For nearly a century, discussions of arithmetic have quietly
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY
But PA itself has no truth predicate, no internal semantics,
and no mechanism for assigning truth values. So what was
called rCLtrue in arithmeticrCY was always meta-theoretic truth
about arithmetic, imported from an external model and never
grounded inside PA.
This conflation was rarely acknowledged, and it shaped the
interpretation of G||delrCOs incompleteness theorems, independence
results like Goodstein and ParisrCoHarrington, and the entire
discourse around rCLtrue but unprovablerCY statements.
My work begins by correcting this foundational error.
PA has no internal truth predicate, so classical claims of
rCLtrue in arithmeticrCY were always meta-theoretic. My system
introduces a truth predicate whose meaning is anchored
entirely in PArCOs axioms and inference rules, not in external
models. Any statement whose meaning requires meta-theoretic
interpretation or non-well-founded self-reference is rejected
as outside the domain of PA. This yields a coherent, internal
notion of truth in arithmetic for the first time.
On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY
But PA itself has no truth predicate, no internal semantics,
and no mechanism for assigning truth values. So what was
called rCLtrue in arithmeticrCY was always meta-theoretic truth
about arithmetic, imported from an external model and never
grounded inside PA.
Nope, just shows you don't understand what TRUTH means.
This conflation was rarely acknowledged, and it shaped the
interpretation of G||delrCOs incompleteness theorems, independence
results like Goodstein and ParisrCoHarrington, and the entire
discourse around rCLtrue but unprovablerCY statements.
WHich Godel proves exsits.
My work begins by correcting this foundational error.
By LYING and destroying the meaninf of truth.
PA has no internal truth predicate, so classical claims of
rCLtrue in arithmeticrCY were always meta-theoretic. My system
introduces a truth predicate whose meaning is anchored
entirely in PArCOs axioms and inference rules, not in external
models. Any statement whose meaning requires meta-theoretic
interpretation or non-well-founded self-reference is rejected
as outside the domain of PA. This yields a coherent, internal
notion of truth in arithmetic for the first time.
Not having a "Predicate" doesn't mean not having a definition of truth.
Your problem is that your "Truth Predicate" forces either you system to--
be "trivial" or "inconsistant".
But, you are too stupid to understand this.
Your own system requires that which you call non-well-founded, so it is itself, by your definition, not-well-founded.
The problem is, except in trivial systems, we can't actually tell if a statement is well founded until we determine its truth, and may
declerations of not-well-founded are themselves not-well-founded.
You can only call Godel G statement not-well-founde by accept that it is true and unprovable, you can not otherwise PROVE that it isn't well- founded.
On 1/17/2026 3:54 PM, Richard Damon wrote:
On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY
But PA itself has no truth predicate, no internal semantics,
and no mechanism for assigning truth values. So what was
called rCLtrue in arithmeticrCY was always meta-theoretic truth
about arithmetic, imported from an external model and never
grounded inside PA.
Nope, just shows you don't understand what TRUTH means.
IrCOm distinguishing internal truth from external truth.
PA has no internal truth predicate, so it cannot express
or evaluate truth internally.
The only notion of truth available for PA is the external,
modelrCatheoretic one rCo which is metarCatheoretic by definition.
This conflation was rarely acknowledged, and it shaped the
interpretation of G||delrCOs incompleteness theorems, independence
results like Goodstein and ParisrCoHarrington, and the entire
discourse around rCLtrue but unprovablerCY statements.
WHich Godel proves exsits.
My work begins by correcting this foundational error.
By LYING and destroying the meaninf of truth.
PA has no internal truth predicate, so classical claims of
rCLtrue in arithmeticrCY were always meta-theoretic. My system
introduces a truth predicate whose meaning is anchored
entirely in PArCOs axioms and inference rules, not in external
models. Any statement whose meaning requires meta-theoretic
interpretation or non-well-founded self-reference is rejected
as outside the domain of PA. This yields a coherent, internal
notion of truth in arithmetic for the first time.
Not having a "Predicate" doesn't mean not having a definition of truth.
A metarCatheoretic definition of truth is not the same
as an internal truth predicate. TarskirCOs definition of
truth for arithmetic is external to PA and cannot be
expressed inside PA. ThatrCOs exactly the distinction
IrCOm drawing.
PA can prove statements, but it cannot assert that
those statements are true. Those are different notions.
Your problem is that your "Truth Predicate" forces either you system
to be "trivial" or "inconsistant".
But, you are too stupid to understand this.
Your own system requires that which you call non-well-founded, so it
is itself, by your definition, not-well-founded.
The problem is, except in trivial systems, we can't actually tell if a
statement is well founded until we determine its truth, and may
declerations of not-well-founded are themselves not-well-founded.
You can only call Godel G statement not-well-founde by accept that it
is true and unprovable, you can not otherwise PROVE that it isn't
well- founded.
On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote:
On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY >>>> But PA itself has no truth predicate, no internal semantics,
and no mechanism for assigning truth values. So what was
called rCLtrue in arithmeticrCY was always meta-theoretic truth
about arithmetic, imported from an external model and never
grounded inside PA.
Nope, just shows you don't understand what TRUTH means.
IrCOm distinguishing internal truth from external truth.
PA has no internal truth predicate, so it cannot express
or evaluate truth internally.
The only notion of truth available for PA is the external,
modelrCatheoretic one rCo which is metarCatheoretic by definition.
But Truth *IS* Truth, or you are just misdefining it.
The fact that a system can't tell you the truth value of a statement
doesn't mean the statement doesn't have a truth value.
And, the problem is that, as was shown, systems with a truth predicate
CAN'T support PA or they are inconsistant.
I guess systems that lie aren't a problem to you since you think lying
is valid logic.
This conflation was rarely acknowledged, and it shaped the
interpretation of G||delrCOs incompleteness theorems, independence
results like Goodstein and ParisrCoHarrington, and the entire
discourse around rCLtrue but unprovablerCY statements.
WHich Godel proves exsits.
My work begins by correcting this foundational error.
By LYING and destroying the meaninf of truth.
PA has no internal truth predicate, so classical claims of
rCLtrue in arithmeticrCY were always meta-theoretic. My system
introduces a truth predicate whose meaning is anchored
entirely in PArCOs axioms and inference rules, not in external
models. Any statement whose meaning requires meta-theoretic
interpretation or non-well-founded self-reference is rejected
as outside the domain of PA. This yields a coherent, internal
notion of truth in arithmetic for the first time.
Not having a "Predicate" doesn't mean not having a definition of truth.
A metarCatheoretic definition of truth is not the same
as an internal truth predicate. TarskirCOs definition of
truth for arithmetic is external to PA and cannot be
expressed inside PA. ThatrCOs exactly the distinction
IrCOm drawing.
No, he shows that any system that support PA and a Truth Predicate is inconstant.
It seems you just want to let your system be inconsistent, as then you
can "prove" whatever you want.
PA can prove statements, but it cannot assert that
those statements are true. Those are different notions.
Right, but statments in PA can be True even without such a predicate.
On 1/17/2026 6:14 PM, Richard Damon wrote:
On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote:
On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY >>>>> But PA itself has no truth predicate, no internal semantics,
and no mechanism for assigning truth values. So what was
called rCLtrue in arithmeticrCY was always meta-theoretic truth
about arithmetic, imported from an external model and never
grounded inside PA.
Nope, just shows you don't understand what TRUTH means.
IrCOm distinguishing internal truth from external truth.
PA has no internal truth predicate, so it cannot express
or evaluate truth internally.
The only notion of truth available for PA is the external,
modelrCatheoretic one rCo which is metarCatheoretic by definition.
But Truth *IS* Truth, or you are just misdefining it.
The fact that a system can't tell you the truth value of a statement
doesn't mean the statement doesn't have a truth value.
And, the problem is that, as was shown, systems with a truth predicate
CAN'T support PA or they are inconsistant.
I guess systems that lie aren't a problem to you since you think lying
is valid logic.
This conflation was rarely acknowledged, and it shaped the
interpretation of G||delrCOs incompleteness theorems, independence
results like Goodstein and ParisrCoHarrington, and the entire
discourse around rCLtrue but unprovablerCY statements.
WHich Godel proves exsits.
My work begins by correcting this foundational error.
By LYING and destroying the meaninf of truth.
PA has no internal truth predicate, so classical claims of
rCLtrue in arithmeticrCY were always meta-theoretic. My system
introduces a truth predicate whose meaning is anchored
entirely in PArCOs axioms and inference rules, not in external
models. Any statement whose meaning requires meta-theoretic
interpretation or non-well-founded self-reference is rejected
as outside the domain of PA. This yields a coherent, internal
notion of truth in arithmetic for the first time.
Not having a "Predicate" doesn't mean not having a definition of truth. >>>>
A metarCatheoretic definition of truth is not the same
as an internal truth predicate. TarskirCOs definition of
truth for arithmetic is external to PA and cannot be
expressed inside PA. ThatrCOs exactly the distinction
IrCOm drawing.
No, he shows that any system that support PA and a Truth Predicate is
inconstant.
It seems you just want to let your system be inconsistent, as then you
can "prove" whatever you want.
PA can prove statements, but it cannot assert that
those statements are true. Those are different notions.
Right, but statments in PA can be True even without such a predicate.
Unless PA can prove it then they never were actually
true in PA. They were true outside of PA in meta-math.
On 1/17/26 7:49 PM, olcott wrote:
On 1/17/2026 6:14 PM, Richard Damon wrote:
On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote:
On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY >>>>>> But PA itself has no truth predicate, no internal semantics,
and no mechanism for assigning truth values. So what was
called rCLtrue in arithmeticrCY was always meta-theoretic truth
about arithmetic, imported from an external model and never
grounded inside PA.
Nope, just shows you don't understand what TRUTH means.
IrCOm distinguishing internal truth from external truth.
PA has no internal truth predicate, so it cannot express
or evaluate truth internally.
The only notion of truth available for PA is the external,
modelrCatheoretic one rCo which is metarCatheoretic by definition.
But Truth *IS* Truth, or you are just misdefining it.
The fact that a system can't tell you the truth value of a statement
doesn't mean the statement doesn't have a truth value.
And, the problem is that, as was shown, systems with a truth
predicate CAN'T support PA or they are inconsistant.
I guess systems that lie aren't a problem to you since you think
lying is valid logic.
This conflation was rarely acknowledged, and it shaped the
interpretation of G||delrCOs incompleteness theorems, independence >>>>>> results like Goodstein and ParisrCoHarrington, and the entire
discourse around rCLtrue but unprovablerCY statements.
WHich Godel proves exsits.
My work begins by correcting this foundational error.
By LYING and destroying the meaninf of truth.
PA has no internal truth predicate, so classical claims of
rCLtrue in arithmeticrCY were always meta-theoretic. My system
introduces a truth predicate whose meaning is anchored
entirely in PArCOs axioms and inference rules, not in external
models. Any statement whose meaning requires meta-theoretic
interpretation or non-well-founded self-reference is rejected
as outside the domain of PA. This yields a coherent, internal
notion of truth in arithmetic for the first time.
Not having a "Predicate" doesn't mean not having a definition of
truth.
A metarCatheoretic definition of truth is not the same
as an internal truth predicate. TarskirCOs definition of
truth for arithmetic is external to PA and cannot be
expressed inside PA. ThatrCOs exactly the distinction
IrCOm drawing.
No, he shows that any system that support PA and a Truth Predicate is
inconstant.
It seems you just want to let your system be inconsistent, as then
you can "prove" whatever you want.
PA can prove statements, but it cannot assert that
those statements are true. Those are different notions.
Right, but statments in PA can be True even without such a predicate.
Unless PA can prove it then they never were actually
true in PA. They were true outside of PA in meta-math.
Sure it is. Truth goes beyond knowledge.
On 1/17/2026 7:20 PM, Richard Damon wrote:
On 1/17/26 7:49 PM, olcott wrote:
On 1/17/2026 6:14 PM, Richard Damon wrote:
On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote:
On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY >>>>>>> But PA itself has no truth predicate, no internal semantics,
and no mechanism for assigning truth values. So what was
called rCLtrue in arithmeticrCY was always meta-theoretic truth
about arithmetic, imported from an external model and never
grounded inside PA.
Nope, just shows you don't understand what TRUTH means.
IrCOm distinguishing internal truth from external truth.
PA has no internal truth predicate, so it cannot express
or evaluate truth internally.
The only notion of truth available for PA is the external,
modelrCatheoretic one rCo which is metarCatheoretic by definition.
But Truth *IS* Truth, or you are just misdefining it.
The fact that a system can't tell you the truth value of a statement
doesn't mean the statement doesn't have a truth value.
And, the problem is that, as was shown, systems with a truth
predicate CAN'T support PA or they are inconsistant.
I guess systems that lie aren't a problem to you since you think
lying is valid logic.
This conflation was rarely acknowledged, and it shaped the
interpretation of G||delrCOs incompleteness theorems, independence >>>>>>> results like Goodstein and ParisrCoHarrington, and the entire
discourse around rCLtrue but unprovablerCY statements.
WHich Godel proves exsits.
My work begins by correcting this foundational error.
By LYING and destroying the meaninf of truth.
PA has no internal truth predicate, so classical claims of
rCLtrue in arithmeticrCY were always meta-theoretic. My system
introduces a truth predicate whose meaning is anchored
entirely in PArCOs axioms and inference rules, not in external
models. Any statement whose meaning requires meta-theoretic
interpretation or non-well-founded self-reference is rejected
as outside the domain of PA. This yields a coherent, internal
notion of truth in arithmetic for the first time.
Not having a "Predicate" doesn't mean not having a definition of
truth.
A metarCatheoretic definition of truth is not the same
as an internal truth predicate. TarskirCOs definition of
truth for arithmetic is external to PA and cannot be
expressed inside PA. ThatrCOs exactly the distinction
IrCOm drawing.
No, he shows that any system that support PA and a Truth Predicate
is inconstant.
It seems you just want to let your system be inconsistent, as then
you can "prove" whatever you want.
PA can prove statements, but it cannot assert that
those statements are true. Those are different notions.
Right, but statments in PA can be True even without such a predicate.
Unless PA can prove it then they never were actually
true in PA. They were true outside of PA in meta-math.
Sure it is. Truth goes beyond knowledge.
You're assuming 'truth in arithmetic' means truth-in-the-standard-model.
But that's a meta-theoretic constructrCoit's truth about arithmetic from outside PA, not truth in arithmetic. PA has no internal truth predicate
and no way to access the standard model from within.
When you say 'truth goes beyond knowledge,' you're really saying 'meta- theoretic truth goes beyond PA-provability.' That's trivially correct,
but it doesn't establish that G||del sentences are 'true in arithmetic'rCo it just shows they're true in a meta-theoretic framework we're smuggling
in and calling 'arithmetic.'
The question isn't whether meta-theoretic truth transcends provability.
It's whether meta-theoretic truth has any claim to being genuinely arithmetical truth, or whether it's a confusion of levels that's been normalized for a century.
On 1/17/26 8:30 PM, olcott wrote:
On 1/17/2026 7:20 PM, Richard Damon wrote:
On 1/17/26 7:49 PM, olcott wrote:
On 1/17/2026 6:14 PM, Richard Damon wrote:
On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote:
On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY
But PA itself has no truth predicate, no internal semantics,
and no mechanism for assigning truth values. So what was
called rCLtrue in arithmeticrCY was always meta-theoretic truth >>>>>>>> about arithmetic, imported from an external model and never
grounded inside PA.
