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

On 4/5/2025 3:15 PM, Richard Damon wrote:

On 4/5/25 1:56 PM, olcott wrote:

On 4/5/2025 2:44 AM, Mikko wrote:

On 2025-04-02 16:03:32 +0000, olcott said:
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On 4/2/2025 4:32 AM, Mikko wrote:

On 2025-04-01 17:56:25 +0000, olcott said:
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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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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.
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The proof is just another proof that some function is not Turing computable.

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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.

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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.
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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.
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 Read Godel's proof.
 Note, this follows from the incompleteness proof, as a proof of consistency yields a proof of completeness and thus any set powerful enough to be incomplete also can not prove its own consistancy.

We are not talking about a proof of consistency
of the whole system, only a proof of consistency
of a finite set of axioms. Simply test them against
each other.
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer