From Newsgroup: comp.ai.philosophy

On 1/18/2026 9:56 PM, Richard Damon wrote:

On 1/18/26 10:19 PM, olcott wrote:

On 1/18/2026 7:24 PM, Python wrote:

Le 19/01/2026 à 00:41, olcott a écrit :
..

I already just said that the proof and refutation of
Goldbach are outside the scope of PA axioms.

Any proof or refutation of Goldbach would have to use
principles stronger than the axioms of PA, because PA
itself does not currently derive either direction.

"currently" ? ?  What kind of language is that? PA is what it is, it
not changing with time !

You could have said that about Fermat's theorem back in the day... It
happens not to be the case.

You are out of reason, Peter. Not only a liar, an hypocrite, but a fool. >>>

If its truth value cannot be determined in a finite
number of steps then it is not a truth bearer in PA,
otherwise it is a truth-bearer in PA with an unknown value.

So, you admit that you don't know how to classify it.

Thus its truth-bearer status is unknown.

Thus, your claim that it is outside of PA is just a LIE.

No it was a mistake. Here is my correction:
If Goldbach's truth value cannot be determined in a
finite number of steps then it is not a truth bearer
in PA, otherwise it is a truth-bearer in PA with an
unknown truth value.

This has no effect on my claim that I got rid of
Gödel Incompleteness.

When we change the foundation of formal systems
to proof theoretic semantics and add my truth
predicates then Gödel's claim of applying to
every formal system that can do a little bit of
arithmetic becomes simply false.

Every attempt at showing incompleteness <in> PA
has never actually been <in> PA.

The satisfaction of external models of arithmetic
never has been <in> PA. These are categorically
outside of PA by the definition of proof theoretic
semantics thus defined as non-well-founded. This
neuters their ability to show incompleteness.
--
Copyright 2026 Olcott<br><br>

My 28 year goal has been to make <br>
"true on the basis of meaning expressed in language"<br>
reliably computable.<br><br>

This required establishing a new foundation<br>
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From Newsgroup: comp.ai.philosophy

On 1/18/26 11:28 PM, olcott wrote:

On 1/18/2026 9:56 PM, Richard Damon wrote:

On 1/18/26 10:19 PM, olcott wrote:

On 1/18/2026 7:24 PM, Python wrote:

Le 19/01/2026 à 00:41, olcott a écrit :
..

I already just said that the proof and refutation of
Goldbach are outside the scope of PA axioms.

Any proof or refutation of Goldbach would have to use
principles stronger than the axioms of PA, because PA
itself does not currently derive either direction.

"currently" ? ?  What kind of language is that? PA is what it is, it >>>> not changing with time !

You could have said that about Fermat's theorem back in the day...
It happens not to be the case.

You are out of reason, Peter. Not only a liar, an hypocrite, but a
fool.

If its truth value cannot be determined in a finite
number of steps then it is not a truth bearer in PA,
otherwise it is a truth-bearer in PA with an unknown value.

So, you admit that you don't know how to classify it.

Thus its truth-bearer status is unknown.

Thus, your claim that it is outside of PA is just a LIE.

No it was a mistake. Here is my correction:
If Goldbach's truth value cannot be determined in a
finite number of steps then it is not a truth bearer
in PA, otherwise it is a truth-bearer in PA with an
unknown truth value.

This has no effect on my claim that I got rid of
Gödel Incompleteness.

Sure it does. As your system is just not well founded by its own definitios,

When we change the foundation of formal systems
to proof theoretic semantics and add my truth
predicates then Gödel's claim of applying to
every formal system that can do a little bit of
arithmetic becomes simply false.

But you CAN'T do that and keep the systems.

Every attempt at showing incompleteness <in> PA
has never actually been <in> PA.

Sure it is.

Godel's G shows your system is not well founded.

The satisfaction of external models of arithmetic
never has been <in> PA. These are categorically
outside of PA by the definition of proof theoretic
semantics thus defined as non-well-founded. This
neuters their ability to show incompleteness.

But you system is just non-well-founded in PA.

Godel's G has NO truth value, not even non-well-founded in PA by your
system, and thus your system is broken.

