Back in 2020 I proved that Wittgenstein was correct
all along. His key essence of grounding truth in
well-founded proof theoretic semantics did not exist
at the time that he made these remarks. Because of
this his remarks were misunderstood to be based
on ignorance instead of the profound insight that
they really were.
According to Wittgenstein:
'True in Russell's system' means, as was said: proved
in Russell's system; and 'false in Russell's system'
means: the opposite has been proved in Russell's system.
(Wittgenstein 1983,118-119)
Formalized by Olcott as:
reCF ree Formal_Systems reCEYAR ree WFF(F) (((FreoEYAR)) rao True(F, EYAR)) reCF ree Formal_Systems reCEYAR ree WFF(F) (((Fre4EYAR)) rao -4True(F, EYAR)) reCF ree Formal_Systems reCEYAR ree WFF(F) (((Freo-4EYAR)) rao False(F, EYAR))
The terminology which has just been used implies that
the elementary statements are not such that their truth
and falsity are known to us without reference to {T}.
(Curry 1977:45)
Simply defining G||del Incompleteness and Tarski Undefinability away V12 olcott Jun 26, 2020, 4:15:48rC>PM comp.theory,comp.ai.philosophy,comp.ai.nat-lang,sci.lang.semantics Message-ID: <tpudnRZeLeDg-GvDnZ2dnUU7-V3NnZ2d@giganews.com> https://groups.google.com/g/comp.ai.nat-lang/c/p_evEnqowPQ/m/0RHg0UjWAAAJ
The Wittgenstein quote above was anchored in what is
now known as well-founded proof theoretic semantics on
the basis of what is now known as Curry's basis of
true in the system. He saw this decades before these
fields were ever established.
The seed of his idea goes all the way back to his
Tractatus (1921)
6.12 The fact that the propositions of logic are
tautologies shows the formal-logical-properties
of language, of the world. That its constituent
parts connected together in this way give a tautology
characterizes the logic of its constituent parts.
In order that propositions connected together in a
definite way may give a tautology they must have
definite properties of structure. That they give a
tautology when so connected shows therefore that they
possess these properties of structure.
On 1/19/26 12:56 PM, olcott wrote:
Back in 2020 I proved that Wittgenstein was correct
all along. His key essence of grounding truth in
well-founded proof theoretic semantics did not exist
at the time that he made these remarks. Because of
this his remarks were misunderstood to be based
on ignorance instead of the profound insight that
they really were.
Nope.
According to Wittgenstein:
'True in Russell's system' means, as was said: proved
in Russell's system; and 'false in Russell's system'
means: the opposite has been proved in Russell's system.
(Wittgenstein 1983,118-119)
Which is only ONE interpretation, (and not a correct one).
Formalized by Olcott as:
reCF ree Formal_Systems reCEYAR ree WFF(F) (((FreoEYAR)) rao True(F, EYAR)) >> reCF ree Formal_Systems reCEYAR ree WFF(F) (((Fre4EYAR)) rao -4True(F, EYAR))
reCF ree Formal_Systems reCEYAR ree WFF(F) (((Freo-4EYAR)) rao False(F, EYAR))
Which can be not-well-founded, as determining *IF* a statement is
proveable or not provable might not be provable, or even knowable.
So, therefore you can't actually evaluate your statement.
The terminology which has just been used implies that
the elementary statements are not such that their truth
and falsity are known to us without reference to {T}.
(Curry 1977:45)
Simply defining G||del Incompleteness and Tarski Undefinability away V12
olcott Jun 26, 2020, 4:15:48rC>PM
comp.theory,comp.ai.philosophy,comp.ai.nat-lang,sci.lang.semantics
Message-ID: <tpudnRZeLeDg-GvDnZ2dnUU7-V3NnZ2d@giganews.com>
https://groups.google.com/g/comp.ai.nat-lang/c/p_evEnqowPQ/m/0RHg0UjWAAAJ
The Wittgenstein quote above was anchored in what is
now known as well-founded proof theoretic semantics on
the basis of what is now known as Curry's basis of
true in the system. He saw this decades before these
fields were ever established.