Nope, just shows you don't understand what TRUTH means.
IrCOm distinguishing internal truth from external truth.
PA has no internal truth predicate, so it cannot express
or evaluate truth internally.
The only notion of truth available for PA is the external,
modelrCatheoretic one rCo which is metarCatheoretic by definition.
But Truth *IS* Truth, or you are just misdefining it.
The fact that a system can't tell you the truth value of a
statement doesn't mean the statement doesn't have a truth value.
And, the problem is that, as was shown, systems with a truth
predicate CAN'T support PA or they are inconsistant.
I guess systems that lie aren't a problem to you since you think
lying is valid logic.
This conflation was rarely acknowledged, and it shaped the
interpretation of G||delrCOs incompleteness theorems, independence >>>>>>>> results like Goodstein and ParisrCoHarrington, and the entire
discourse around rCLtrue but unprovablerCY statements.
WHich Godel proves exsits.
My work begins by correcting this foundational error.
By LYING and destroying the meaninf of truth.
PA has no internal truth predicate, so classical claims of
rCLtrue in arithmeticrCY were always meta-theoretic. My system >>>>>>>> introduces a truth predicate whose meaning is anchored
entirely in PArCOs axioms and inference rules, not in external >>>>>>>> models. Any statement whose meaning requires meta-theoretic
interpretation or non-well-founded self-reference is rejected
as outside the domain of PA. This yields a coherent, internal
notion of truth in arithmetic for the first time.
Not having a "Predicate" doesn't mean not having a definition of >>>>>>> truth.
A metarCatheoretic definition of truth is not the same
as an internal truth predicate. TarskirCOs definition of
truth for arithmetic is external to PA and cannot be
expressed inside PA. ThatrCOs exactly the distinction
IrCOm drawing.
No, he shows that any system that support PA and a Truth Predicate
is inconstant.
It seems you just want to let your system be inconsistent, as then
you can "prove" whatever you want.
PA can prove statements, but it cannot assert that
those statements are true. Those are different notions.
Right, but statments in PA can be True even without such a predicate. >>>>>
Unless PA can prove it then they never were actually
true in PA. They were true outside of PA in meta-math.
Sure it is. Truth goes beyond knowledge.
You're assuming 'truth in arithmetic' means truth-in-the-standard-
model. But that's a meta-theoretic constructrCoit's truth about
arithmetic from outside PA, not truth in arithmetic. PA has no
internal truth predicate and no way to access the standard model from
within.
No, PA (Peano Arithmetic) itself defines the numbers and the arithmatic.
Why do you think otherwise?
And why does it NEED to access the model from within?
On 1/17/2026 7:46 PM, Richard Damon wrote:
On 1/17/26 8:30 PM, olcott wrote:
On 1/17/2026 7:20 PM, Richard Damon wrote:
On 1/17/26 7:49 PM, olcott wrote:
On 1/17/2026 6:14 PM, Richard Damon wrote:
On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote:But Truth *IS* Truth, or you are just misdefining it.
On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly >>>>>>>>> relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY
But PA itself has no truth predicate, no internal semantics, >>>>>>>>> and no mechanism for assigning truth values. So what was
called rCLtrue in arithmeticrCY was always meta-theoretic truth >>>>>>>>> about arithmetic, imported from an external model and never
grounded inside PA.
Nope, just shows you don't understand what TRUTH means.
IrCOm distinguishing internal truth from external truth.
PA has no internal truth predicate, so it cannot express
or evaluate truth internally.
The only notion of truth available for PA is the external,
modelrCatheoretic one rCo which is metarCatheoretic by definition. >>>>>>
The fact that a system can't tell you the truth value of a
statement doesn't mean the statement doesn't have a truth value.
And, the problem is that, as was shown, systems with a truth
predicate CAN'T support PA or they are inconsistant.
I guess systems that lie aren't a problem to you since you think
lying is valid logic.
This conflation was rarely acknowledged, and it shaped the
interpretation of G||delrCOs incompleteness theorems, independence >>>>>>>>> results like Goodstein and ParisrCoHarrington, and the entire >>>>>>>>> discourse around rCLtrue but unprovablerCY statements.
WHich Godel proves exsits.
My work begins by correcting this foundational error.
By LYING and destroying the meaninf of truth.
PA has no internal truth predicate, so classical claims of
rCLtrue in arithmeticrCY were always meta-theoretic. My system >>>>>>>>> introduces a truth predicate whose meaning is anchored
entirely in PArCOs axioms and inference rules, not in external >>>>>>>>> models. Any statement whose meaning requires meta-theoretic
interpretation or non-well-founded self-reference is rejected >>>>>>>>> as outside the domain of PA. This yields a coherent, internal >>>>>>>>> notion of truth in arithmetic for the first time.
Not having a "Predicate" doesn't mean not having a definition of >>>>>>>> truth.
A metarCatheoretic definition of truth is not the same
as an internal truth predicate. TarskirCOs definition of
truth for arithmetic is external to PA and cannot be
expressed inside PA. ThatrCOs exactly the distinction
IrCOm drawing.
No, he shows that any system that support PA and a Truth Predicate >>>>>> is inconstant.
It seems you just want to let your system be inconsistent, as then >>>>>> you can "prove" whatever you want.
PA can prove statements, but it cannot assert that
those statements are true. Those are different notions.
Right, but statments in PA can be True even without such a predicate. >>>>>>
Unless PA can prove it then they never were actually
true in PA. They were true outside of PA in meta-math.
Sure it is. Truth goes beyond knowledge.
You're assuming 'truth in arithmetic' means truth-in-the-standard-
model. But that's a meta-theoretic constructrCoit's truth about
arithmetic from outside PA, not truth in arithmetic. PA has no
internal truth predicate and no way to access the standard model from
within.
No, PA (Peano Arithmetic) itself defines the numbers and the arithmatic.
Why do you think otherwise?
And why does it NEED to access the model from within?
G||delrCastyle incompleteness only appears when rCLtruthrCY is
defined using an outside model of the natural numbers.
If you stop using modelrCatheoretic truth and rely only
on the meanings that come from the rules of the system
itself, then rCLtruerCY and rCLprovablerCY coincide rCo so the
incompleteness gap never arises.
ThatrCOs exactly how rCLtrue on the basis of meaning
expressed in languagerCY has always worked.
On 1/17/26 8:59 PM, olcott wrote:
On 1/17/2026 7:46 PM, Richard Damon wrote:
On 1/17/26 8:30 PM, olcott wrote:
On 1/17/2026 7:20 PM, Richard Damon wrote:
On 1/17/26 7:49 PM, olcott wrote:
On 1/17/2026 6:14 PM, Richard Damon wrote:
On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote:But Truth *IS* Truth, or you are just misdefining it.
On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly >>>>>>>>>> relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY
But PA itself has no truth predicate, no internal semantics, >>>>>>>>>> and no mechanism for assigning truth values. So what was
called rCLtrue in arithmeticrCY was always meta-theoretic truth >>>>>>>>>> about arithmetic, imported from an external model and never >>>>>>>>>> grounded inside PA.
Nope, just shows you don't understand what TRUTH means.
IrCOm distinguishing internal truth from external truth.
PA has no internal truth predicate, so it cannot express
or evaluate truth internally.
The only notion of truth available for PA is the external,
modelrCatheoretic one rCo which is metarCatheoretic by definition. >>>>>>>
The fact that a system can't tell you the truth value of a
statement doesn't mean the statement doesn't have a truth value. >>>>>>>
And, the problem is that, as was shown, systems with a truth
predicate CAN'T support PA or they are inconsistant.
I guess systems that lie aren't a problem to you since you think >>>>>>> lying is valid logic.
This conflation was rarely acknowledged, and it shaped the >>>>>>>>>> interpretation of G||delrCOs incompleteness theorems, independence >>>>>>>>>> results like Goodstein and ParisrCoHarrington, and the entire >>>>>>>>>> discourse around rCLtrue but unprovablerCY statements.
WHich Godel proves exsits.
My work begins by correcting this foundational error.
By LYING and destroying the meaninf of truth.
PA has no internal truth predicate, so classical claims of >>>>>>>>>> rCLtrue in arithmeticrCY were always meta-theoretic. My system >>>>>>>>>> introduces a truth predicate whose meaning is anchored
entirely in PArCOs axioms and inference rules, not in external >>>>>>>>>> models. Any statement whose meaning requires meta-theoretic >>>>>>>>>> interpretation or non-well-founded self-reference is rejected >>>>>>>>>> as outside the domain of PA. This yields a coherent, internal >>>>>>>>>> notion of truth in arithmetic for the first time.
Not having a "Predicate" doesn't mean not having a definition >>>>>>>>> of truth.
A metarCatheoretic definition of truth is not the same
as an internal truth predicate. TarskirCOs definition of
truth for arithmetic is external to PA and cannot be
expressed inside PA. ThatrCOs exactly the distinction
IrCOm drawing.
No, he shows that any system that support PA and a Truth
Predicate is inconstant.
It seems you just want to let your system be inconsistent, as
then you can "prove" whatever you want.
PA can prove statements, but it cannot assert that
those statements are true. Those are different notions.
Right, but statments in PA can be True even without such a
predicate.
Unless PA can prove it then they never were actually
true in PA. They were true outside of PA in meta-math.
Sure it is. Truth goes beyond knowledge.
You're assuming 'truth in arithmetic' means truth-in-the-standard-
model. But that's a meta-theoretic constructrCoit's truth about
arithmetic from outside PA, not truth in arithmetic. PA has no
internal truth predicate and no way to access the standard model
from within.
No, PA (Peano Arithmetic) itself defines the numbers and the arithmatic. >>>
Why do you think otherwise?
And why does it NEED to access the model from within?
G||delrCastyle incompleteness only appears when rCLtruthrCY is
defined using an outside model of the natural numbers.
No, it uses the innate properties of the Natural Nubmers.
If you stop using modelrCatheoretic truth and rely only
on the meanings that come from the rules of the system
itself, then rCLtruerCY and rCLprovablerCY coincide rCo so the
incompleteness gap never arises.
That doesn't make sense. The answer to the arithmatic doesn't depend on anything outside the rules, as numbers mean themselves.
That a number statisfies the relationship derived doesn't depend on
anything outside of that arithmatic.
Thus, the FACT that no number will statisfy the relationsip doesn't
depend on anythign outside of that arithmatic.
Thus, G is TRUE in the system, based on nothing but the basic rules of arithmatic in the system.
It also turns out that there can not be a proof in the system, as even
if the system can't understand the meaning if the statement in the meta- system, it still follows the results of that.
You seem to think that the proof is based on some complicated logic that
you can make not true. No, it is based on the fundamental properties of Mathematics, and the fact that Mathematics creates a truth-conditional system, even if you want to try to limit what you can understand of it.
If you try to deny that ability to operate, then you lose *ALL* ability
to talk about "computability" of truth.
So, do you think the Pythagorean Theorem is True based on the meaning of
ThatrCOs exactly how rCLtrue on the basis of meaning
expressed in languagerCY has always worked.
the language?
Or, are you admitting that you system can't handle systems like
mathematics.
On 1/17/2026 9:20 PM, Richard Damon wrote:
On 1/17/26 8:59 PM, olcott wrote:
On 1/17/2026 7:46 PM, Richard Damon wrote:
On 1/17/26 8:30 PM, olcott wrote:
On 1/17/2026 7:20 PM, Richard Damon wrote:
On 1/17/26 7:49 PM, olcott wrote:
On 1/17/2026 6:14 PM, Richard Damon wrote:
On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote:But Truth *IS* Truth, or you are just misdefining it.
On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly >>>>>>>>>>> relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY
But PA itself has no truth predicate, no internal semantics, >>>>>>>>>>> and no mechanism for assigning truth values. So what was >>>>>>>>>>> called rCLtrue in arithmeticrCY was always meta-theoretic truth >>>>>>>>>>> about arithmetic, imported from an external model and never >>>>>>>>>>> grounded inside PA.
Nope, just shows you don't understand what TRUTH means.
IrCOm distinguishing internal truth from external truth.
PA has no internal truth predicate, so it cannot express
or evaluate truth internally.
The only notion of truth available for PA is the external,
modelrCatheoretic one rCo which is metarCatheoretic by definition. >>>>>>>>
The fact that a system can't tell you the truth value of a
statement doesn't mean the statement doesn't have a truth value. >>>>>>>>
And, the problem is that, as was shown, systems with a truth
predicate CAN'T support PA or they are inconsistant.
I guess systems that lie aren't a problem to you since you think >>>>>>>> lying is valid logic.
This conflation was rarely acknowledged, and it shaped the >>>>>>>>>>> interpretation of G||delrCOs incompleteness theorems, independence >>>>>>>>>>> results like Goodstein and ParisrCoHarrington, and the entire >>>>>>>>>>> discourse around rCLtrue but unprovablerCY statements.
WHich Godel proves exsits.
My work begins by correcting this foundational error.
By LYING and destroying the meaninf of truth.
PA has no internal truth predicate, so classical claims of >>>>>>>>>>> rCLtrue in arithmeticrCY were always meta-theoretic. My system >>>>>>>>>>> introduces a truth predicate whose meaning is anchored
entirely in PArCOs axioms and inference rules, not in external >>>>>>>>>>> models. Any statement whose meaning requires meta-theoretic >>>>>>>>>>> interpretation or non-well-founded self-reference is rejected >>>>>>>>>>> as outside the domain of PA. This yields a coherent, internal >>>>>>>>>>> notion of truth in arithmetic for the first time.
Not having a "Predicate" doesn't mean not having a definition >>>>>>>>>> of truth.
A metarCatheoretic definition of truth is not the same
as an internal truth predicate. TarskirCOs definition of
truth for arithmetic is external to PA and cannot be
expressed inside PA. ThatrCOs exactly the distinction
IrCOm drawing.
No, he shows that any system that support PA and a Truth
Predicate is inconstant.
It seems you just want to let your system be inconsistent, as >>>>>>>> then you can "prove" whatever you want.
PA can prove statements, but it cannot assert that
those statements are true. Those are different notions.
Right, but statments in PA can be True even without such a
predicate.
Unless PA can prove it then they never were actually
true in PA. They were true outside of PA in meta-math.
Sure it is. Truth goes beyond knowledge.
You're assuming 'truth in arithmetic' means truth-in-the-standard-
model. But that's a meta-theoretic constructrCoit's truth about
arithmetic from outside PA, not truth in arithmetic. PA has no
internal truth predicate and no way to access the standard model
from within.
No, PA (Peano Arithmetic) itself defines the numbers and the
arithmatic.
Why do you think otherwise?
And why does it NEED to access the model from within?
G||delrCastyle incompleteness only appears when rCLtruthrCY is
defined using an outside model of the natural numbers.
No, it uses the innate properties of the Natural Nubmers.
meta-math is outside of math.
If you stop using modelrCatheoretic truth and rely only
on the meanings that come from the rules of the system
itself, then rCLtruerCY and rCLprovablerCY coincide rCo so the
incompleteness gap never arises.
That doesn't make sense. The answer to the arithmatic doesn't depend
on anything outside the rules, as numbers mean themselves.
That a number statisfies the relationship derived doesn't depend on
anything outside of that arithmatic.
meta-math is outside of math.
Inside PA, all you have are symbols and rules.
Whether a statement is rCLtruerCY in the standard
model is something we say from outside PA.
ThatrCOs why G||del can separate rCytruerCO from
rCyprovablerCOrCobecause truth is defined externally.