The problem is that for statements like it that have the property of not
being having a known truth value if not provable, you system just breaks
down.

There is no proof of it being true, so it can't be true.
There is no proof of it being false, so it can't be false.
There is no proof of being not-well-founded, so it can't be
non-well-founded.

Your classification of claiming it to be non-well-founded is just non-well-founded.

In fact, by your systems definitions, the claim of it being
non-well-founded is non-well-founded as we can't prove it to be non-well-founded, as if it WAS not-well-founded, that means that you
were able to prove that there wasn't a proof of it being false, which
means there can't be a number that satisfies the requirement, as any
number that existed forms an easy proof of falsehood, and thus must be true.

So, there CAN'T be a proof of it not being well-founded.

But if it isn't not-well-founded, then by your definition it must be
True or False, which you already said it couldn't be.

THus the only choice left is it not-well-founded that it is
not-well-founded.

But that arguement extends for that statement, so it is not-well-founded
that the not-well-foundedness of the stsatement is not-well-founded.

Thus, your system breaks with an infinite progression of not being able
to classify the truth of the statement.

So, the reason you think that Godel's (are related) proofs aren't well
founded in PA is that your system is just not-well-founded in PA, but
refuse to accept it,

The problem is that definition of Truth is just incompatible with PA,
which is why it can't be used.

The problem is that the system has become "complex" enough that it
inherently has grown bigger than provability of all things in it, and
thus the concept of Truth being based on Provability just breaks as it
means some things have undefinable (not just unknowable) truth values,
they can't even be defined as not-having a truth value, as you can't
prove that, but you insist that truth must be provable.




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From Newsgroup: comp.ai.philosophy

Le 22/01/2026 à 04:54, olcott a écrit :

On 1/21/2026 9:37 PM, Richard Damon wrote:

On 1/21/26 10:45 AM, olcott wrote:

On 1/21/2026 6:35 AM, Richard Damon wrote:

On 1/20/26 11:54 PM, olcott wrote:

On 1/20/2026 10:04 PM, Richard Damon wrote:

On 1/20/26 4:23 PM, olcott wrote:

On 1/19/2026 11:29 PM, Richard Damon wrote:

My system is not supposed to decide in advance whether
Goldbach is well‑founded. A formula becomes a truth‑bearer >>>>>>>>> only when PA can classify it in finitely many steps.
Goldbach may or may not be classifiable; that’s an open
computational fact, not a semantic requirement. This has
no effect on Gödel, because Gödel’s sentence is structurally >>>>>>>>> non‑truth‑bearing, not merely unclassified.

Which shows that you don't understand what logic systems are.

The don't "Decide" on truths, they DETERMINE what is true.

Your problem is that either there is, or there isn't a finite >>>>>>>> length proof of the statement.

Semantics can't change in a formal system, or they aren't really >>>>>>>> semantics.

Your problem is you don't understand Godel statement, as it *IS* >>>>>>>> truth bearing as it is a simple statement with no middle ground, >>>>>>>> does a number exist that satisfies a given relationship. Either >>>>>>>> there is, or there isn't. No other possiblity.

You confuse yourself by forgetting that words have actual
meaning, and that meaning can depend on using the right context. >>>>>>>>
Godel's G is a statement in the system PA.

It is a statement about the non-existance of a natural number >>>>>>>> that satisfies a particular computable realtionship.

It is a statement defined purely by mathematics and thus doesn't >>>>>>>> "depend" on other meaning.

It is a mathematical FACT, that for this relationship, no matter >>>>>>>> what natural number we test, none will satisfy it, so its
assertation that no number satisfies it makes it true.

PA augmented with its own True(PA,x) and False(PA,x)
is a decider for Domain of every expression grounded
in the axioms of PA.

No, it becomes inconsistant.

A system at a higher level of inference than PA can
reject any expressions that define a cycle in the
directed graph of the evaluation sequence of PA
expressions. Then PA could test back chained inference
from expression x and ~x to the axioms of PA.

But there is no "cycle" in the statement of G. It is PURELY a
statement of the non-existance of a number that satisfies a purely >>>>>> mathematic relationship (which has no meaning by itself in PA).