The seed of his idea goes all the way back to his
Tractatus (1921)
6.12 The fact that the propositions of logic are
tautologies shows the formal-logical-properties
of language, of the world. That its constituent
parts connected together in this way give a tautology
characterizes the logic of its constituent parts.
In order that propositions connected together in a
definite way may give a tautology they must have
definite properties of structure. That they give a
tautology when so connected shows therefore that they
possess these properties of structure.
On 1/19/2026 11:29 PM, Richard Damon wrote:
On 1/19/26 12:56 PM, olcott wrote:
Back in 2020 I proved that Wittgenstein was correct
all along. His key essence of grounding truth in
well-founded proof theoretic semantics did not exist
at the time that he made these remarks. Because of
this his remarks were misunderstood to be based
on ignorance instead of the profound insight that
they really were.
Nope.
According to Wittgenstein:
'True in Russell's system' means, as was said: proved
in Russell's system; and 'false in Russell's system'
means: the opposite has been proved in Russell's system.
(Wittgenstein 1983,118-119)
Which is only ONE interpretation, (and not a correct one).
All we need to do to make PA complete
is replace model theoretic semantics
with wellfounded proof theoretic sematics
and ground true in OA the way Haskell
Curry defines it entirely on the basis
of the axioms of PA,
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~WellFounded(PA, x) rei (~True(PA, x) reo (~False(PA, x))
Then PA becomes complete.
This is very similar to my work 8 years ago
where the axioms are construed as BaseFacts.
It was pure proof theoretic even way back then.
The ultimate foundation of [a priori] Truth
Olcott Feb 17, 2018, 12:42:55rC>AM https://groups.google.com/g/sci.logic/c/dbk5vsDzZbQ/m/4ajW9R08CQAJ
Formalized by Olcott as:
reCF ree Formal_Systems reCEYAR ree WFF(F) (((FreoEYAR)) rao True(F, EYAR)) >>> reCF ree Formal_Systems reCEYAR ree WFF(F) (((Fre4EYAR)) rao -4True(F, EYAR))
reCF ree Formal_Systems reCEYAR ree WFF(F) (((Freo-4EYAR)) rao False(F, EYAR))
Which can be not-well-founded, as determining *IF* a statement is
proveable or not provable might not be provable, or even knowable.
So, therefore you can't actually evaluate your statement.
All meta-math is defined to be outside the scope of PA.
The terminology which has just been used implies that
the elementary statements are not such that their truth
and falsity are known to us without reference to {T}.
(Curry 1977:45)
Simply defining G||del Incompleteness and Tarski Undefinability away V12 >>> olcott Jun 26, 2020, 4:15:48rC>PM
comp.theory,comp.ai.philosophy,comp.ai.nat-lang,sci.lang.semantics
Message-ID: <tpudnRZeLeDg-GvDnZ2dnUU7-V3NnZ2d@giganews.com>
https://groups.google.com/g/comp.ai.nat-lang/c/p_evEnqowPQ/
m/0RHg0UjWAAAJ
The Wittgenstein quote above was anchored in what is
now known as well-founded proof theoretic semantics on
the basis of what is now known as Curry's basis of
true in the system. He saw this decades before these
fields were ever established.
The seed of his idea goes all the way back to his
Tractatus (1921)
6.12 The fact that the propositions of logic are
tautologies shows the formal-logical-properties
of language, of the world. That its constituent
parts connected together in this way give a tautology
characterizes the logic of its constituent parts.
In order that propositions connected together in a
definite way may give a tautology they must have
definite properties of structure. That they give a
tautology when so connected shows therefore that they
possess these properties of structure.
On 1/20/2026 10:00 PM, Richard Damon wrote:
On 1/20/26 1:13 PM, olcott wrote:
On 1/19/2026 11:29 PM, Richard Damon wrote:
On 1/19/26 12:56 PM, olcott wrote:
Back in 2020 I proved that Wittgenstein was correct
all along. His key essence of grounding truth in
well-founded proof theoretic semantics did not exist
at the time that he made these remarks. Because of
this his remarks were misunderstood to be based
on ignorance instead of the profound insight that
they really were.
Nope.
According to Wittgenstein:
'True in Russell's system' means, as was said: proved
in Russell's system; and 'false in Russell's system'
means: the opposite has been proved in Russell's system.
(Wittgenstein 1983,118-119)
Which is only ONE interpretation, (and not a correct one).
All we need to do to make PA complete
is replace model theoretic semantics
with wellfounded proof theoretic sematics
and ground true in OA the way Haskell
Curry defines it entirely on the basis
of the axioms of PA,
Nope, doesn't work.
THe system breaks as it can't consistantly determine the truth value
of some statements.
Just to make it simpler for you to understand think
of it as a truth and falsity recognizer that gets
stuck in an infinite loop on anything else.
So PA is complete for its domain.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~WellFounded(PA, x) rei (~True(PA, x) reo (~False(PA, x))
Then PA becomes complete.
And, in proof-theoretic semantics, this is just not-well-founded as
there are statements that you can not determine if any of these are
applicable or not.
This is very similar to my work 8 years ago
where the axioms are construed as BaseFacts.
It was pure proof theoretic even way back then.
The ultimate foundation of [a priori] Truth
Olcott Feb 17, 2018, 12:42:55rC>AM
https://groups.google.com/g/sci.logic/c/dbk5vsDzZbQ/m/4ajW9R08CQAJ
At least that accepted that there were statement that it couldn't
handle as they were neiteher true or false.
With your addition, we get that there are statements that can be none
of True, False, or ~WellFounded.
This was the earliest documented work that
can be classified as well-founded proof theoretic semantics.
My actual work is documented to go back to 1998.
Formalized by Olcott as:
reCF ree Formal_Systems reCEYAR ree WFF(F) (((FreoEYAR)) rao True(F, EYAR))
reCF ree Formal_Systems reCEYAR ree WFF(F) (((Fre4EYAR)) rao -4True(F, EYAR))
reCF ree Formal_Systems reCEYAR ree WFF(F) (((Freo-4EYAR)) rao False(F, EYAR))
Which can be not-well-founded, as determining *IF* a statement is
proveable or not provable might not be provable, or even knowable.
So, therefore you can't actually evaluate your statement.
All meta-math is defined to be outside the scope of PA.
But we don't need "meta-math" to establish the answer.
It is a FACT that no number will satisfy the Relationship,
That relationship does not even exist outside of meta-math
On 1/20/26 11:49 PM, olcott wrote:
On 1/20/2026 10:00 PM, Richard Damon wrote:
On 1/20/26 1:13 PM, olcott wrote:
On 1/19/2026 11:29 PM, Richard Damon wrote:
On 1/19/26 12:56 PM, olcott wrote:
Back in 2020 I proved that Wittgenstein was correct
all along. His key essence of grounding truth in
well-founded proof theoretic semantics did not exist
at the time that he made these remarks. Because of
this his remarks were misunderstood to be based
on ignorance instead of the profound insight that
they really were.
Nope.
According to Wittgenstein:
'True in Russell's system' means, as was said: proved
in Russell's system; and 'false in Russell's system'
means: the opposite has been proved in Russell's system.
(Wittgenstein 1983,118-119)
Which is only ONE interpretation, (and not a correct one).
All we need to do to make PA complete
is replace model theoretic semantics
with wellfounded proof theoretic sematics
and ground true in OA the way Haskell
Curry defines it entirely on the basis
of the axioms of PA,
Nope, doesn't work.
THe system breaks as it can't consistantly determine the truth value
of some statements.
Just to make it simpler for you to understand think
of it as a truth and falsity recognizer that gets
stuck in an infinite loop on anything else.
So PA is complete for its domain.
Nope, as your idea to make it complete breaks everything.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~WellFounded(PA, x) rei (~True(PA, x) reo (~False(PA, x)) >>>> Then PA becomes complete.
And, in proof-theoretic semantics, this is just not-well-founded as
there are statements that you can not determine if any of these are
applicable or not.
This is very similar to my work 8 years ago
where the axioms are construed as BaseFacts.
It was pure proof theoretic even way back then.
The ultimate foundation of [a priori] Truth
Olcott Feb 17, 2018, 12:42:55rC>AM
https://groups.google.com/g/sci.logic/c/dbk5vsDzZbQ/m/4ajW9R08CQAJ
At least that accepted that there were statement that it couldn't
handle as they were neiteher true or false.