If you stop using that external notion of truth
and rely only on the rules inside PA, then rCLtruerCY
just means rCLprovable,rCY and the incompleteness
gap disappears.
Thus, the FACT that no number will statisfy the relationsip doesn't
depend on anythign outside of that arithmatic.
Thus, G is TRUE in the system, based on nothing but the basic rules of
arithmatic in the system.
It also turns out that there can not be a proof in the system, as even
if the system can't understand the meaning if the statement in the
meta- system, it still follows the results of that.
You seem to think that the proof is based on some complicated logic
that you can make not true. No, it is based on the fundamental
properties of Mathematics, and the fact that Mathematics creates a
truth-conditional system, even if you want to try to limit what you
can understand of it.
If you try to deny that ability to operate, then you lose *ALL*
ability to talk about "computability" of truth.
So, do you think the Pythagorean Theorem is True based on the meaning
ThatrCOs exactly how rCLtrue on the basis of meaning
expressed in languagerCY has always worked.
of the language?
Or, are you admitting that you system can't handle systems like
mathematics.
On 1/17/26 10:59 PM, olcott wrote:
On 1/17/2026 9:20 PM, Richard Damon wrote:
On 1/17/26 8:59 PM, olcott wrote:
On 1/17/2026 7:46 PM, Richard Damon wrote:
On 1/17/26 8:30 PM, olcott wrote:
On 1/17/2026 7:20 PM, Richard Damon wrote:
On 1/17/26 7:49 PM, olcott wrote:
On 1/17/2026 6:14 PM, Richard Damon wrote:
On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote:But Truth *IS* Truth, or you are just misdefining it.
On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly >>>>>>>>>>>> relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY
But PA itself has no truth predicate, no internal semantics, >>>>>>>>>>>> and no mechanism for assigning truth values. So what was >>>>>>>>>>>> called rCLtrue in arithmeticrCY was always meta-theoretic truth >>>>>>>>>>>> about arithmetic, imported from an external model and never >>>>>>>>>>>> grounded inside PA.
Nope, just shows you don't understand what TRUTH means.
IrCOm distinguishing internal truth from external truth.
PA has no internal truth predicate, so it cannot express
or evaluate truth internally.
The only notion of truth available for PA is the external, >>>>>>>>>> modelrCatheoretic one rCo which is metarCatheoretic by definition. >>>>>>>>>
The fact that a system can't tell you the truth value of a
statement doesn't mean the statement doesn't have a truth value. >>>>>>>>>
And, the problem is that, as was shown, systems with a truth >>>>>>>>> predicate CAN'T support PA or they are inconsistant.
I guess systems that lie aren't a problem to you since you
think lying is valid logic.
This conflation was rarely acknowledged, and it shaped the >>>>>>>>>>>> interpretation of G||delrCOs incompleteness theorems, independence >>>>>>>>>>>> results like Goodstein and ParisrCoHarrington, and the entire >>>>>>>>>>>> discourse around rCLtrue but unprovablerCY statements.
WHich Godel proves exsits.
My work begins by correcting this foundational error.
By LYING and destroying the meaninf of truth.
PA has no internal truth predicate, so classical claims of >>>>>>>>>>>> rCLtrue in arithmeticrCY were always meta-theoretic. My system >>>>>>>>>>>> introduces a truth predicate whose meaning is anchored >>>>>>>>>>>> entirely in PArCOs axioms and inference rules, not in external >>>>>>>>>>>> models. Any statement whose meaning requires meta-theoretic >>>>>>>>>>>> interpretation or non-well-founded self-reference is rejected >>>>>>>>>>>> as outside the domain of PA. This yields a coherent, internal >>>>>>>>>>>> notion of truth in arithmetic for the first time.
Not having a "Predicate" doesn't mean not having a definition >>>>>>>>>>> of truth.
A metarCatheoretic definition of truth is not the same
as an internal truth predicate. TarskirCOs definition of
truth for arithmetic is external to PA and cannot be
expressed inside PA. ThatrCOs exactly the distinction
IrCOm drawing.
No, he shows that any system that support PA and a Truth
Predicate is inconstant.
It seems you just want to let your system be inconsistent, as >>>>>>>>> then you can "prove" whatever you want.
PA can prove statements, but it cannot assert that
those statements are true. Those are different notions.
Right, but statments in PA can be True even without such a
predicate.
Unless PA can prove it then they never were actually
true in PA. They were true outside of PA in meta-math.
Sure it is. Truth goes beyond knowledge.
You're assuming 'truth in arithmetic' means truth-in-the-standard- >>>>>> model. But that's a meta-theoretic constructrCoit's truth about
arithmetic from outside PA, not truth in arithmetic. PA has no
internal truth predicate and no way to access the standard model
from within.
No, PA (Peano Arithmetic) itself defines the numbers and the
arithmatic.
Why do you think otherwise?
And why does it NEED to access the model from within?
G||delrCastyle incompleteness only appears when rCLtruthrCY is
defined using an outside model of the natural numbers.
No, it uses the innate properties of the Natural Nubmers.
meta-math is outside of math.
If you stop using modelrCatheoretic truth and rely only
on the meanings that come from the rules of the system
itself, then rCLtruerCY and rCLprovablerCY coincide rCo so the
incompleteness gap never arises.
That doesn't make sense. The answer to the arithmatic doesn't depend
on anything outside the rules, as numbers mean themselves.
That a number statisfies the relationship derived doesn't depend on
anything outside of that arithmatic.
meta-math is outside of math.
No, it uses just the math of PA.
The meta-system just embues some additional meaning into the numbers.
For nearly a century, discussions of arithmetic have quietly
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY
But PA itself has no truth predicate, no internal semantics,
and no mechanism for assigning truth values. So what was
called rCLtrue in arithmeticrCY was always meta-theoretic truth
about arithmetic, imported from an external model and never
grounded inside PA.
For nearly a century, discussions of arithmetic have quietly--
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY
But PA itself has no truth predicate, no internal semantics,
and no mechanism for assigning truth values. So what was
called rCLtrue in arithmeticrCY was always meta-theoretic truth
about arithmetic, imported from an external model and never
grounded inside PA.
This conflation was rarely acknowledged, and it shaped the
interpretation of G||delrCOs incompleteness theorems, independence
results like Goodstein and ParisrCoHarrington, and the entire
discourse around rCLtrue but unprovablerCY statements.
My work begins by correcting this foundational error.
PA has no internal truth predicate, so classical claims of
rCLtrue in arithmeticrCY were always meta-theoretic
1 + 1 = 2 IS true
The problem is you don't understand that there is an internal truth,
that comes just out of the base properties of mathematics.
On 1/17/2026 10:13 PM, Richard Damon wrote:
On 1/17/26 10:59 PM, olcott wrote:
On 1/17/2026 9:20 PM, Richard Damon wrote:
On 1/17/26 8:59 PM, olcott wrote:
On 1/17/2026 7:46 PM, Richard Damon wrote:
On 1/17/26 8:30 PM, olcott wrote:
On 1/17/2026 7:20 PM, Richard Damon wrote:
On 1/17/26 7:49 PM, olcott wrote:
On 1/17/2026 6:14 PM, Richard Damon wrote:
On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote:But Truth *IS* Truth, or you are just misdefining it.
On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly >>>>>>>>>>>>> relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY
But PA itself has no truth predicate, no internal semantics, >>>>>>>>>>>>> and no mechanism for assigning truth values. So what was >>>>>>>>>>>>> called rCLtrue in arithmeticrCY was always meta-theoretic truth >>>>>>>>>>>>> about arithmetic, imported from an external model and never >>>>>>>>>>>>> grounded inside PA.
Nope, just shows you don't understand what TRUTH means. >>>>>>>>>>>>
IrCOm distinguishing internal truth from external truth. >>>>>>>>>>> PA has no internal truth predicate, so it cannot express >>>>>>>>>>> or evaluate truth internally.
The only notion of truth available for PA is the external, >>>>>>>>>>> modelrCatheoretic one rCo which is metarCatheoretic by definition. >>>>>>>>>>
The fact that a system can't tell you the truth value of a >>>>>>>>>> statement doesn't mean the statement doesn't have a truth value. >>>>>>>>>>
And, the problem is that, as was shown, systems with a truth >>>>>>>>>> predicate CAN'T support PA or they are inconsistant.
I guess systems that lie aren't a problem to you since you >>>>>>>>>> think lying is valid logic.
WHich Godel proves exsits.
This conflation was rarely acknowledged, and it shaped the >>>>>>>>>>>>> interpretation of G||delrCOs incompleteness theorems, >>>>>>>>>>>>> independence
results like Goodstein and ParisrCoHarrington, and the entire >>>>>>>>>>>>> discourse around rCLtrue but unprovablerCY statements. >>>>>>>>>>>>
By LYING and destroying the meaninf of truth.
My work begins by correcting this foundational error. >>>>>>>>>>>>
PA has no internal truth predicate, so classical claims of >>>>>>>>>>>>> rCLtrue in arithmeticrCY were always meta-theoretic. My system >>>>>>>>>>>>> introduces a truth predicate whose meaning is anchored >>>>>>>>>>>>> entirely in PArCOs axioms and inference rules, not in external >>>>>>>>>>>>> models. Any statement whose meaning requires meta-theoretic >>>>>>>>>>>>> interpretation or non-well-founded self-reference is rejected >>>>>>>>>>>>> as outside the domain of PA. This yields a coherent, internal >>>>>>>>>>>>> notion of truth in arithmetic for the first time.
Not having a "Predicate" doesn't mean not having a
definition of truth.
A metarCatheoretic definition of truth is not the same
as an internal truth predicate. TarskirCOs definition of >>>>>>>>>>> truth for arithmetic is external to PA and cannot be
expressed inside PA. ThatrCOs exactly the distinction
IrCOm drawing.
No, he shows that any system that support PA and a Truth
Predicate is inconstant.
It seems you just want to let your system be inconsistent, as >>>>>>>>>> then you can "prove" whatever you want.
PA can prove statements, but it cannot assert that
those statements are true. Those are different notions.
Right, but statments in PA can be True even without such a >>>>>>>>>> predicate.
Unless PA can prove it then they never were actually
true in PA. They were true outside of PA in meta-math.
Sure it is. Truth goes beyond knowledge.
You're assuming 'truth in arithmetic' means truth-in-the-
standard- model. But that's a meta-theoretic constructrCoit's truth >>>>>>> about arithmetic from outside PA, not truth in arithmetic. PA has >>>>>>> no internal truth predicate and no way to access the standard
model from within.
No, PA (Peano Arithmetic) itself defines the numbers and the
arithmatic.
Why do you think otherwise?
And why does it NEED to access the model from within?
G||delrCastyle incompleteness only appears when rCLtruthrCY is
defined using an outside model of the natural numbers.
No, it uses the innate properties of the Natural Nubmers.
meta-math is outside of math.
If you stop using modelrCatheoretic truth and rely only
on the meanings that come from the rules of the system
itself, then rCLtruerCY and rCLprovablerCY coincide rCo so the
incompleteness gap never arises.
That doesn't make sense. The answer to the arithmatic doesn't depend
on anything outside the rules, as numbers mean themselves.
That a number statisfies the relationship derived doesn't depend on
anything outside of that arithmatic.
meta-math is outside of math.
No, it uses just the math of PA.
The meta-system just embues some additional meaning into the numbers.
That is where it steps outside of math
On 1/17/26 11:38 PM, olcott wrote:
On 1/17/2026 10:13 PM, Richard Damon wrote:
On 1/17/26 10:59 PM, olcott wrote:
On 1/17/2026 9:20 PM, Richard Damon wrote:
On 1/17/26 8:59 PM, olcott wrote:
On 1/17/2026 7:46 PM, Richard Damon wrote:
On 1/17/26 8:30 PM, olcott wrote:
On 1/17/2026 7:20 PM, Richard Damon wrote:
On 1/17/26 7:49 PM, olcott wrote:
On 1/17/2026 6:14 PM, Richard Damon wrote:
On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote:But Truth *IS* Truth, or you are just misdefining it.
On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly >>>>>>>>>>>>>> relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY
But PA itself has no truth predicate, no internal semantics, >>>>>>>>>>>>>> and no mechanism for assigning truth values. So what was >>>>>>>>>>>>>> called rCLtrue in arithmeticrCY was always meta-theoretic truth >>>>>>>>>>>>>> about arithmetic, imported from an external model and never >>>>>>>>>>>>>> grounded inside PA.
Nope, just shows you don't understand what TRUTH means. >>>>>>>>>>>>>
IrCOm distinguishing internal truth from external truth. >>>>>>>>>>>> PA has no internal truth predicate, so it cannot express >>>>>>>>>>>> or evaluate truth internally.
The only notion of truth available for PA is the external, >>>>>>>>>>>> modelrCatheoretic one rCo which is metarCatheoretic by definition. >>>>>>>>>>>
The fact that a system can't tell you the truth value of a >>>>>>>>>>> statement doesn't mean the statement doesn't have a truth value. >>>>>>>>>>>
And, the problem is that, as was shown, systems with a truth >>>>>>>>>>> predicate CAN'T support PA or they are inconsistant.
I guess systems that lie aren't a problem to you since you >>>>>>>>>>> think lying is valid logic.
WHich Godel proves exsits.
This conflation was rarely acknowledged, and it shaped the >>>>>>>>>>>>>> interpretation of G||delrCOs incompleteness theorems, >>>>>>>>>>>>>> independence
results like Goodstein and ParisrCoHarrington, and the entire >>>>>>>>>>>>>> discourse around rCLtrue but unprovablerCY statements. >>>>>>>>>>>>>
By LYING and destroying the meaninf of truth.
My work begins by correcting this foundational error. >>>>>>>>>>>>>
PA has no internal truth predicate, so classical claims of >>>>>>>>>>>>>> rCLtrue in arithmeticrCY were always meta-theoretic. My system >>>>>>>>>>>>>> introduces a truth predicate whose meaning is anchored >>>>>>>>>>>>>> entirely in PArCOs axioms and inference rules, not in external >>>>>>>>>>>>>> models. Any statement whose meaning requires meta-theoretic >>>>>>>>>>>>>> interpretation or non-well-founded self-reference is rejected >>>>>>>>>>>>>> as outside the domain of PA. This yields a coherent, internal >>>>>>>>>>>>>> notion of truth in arithmetic for the first time.
Not having a "Predicate" doesn't mean not having a
definition of truth.
A metarCatheoretic definition of truth is not the same >>>>>>>>>>>> as an internal truth predicate. TarskirCOs definition of >>>>>>>>>>>> truth for arithmetic is external to PA and cannot be
expressed inside PA. ThatrCOs exactly the distinction
IrCOm drawing.
No, he shows that any system that support PA and a Truth >>>>>>>>>>> Predicate is inconstant.
It seems you just want to let your system be inconsistent, as >>>>>>>>>>> then you can "prove" whatever you want.
Right, but statments in PA can be True even without such a >>>>>>>>>>> predicate.
PA can prove statements, but it cannot assert that
those statements are true. Those are different notions. >>>>>>>>>>>
Unless PA can prove it then they never were actually
true in PA. They were true outside of PA in meta-math.
Sure it is. Truth goes beyond knowledge.
You're assuming 'truth in arithmetic' means truth-in-the-
standard- model. But that's a meta-theoretic constructrCoit's >>>>>>>> truth about arithmetic from outside PA, not truth in arithmetic. >>>>>>>> PA has no internal truth predicate and no way to access the
standard model from within.
