apr 2024 12:49:28 +0300
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On 2024-04-23 14:54:09 +0000, olcott said:

> 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'.

That is how a sentence must be quoted. The proof that the quoted
sentence can be understood is that Richard Damon undesstood it.

>>> 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.

No, it doesn't. There is no reference to an epistemological
anitnomy in "undecidable".

> 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.

To say about an expression that neither it nor its negation cannot be
proven is not the same as to say about a formal system that it contains
expressions that can neither be proven or disproven.

-- 
Mikko