With your addition, we get that there are statements that can be none
of True, False, or ~WellFounded.
This was the earliest documented work that
can be classified as well-founded proof theoretic semantics.
My actual work is documented to go back to 1998.
But it isn't well-founded, as it isn't actualy based on proof.
Formalized by Olcott as:
reCF ree Formal_Systems reCEYAR ree WFF(F) (((FreoEYAR)) rao True(F, EYAR))
reCF ree Formal_Systems reCEYAR ree WFF(F) (((Fre4EYAR)) rao -4True(F, EYAR))
reCF ree Formal_Systems reCEYAR ree WFF(F) (((Freo-4EYAR)) rao False(F, EYAR))
Which can be not-well-founded, as determining *IF* a statement is
proveable or not provable might not be provable, or even knowable.
So, therefore you can't actually evaluate your statement.
All meta-math is defined to be outside the scope of PA.
But we don't need "meta-math" to establish the answer.
It is a FACT that no number will satisfy the Relationship,
That relationship does not even exist outside of meta-math
So, numbers don't exsist?
OR is it the "for all" part that doesn't exist, and thus your proof- theoretic logic can't exist either?
Sorry, you are just stuck trying to outlaw that which you need.
On 1/20/26 1:13 PM, olcott wrote:
All meta-math is defined to be outside the scope of PA.
But we don't need "meta-math" to establish the answer.
It is a FACT that no number will satisfy the Relationship, but this is
not provable, and the not being provable is not provable.
Thus it is NOT
An BaseFact is an expression X of (natural or formal)
language L that has been assigned the semantic property
of True. (Similar to a math Axiom).
On 21/01/2026 15:14, olcott wrote:
An BaseFact is an expression X of (natural or formal)
language L that has been assigned the semantic property
of True. (Similar to a math Axiom).
Do you mean "statement" or really mean "expression" - understanding that
the term "expression" (of a language) includes things like "ngs lik"
from this statement.
On 21/01/2026 20:14, olcott wrote:
On 1/21/2026 1:02 PM, Tristan Wibberley wrote:
On 21/01/2026 15:14, olcott wrote:
An BaseFact is an expression X of (natural or formal)
language L that has been assigned the semantic property
of True. (Similar to a math Axiom).
Do you mean "statement" or really mean "expression" - understanding that >>> the term "expression" (of a language) includes things like "ngs lik"
from this statement.
It turns out that I must have remembered this from:
*RussellrCOs Logical Atomism*
-a the claim that the world consists of a plurality of
-a independently existing things exhibiting qualities
-a and standing in relations. According to logical
-a atomism, all truths are ultimately dependent upon
-a a layer of atomic facts, which consist either of a
-a simple particular exhibiting a quality, or multiple
-a simple particulars standing in a relation.
-a https://plato.stanford.edu/entries/logical-atomism/
I bought his book a few decades ago.
According to that link, Russel's 'expression' is more restrictive that Haskell Curry's, Curry considered "ngs lik" to be an expression while
Russel is there said to consider "the King of France" to be an
expression, I suppose giving the two noun phrases as the only two
examples, as they do, is intended to exclude the supposition that "ngs
lik" is an expression, which Curry allows. I wonder whether Curry misunderstood or adopted/formed a second school.
On 1/21/2026 6:38 AM, Richard Damon wrote:
On 1/20/26 11:49 PM, olcott wrote:
On 1/20/2026 10:00 PM, Richard Damon wrote:
On 1/20/26 1:13 PM, olcott wrote:
On 1/19/2026 11:29 PM, Richard Damon wrote:
On 1/19/26 12:56 PM, olcott wrote:
Back in 2020 I proved that Wittgenstein was correct
all along. His key essence of grounding truth in
well-founded proof theoretic semantics did not exist
at the time that he made these remarks. Because of
this his remarks were misunderstood to be based
on ignorance instead of the profound insight that
they really were.
Nope.
According to Wittgenstein:
'True in Russell's system' means, as was said: proved
in Russell's system; and 'false in Russell's system'
means: the opposite has been proved in Russell's system.