No, PA (Peano Arithmetic) itself defines the numbers and the
arithmatic.
Why do you think otherwise?
And why does it NEED to access the model from within?
G||delrCastyle incompleteness only appears when rCLtruthrCY is
defined using an outside model of the natural numbers.
No, it uses the innate properties of the Natural Nubmers.
meta-math is outside of math.
If you stop using modelrCatheoretic truth and rely only
on the meanings that come from the rules of the system
itself, then rCLtruerCY and rCLprovablerCY coincide rCo so the
incompleteness gap never arises.
That doesn't make sense. The answer to the arithmatic doesn't
depend on anything outside the rules, as numbers mean themselves.
That a number statisfies the relationship derived doesn't depend on >>>>> anything outside of that arithmatic.
meta-math is outside of math.
No, it uses just the math of PA.
The meta-system just embues some additional meaning into the numbers.
That is where it steps outside of math
But that meaning doesn't actually affect the results in the system, only
to let us KNOW the results.
On 1/18/2026 11:37 AM, Richard Damon wrote:
On 1/17/26 11:38 PM, olcott wrote:
On 1/17/2026 10:13 PM, Richard Damon wrote:
On 1/17/26 10:59 PM, olcott wrote:
On 1/17/2026 9:20 PM, Richard Damon wrote:
On 1/17/26 8:59 PM, olcott wrote:
On 1/17/2026 7:46 PM, Richard Damon wrote:
On 1/17/26 8:30 PM, olcott wrote:
On 1/17/2026 7:20 PM, Richard Damon wrote:
On 1/17/26 7:49 PM, olcott wrote:
On 1/17/2026 6:14 PM, Richard Damon wrote:
On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote:
On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly >>>>>>>>>>>>>>> relied on a fundamental conflation: the idea that >>>>>>>>>>>>>>> rCLtrue in arithmeticrCY meant rCLtrue in the standard model of
rao.rCY
But PA itself has no truth predicate, no internal semantics, >>>>>>>>>>>>>>> and no mechanism for assigning truth values. So what was >>>>>>>>>>>>>>> called rCLtrue in arithmeticrCY was always meta-theoretic truth >>>>>>>>>>>>>>> about arithmetic, imported from an external model and never >>>>>>>>>>>>>>> grounded inside PA.
Nope, just shows you don't understand what TRUTH means. >>>>>>>>>>>>>>
IrCOm distinguishing internal truth from external truth. >>>>>>>>>>>>> PA has no internal truth predicate, so it cannot express >>>>>>>>>>>>> or evaluate truth internally.
The only notion of truth available for PA is the external, >>>>>>>>>>>>> modelrCatheoretic one rCo which is metarCatheoretic by definition.
But Truth *IS* Truth, or you are just misdefining it.
The fact that a system can't tell you the truth value of a >>>>>>>>>>>> statement doesn't mean the statement doesn't have a truth >>>>>>>>>>>> value.
And, the problem is that, as was shown, systems with a truth >>>>>>>>>>>> predicate CAN'T support PA or they are inconsistant.
I guess systems that lie aren't a problem to you since you >>>>>>>>>>>> think lying is valid logic.
WHich Godel proves exsits.
This conflation was rarely acknowledged, and it shaped the >>>>>>>>>>>>>>> interpretation of G||delrCOs incompleteness theorems, >>>>>>>>>>>>>>> independence
results like Goodstein and ParisrCoHarrington, and the entire >>>>>>>>>>>>>>> discourse around rCLtrue but unprovablerCY statements. >>>>>>>>>>>>>>
By LYING and destroying the meaninf of truth.
My work begins by correcting this foundational error. >>>>>>>>>>>>>>
PA has no internal truth predicate, so classical claims of >>>>>>>>>>>>>>> rCLtrue in arithmeticrCY were always meta-theoretic. My system >>>>>>>>>>>>>>> introduces a truth predicate whose meaning is anchored >>>>>>>>>>>>>>> entirely in PArCOs axioms and inference rules, not in external >>>>>>>>>>>>>>> models. Any statement whose meaning requires meta-theoretic >>>>>>>>>>>>>>> interpretation or non-well-founded self-reference is >>>>>>>>>>>>>>> rejected
as outside the domain of PA. This yields a coherent, >>>>>>>>>>>>>>> internal
notion of truth in arithmetic for the first time. >>>>>>>>>>>>>>>
Not having a "Predicate" doesn't mean not having a >>>>>>>>>>>>>> definition of truth.
A metarCatheoretic definition of truth is not the same >>>>>>>>>>>>> as an internal truth predicate. TarskirCOs definition of >>>>>>>>>>>>> truth for arithmetic is external to PA and cannot be >>>>>>>>>>>>> expressed inside PA. ThatrCOs exactly the distinction >>>>>>>>>>>>> IrCOm drawing.
No, he shows that any system that support PA and a Truth >>>>>>>>>>>> Predicate is inconstant.
It seems you just want to let your system be inconsistent, >>>>>>>>>>>> as then you can "prove" whatever you want.
Right, but statments in PA can be True even without such a >>>>>>>>>>>> predicate.
PA can prove statements, but it cannot assert that
those statements are true. Those are different notions. >>>>>>>>>>>>
Unless PA can prove it then they never were actually
true in PA. They were true outside of PA in meta-math.
Sure it is. Truth goes beyond knowledge.
You're assuming 'truth in arithmetic' means truth-in-the-
standard- model. But that's a meta-theoretic constructrCoit's >>>>>>>>> truth about arithmetic from outside PA, not truth in
arithmetic. PA has no internal truth predicate and no way to >>>>>>>>> access the standard model from within.
No, PA (Peano Arithmetic) itself defines the numbers and the
arithmatic.
Why do you think otherwise?
And why does it NEED to access the model from within?
G||delrCastyle incompleteness only appears when rCLtruthrCY is
defined using an outside model of the natural numbers.
No, it uses the innate properties of the Natural Nubmers.
meta-math is outside of math.
If you stop using modelrCatheoretic truth and rely only
on the meanings that come from the rules of the system
itself, then rCLtruerCY and rCLprovablerCY coincide rCo so the
incompleteness gap never arises.
That doesn't make sense. The answer to the arithmatic doesn't
depend on anything outside the rules, as numbers mean themselves.
That a number statisfies the relationship derived doesn't depend
on anything outside of that arithmatic.
meta-math is outside of math.
No, it uses just the math of PA.
The meta-system just embues some additional meaning into the numbers.
That is where it steps outside of math
But that meaning doesn't actually affect the results in the system,
only to let us KNOW the results.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~TruthBearer(PA, x) rei (~True(PA, x) reo (~False(PA, x))
When we look at what is actually true directly in PA
and not what is true about PA in meta-math then G||del
Incompleteness cannot arise. The nearly century long
mistake was conflating true about PA in meta-math for
what is actually true in PA.
On 1/18/26 1:38 PM, olcott wrote:
On 1/18/2026 11:37 AM, Richard Damon wrote:
On 1/17/26 11:38 PM, olcott wrote:
On 1/17/2026 10:13 PM, Richard Damon wrote:
On 1/17/26 10:59 PM, olcott wrote:
On 1/17/2026 9:20 PM, Richard Damon wrote:
On 1/17/26 8:59 PM, olcott wrote:
On 1/17/2026 7:46 PM, Richard Damon wrote:
On 1/17/26 8:30 PM, olcott wrote:
On 1/17/2026 7:20 PM, Richard Damon wrote:
On 1/17/26 7:49 PM, olcott wrote:
On 1/17/2026 6:14 PM, Richard Damon wrote:
On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote:
On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have >>>>>>>>>>>>>>>> quietly
relied on a fundamental conflation: the idea that >>>>>>>>>>>>>>>> rCLtrue in arithmeticrCY meant rCLtrue in the standard model >>>>>>>>>>>>>>>> of rao.rCY
But PA itself has no truth predicate, no internal >>>>>>>>>>>>>>>> semantics,
and no mechanism for assigning truth values. So what was >>>>>>>>>>>>>>>> called rCLtrue in arithmeticrCY was always meta-theoretic truth
about arithmetic, imported from an external model and never >>>>>>>>>>>>>>>> grounded inside PA.
Nope, just shows you don't understand what TRUTH means. >>>>>>>>>>>>>>>
IrCOm distinguishing internal truth from external truth. >>>>>>>>>>>>>> PA has no internal truth predicate, so it cannot express >>>>>>>>>>>>>> or evaluate truth internally.
The only notion of truth available for PA is the external, >>>>>>>>>>>>>> modelrCatheoretic one rCo which is metarCatheoretic by definition.
But Truth *IS* Truth, or you are just misdefining it. >>>>>>>>>>>>>
The fact that a system can't tell you the truth value of a >>>>>>>>>>>>> statement doesn't mean the statement doesn't have a truth >>>>>>>>>>>>> value.
And, the problem is that, as was shown, systems with a >>>>>>>>>>>>> truth predicate CAN'T support PA or they are inconsistant. >>>>>>>>>>>>>
I guess systems that lie aren't a problem to you since you >>>>>>>>>>>>> think lying is valid logic.
WHich Godel proves exsits.
This conflation was rarely acknowledged, and it shaped the >>>>>>>>>>>>>>>> interpretation of G||delrCOs incompleteness theorems, >>>>>>>>>>>>>>>> independence
results like Goodstein and ParisrCoHarrington, and the entire >>>>>>>>>>>>>>>> discourse around rCLtrue but unprovablerCY statements. >>>>>>>>>>>>>>>
By LYING and destroying the meaninf of truth.
My work begins by correcting this foundational error. >>>>>>>>>>>>>>>
PA has no internal truth predicate, so classical claims of >>>>>>>>>>>>>>>> rCLtrue in arithmeticrCY were always meta-theoretic. My system >>>>>>>>>>>>>>>> introduces a truth predicate whose meaning is anchored >>>>>>>>>>>>>>>> entirely in PArCOs axioms and inference rules, not in >>>>>>>>>>>>>>>> external
models. Any statement whose meaning requires meta-theoretic >>>>>>>>>>>>>>>> interpretation or non-well-founded self-reference is >>>>>>>>>>>>>>>> rejected
as outside the domain of PA. This yields a coherent, >>>>>>>>>>>>>>>> internal
notion of truth in arithmetic for the first time. >>>>>>>>>>>>>>>>
Not having a "Predicate" doesn't mean not having a >>>>>>>>>>>>>>> definition of truth.
A metarCatheoretic definition of truth is not the same >>>>>>>>>>>>>> as an internal truth predicate. TarskirCOs definition of >>>>>>>>>>>>>> truth for arithmetic is external to PA and cannot be >>>>>>>>>>>>>> expressed inside PA. ThatrCOs exactly the distinction >>>>>>>>>>>>>> IrCOm drawing.
No, he shows that any system that support PA and a Truth >>>>>>>>>>>>> Predicate is inconstant.
It seems you just want to let your system be inconsistent, >>>>>>>>>>>>> as then you can "prove" whatever you want.
Right, but statments in PA can be True even without such a >>>>>>>>>>>>> predicate.
PA can prove statements, but it cannot assert that >>>>>>>>>>>>>> those statements are true. Those are different notions. >>>>>>>>>>>>>
Unless PA can prove it then they never were actually
true in PA. They were true outside of PA in meta-math. >>>>>>>>>>>>
Sure it is. Truth goes beyond knowledge.
You're assuming 'truth in arithmetic' means truth-in-the- >>>>>>>>>> standard- model. But that's a meta-theoretic constructrCoit's >>>>>>>>>> truth about arithmetic from outside PA, not truth in
arithmetic. PA has no internal truth predicate and no way to >>>>>>>>>> access the standard model from within.
No, PA (Peano Arithmetic) itself defines the numbers and the >>>>>>>>> arithmatic.
Why do you think otherwise?
And why does it NEED to access the model from within?
G||delrCastyle incompleteness only appears when rCLtruthrCY is >>>>>>>> defined using an outside model of the natural numbers.
No, it uses the innate properties of the Natural Nubmers.
meta-math is outside of math.
If you stop using modelrCatheoretic truth and rely only
on the meanings that come from the rules of the system
itself, then rCLtruerCY and rCLprovablerCY coincide rCo so the >>>>>>>> incompleteness gap never arises.
That doesn't make sense. The answer to the arithmatic doesn't
depend on anything outside the rules, as numbers mean themselves. >>>>>>>
That a number statisfies the relationship derived doesn't depend >>>>>>> on anything outside of that arithmatic.
meta-math is outside of math.
No, it uses just the math of PA.
The meta-system just embues some additional meaning into the numbers. >>>>>
That is where it steps outside of math
But that meaning doesn't actually affect the results in the system,
only to let us KNOW the results.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~TruthBearer(PA, x) rei (~True(PA, x) reo (~False(PA, x))
When we look at what is actually true directly in PA
and not what is true about PA in meta-math then G||del
Incompleteness cannot arise. The nearly century long
mistake was conflating true about PA in meta-math for
what is actually true in PA.
Except that none of the those statements are well-formed for all x,
since we can't check ALL possible proofs (since there is an infinite
number of them) to determine if a given statement is True, False, or Not
a TruthBearer.
You criteria only works in a system with only a finite number of
possible proofs, of which PA doesn't fit.
For instance, Which is the Goldbach conjecture?
We think it is likely true, but don't have a proof YET.
There COULD be a counter example, but we haven't found it.
It might not be provable, but we don't know that either.
Thus, your system can't even classify a simple problem, because your criteria are not well-founded.
Since Goldbach is undecidable in PA
On 1/18/2026 2:55 PM, Richard Damon wrote:
On 1/18/26 1:38 PM, olcott wrote:
On 1/18/2026 11:37 AM, Richard Damon wrote:
On 1/17/26 11:38 PM, olcott wrote:
On 1/17/2026 10:13 PM, Richard Damon wrote:
On 1/17/26 10:59 PM, olcott wrote:
On 1/17/2026 9:20 PM, Richard Damon wrote:
On 1/17/26 8:59 PM, olcott wrote:
On 1/17/2026 7:46 PM, Richard Damon wrote:
On 1/17/26 8:30 PM, olcott wrote:
On 1/17/2026 7:20 PM, Richard Damon wrote:
On 1/17/26 7:49 PM, olcott wrote:
On 1/17/2026 6:14 PM, Richard Damon wrote:
On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote:
On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have >>>>>>>>>>>>>>>>> quietly
relied on a fundamental conflation: the idea that >>>>>>>>>>>>>>>>> rCLtrue in arithmeticrCY meant rCLtrue in the standard model >>>>>>>>>>>>>>>>> of rao.rCY
But PA itself has no truth predicate, no internal >>>>>>>>>>>>>>>>> semantics,
and no mechanism for assigning truth values. So what was >>>>>>>>>>>>>>>>> called rCLtrue in arithmeticrCY was always meta-theoretic >>>>>>>>>>>>>>>>> truth
about arithmetic, imported from an external model and >>>>>>>>>>>>>>>>> never
grounded inside PA.
Nope, just shows you don't understand what TRUTH means. >>>>>>>>>>>>>>>>
IrCOm distinguishing internal truth from external truth. >>>>>>>>>>>>>>> PA has no internal truth predicate, so it cannot express >>>>>>>>>>>>>>> or evaluate truth internally.
The only notion of truth available for PA is the external, >>>>>>>>>>>>>>> modelrCatheoretic one rCo which is metarCatheoretic by definition.