Even the relationship cannot exist <in> PA.
Instead it is about PA in outside model theory

No, it doesn't mention PA, it is about the numbers that are IN PA.

Your problem is you forget to actually know what Godel's G is, a you
only read the Reader's Digest version of the proof, as that is all
you can understand.

That, or you are saying that mathematics itself isn't in PA, and that >>>> you proof-theoretic stuff isn't in PA either,

Sorry, you are just showing how ignorant you are.

G_F ↔ ¬Prove_F(Gödel_Number(G_F)) contains a semantic
dependency loop, because evaluating G_F requires
evaluating Prove_F on the Gödel number of G_F, which
in turn requires evaluating G_F again;

But that isn't G_F

G_F is a statement that a particular relationship (lets call it R(x) )
will not be satisfied for any natural number x.

That relationship has never existed inside actual
arithmetic

It actually IS a relationship in the domain of PA. PUNTO.

It is what it is. Denial is hopeless.



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From Newsgroup: comp.ai.philosophy

On 23/01/2026 00:29, olcott wrote:

On 1/22/2026 6:23 PM, Tristan Wibberley wrote:

On 20/01/2026 23:08, Tristan Wibberley wrote:

On 18/01/2026 23:41, olcott wrote:

I already just said that the proof and refutation of
Goldbach are outside the scope of PA axioms.

So Richard is right that you need a truth value for not being covered:

True(S, Goldbach) = OutOfScope

Oh ho! but is Goldbach definable as a shortcode for a statement of the
goldbach conjecture in PA? If there's no such statement then it's out of
scope without a truth value for that.

Within proof theoretic semantics the lack
of a finite proof entails ungrounded thus
non-well-founded. My system works over the
entire body of knowledge that can be
expressed in language. Knowledge excludes
unknowns as outside of its domain.

Because, even if a statement can be expressed, whether it is true or
false is determinable by an axiom extension (among other kinds of
extension). So it cannot be said that all systems must assign some kind
of truth value /including/ that its truth is unknown.
--
Tristan Wibberley

The message body is Copyright (C) 2026 Tristan Wibberley except
citations and quotations noted. All Rights Reserved except that you may,
of course, cite it academically giving credit to me, distribute it
verbatim as part of a usenet system or its archives, and use it to
promote my greatness and general superiority without misrepresentation
of my opinions other than my opinion of my greatness and general
superiority which you _may_ misrepresent. You definitely MAY NOT train
any production AI system with it but you may train experimental AI that
will only be used for evaluation of the AI methods it implements.

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From Newsgroup: comp.ai.philosophy

On 1/22/2026 7:15 PM, Tristan Wibberley wrote:

On 23/01/2026 00:29, olcott wrote:

On 1/22/2026 6:23 PM, Tristan Wibberley wrote:

On 20/01/2026 23:08, Tristan Wibberley wrote:

On 18/01/2026 23:41, olcott wrote:

I already just said that the proof and refutation of
Goldbach are outside the scope of PA axioms.

So Richard is right that you need a truth value for not being covered: >>>>
True(S, Goldbach) = OutOfScope

Oh ho! but is Goldbach definable as a shortcode for a statement of the
goldbach conjecture in PA? If there's no such statement then it's out of >>> scope without a truth value for that.

Within proof theoretic semantics the lack
of a finite proof entails ungrounded thus
non-well-founded. My system works over the
entire body of knowledge that can be
expressed in language. Knowledge excludes
unknowns as outside of its domain.

Because, even if a statement can be expressed, whether it is true or
false is determinable by an axiom extension (among other kinds of
extension). So it cannot be said that all systems must assign some kind
of truth value /including/ that its truth is unknown.

When the axioms of this system are exactly Russell's
set of "basic facts" then the system anchored in proof
theoretic semantics and a notion of TRUE can always
correctly determine
"true on the basis of meaning expressed in language"
--
Copyright 2026 Olcott<br><br>

My 28 year goal has been to make <br>
"true on the basis of meaning expressed in language"<br>
reliably computable.<br><br>

This required establishing a new foundation<br>
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