(Wittgenstein 1983,118-119)
Which is only ONE interpretation, (and not a correct one).
All we need to do to make PA complete
is replace model theoretic semantics
with wellfounded proof theoretic sematics
and ground true in OA the way Haskell
Curry defines it entirely on the basis
of the axioms of PA,
Nope, doesn't work.
THe system breaks as it can't consistantly determine the truth value
of some statements.
Just to make it simpler for you to understand think
of it as a truth and falsity recognizer that gets
stuck in an infinite loop on anything else.
So PA is complete for its domain.
Nope, as your idea to make it complete breaks everything.
You keep asserting that it rCLbreaks everything,rCY
but you havenrCOt identified a single axiom of
PA, rule of inference, or valid derivation that fails.
The recognizer does exactly what itrCOs supposed to:
rCo returns true when PA proves -o
rCo returns false when PA proves -4-o
rCo diverges on anything PA cannot settle
ThatrCOs not breaking anything.
ThatrCOs the definition of a recognizer.
So what, specifically, do you think is broken?
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~WellFounded(PA, x) rei (~True(PA, x) reo (~False(PA, x)) >>>>> Then PA becomes complete.
And, in proof-theoretic semantics, this is just not-well-founded as
there are statements that you can not determine if any of these are
applicable or not.
This is very similar to my work 8 years ago
where the axioms are construed as BaseFacts.
It was pure proof theoretic even way back then.
The ultimate foundation of [a priori] Truth
Olcott Feb 17, 2018, 12:42:55rC>AM
https://groups.google.com/g/sci.logic/c/dbk5vsDzZbQ/m/4ajW9R08CQAJ
At least that accepted that there were statement that it couldn't
handle as they were neiteher true or false.
With your addition, we get that there are statements that can be
none of True, False, or ~WellFounded.
This was the earliest documented work that
can be classified as well-founded proof theoretic semantics.
My actual work is documented to go back to 1998.
An BaseFact is an expression X of (natural or formal)
language L that has been assigned the semantic property
of True. (Similar to a math Axiom).
A Collection T of BaseFacts of language L forms the
ultimate foundation of the notion of Truth in language L.
To verify that an expression X of language L is True or
False only requires a syntactic logical consequence
inference chain (formal proof) from one or more elements
of T to X or ~X.
True(L, X) rao rea+o rea BaseFact(L) Provable(+o, X)
False(L, X) rao rea+o rea BaseFact(L) Provable(+o, ~X)
But it isn't well-founded, as it isn't actualy based on proof.
True(L, X) means: there exists a proof of X from the base facts
False(L, X) means: there exists a proof of -4X from the base facts
Everything else raA the recognizer diverges (no proof either way)
That is proofrCatheoretic semantics.
It is literally the definition of truth in a proofrCatheoretic framework.
Formalized by Olcott as:
reCF ree Formal_Systems reCEYAR ree WFF(F) (((FreoEYAR)) rao True(F, EYAR))
reCF ree Formal_Systems reCEYAR ree WFF(F) (((Fre4EYAR)) rao -4True(F, EYAR))
reCF ree Formal_Systems reCEYAR ree WFF(F) (((Freo-4EYAR)) rao False(F, EYAR))
Which can be not-well-founded, as determining *IF* a statement is >>>>>> proveable or not provable might not be provable, or even knowable. >>>>>>
So, therefore you can't actually evaluate your statement.
All meta-math is defined to be outside the scope of PA.
But we don't need "meta-math" to establish the answer.
It is a FACT that no number will satisfy the Relationship,
That relationship does not even exist outside of meta-math
So, numbers don't exsist?
OR is it the "for all" part that doesn't exist, and thus your proof-
theoretic logic can't exist either?
Sorry, you are just stuck trying to outlaw that which you need.
PA contains arithmetic relations about numbers.
It does not contain metarCamathematical relations about PA itself.
G||delrCOs construction uses the latter, not the former.