But Truth *IS* Truth, or you are just misdefining it. >>>>>>>>>>>>>>
The fact that a system can't tell you the truth value of a >>>>>>>>>>>>>> statement doesn't mean the statement doesn't have a truth >>>>>>>>>>>>>> value.
And, the problem is that, as was shown, systems with a >>>>>>>>>>>>>> truth predicate CAN'T support PA or they are inconsistant. >>>>>>>>>>>>>>
I guess systems that lie aren't a problem to you since you >>>>>>>>>>>>>> think lying is valid logic.
WHich Godel proves exsits.
This conflation was rarely acknowledged, and it shaped the >>>>>>>>>>>>>>>>> interpretation of G||delrCOs incompleteness theorems, >>>>>>>>>>>>>>>>> independence
results like Goodstein and ParisrCoHarrington, and the >>>>>>>>>>>>>>>>> entire
discourse around rCLtrue but unprovablerCY statements. >>>>>>>>>>>>>>>>
By LYING and destroying the meaninf of truth.
My work begins by correcting this foundational error. >>>>>>>>>>>>>>>>
PA has no internal truth predicate, so classical claims of >>>>>>>>>>>>>>>>> rCLtrue in arithmeticrCY were always meta-theoretic. My system
introduces a truth predicate whose meaning is anchored >>>>>>>>>>>>>>>>> entirely in PArCOs axioms and inference rules, not in >>>>>>>>>>>>>>>>> external
models. Any statement whose meaning requires meta- >>>>>>>>>>>>>>>>> theoretic
interpretation or non-well-founded self-reference is >>>>>>>>>>>>>>>>> rejected
as outside the domain of PA. This yields a coherent, >>>>>>>>>>>>>>>>> internal
notion of truth in arithmetic for the first time. >>>>>>>>>>>>>>>>>
Not having a "Predicate" doesn't mean not having a >>>>>>>>>>>>>>>> definition of truth.
A metarCatheoretic definition of truth is not the same >>>>>>>>>>>>>>> as an internal truth predicate. TarskirCOs definition of >>>>>>>>>>>>>>> truth for arithmetic is external to PA and cannot be >>>>>>>>>>>>>>> expressed inside PA. ThatrCOs exactly the distinction >>>>>>>>>>>>>>> IrCOm drawing.
No, he shows that any system that support PA and a Truth >>>>>>>>>>>>>> Predicate is inconstant.
It seems you just want to let your system be inconsistent, >>>>>>>>>>>>>> as then you can "prove" whatever you want.
Right, but statments in PA can be True even without such a >>>>>>>>>>>>>> predicate.
PA can prove statements, but it cannot assert that >>>>>>>>>>>>>>> those statements are true. Those are different notions. >>>>>>>>>>>>>>
Unless PA can prove it then they never were actually >>>>>>>>>>>>> true in PA. They were true outside of PA in meta-math. >>>>>>>>>>>>>
Sure it is. Truth goes beyond knowledge.
You're assuming 'truth in arithmetic' means truth-in-the- >>>>>>>>>>> standard- model. But that's a meta-theoretic constructrCoit's >>>>>>>>>>> truth about arithmetic from outside PA, not truth in
arithmetic. PA has no internal truth predicate and no way to >>>>>>>>>>> access the standard model from within.
No, PA (Peano Arithmetic) itself defines the numbers and the >>>>>>>>>> arithmatic.
Why do you think otherwise?
And why does it NEED to access the model from within?
G||delrCastyle incompleteness only appears when rCLtruthrCY is >>>>>>>>> defined using an outside model of the natural numbers.
No, it uses the innate properties of the Natural Nubmers.
meta-math is outside of math.
If you stop using modelrCatheoretic truth and rely only
on the meanings that come from the rules of the system
itself, then rCLtruerCY and rCLprovablerCY coincide rCo so the >>>>>>>>> incompleteness gap never arises.
That doesn't make sense. The answer to the arithmatic doesn't >>>>>>>> depend on anything outside the rules, as numbers mean themselves. >>>>>>>>
That a number statisfies the relationship derived doesn't depend >>>>>>>> on anything outside of that arithmatic.
meta-math is outside of math.
No, it uses just the math of PA.
The meta-system just embues some additional meaning into the numbers. >>>>>>
That is where it steps outside of math
But that meaning doesn't actually affect the results in the system,
only to let us KNOW the results.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~TruthBearer(PA, x) rei (~True(PA, x) reo (~False(PA, x))
When we look at what is actually true directly in PA
and not what is true about PA in meta-math then G||del
Incompleteness cannot arise. The nearly century long
mistake was conflating true about PA in meta-math for
what is actually true in PA.
Except that none of the those statements are well-formed for all x,
since we can't check ALL possible proofs (since there is an infinite
number of them) to determine if a given statement is True, False, or
Not a TruthBearer.
True(PA, x) rei PA reo x
does not require PA to search all proofs. It simply states:
---If PA proves x, then True(PA, x) holds.
---If PA does not prove x, then True(PA, x) does not hold.
You criteria only works in a system with only a finite number of
possible proofs, of which PA doesn't fit.
For instance, Which is the Goldbach conjecture?
We think it is likely true, but don't have a proof YET.
There COULD be a counter example, but we haven't found it.
It might not be provable, but we don't know that either.
Thus, your system can't even classify a simple problem, because your
criteria are not well-founded.
Goldbach is outside PA because PA neither proves
it nor refutes it. In a proofrCatheoretic framework,
a statement belongs to PArCOs inferential domain only
if it is derivable from PArCOs axioms. Since Goldbach
is undecidable in PA, it has no inferential grounding
there. Therefore, if a proof of Goldbach exists at
all, it must lie outside PArCOs deductive power.
On 1/18/26 4:49 PM, olcott wrote:
On 1/18/2026 2:55 PM, Richard Damon wrote:
On 1/18/26 1:38 PM, olcott wrote:
On 1/18/2026 11:37 AM, Richard Damon wrote:
On 1/17/26 11:38 PM, olcott wrote:
On 1/17/2026 10:13 PM, Richard Damon wrote:
On 1/17/26 10:59 PM, olcott wrote:
On 1/17/2026 9:20 PM, Richard Damon wrote:
On 1/17/26 8:59 PM, olcott wrote:
On 1/17/2026 7:46 PM, Richard Damon wrote:
On 1/17/26 8:30 PM, olcott wrote:
On 1/17/2026 7:20 PM, Richard Damon wrote:
On 1/17/26 7:49 PM, olcott wrote:
On 1/17/2026 6:14 PM, Richard Damon wrote:
On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote:
On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have >>>>>>>>>>>>>>>>>> quietly
relied on a fundamental conflation: the idea that >>>>>>>>>>>>>>>>>> rCLtrue in arithmeticrCY meant rCLtrue in the standard model
of rao.rCY
But PA itself has no truth predicate, no internal >>>>>>>>>>>>>>>>>> semantics,
and no mechanism for assigning truth values. So what was >>>>>>>>>>>>>>>>>> called rCLtrue in arithmeticrCY was always meta-theoretic >>>>>>>>>>>>>>>>>> truth
about arithmetic, imported from an external model and >>>>>>>>>>>>>>>>>> never
grounded inside PA.
Nope, just shows you don't understand what TRUTH means. >>>>>>>>>>>>>>>>>
IrCOm distinguishing internal truth from external truth. >>>>>>>>>>>>>>>> PA has no internal truth predicate, so it cannot express >>>>>>>>>>>>>>>> or evaluate truth internally.
The only notion of truth available for PA is the external, >>>>>>>>>>>>>>>> modelrCatheoretic one rCo which is metarCatheoretic by >>>>>>>>>>>>>>>> definition.
But Truth *IS* Truth, or you are just misdefining it. >>>>>>>>>>>>>>>
The fact that a system can't tell you the truth value of >>>>>>>>>>>>>>> a statement doesn't mean the statement doesn't have a >>>>>>>>>>>>>>> truth value.
And, the problem is that, as was shown, systems with a >>>>>>>>>>>>>>> truth predicate CAN'T support PA or they are inconsistant. >>>>>>>>>>>>>>>
I guess systems that lie aren't a problem to you since >>>>>>>>>>>>>>> you think lying is valid logic.
WHich Godel proves exsits.
This conflation was rarely acknowledged, and it shaped >>>>>>>>>>>>>>>>>> the
interpretation of G||delrCOs incompleteness theorems, >>>>>>>>>>>>>>>>>> independence
results like Goodstein and ParisrCoHarrington, and the >>>>>>>>>>>>>>>>>> entire
discourse around rCLtrue but unprovablerCY statements. >>>>>>>>>>>>>>>>>
By LYING and destroying the meaninf of truth. >>>>>>>>>>>>>>>>>
My work begins by correcting this foundational error. >>>>>>>>>>>>>>>>>
PA has no internal truth predicate, so classical >>>>>>>>>>>>>>>>>> claims of
rCLtrue in arithmeticrCY were always meta-theoretic. My >>>>>>>>>>>>>>>>>> system
introduces a truth predicate whose meaning is anchored >>>>>>>>>>>>>>>>>> entirely in PArCOs axioms and inference rules, not in >>>>>>>>>>>>>>>>>> external
models. Any statement whose meaning requires meta- >>>>>>>>>>>>>>>>>> theoretic
interpretation or non-well-founded self-reference is >>>>>>>>>>>>>>>>>> rejected
as outside the domain of PA. This yields a coherent, >>>>>>>>>>>>>>>>>> internal
notion of truth in arithmetic for the first time. >>>>>>>>>>>>>>>>>>
Not having a "Predicate" doesn't mean not having a >>>>>>>>>>>>>>>>> definition of truth.
A metarCatheoretic definition of truth is not the same >>>>>>>>>>>>>>>> as an internal truth predicate. TarskirCOs definition of >>>>>>>>>>>>>>>> truth for arithmetic is external to PA and cannot be >>>>>>>>>>>>>>>> expressed inside PA. ThatrCOs exactly the distinction >>>>>>>>>>>>>>>> IrCOm drawing.
No, he shows that any system that support PA and a Truth >>>>>>>>>>>>>>> Predicate is inconstant.
It seems you just want to let your system be
inconsistent, as then you can "prove" whatever you want. >>>>>>>>>>>>>>>
Right, but statments in PA can be True even without such >>>>>>>>>>>>>>> a predicate.
PA can prove statements, but it cannot assert that >>>>>>>>>>>>>>>> those statements are true. Those are different notions. >>>>>>>>>>>>>>>
Unless PA can prove it then they never were actually >>>>>>>>>>>>>> true in PA. They were true outside of PA in meta-math. >>>>>>>>>>>>>>
Sure it is. Truth goes beyond knowledge.
You're assuming 'truth in arithmetic' means truth-in-the- >>>>>>>>>>>> standard- model. But that's a meta-theoretic constructrCoit's >>>>>>>>>>>> truth about arithmetic from outside PA, not truth in
arithmetic. PA has no internal truth predicate and no way to >>>>>>>>>>>> access the standard model from within.
No, PA (Peano Arithmetic) itself defines the numbers and the >>>>>>>>>>> arithmatic.
Why do you think otherwise?
And why does it NEED to access the model from within?
G||delrCastyle incompleteness only appears when rCLtruthrCY is >>>>>>>>>> defined using an outside model of the natural numbers.
No, it uses the innate properties of the Natural Nubmers.
meta-math is outside of math.
If you stop using modelrCatheoretic truth and rely only
on the meanings that come from the rules of the system
itself, then rCLtruerCY and rCLprovablerCY coincide rCo so the >>>>>>>>>> incompleteness gap never arises.
That doesn't make sense. The answer to the arithmatic doesn't >>>>>>>>> depend on anything outside the rules, as numbers mean themselves. >>>>>>>>>
That a number statisfies the relationship derived doesn't
depend on anything outside of that arithmatic.
meta-math is outside of math.
No, it uses just the math of PA.
The meta-system just embues some additional meaning into the
numbers.
That is where it steps outside of math
But that meaning doesn't actually affect the results in the system, >>>>> only to let us KNOW the results.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~TruthBearer(PA, x) rei (~True(PA, x) reo (~False(PA, x)) >>>>
When we look at what is actually true directly in PA
and not what is true about PA in meta-math then G||del
Incompleteness cannot arise. The nearly century long
mistake was conflating true about PA in meta-math for
what is actually true in PA.
Except that none of the those statements are well-formed for all x,
since we can't check ALL possible proofs (since there is an infinite
number of them) to determine if a given statement is True, False, or
Not a TruthBearer.
True(PA, x) rei PA reo x
does not require PA to search all proofs. It simply states:
---If PA proves x, then True(PA, x) holds.
---If PA does not prove x, then True(PA, x) does not hold.
And how can you tell if PA proves something?
You might know of the proof, but there might be one you don't know.
THus, you STILL need a state for Truth Value exists but is unknown.
You criteria only works in a system with only a finite number of
possible proofs, of which PA doesn't fit.
For instance, Which is the Goldbach conjecture?
We think it is likely true, but don't have a proof YET.
There COULD be a counter example, but we haven't found it.
It might not be provable, but we don't know that either.
Thus, your system can't even classify a simple problem, because your
criteria are not well-founded.
Goldbach is outside PA because PA neither proves
it nor refutes it. In a proofrCatheoretic framework,
a statement belongs to PArCOs inferential domain only
if it is derivable from PArCOs axioms. Since Goldbach
is undecidable in PA, it has no inferential grounding
there. Therefore, if a proof of Goldbach exists at
all, it must lie outside PArCOs deductive power.
DO you KNOW that PA can't prove it? or is it you just don't know of a
way to prove it in PA.
Do you KNOW that PA can't refute it? or is it you just haven't found a refuation.
If you can actually prove one of those statement then you will be famous.
Actually, if you can prove that PA can't refute Goldbach, then you have proven Goldbach, as a refutation is simple, it is a single even number--
that can not be the sum of two primes.
Thus, if Goldbach isn't true, it is "easily" refuted by finding one of
the numbers that doesn't work.
This is your problem, You just made an assertion without proof, because
you are just a pathological liar that doesn't understand the difference between truth and knowldege.
On 1/18/2026 5:28 PM, Richard Damon wrote:
On 1/18/26 4:49 PM, olcott wrote:
On 1/18/2026 2:55 PM, Richard Damon wrote:
On 1/18/26 1:38 PM, olcott wrote:
On 1/18/2026 11:37 AM, Richard Damon wrote:
On 1/17/26 11:38 PM, olcott wrote:
On 1/17/2026 10:13 PM, Richard Damon wrote:
On 1/17/26 10:59 PM, olcott wrote:
On 1/17/2026 9:20 PM, Richard Damon wrote:
On 1/17/26 8:59 PM, olcott wrote:
On 1/17/2026 7:46 PM, Richard Damon wrote:
On 1/17/26 8:30 PM, olcott wrote:
On 1/17/2026 7:20 PM, Richard Damon wrote:
On 1/17/26 7:49 PM, olcott wrote:
On 1/17/2026 6:14 PM, Richard Damon wrote:
On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote:
On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have >>>>>>>>>>>>>>>>>>> quietly
relied on a fundamental conflation: the idea that >>>>>>>>>>>>>>>>>>> rCLtrue in arithmeticrCY meant rCLtrue in the standard >>>>>>>>>>>>>>>>>>> model of rao.rCY
But PA itself has no truth predicate, no internal >>>>>>>>>>>>>>>>>>> semantics,
and no mechanism for assigning truth values. So what was >>>>>>>>>>>>>>>>>>> called rCLtrue in arithmeticrCY was always meta-theoretic >>>>>>>>>>>>>>>>>>> truth
about arithmetic, imported from an external model and >>>>>>>>>>>>>>>>>>> never
grounded inside PA.
