Re: How a True(X) predicate can be defined for the set of analytic knowledge

On 2025-04-05 17:56:05 +0000, olcott said:

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.

Every particular case can be detected with some partial method but
no method can detect all cases. There are infinitely many partial
methods so you can't try them all. Some partial methods fail to
terminate if no contradiction is detected, so can't prove
non-contracition.
--
Mikko