On 1/21/26 10:14 AM, olcott wrote:
On 1/21/2026 6:38 AM, Richard Damon wrote:
On 1/20/26 11:49 PM, olcott wrote:
On 1/20/2026 10:00 PM, Richard Damon wrote:
On 1/20/26 1:13 PM, olcott wrote:
On 1/19/2026 11:29 PM, Richard Damon wrote:
On 1/19/26 12:56 PM, olcott wrote:
Back in 2020 I proved that Wittgenstein was correct
all along. His key essence of grounding truth in
well-founded proof theoretic semantics did not exist
at the time that he made these remarks. Because of
this his remarks were misunderstood to be based
on ignorance instead of the profound insight that
they really were.
Nope.
According to Wittgenstein:
'True in Russell's system' means, as was said: proved
in Russell's system; and 'false in Russell's system'
means: the opposite has been proved in Russell's system.
(Wittgenstein 1983,118-119)
Which is only ONE interpretation, (and not a correct one).
All we need to do to make PA complete
is replace model theoretic semantics
with wellfounded proof theoretic sematics
and ground true in OA the way Haskell
Curry defines it entirely on the basis
of the axioms of PA,
Nope, doesn't work.
THe system breaks as it can't consistantly determine the truth
value of some statements.
Just to make it simpler for you to understand think
of it as a truth and falsity recognizer that gets
stuck in an infinite loop on anything else.
So PA is complete for its domain.
Nope, as your idea to make it complete breaks everything.
You keep asserting that it rCLbreaks everything,rCY
but you havenrCOt identified a single axiom of
PA, rule of inference, or valid derivation that fails.
What fails, is your definition of truth.
The recognizer does exactly what itrCOs supposed to:
rCo returns true when PA proves -o
rCo returns false when PA proves -4-o
rCo diverges on anything PA cannot settle
But your "not-well-founded" isn't a REcOGNIZER, it is a PREDICATE, which ALWAYS needs to return a value.
ThatrCOs not breaking anything.
ThatrCOs the definition of a recognizer.
So what, specifically, do you think is broken?
You definition of "Truth", which can't have a value by your logic.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~WellFounded(PA, x) rei (~True(PA, x) reo (~False(PA, x)) >>>>>> Then PA becomes complete.
And, in proof-theoretic semantics, this is just not-well-founded as >>>>> there are statements that you can not determine if any of these are >>>>> applicable or not.
This is very similar to my work 8 years ago
where the axioms are construed as BaseFacts.
It was pure proof theoretic even way back then.
The ultimate foundation of [a priori] Truth
Olcott Feb 17, 2018, 12:42:55rC>AM
https://groups.google.com/g/sci.logic/c/dbk5vsDzZbQ/m/4ajW9R08CQAJ
At least that accepted that there were statement that it couldn't
handle as they were neiteher true or false.
With your addition, we get that there are statements that can be
none of True, False, or ~WellFounded.
This was the earliest documented work that
can be classified as well-founded proof theoretic semantics.
My actual work is documented to go back to 1998.
An BaseFact is an expression X of (natural or formal)
language L that has been assigned the semantic property
of True. (Similar to a math Axiom).
A Collection T of BaseFacts of language L forms the
ultimate foundation of the notion of Truth in language L.
To verify that an expression X of language L is True or
False only requires a syntactic logical consequence
inference chain (formal proof) from one or more elements
of T to X or ~X.
True(L, X) rao rea+o rea BaseFact(L) Provable(+o, X)
False(L, X) rao rea+o rea BaseFact(L) Provable(+o, ~X)
And what it the provable truth value of Godel's G statement?
It can't be True, since it turns out to not be provable.
It can't be False, as no number exists to make it false.
It can't be Proven Not-Well-Founded, as proving that it can't be false, establishes that no such number exists, which makes it true in the system.
Thus, your definition of "Truth" as being True/False/Not-Well-Founded is just self-contradictory.
All you are doing is your normal back-pedeling and dupliciously changing
you claim that actually negates your other position.
But it isn't well-founded, as it isn't actualy based on proof.
True(L, X) means: there exists a proof of X from the base facts
False(L, X) means: there exists a proof of -4X from the base facts
Everything else raA the recognizer diverges (no proof either way)
In other words, your "Proff-Theoretic" system is actually Truth- Conditional, and thus you can't use it.
framework.
That is proofrCatheoretic semantics.
