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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic
Subject: Re: Undecidability based on epistemological antinomies V2
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Date: Mon, 29 Apr 2024 10:26:23 -0500
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On 4/29/2024 10:13 AM, Mikko wrote:
> On 2024-04-29 14:22:36 +0000, olcott said:
> 
>> On 4/29/2024 4:09 AM, Mikko wrote:
>>> On 2024-04-28 13:41:50 +0000, olcott said:
>>>
>>>> On 4/28/2024 4:34 AM, Mikko wrote:
>>>>> On 2024-04-27 13:36:56 +0000, olcott said:
>>>>>
>>>>>> On 4/27/2024 3:18 AM, Mikko wrote:
>>>>>>> On 2024-04-26 15:28:08 +0000, olcott said:
>>>>>>>
>>>>>>>> On 4/26/2024 3:42 AM, Mikko wrote:
>>>>>>>>> On 2024-04-25 14:27:23 +0000, olcott said:
>>>>>>>>>
>>>>>>>>>> On 4/25/2024 3:26 AM, Mikko wrote:
>>>>>>>>>>> epistemological antinomy
>>>>>>>>>>
>>>>>>>>>> It <is> part of the current (thus incorrect) definition
>>>>>>>>>> of undecidability because expressions of language that
>>>>>>>>>> are neither true nor false (epistemological antinomies)
>>>>>>>>>> do prove undecidability even though these expressions
>>>>>>>>>> are not truth bearers thus not propositions.
>>>>>>>>>
>>>>>>>>> That a definition is current does not mean that is incorrect.
>>>>>>>>>
>>>>>>>>
>>>>>>>> ...14 Every epistemological antinomy can likewise be used for a 
>>>>>>>> similar
>>>>>>>> undecidability proof...(Gödel 1931:43-44)
>>>>>>>>
>>>>>>>>> An epistemological antinomy can only be an undecidable sentence
>>>>>>>>> if it can be a sentence. What epistemological antinomies you
>>>>>>>>> can find that can be expressed in, say, first order goup theory
>>>>>>>>> or first order arithmetic or first order set tehory?
>>>>>>>>>
>>>>>>>>
>>>>>>>> It only matters that they can be expressed in some formal system.
>>>>>>>> If they cannot be expressed in any formal system then Gödel is
>>>>>>>> wrong for a different reason.
>>>>>>>
>>>>>>> How is it relevant to the incompleteness of a theory whether an
>>>>>>> epistemological antińomy can be expressed in some other formal
>>>>>>> system?
>>>>>>
>>>>>> When an expression of language cannot be proved in a formal system 
>>>>>> only
>>>>>> because it is contradictory in this formal system then the 
>>>>>> inability to
>>>>>> prove this expression does not place any actual limit on what can be
>>>>>> proven because formal system are not supposed to prove 
>>>>>> contradictions.
>>>>>
>>>>> The first order theories of Peano arithmetic, ZFC set theory, and
>>>>> group theroy are said to be incomplete but you have not shown any
>>>>> fromula of any of them that could be called an epistemoloigcal
>>>>> antinomy.
>>>>>
>>>>
>>>> The details of the semantics of the inference steps are hidden behind
>>>> arithmetization and diagonalization in Gödel's actual proof.
>>>
>>> The correctness of a proof can be checked without any consideration of
>>> semantics. If the proof is fully formal there is an algorithm to check
>>> the correctness.
>>>
>>>> ($)   ⊢k G ⇔ (∀x2) ¬𝒫𝑓 (x2, ⌜G⌝)
>>>>
>>>> Observe that, in terms of the standard interpretation (∀x2) ¬𝒫𝑓 (x2,
>>>> ⌜G⌝) says that there is no natural number that is the Gödel number of a
>>>> proof in K of the wf G, which is equivalent to asserting that there is
>>>> no proof in K of G.
>>>
>>> The standard interpretation of artihmetic does not say anything about
>>> proofs and Gödel numbers.
>>>
>>
>> That was a direct quote from a math textbook, here it is again:
> 
> That quote didn't define "standard semantics".
> 

It need not define standard semantics once is has summed of the essence 
of that whole proof as: G says “I am not provable in K”.



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