Nope, just shows you don't understand what TRUTH means. >>>>>>>>>>>>>>>>>>
IrCOm distinguishing internal truth from external truth. >>>>>>>>>>>>>>>>> PA has no internal truth predicate, so it cannot express >>>>>>>>>>>>>>>>> or evaluate truth internally.
The only notion of truth available for PA is the external, >>>>>>>>>>>>>>>>> modelrCatheoretic one rCo which is metarCatheoretic by >>>>>>>>>>>>>>>>> definition.
But Truth *IS* Truth, or you are just misdefining it. >>>>>>>>>>>>>>>>
The fact that a system can't tell you the truth value of >>>>>>>>>>>>>>>> a statement doesn't mean the statement doesn't have a >>>>>>>>>>>>>>>> truth value.
And, the problem is that, as was shown, systems with a >>>>>>>>>>>>>>>> truth predicate CAN'T support PA or they are inconsistant. >>>>>>>>>>>>>>>>
I guess systems that lie aren't a problem to you since >>>>>>>>>>>>>>>> you think lying is valid logic.
WHich Godel proves exsits.
This conflation was rarely acknowledged, and it >>>>>>>>>>>>>>>>>>> shaped the
interpretation of G||delrCOs incompleteness theorems, >>>>>>>>>>>>>>>>>>> independence
results like Goodstein and ParisrCoHarrington, and the >>>>>>>>>>>>>>>>>>> entire
discourse around rCLtrue but unprovablerCY statements. >>>>>>>>>>>>>>>>>>
By LYING and destroying the meaninf of truth. >>>>>>>>>>>>>>>>>>
My work begins by correcting this foundational error. >>>>>>>>>>>>>>>>>>
PA has no internal truth predicate, so classical >>>>>>>>>>>>>>>>>>> claims of
rCLtrue in arithmeticrCY were always meta-theoretic. My >>>>>>>>>>>>>>>>>>> system
introduces a truth predicate whose meaning is anchored >>>>>>>>>>>>>>>>>>> entirely in PArCOs axioms and inference rules, not in >>>>>>>>>>>>>>>>>>> external
models. Any statement whose meaning requires meta- >>>>>>>>>>>>>>>>>>> theoretic
interpretation or non-well-founded self-reference is >>>>>>>>>>>>>>>>>>> rejected
as outside the domain of PA. This yields a coherent, >>>>>>>>>>>>>>>>>>> internal
notion of truth in arithmetic for the first time. >>>>>>>>>>>>>>>>>>>
Not having a "Predicate" doesn't mean not having a >>>>>>>>>>>>>>>>>> definition of truth.
A metarCatheoretic definition of truth is not the same >>>>>>>>>>>>>>>>> as an internal truth predicate. TarskirCOs definition of >>>>>>>>>>>>>>>>> truth for arithmetic is external to PA and cannot be >>>>>>>>>>>>>>>>> expressed inside PA. ThatrCOs exactly the distinction >>>>>>>>>>>>>>>>> IrCOm drawing.
No, he shows that any system that support PA and a Truth >>>>>>>>>>>>>>>> Predicate is inconstant.
It seems you just want to let your system be
inconsistent, as then you can "prove" whatever you want. >>>>>>>>>>>>>>>>
Right, but statments in PA can be True even without such >>>>>>>>>>>>>>>> a predicate.
PA can prove statements, but it cannot assert that >>>>>>>>>>>>>>>>> those statements are true. Those are different notions. >>>>>>>>>>>>>>>>
Unless PA can prove it then they never were actually >>>>>>>>>>>>>>> true in PA. They were true outside of PA in meta-math. >>>>>>>>>>>>>>>
Sure it is. Truth goes beyond knowledge.
You're assuming 'truth in arithmetic' means truth-in-the- >>>>>>>>>>>>> standard- model. But that's a meta-theoretic constructrCoit's >>>>>>>>>>>>> truth about arithmetic from outside PA, not truth in >>>>>>>>>>>>> arithmetic. PA has no internal truth predicate and no way >>>>>>>>>>>>> to access the standard model from within.
No, PA (Peano Arithmetic) itself defines the numbers and the >>>>>>>>>>>> arithmatic.
Why do you think otherwise?
And why does it NEED to access the model from within?
G||delrCastyle incompleteness only appears when rCLtruthrCY is >>>>>>>>>>> defined using an outside model of the natural numbers.
No, it uses the innate properties of the Natural Nubmers.
meta-math is outside of math.
If you stop using modelrCatheoretic truth and rely only
on the meanings that come from the rules of the system
itself, then rCLtruerCY and rCLprovablerCY coincide rCo so the >>>>>>>>>>> incompleteness gap never arises.
That doesn't make sense. The answer to the arithmatic doesn't >>>>>>>>>> depend on anything outside the rules, as numbers mean themselves. >>>>>>>>>>
That a number statisfies the relationship derived doesn't >>>>>>>>>> depend on anything outside of that arithmatic.
meta-math is outside of math.
No, it uses just the math of PA.
The meta-system just embues some additional meaning into the
numbers.
That is where it steps outside of math
But that meaning doesn't actually affect the results in the
system, only to let us KNOW the results.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~TruthBearer(PA, x) rei (~True(PA, x) reo (~False(PA, x)) >>>>>
When we look at what is actually true directly in PA
and not what is true about PA in meta-math then G||del
Incompleteness cannot arise. The nearly century long
mistake was conflating true about PA in meta-math for
what is actually true in PA.
Except that none of the those statements are well-formed for all x,
since we can't check ALL possible proofs (since there is an infinite
number of them) to determine if a given statement is True, False, or
Not a TruthBearer.
True(PA, x) rei PA reo x
does not require PA to search all proofs. It simply states:
---If PA proves x, then True(PA, x) holds.
---If PA does not prove x, then True(PA, x) does not hold.
And how can you tell if PA proves something?
Every expression such as "2 + 3 = 5" that can be verified
entirely on the basis of PA axioms is provable in PA.
You might know of the proof, but there might be one you don't know.
THus, you STILL need a state for Truth Value exists but is unknown.
You criteria only works in a system with only a finite number of
possible proofs, of which PA doesn't fit.
For instance, Which is the Goldbach conjecture?
We think it is likely true, but don't have a proof YET.
There COULD be a counter example, but we haven't found it.
It might not be provable, but we don't know that either.
Thus, your system can't even classify a simple problem, because your
criteria are not well-founded.
Goldbach is outside PA because PA neither proves
it nor refutes it. In a proofrCatheoretic framework,
a statement belongs to PArCOs inferential domain only
if it is derivable from PArCOs axioms. Since Goldbach
is undecidable in PA, it has no inferential grounding
there. Therefore, if a proof of Goldbach exists at
all, it must lie outside PArCOs deductive power.
DO you KNOW that PA can't prove it? or is it you just don't know of a
way to prove it in PA.
Do you KNOW that PA can't refute it? or is it you just haven't found a
refuation.
If you can actually prove one of those statement then you will be famous.
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.
Actually, if you can prove that PA can't refute Goldbach, then you
have proven Goldbach, as a refutation is simple, it is a single even
number that can not be the sum of two primes.
Thus, if Goldbach isn't true, it is "easily" refuted by finding one of
the numbers that doesn't work.
This is your problem, You just made an assertion without proof,
because you are just a pathological liar that doesn't understand the
difference between truth and knowldege.
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.
On 1/18/26 6:41 PM, olcott wrote:
On 1/18/2026 5:28 PM, Richard Damon wrote:
On 1/18/26 4:49 PM, olcott wrote:
On 1/18/2026 2:55 PM, Richard Damon wrote:
On 1/18/26 1:38 PM, olcott wrote:
On 1/18/2026 11:37 AM, Richard Damon wrote:
On 1/17/26 11:38 PM, olcott wrote:
On 1/17/2026 10:13 PM, Richard Damon wrote:
On 1/17/26 10:59 PM, olcott wrote:
On 1/17/2026 9:20 PM, Richard Damon wrote:
On 1/17/26 8:59 PM, olcott wrote:
On 1/17/2026 7:46 PM, Richard Damon wrote:
On 1/17/26 8:30 PM, olcott wrote:
On 1/17/2026 7:20 PM, Richard Damon wrote:
On 1/17/26 7:49 PM, olcott wrote:
On 1/17/2026 6:14 PM, Richard Damon wrote:
On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>> On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have >>>>>>>>>>>>>>>>>>>> quietly
relied on a fundamental conflation: the idea that >>>>>>>>>>>>>>>>>>>> rCLtrue in arithmeticrCY meant rCLtrue in the standard >>>>>>>>>>>>>>>>>>>> model of rao.rCY
But PA itself has no truth predicate, no internal >>>>>>>>>>>>>>>>>>>> semantics,
and no mechanism for assigning truth values. So what >>>>>>>>>>>>>>>>>>>> was
called rCLtrue in arithmeticrCY was always meta- >>>>>>>>>>>>>>>>>>>> theoretic truth
about arithmetic, imported from an external model >>>>>>>>>>>>>>>>>>>> and never
grounded inside PA.
Nope, just shows you don't understand what TRUTH means. >>>>>>>>>>>>>>>>>>>
IrCOm distinguishing internal truth from external truth. >>>>>>>>>>>>>>>>>> PA has no internal truth predicate, so it cannot express >>>>>>>>>>>>>>>>>> or evaluate truth internally.
The only notion of truth available for PA is the >>>>>>>>>>>>>>>>>> external,
modelrCatheoretic one rCo which is metarCatheoretic by >>>>>>>>>>>>>>>>>> definition.
But Truth *IS* Truth, or you are just misdefining it. >>>>>>>>>>>>>>>>>
The fact that a system can't tell you the truth value >>>>>>>>>>>>>>>>> of a statement doesn't mean the statement doesn't have >>>>>>>>>>>>>>>>> a truth value.
And, the problem is that, as was shown, systems with a >>>>>>>>>>>>>>>>> truth predicate CAN'T support PA or they are inconsistant. >>>>>>>>>>>>>>>>>
I guess systems that lie aren't a problem to you since >>>>>>>>>>>>>>>>> you think lying is valid logic.
WHich Godel proves exsits.
This conflation was rarely acknowledged, and it >>>>>>>>>>>>>>>>>>>> shaped the
interpretation of G||delrCOs incompleteness theorems, >>>>>>>>>>>>>>>>>>>> independence
results like Goodstein and ParisrCoHarrington, and the >>>>>>>>>>>>>>>>>>>> entire
discourse around rCLtrue but unprovablerCY statements. >>>>>>>>>>>>>>>>>>>
By LYING and destroying the meaninf of truth. >>>>>>>>>>>>>>>>>>>
My work begins by correcting this foundational error. >>>>>>>>>>>>>>>>>>>
PA has no internal truth predicate, so classical >>>>>>>>>>>>>>>>>>>> claims of
rCLtrue in arithmeticrCY were always meta-theoretic. My >>>>>>>>>>>>>>>>>>>> system
introduces a truth predicate whose meaning is anchored >>>>>>>>>>>>>>>>>>>> entirely in PArCOs axioms and inference rules, not in >>>>>>>>>>>>>>>>>>>> external
models. Any statement whose meaning requires meta- >>>>>>>>>>>>>>>>>>>> theoretic
interpretation or non-well-founded self-reference is >>>>>>>>>>>>>>>>>>>> rejected
as outside the domain of PA. This yields a coherent, >>>>>>>>>>>>>>>>>>>> internal
notion of truth in arithmetic for the first time. >>>>>>>>>>>>>>>>>>>>
Not having a "Predicate" doesn't mean not having a >>>>>>>>>>>>>>>>>>> definition of truth.
A metarCatheoretic definition of truth is not the same >>>>>>>>>>>>>>>>>> as an internal truth predicate. TarskirCOs definition of >>>>>>>>>>>>>>>>>> truth for arithmetic is external to PA and cannot be >>>>>>>>>>>>>>>>>> expressed inside PA. ThatrCOs exactly the distinction >>>>>>>>>>>>>>>>>> IrCOm drawing.
No, he shows that any system that support PA and a >>>>>>>>>>>>>>>>> Truth Predicate is inconstant.
It seems you just want to let your system be >>>>>>>>>>>>>>>>> inconsistent, as then you can "prove" whatever you want. >>>>>>>>>>>>>>>>>
Right, but statments in PA can be True even without >>>>>>>>>>>>>>>>> such a predicate.
PA can prove statements, but it cannot assert that >>>>>>>>>>>>>>>>>> those statements are true. Those are different notions. >>>>>>>>>>>>>>>>>
Unless PA can prove it then they never were actually >>>>>>>>>>>>>>>> true in PA. They were true outside of PA in meta-math. >>>>>>>>>>>>>>>>
Sure it is. Truth goes beyond knowledge.
You're assuming 'truth in arithmetic' means truth-in-the- >>>>>>>>>>>>>> standard- model. But that's a meta-theoretic constructrCo >>>>>>>>>>>>>> it's truth about arithmetic from outside PA, not truth in >>>>>>>>>>>>>> arithmetic. PA has no internal truth predicate and no way >>>>>>>>>>>>>> to access the standard model from within.
No, PA (Peano Arithmetic) itself defines the numbers and >>>>>>>>>>>>> the arithmatic.
Why do you think otherwise?
And why does it NEED to access the model from within? >>>>>>>>>>>>>
G||delrCastyle incompleteness only appears when rCLtruthrCY is >>>>>>>>>>>> defined using an outside model of the natural numbers.
No, it uses the innate properties of the Natural Nubmers. >>>>>>>>>>>
meta-math is outside of math.
If you stop using modelrCatheoretic truth and rely only >>>>>>>>>>>> on the meanings that come from the rules of the system >>>>>>>>>>>> itself, then rCLtruerCY and rCLprovablerCY coincide rCo so the >>>>>>>>>>>> incompleteness gap never arises.
That doesn't make sense. The answer to the arithmatic doesn't >>>>>>>>>>> depend on anything outside the rules, as numbers mean
themselves.
That a number statisfies the relationship derived doesn't >>>>>>>>>>> depend on anything outside of that arithmatic.
meta-math is outside of math.
No, it uses just the math of PA.
The meta-system just embues some additional meaning into the >>>>>>>>> numbers.
That is where it steps outside of math
But that meaning doesn't actually affect the results in the
system, only to let us KNOW the results.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~TruthBearer(PA, x) rei (~True(PA, x) reo (~False(PA, x)) >>>>>>
When we look at what is actually true directly in PA
and not what is true about PA in meta-math then G||del
Incompleteness cannot arise. The nearly century long
mistake was conflating true about PA in meta-math for
what is actually true in PA.
Except that none of the those statements are well-formed for all x, >>>>> since we can't check ALL possible proofs (since there is an
infinite number of them) to determine if a given statement is True, >>>>> False, or Not a TruthBearer.
True(PA, x) rei PA reo x
does not require PA to search all proofs. It simply states:
---If PA proves x, then True(PA, x) holds.
---If PA does not prove x, then True(PA, x) does not hold.
And how can you tell if PA proves something?
Every expression such as "2 + 3 = 5" that can be verified
entirely on the basis of PA axioms is provable in PA.
You might know of the proof, but there might be one you don't know.
THus, you STILL need a state for Truth Value exists but is unknown.
You criteria only works in a system with only a finite number of
possible proofs, of which PA doesn't fit.
