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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 22:14:04 -0500
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In-Reply-To: <0ca9fab9e728838e60c8a9ff1a657d7644a6c188@i2pn2.org>

On 4/5/2025 5:24 PM, Richard Damon wrote:
> On 4/5/25 5:03 PM, olcott wrote:
>> 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.
>>
> 
> But the test of consistency of the axioms is the test of the consistency 
> of the logic system they create.
> 
No it is not. The axioms can be consistent and create
an inconsistent system because the inference steps
are not truth preserving.

> You just don't understand the meaning of the words you are using.



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