It is literally the definition of truth in a proofrCatheoretic
Which means proof-theoretic needs truth-conditional to be accepted by
your logic.
Proof-Theoretic can work if it says that it just can't handle some statements like G.
Which is an admission of its own limitations.
Proof-Theoretic ADMITS it is incomplete in PA, as there are statements
it can not determine if they are true, false, or neither in the system, because a "proof" on not being true or false actually establishes the statement as true (or for other statments, that they are false).
Formalized by Olcott as:
reCF ree Formal_Systems reCEYAR ree WFF(F) (((FreoEYAR)) rao True(F, EYAR))
reCF ree Formal_Systems reCEYAR ree WFF(F) (((Fre4EYAR)) rao -4True(F, EYAR))
reCF ree Formal_Systems reCEYAR ree WFF(F) (((Freo-4EYAR)) rao False(F, EYAR))
Which can be not-well-founded, as determining *IF* a statement is >>>>>>> proveable or not provable might not be provable, or even knowable. >>>>>>>
So, therefore you can't actually evaluate your statement.
All meta-math is defined to be outside the scope of PA.
But we don't need "meta-math" to establish the answer.
It is a FACT that no number will satisfy the Relationship,
That relationship does not even exist outside of meta-math
So, numbers don't exsist?
OR is it the "for all" part that doesn't exist, and thus your proof-
theoretic logic can't exist either?
Sorry, you are just stuck trying to outlaw that which you need.
PA contains arithmetic relations about numbers.
It does not contain metarCamathematical relations about PA itself.
G||delrCOs construction uses the latter, not the former.
But G doesn't dp that. It just asserts that a given relationship is
never satisfied.
Or, is your claim that you can't do qualification over statements in PA,
and thus your definition of truth doesn't actually exist in PA, so your truth is still external to PA.
Your problem is you don't understand what the relationship in G actually
is, because you don't understand how context works, so you don't
actually understand how semantics work.
On 1/22/2026 6:42 AM, Richard Damon wrote:
On 1/21/26 10:14 AM, olcott wrote:
On 1/21/2026 6:38 AM, Richard Damon wrote:
On 1/20/26 11:49 PM, olcott wrote:
On 1/20/2026 10:00 PM, Richard Damon wrote:
On 1/20/26 1:13 PM, olcott wrote:
On 1/19/2026 11:29 PM, Richard Damon wrote:
On 1/19/26 12:56 PM, olcott wrote:
Back in 2020 I proved that Wittgenstein was correct
all along. His key essence of grounding truth in
well-founded proof theoretic semantics did not exist
at the time that he made these remarks. Because of
this his remarks were misunderstood to be based
on ignorance instead of the profound insight that
they really were.
Nope.
According to Wittgenstein:
'True in Russell's system' means, as was said: proved
in Russell's system; and 'false in Russell's system'
means: the opposite has been proved in Russell's system.
(Wittgenstein 1983,118-119)
Which is only ONE interpretation, (and not a correct one).
All we need to do to make PA complete
is replace model theoretic semantics
with wellfounded proof theoretic sematics
and ground true in OA the way Haskell
Curry defines it entirely on the basis
of the axioms of PA,
Nope, doesn't work.
THe system breaks as it can't consistantly determine the truth
value of some statements.
Just to make it simpler for you to understand think
of it as a truth and falsity recognizer that gets
stuck in an infinite loop on anything else.
So PA is complete for its domain.
Nope, as your idea to make it complete breaks everything.
You keep asserting that it rCLbreaks everything,rCY
but you havenrCOt identified a single axiom of
PA, rule of inference, or valid derivation that fails.
What fails, is your definition of truth.
The recognizer does exactly what itrCOs supposed to:
rCo returns true when PA proves -o
rCo returns false when PA proves -4-o
rCo diverges on anything PA cannot settle
But your "not-well-founded" isn't a REcOGNIZER, it is a PREDICATE,
which ALWAYS needs to return a value.
ThatrCOs not breaking anything.
ThatrCOs the definition of a recognizer.
So what, specifically, do you think is broken?
You definition of "Truth", which can't have a value by your logic.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~WellFounded(PA, x) rei (~True(PA, x) reo (~False(PA, x)) >>>>>>> Then PA becomes complete.
