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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: Proving the consistency of the body of knowledge expressed in language
Date: Wed, 2 Apr 2025 12:37:02 +0300
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On 2025-04-02 02:13:36 +0000, olcott said:

> On 4/1/2025 8:03 PM, Richard Damon wrote:
>> On 4/1/25 7:22 PM, olcott wrote:
>>> On 4/1/2025 5:30 PM, Richard Damon wrote:
>>>> On 4/1/25 1:56 PM, olcott wrote:
>>>>> 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).
>>>>> 
>>>> 
>>>> No, the proof is that it is impossible to prove that a system is 
>>>> consistant. (sort of the opposite of what you are thinking of).
>>>> 
>>>> Proving inconsistancy is easy, you just need one example.
>>>> 
>>>> Proving the non-existance isn't as easy, and for a complicated enough 
>>>> system, can't be done, as you need to search an infinite space for the 
>>>> problem, which we can't be sure we have finished,
>>>> 
>>> 
>>> I have always only been referring to the consistency
>>> of a finite set of axioms. Just test each one against
>>> all the others. When we use a type hierarchy we only
>>> have to do this for axioms with compatible types.
>> 
>> And, if they can support the needed level of logic, Godel has shown 
>> that they can not prove their own consistancy.
>> 
> 
> How is it that each element of a finite set of axioms
> can not simply be tested against all of the others?

You can't do so if there is no test method.

-- 
Mikko