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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: comp.theory
Subject: Re: The philosophy of logic reformulates existing ideas on a new basis --- infallibly correct
Date: Thu, 14 Nov 2024 10:39:25 +0200
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On 2024-11-13 23:01:50 +0000, olcott said:

> On 11/13/2024 4:45 AM, Mikko wrote:
>> On 2024-11-12 23:17:20 +0000, olcott said:
>> 
>>> On 11/10/2024 2:36 PM, Alan Mackenzie wrote:
>>>> olcott <polcott333@gmail.com> wrote:
>>>>> On 11/10/2024 1:04 PM, Alan Mackenzie wrote:
>>>> 
>>>> [ .... ]
>>>> 
>>>>>> I have addressed your point perfectly well.  Gödel's theorem is correct,
>>>>>> therefore you are wrong.  What part of that don't you understand?
>>>> 
>>>>> YOU FAIL TO SHOW THE DETAILS OF HOW THIS DOES
>>>>> NOT GET RID OF INCOMPLETENESS.
>>>> 
>>>> The details are unimportant.  Gödel's theorem is correct.  Your ideas
>>>> contradict that theorem.  Therefore your ideas are incorrect.  Again, the
>>>> precise details are unimportant, and you wouldn't understand them
>>>> anyway.  Your ideas are as coherent as 2 + 2 = 5.
>>>> 
>>> 
>>> Incomplete(L) ≡  ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
>> 
>> That's correct (although T is usually used instead of L).
>> Per this definition the first order group theory and the first order
>> Peano arithmetic are incomplete.
> 
> Every language that can by any means express self-contradiction
> incorrectly shows that its formal system is incomplete.

That "incorrectly shows" is non-sense. A language does not show,
incorrectly or otherwise. A proof shows but not incorrectly. But
for a proof you need a theory, i.e. more than just a language.

That a theory can't prove something is usually not provable in the
theory itself but usually needs be proven in another theory, one
that can be interpreted as a metatheory.

-- 
Mikko