Proving the consistency of the body of knowledge expressed in language

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).
 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.
If we are only allowed to apply the single truth
preserving operation of semantic logical entailment
then we know the whole system must be consistent.
https://en.wikipedia.org/wiki/Logical_consequence#Semantic_consequence
We bypass any need for model theory by having the full
semantics embedded directly in the formal language.

Sort of like we can easily prove that a machine halts, but simulating it to that point (like a real emulator can do for DDD), but showing that a machine is non-halting can be more of a problem. Sometimes we can find an induction property to let us prove it, but not always.

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