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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic
Subject: Re: How a True(X) predicate can be defined for the set of analytic
 knowledge
Date: Sat, 5 Apr 2025 16:03:28 -0500
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On 4/5/2025 3:15 PM, Richard Damon wrote:
> On 4/5/25 1:56 PM, olcott wrote:
>> On 4/5/2025 2:44 AM, Mikko wrote:
>>> On 2025-04-02 16:03:32 +0000, olcott said:
>>>
>>>> On 4/2/2025 4:32 AM, Mikko wrote:
>>>>> On 2025-04-01 17:56:25 +0000, olcott said:
>>>>>
>>>>>> On 4/1/2025 1:33 AM, Mikko wrote:
>>>>>>> On 2025-03-31 18:33:26 +0000, olcott said:
>>>>>>>
>>>>>>>>
>>>>>>>> Anything the contradicts basic facts or expressions
>>>>>>>> semantically entailed from these basic facts is proven
>>>>>>>> false.
>>>>>>>
>>>>>>> Anything that follows from true sentences by a truth preserving
>>>>>>> transformations is true. If you can prove that a true sentence
>>>>>>> is false your system is unsound.
>>>>>>
>>>>>> Ah so we finally agree on something.
>>>>>> What about the "proof" that detecting inconsistent
>>>>>> axioms is impossible? (I thought that I remebered this).
>>>>>
>>>>> A method that can always determine whether a set of axioms is 
>>>>> inconsistent
>>>>> does not exist. However, there are methods that can correctly 
>>>>> determine
>>>>> about some axiom systems that they are inconsistent and fail on 
>>>>> others.
>>>>>
>>>>> The proof is just another proof that some function is not Turing 
>>>>> computable.
>>>>
>>>> A finite set of axioms would seem to always be verifiable
>>>> as consistent or inconsistent.  This may be the same for
>>>> a finite list of axiom schemas.
>>>
>>> If ordinary logic is used it is sufficient to prove that there is
>>> a sentence that cannot be proven in order to prove consistency or
>>> to prove two sentences that contradict each other in order to prove 
>>> inconsistency. But if neither proof is known there is no method to
>>> find one.
>>>
>>
>> We are only talking about the inability to detect
>> that basic facts contradict each other. I need a
>> 100% concrete example proving this that this is
>> sometimes impossible.
>>
> 
> Read Godel's proof.
> 
> Note, this follows from the incompleteness proof, as a proof of 
> consistency yields a proof of completeness and thus any set powerful 
> enough to be incomplete also can not prove its own consistancy.

We are not talking about a proof of consistency
of the whole system, only a proof of consistency
of a finite set of axioms. Simply test them against
each other.

-- 
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer