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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: Sat, 5 Apr 2025 10:33:32 +0300
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On 2025-04-02 16:04:33 +0000, olcott said:

> On 4/2/2025 4:37 AM, Mikko wrote:
>> 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.
> 
> I need a concrete example.

A concerete example of what?

-- 
Mikko