For instance, Which is the Goldbach conjecture?
We think it is likely true, but don't have a proof YET.
There COULD be a counter example, but we haven't found it.
It might not be provable, but we don't know that either.
Thus, your system can't even classify a simple problem, because
your criteria are not well-founded.
Goldbach is outside PA because PA neither proves
it nor refutes it. In a proofrCatheoretic framework,
a statement belongs to PArCOs inferential domain only
if it is derivable from PArCOs axioms. Since Goldbach
is undecidable in PA, it has no inferential grounding
there. Therefore, if a proof of Goldbach exists at
all, it must lie outside PArCOs deductive power.
DO you KNOW that PA can't prove it? or is it you just don't know of a
way to prove it in PA.
Do you KNOW that PA can't refute it? or is it you just haven't found
a refuation.
If you can actually prove one of those statement then you will be
famous.
I already just said that the proof and refutation of
Goldbach are outside the scope of PA axioms.
But you didn't PROVE it, you just claim it based on lack of knowledge.
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.
Why do you say that?
Can you PROVE it?
Le 19/01/2026 |a 00:41, olcott a |-crit :
..
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" ? ?-a 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.
On 1/18/2026 7:24 PM, Python wrote:
Le 19/01/2026 |a 00:41, olcott a |-crit :
..
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" ? ?-a 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.
On 1/18/26 10:19 PM, olcott wrote:
On 1/18/2026 7:24 PM, Python wrote:
Le 19/01/2026 |a 00:41, olcott a |-crit :
..
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" ? ?-a 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.
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 |a 00:41, olcott a |-crit :
..
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" ? ?-a 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
G||del Incompleteness.
When we change the foundation of formal systems
to proof theoretic semantics and add my truth
predicates then G||del'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.
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.
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 |a 00:41, olcott a |-crit :
..
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" ? ?-a 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
G||del 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 G||del'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.
On 19/01/2026 11:49, Richard Damon wrote:
...
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
I'm pretty sure that's not the right word.
prove that, but you insist that truth must be provable.
Unless you're lucky enough to make a statement about them be an axiom of
the system. Then you are hoping you've defined a consistent system but perhaps you got lucky.
Is it really true, though, that truth based on provability always breaks
so? It looks like falsity based on non-provability is the problem and
then only in conjunction with some notions of negation and maybe some
notions of conjunction too (obviously the Quine might be the problem but
we know fixed points give us Quines and vice-versa and they're so
important we don't want to lose them).
What is the negation of "go to the shop" ?
What is the negation of "is so! is not! is so! is not! ..." but "is not!
is so! is not! is so! ..."
Given positive intuitionist systems (where a system has unprovable
things that are provable in extensions) our truth predicate must leave anything unprovable that could be an axiom of an extension as neither
true nor false but rather be inapplicable.
A binary Truth predicate (at
minimum) is required to even make sense and maybe it requires a further restriction argument (a 2nd order logic, then), which Tarski's
indefinability theorem doesn't cover, not by a long way.
For nearly a century, discussions of arithmetic have quietly
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY
But PA itself has no truth predicate, no internal semantics,
and no mechanism for assigning truth values. So what was
called rCLtrue in arithmeticrCY was always meta-theoretic truth
about arithmetic, imported from an external model and never
grounded inside PA.
This conflation was rarely acknowledged, and it shaped the
interpretation of G||delrCOs incompleteness theorems, independence
results like Goodstein and ParisrCoHarrington, and the entire
discourse around rCLtrue but unprovablerCY statements.
My work begins by correcting this foundational error.
PA has no internal truth predicate, so classical claims of
rCLtrue in arithmeticrCY were always meta-theoretic. My system
introduces a truth predicate whose meaning is anchored
entirely in PArCOs axioms and inference rules, not in external
models. Any statement whose meaning requires meta-theoretic
interpretation or non-well-founded self-reference is rejected
as outside the domain of PA. This yields a coherent, internal
notion of truth in arithmetic for the first time.
On 1/18/2026 6:28 PM, Richard Damon wrote:
On 1/18/26 6:41 PM, olcott wrote:
On 1/18/2026 5:28 PM, Richard Damon wrote:
On 1/18/26 4:49 PM, olcott wrote:
On 1/18/2026 2:55 PM, Richard Damon wrote:
On 1/18/26 1:38 PM, olcott wrote:
On 1/18/2026 11:37 AM, Richard Damon wrote:
On 1/17/26 11:38 PM, olcott wrote:
On 1/17/2026 10:13 PM, Richard Damon wrote:
On 1/17/26 10:59 PM, olcott wrote:
On 1/17/2026 9:20 PM, Richard Damon wrote:
On 1/17/26 8:59 PM, olcott wrote:
On 1/17/2026 7:46 PM, Richard Damon wrote:No, it uses the innate properties of the Natural Nubmers. >>>>>>>>>>>>
On 1/17/26 8:30 PM, olcott wrote:
On 1/17/2026 7:20 PM, Richard Damon wrote:
On 1/17/26 7:49 PM, olcott wrote:
On 1/17/2026 6:14 PM, Richard Damon wrote:
On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>> On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic >>>>>>>>>>>>>>>>>>>>> have quietly
relied on a fundamental conflation: the idea that >>>>>>>>>>>>>>>>>>>>> rCLtrue in arithmeticrCY meant rCLtrue in the standard >>>>>>>>>>>>>>>>>>>>> model of rao.rCY
But PA itself has no truth predicate, no internal >>>>>>>>>>>>>>>>>>>>> semantics,
and no mechanism for assigning truth values. So >>>>>>>>>>>>>>>>>>>>> what was
called rCLtrue in arithmeticrCY was always meta- >>>>>>>>>>>>>>>>>>>>> theoretic truth
about arithmetic, imported from an external model >>>>>>>>>>>>>>>>>>>>> and never
grounded inside PA.
Nope, just shows you don't understand what TRUTH means. >>>>>>>>>>>>>>>>>>>>
IrCOm distinguishing internal truth from external truth. >>>>>>>>>>>>>>>>>>> PA has no internal truth predicate, so it cannot express >>>>>>>>>>>>>>>>>>> or evaluate truth internally.
The only notion of truth available for PA is the >>>>>>>>>>>>>>>>>>> external,
modelrCatheoretic one rCo which is metarCatheoretic by >>>>>>>>>>>>>>>>>>> definition.
But Truth *IS* Truth, or you are just misdefining it. >>>>>>>>>>>>>>>>>>
The fact that a system can't tell you the truth value >>>>>>>>>>>>>>>>>> of a statement doesn't mean the statement doesn't have >>>>>>>>>>>>>>>>>> a truth value.
And, the problem is that, as was shown, systems with a >>>>>>>>>>>>>>>>>> truth predicate CAN'T support PA or they are >>>>>>>>>>>>>>>>>> inconsistant.
I guess systems that lie aren't a problem to you since >>>>>>>>>>>>>>>>>> you think lying is valid logic.
WHich Godel proves exsits.
This conflation was rarely acknowledged, and it >>>>>>>>>>>>>>>>>>>>> shaped the
interpretation of G||delrCOs incompleteness theorems, >>>>>>>>>>>>>>>>>>>>> independence
results like Goodstein and ParisrCoHarrington, and >>>>>>>>>>>>>>>>>>>>> the entire
discourse around rCLtrue but unprovablerCY statements. >>>>>>>>>>>>>>>>>>>>
By LYING and destroying the meaninf of truth. >>>>>>>>>>>>>>>>>>>>
My work begins by correcting this foundational error. >>>>>>>>>>>>>>>>>>>>
PA has no internal truth predicate, so classical >>>>>>>>>>>>>>>>>>>>> claims of
rCLtrue in arithmeticrCY were always meta-theoretic. My >>>>>>>>>>>>>>>>>>>>> system
introduces a truth predicate whose meaning is anchored >>>>>>>>>>>>>>>>>>>>> entirely in PArCOs axioms and inference rules, not in >>>>>>>>>>>>>>>>>>>>> external
models. Any statement whose meaning requires meta- >>>>>>>>>>>>>>>>>>>>> theoretic
interpretation or non-well-founded self-reference >>>>>>>>>>>>>>>>>>>>> is rejected
as outside the domain of PA. This yields a >>>>>>>>>>>>>>>>>>>>> coherent, internal
notion of truth in arithmetic for the first time. >>>>>>>>>>>>>>>>>>>>>
Not having a "Predicate" doesn't mean not having a >>>>>>>>>>>>>>>>>>>> definition of truth.
A metarCatheoretic definition of truth is not the same >>>>>>>>>>>>>>>>>>> as an internal truth predicate. TarskirCOs definition of >>>>>>>>>>>>>>>>>>> truth for arithmetic is external to PA and cannot be >>>>>>>>>>>>>>>>>>> expressed inside PA. ThatrCOs exactly the distinction >>>>>>>>>>>>>>>>>>> IrCOm drawing.
No, he shows that any system that support PA and a >>>>>>>>>>>>>>>>>> Truth Predicate is inconstant.
It seems you just want to let your system be >>>>>>>>>>>>>>>>>> inconsistent, as then you can "prove" whatever you want. >>>>>>>>>>>>>>>>>>
Right, but statments in PA can be True even without >>>>>>>>>>>>>>>>>> such a predicate.
PA can prove statements, but it cannot assert that >>>>>>>>>>>>>>>>>>> those statements are true. Those are different notions. >>>>>>>>>>>>>>>>>>
Unless PA can prove it then they never were actually >>>>>>>>>>>>>>>>> true in PA. They were true outside of PA in meta-math. >>>>>>>>>>>>>>>>>
Sure it is. Truth goes beyond knowledge.
You're assuming 'truth in arithmetic' means truth-in-the- >>>>>>>>>>>>>>> standard- model. But that's a meta-theoretic constructrCo >>>>>>>>>>>>>>> it's truth about arithmetic from outside PA, not truth in >>>>>>>>>>>>>>> arithmetic. PA has no internal truth predicate and no way >>>>>>>>>>>>>>> to access the standard model from within.
No, PA (Peano Arithmetic) itself defines the numbers and >>>>>>>>>>>>>> the arithmatic.
Why do you think otherwise?
And why does it NEED to access the model from within? >>>>>>>>>>>>>>
G||delrCastyle incompleteness only appears when rCLtruthrCY is >>>>>>>>>>>>> defined using an outside model of the natural numbers. >>>>>>>>>>>>
meta-math is outside of math.
If you stop using modelrCatheoretic truth and rely only >>>>>>>>>>>>> on the meanings that come from the rules of the system >>>>>>>>>>>>> itself, then rCLtruerCY and rCLprovablerCY coincide rCo so the >>>>>>>>>>>>> incompleteness gap never arises.
That doesn't make sense. The answer to the arithmatic >>>>>>>>>>>> doesn't depend on anything outside the rules, as numbers >>>>>>>>>>>> mean themselves.
That a number statisfies the relationship derived doesn't >>>>>>>>>>>> depend on anything outside of that arithmatic.
meta-math is outside of math.
No, it uses just the math of PA.
The meta-system just embues some additional meaning into the >>>>>>>>>> numbers.
That is where it steps outside of math
But that meaning doesn't actually affect the results in the
system, only to let us KNOW the results.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~TruthBearer(PA, x) rei (~True(PA, x) reo (~False(PA, x)) >>>>>>>
When we look at what is actually true directly in PA
and not what is true about PA in meta-math then G||del
Incompleteness cannot arise. The nearly century long
mistake was conflating true about PA in meta-math for
what is actually true in PA.
Except that none of the those statements are well-formed for all
x, since we can't check ALL possible proofs (since there is an
infinite number of them) to determine if a given statement is
True, False, or Not a TruthBearer.
True(PA, x) rei PA reo x
does not require PA to search all proofs. It simply states:
---If PA proves x, then True(PA, x) holds.
---If PA does not prove x, then True(PA, x) does not hold.
And how can you tell if PA proves something?
Every expression such as "2 + 3 = 5" that can be verified
entirely on the basis of PA axioms is provable in PA.
You might know of the proof, but there might be one you don't know.
THus, you STILL need a state for Truth Value exists but is unknown.
You criteria only works in a system with only a finite number of
possible proofs, of which PA doesn't fit.
For instance, Which is the Goldbach conjecture?
We think it is likely true, but don't have a proof YET.
There COULD be a counter example, but we haven't found it.
It might not be provable, but we don't know that either.
Thus, your system can't even classify a simple problem, because
your criteria are not well-founded.
Goldbach is outside PA because PA neither proves
it nor refutes it. In a proofrCatheoretic framework,
a statement belongs to PArCOs inferential domain only
if it is derivable from PArCOs axioms. Since Goldbach
is undecidable in PA, it has no inferential grounding
there. Therefore, if a proof of Goldbach exists at
all, it must lie outside PArCOs deductive power.
DO you KNOW that PA can't prove it? or is it you just don't know of
a way to prove it in PA.
Do you KNOW that PA can't refute it? or is it you just haven't found
a refuation.
If you can actually prove one of those statement then you will be
famous.
I already just said that the proof and refutation of
Goldbach are outside the scope of PA axioms.
But you didn't PROVE it, you just claim it based on lack of knowledge.
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.
Why do you say that?
Can you PROVE it?
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.
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 |a 00:41, olcott a |-crit :
..
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" ? ?-a 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
G||del Incompleteness.
When we change the foundation of formal systems
to proof theoretic semantics and add my truth
predicates then G||del'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.
On 1/19/2026 5:49 AM, Richard Damon wrote:
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 |a 00:41, olcott a |-crit :
..
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" ? ?-a 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
G||del Incompleteness.
Sure it does. As your system is just not well founded by its own
definitios,
My system is not supposed to decide in advance whether
Goldbach is wellrCafounded. A formula becomes a truthrCabearer
only when PA can classify it in finitely many steps.
Goldbach may or may not be classifiable; thatrCOs an open
computational fact, not a semantic requirement. This has
no effect on G||del, because G||delrCOs sentence is structurally nonrCatruthrCabearing, not merely unclassified.
When we change the foundation of formal systems
to proof theoretic semantics and add my truth
predicates then G||del'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.
I am keeping the systems.
IrCOm changing the semantics.
PArCOs syntax, axioms, and rules stay exactly
the same. What changes is that truth is internal
- finite derivability - and external modelrCatheoretic
satisfaction is no longer imported into PA. G||delrCOs
claim depends on that external semantics, so once itrCOs
removed, his universal claim simply doesnrCOt apply.\
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.
G||delrCOs G only rCLshowsrCY anything if you assume
classical semantic truth in an external model.
My system does not use that semantics. Truth in
PA is finite derivability; anything PA cannot
classify is not a truthrCabearer.
G||delrCOs G is therefore not a truthrCabearer, not
a counterexample, and not evidence of illrCafoundedness.
YourCOre evaluating my system using assumptions it
does not adopt.
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.
G is a truth bearer outside of PA in meta-math in the
same way that the Liar Paradox becomes true when it
refers to a different instance of itself.
This sentence is not true: "This sentence is not true"
is true because the inner sentence is not a truth bearer.
On 1/17/2026 3:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY
But PA itself has no truth predicate, no internal semantics,
and no mechanism for assigning truth values. So what was
called rCLtrue in arithmeticrCY was always meta-theoretic truth
about arithmetic, imported from an external model and never
grounded inside PA.
This conflation was rarely acknowledged, and it shaped the
interpretation of G||delrCOs incompleteness theorems, independence
results like Goodstein and ParisrCoHarrington, and the entire
discourse around rCLtrue but unprovablerCY statements.
