Re: Proving the consistency of the body of knowledge expressed in language

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:
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Anything the contradicts basic facts or expressions
semantically entailed from these basic facts is proven
false.

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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.
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Ah so we finally agree on something.
What about the "proof" that detecting inconsistent
axioms is impossible? (I thought that I remebered this).
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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).
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Proving inconsistancy is easy, you just need one example.
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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,
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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.

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And, if they can support the needed level of logic, Godel has shown that they can not prove their own consistancy.
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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.