And, in proof-theoretic semantics, this is just not-well-founded
as there are statements that you can not determine if any of these >>>>>> are applicable or not.
At least that accepted that there were statement that it couldn't >>>>>> handle as they were neiteher true or false.
This is very similar to my work 8 years ago
where the axioms are construed as BaseFacts.
It was pure proof theoretic even way back then.
The ultimate foundation of [a priori] Truth
Olcott Feb 17, 2018, 12:42:55rC>AM
https://groups.google.com/g/sci.logic/c/dbk5vsDzZbQ/m/4ajW9R08CQAJ >>>>>>
With your addition, we get that there are statements that can be
none of True, False, or ~WellFounded.
This was the earliest documented work that
can be classified as well-founded proof theoretic semantics.
My actual work is documented to go back to 1998.
An BaseFact is an expression X of (natural or formal)
language L that has been assigned the semantic property
of True. (Similar to a math Axiom).
A Collection T of BaseFacts of language L forms the
ultimate foundation of the notion of Truth in language L.
To verify that an expression X of language L is True or
False only requires a syntactic logical consequence
inference chain (formal proof) from one or more elements
of T to X or ~X.
True(L, X) rao rea+o rea BaseFact(L) Provable(+o, X)
False(L, X) rao rea+o rea BaseFact(L) Provable(+o, ~X)
And what it the provable truth value of Godel's G statement?
It can't be True, since it turns out to not be provable.
It can't be False, as no number exists to make it false.
It can't be Proven Not-Well-Founded, as proving that it can't be
false, establishes that no such number exists, which makes it true in
the system.
Thus, your definition of "Truth" as being True/False/Not-Well-Founded
is just self-contradictory.
All you are doing is your normal back-pedeling and dupliciously
changing you claim that actually negates your other position.
But it isn't well-founded, as it isn't actualy based on proof.
True(L, X) means: there exists a proof of X from the base facts
False(L, X) means: there exists a proof of -4X from the base facts
Everything else raA the recognizer diverges (no proof either way)
In other words, your "Proff-Theoretic" system is actually Truth-
Conditional, and thus you can't use it.
framework.
That is proofrCatheoretic semantics.
It is literally the definition of truth in a proofrCatheoretic
Which means proof-theoretic needs truth-conditional to be accepted by
your logic.
Proof-Theoretic can work if it says that it just can't handle some
statements like G.
Which is an admission of its own limitations.
Proof-Theoretic ADMITS it is incomplete in PA, as there are statements
it can not determine if they are true, false, or neither in the
system, because a "proof" on not being true or false actually
establishes the statement as true (or for other statments, that they
are false).
Formalized by Olcott as:
reCF ree Formal_Systems reCEYAR ree WFF(F) (((FreoEYAR)) rao True(F, EYAR))
reCF ree Formal_Systems reCEYAR ree WFF(F) (((Fre4EYAR)) rao -4True(F, EYAR))
reCF ree Formal_Systems reCEYAR ree WFF(F) (((Freo-4EYAR)) rao False(F, EYAR))
Which can be not-well-founded, as determining *IF* a statement >>>>>>>> is proveable or not provable might not be provable, or even
knowable.
So, therefore you can't actually evaluate your statement.
All meta-math is defined to be outside the scope of PA.
But we don't need "meta-math" to establish the answer.
It is a FACT that no number will satisfy the Relationship,
That relationship does not even exist outside of meta-math
So, numbers don't exsist?
OR is it the "for all" part that doesn't exist, and thus your proof-
theoretic logic can't exist either?
Sorry, you are just stuck trying to outlaw that which you need.
PA contains arithmetic relations about numbers.
It does not contain metarCamathematical relations about PA itself.
G||delrCOs construction uses the latter, not the former.
But G doesn't dp that. It just asserts that a given relationship is
never satisfied.
Or, is your claim that you can't do qualification over statements in
PA, and thus your definition of truth doesn't actually exist in PA, so
your truth is still external to PA.
Your problem is you don't understand what the relationship in G
actually is, because you don't understand how context works, so you
don't actually understand how semantics work.
See my reply to Mikko
[The Halting Problem asks for too much]
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