My work begins by correcting this foundational error.
PA has no internal truth predicate, so classical claims of
rCLtrue in arithmeticrCY were always meta-theoretic. My system
introduces a truth predicate whose meaning is anchored
entirely in PArCOs axioms and inference rules, not in external
models. Any statement whose meaning requires meta-theoretic
interpretation or non-well-founded self-reference is rejected
as outside the domain of PA. This yields a coherent, internal
notion of truth in arithmetic for the first time.
The only reason that anyone ever suggested an external measure of truth
as a proxy for actual truth <in> PA is because PA did not have its own
truth predicate. I fixed that anchored in PA's own axioms. Now we can
see that an external measure of true <in> PA was never actually true
<in> PA at all. It was true about PA one level of indirect reference
away from true in PA. It was incorrectly conflated with true in PA
because no one saw any other alternatives.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~TruthBearer(PA, x) rei (~True(PA, x) reo (~False(PA, x))
On 1/18/26 10:17 PM, olcott wrote:
On 1/18/2026 6:28 PM, Richard Damon wrote:
On 1/18/26 6:41 PM, olcott wrote:
On 1/18/2026 5:28 PM, Richard Damon wrote:
On 1/18/26 4:49 PM, olcott wrote:
On 1/18/2026 2:55 PM, Richard Damon wrote:
On 1/18/26 1:38 PM, olcott wrote:
On 1/18/2026 11:37 AM, Richard Damon wrote:
On 1/17/26 11:38 PM, olcott wrote:
On 1/17/2026 10:13 PM, Richard Damon wrote:
On 1/17/26 10:59 PM, olcott wrote:
On 1/17/2026 9:20 PM, Richard Damon wrote:
On 1/17/26 8:59 PM, olcott wrote:
On 1/17/2026 7:46 PM, Richard Damon wrote:No, it uses the innate properties of the Natural Nubmers. >>>>>>>>>>>>>
On 1/17/26 8:30 PM, olcott wrote:
On 1/17/2026 7:20 PM, Richard Damon wrote:
On 1/17/26 7:49 PM, olcott wrote:
On 1/17/2026 6:14 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>> On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>> On 1/17/26 4:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic >>>>>>>>>>>>>>>>>>>>>> have quietly
relied on a fundamental conflation: the idea that >>>>>>>>>>>>>>>>>>>>>> rCLtrue in arithmeticrCY meant rCLtrue in the standard >>>>>>>>>>>>>>>>>>>>>> model of rao.rCY
But PA itself has no truth predicate, no internal >>>>>>>>>>>>>>>>>>>>>> semantics,
and no mechanism for assigning truth values. So >>>>>>>>>>>>>>>>>>>>>> what was
called rCLtrue in arithmeticrCY was always meta- >>>>>>>>>>>>>>>>>>>>>> theoretic truth
about arithmetic, imported from an external model >>>>>>>>>>>>>>>>>>>>>> and never
grounded inside PA.
Nope, just shows you don't understand what TRUTH >>>>>>>>>>>>>>>>>>>>> means.
IrCOm distinguishing internal truth from external truth. >>>>>>>>>>>>>>>>>>>> PA has no internal truth predicate, so it cannot >>>>>>>>>>>>>>>>>>>> express
or evaluate truth internally.
The only notion of truth available for PA is the >>>>>>>>>>>>>>>>>>>> external,
modelrCatheoretic one rCo which is metarCatheoretic by >>>>>>>>>>>>>>>>>>>> definition.
But Truth *IS* Truth, or you are just misdefining it. >>>>>>>>>>>>>>>>>>>
The fact that a system can't tell you the truth value >>>>>>>>>>>>>>>>>>> of a statement doesn't mean the statement doesn't >>>>>>>>>>>>>>>>>>> have a truth value.
And, the problem is that, as was shown, systems with >>>>>>>>>>>>>>>>>>> a truth predicate CAN'T support PA or they are >>>>>>>>>>>>>>>>>>> inconsistant.
I guess systems that lie aren't a problem to you >>>>>>>>>>>>>>>>>>> since you think lying is valid logic.
WHich Godel proves exsits.
This conflation was rarely acknowledged, and it >>>>>>>>>>>>>>>>>>>>>> shaped the
interpretation of G||delrCOs incompleteness theorems, >>>>>>>>>>>>>>>>>>>>>> independence
results like Goodstein and ParisrCoHarrington, and >>>>>>>>>>>>>>>>>>>>>> the entire
discourse around rCLtrue but unprovablerCY statements. >>>>>>>>>>>>>>>>>>>>>
By LYING and destroying the meaninf of truth. >>>>>>>>>>>>>>>>>>>>>
My work begins by correcting this foundational error. >>>>>>>>>>>>>>>>>>>>>
PA has no internal truth predicate, so classical >>>>>>>>>>>>>>>>>>>>>> claims of
rCLtrue in arithmeticrCY were always meta-theoretic. >>>>>>>>>>>>>>>>>>>>>> My system
introduces a truth predicate whose meaning is >>>>>>>>>>>>>>>>>>>>>> anchored
entirely in PArCOs axioms and inference rules, not >>>>>>>>>>>>>>>>>>>>>> in external
models. Any statement whose meaning requires meta- >>>>>>>>>>>>>>>>>>>>>> theoretic
interpretation or non-well-founded self-reference >>>>>>>>>>>>>>>>>>>>>> is rejected
as outside the domain of PA. This yields a >>>>>>>>>>>>>>>>>>>>>> coherent, internal
notion of truth in arithmetic for the first time. >>>>>>>>>>>>>>>>>>>>>>
Not having a "Predicate" doesn't mean not having a >>>>>>>>>>>>>>>>>>>>> definition of truth.
A metarCatheoretic definition of truth is not the same >>>>>>>>>>>>>>>>>>>> as an internal truth predicate. TarskirCOs definition of >>>>>>>>>>>>>>>>>>>> truth for arithmetic is external to PA and cannot be >>>>>>>>>>>>>>>>>>>> expressed inside PA. ThatrCOs exactly the distinction >>>>>>>>>>>>>>>>>>>> IrCOm drawing.
No, he shows that any system that support PA and a >>>>>>>>>>>>>>>>>>> Truth Predicate is inconstant.
It seems you just want to let your system be >>>>>>>>>>>>>>>>>>> inconsistent, as then you can "prove" whatever you want. >>>>>>>>>>>>>>>>>>>
Right, but statments in PA can be True even without >>>>>>>>>>>>>>>>>>> such a predicate.
PA can prove statements, but it cannot assert that >>>>>>>>>>>>>>>>>>>> those statements are true. Those are different notions. >>>>>>>>>>>>>>>>>>>
Unless PA can prove it then they never were actually >>>>>>>>>>>>>>>>>> true in PA. They were true outside of PA in meta-math. >>>>>>>>>>>>>>>>>>
Sure it is. Truth goes beyond knowledge.
You're assuming 'truth in arithmetic' means truth-in- >>>>>>>>>>>>>>>> the- standard- model. But that's a meta-theoretic >>>>>>>>>>>>>>>> constructrCo it's truth about arithmetic from outside PA, >>>>>>>>>>>>>>>> not truth in arithmetic. PA has no internal truth >>>>>>>>>>>>>>>> predicate and no way to access the standard model from >>>>>>>>>>>>>>>> within.
No, PA (Peano Arithmetic) itself defines the numbers and >>>>>>>>>>>>>>> the arithmatic.
Why do you think otherwise?
And why does it NEED to access the model from within? >>>>>>>>>>>>>>>
G||delrCastyle incompleteness only appears when rCLtruthrCY is >>>>>>>>>>>>>> defined using an outside model of the natural numbers. >>>>>>>>>>>>>
meta-math is outside of math.
If you stop using modelrCatheoretic truth and rely only >>>>>>>>>>>>>> on the meanings that come from the rules of the system >>>>>>>>>>>>>> itself, then rCLtruerCY and rCLprovablerCY coincide rCo so the >>>>>>>>>>>>>> incompleteness gap never arises.
That doesn't make sense. The answer to the arithmatic >>>>>>>>>>>>> doesn't depend on anything outside the rules, as numbers >>>>>>>>>>>>> mean themselves.
That a number statisfies the relationship derived doesn't >>>>>>>>>>>>> depend on anything outside of that arithmatic.
meta-math is outside of math.
No, it uses just the math of PA.
The meta-system just embues some additional meaning into the >>>>>>>>>>> numbers.
That is where it steps outside of math
But that meaning doesn't actually affect the results in the >>>>>>>>> system, only to let us KNOW the results.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~TruthBearer(PA, x) rei (~True(PA, x) reo (~False(PA, x)) >>>>>>>>
When we look at what is actually true directly in PA
and not what is true about PA in meta-math then G||del
Incompleteness cannot arise. The nearly century long
mistake was conflating true about PA in meta-math for
what is actually true in PA.
Except that none of the those statements are well-formed for all >>>>>>> x, since we can't check ALL possible proofs (since there is an
infinite number of them) to determine if a given statement is
True, False, or Not a TruthBearer.
True(PA, x) rei PA reo x
does not require PA to search all proofs. It simply states:
---If PA proves x, then True(PA, x) holds.
---If PA does not prove x, then True(PA, x) does not hold.
And how can you tell if PA proves something?
Every expression such as "2 + 3 = 5" that can be verified
entirely on the basis of PA axioms is provable in PA.
You might know of the proof, but there might be one you don't know.
THus, you STILL need a state for Truth Value exists but is unknown.
You criteria only works in a system with only a finite number of >>>>>>> possible proofs, of which PA doesn't fit.
For instance, Which is the Goldbach conjecture?
We think it is likely true, but don't have a proof YET.
There COULD be a counter example, but we haven't found it.
It might not be provable, but we don't know that either.
Thus, your system can't even classify a simple problem, because >>>>>>> your criteria are not well-founded.
Goldbach is outside PA because PA neither proves
it nor refutes it. In a proofrCatheoretic framework,
a statement belongs to PArCOs inferential domain only
if it is derivable from PArCOs axioms. Since Goldbach
is undecidable in PA, it has no inferential grounding
there. Therefore, if a proof of Goldbach exists at
all, it must lie outside PArCOs deductive power.
DO you KNOW that PA can't prove it? or is it you just don't know of >>>>> a way to prove it in PA.
Do you KNOW that PA can't refute it? or is it you just haven't
found a refuation.
If you can actually prove one of those statement then you will be
famous.
I already just said that the proof and refutation of
Goldbach are outside the scope of PA axioms.
But you didn't PROVE it, you just claim it based on lack of knowledge.
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.
Why do you say that?
Can you PROVE it?
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.
And how do you determine if its truth value cannot be determined in a
finite number of steps?
Your proof-theoretic definitions still require truth-conditional logic
to be used.
The bigger problem is that we have statements that can not be shown by proof-theoretic means to be one of True, False, or Not-Well-Founded, and
in fact forcing that makes a contradiction.
For instance, look at Godel's G, which states that there is no natural number g that satisfies a given computable relationship, which is a pure mathematical operation, so thus totally determined.
This statement can NOT be proven to be not-well-founded, as to do so
means we can prove that its converse isn't true. which means we can
prove that no number g can exist that meets the requirement, (as if it could, we couldn't prove that the statement can't be false), and thus we
now HAVE a proof that it is true, as that condition is EXACTLY what the statement claims.
The problem comes because some problem are inhenently following the laws
of the excluded middle, and thus MUST be True or False. But while they
must be true or false, there doesn't need to be a finite proof that
makes that true.
Godel's statement is an example of this, as mathementics, because of it correlation to programming, is able to create "computations" that embue meaning into the numbers, a meaning that can't be seen in the base
number system, but is "understood" by the program/relationship that was created with it. This means that while PA doesn't understand the
meaning, the determinism of mathematics brings the results of that
meaning into the system.
The relationship turns out to be a proof checker, in particular, a proof checker for the statement of G. a number represents a "statement" (or
seires of statements) in PA, and a number that satisfies it will
represent a valid proof of G.
Thus, if a number existed, then the statements it represented exist in
PA, and those statements become a proof that no such number could exist.
Since this is logically impossible it can not be, thus the statement
must be true.
But, if a proof existed of this fact, then we could compute in the meta system the number that proof represents, and that number would by the construciton of the relationship statisfy the relationship, makeing the statement false.
So, unless you think that it is possible for there to be a proof that a false statement is true, the statement MUST be true but unprovable.
That, or you get crasyness like mathematics is inconsistant, that some
basic mathematical operation of two natural numbers can give different results at different times. As in while we THINK that 1 + 2 = 3, it
might be that sometimes 1 + 2 = 4.
That, or you think that it is impossible to create a program that given
a proof in a specified system, checks that the proof is valid with 100% certainty in that system.
Sorry, your problem is that your concept just can't work in PA and
similar systems.
This is one way to interprete Godel's Incompleteness proof.
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 |a 00:41, olcott a |-crit :
..
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" ? ?-a 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.
But,
This has no effect on my claim that I got rid of
G||del Incompleteness.
Sure it does, because it shows your system is not well founded.
When we change the foundation of formal systems
to proof theoretic semantics and add my truth
predicates then G||del's claim of applying to
every formal system that can do a little bit of
arithmetic becomes simply false.
But proof-theoretic semantics are not-well-founded when applied to
systems like PA, as they need to use truth-conditional logic to
determine their proof-theoretic fvalues.
Every attempt at showing incompleteness <in> PA
has never actually been <in> PA.
Sure they were in PA. PA as a system defines the basics of mathematics.
It DEFINES a version of the Natural Numbers with a set of properties.
These properties can not all be resloved with the finite proofs that
proof theoretic semantics allows.
In particular, you often can't determine that no proof exists (except by finding the proof of the negation of the statement) as there are an
infinte number of possible proofs to rule out.
This means that actually PROVING that a statement is not-well-founded
can't be done in a proof-theoretic manner.
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.
No, proof-theoretic semantics are just not well founded in PA.
As you can't determine a proof-theoretic truth value for some statements.
it isn't that the value is unknown, as that just means that further
search can find the answer, but that literally there is NO valid proof- theoretic truth value by your definition.
There is no finite proof that it is true.
There is no finite proof that it is false.
There is no finite proof of the above two statements.
Thus, there is no proof-theoretic "truth value" for the statement, not
even not-well-founded, so the definition creates a system that is not
well founded.
My system is not supposed to decide in advance whether
Goldbach is wellrCafounded. A formula becomes a truthrCabearer
only when PA can classify it in finitely many steps.
Goldbach may or may not be classifiable; thatrCOs an open
computational fact, not a semantic requirement. This has
no effect on G||del, because G||delrCOs sentence is structurally
nonrCatruthrCabearing, 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.
On 1/19/26 9:39 PM, olcott wrote:
On 1/17/2026 3:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY
But PA itself has no truth predicate, no internal semantics,
and no mechanism for assigning truth values. So what was
called rCLtrue in arithmeticrCY was always meta-theoretic truth
about arithmetic, imported from an external model and never
grounded inside PA.
This conflation was rarely acknowledged, and it shaped the
interpretation of G||delrCOs incompleteness theorems, independence
results like Goodstein and ParisrCoHarrington, and the entire
discourse around rCLtrue but unprovablerCY statements.
My work begins by correcting this foundational error.
PA has no internal truth predicate, so classical claims of
rCLtrue in arithmeticrCY were always meta-theoretic. My system
introduces a truth predicate whose meaning is anchored
entirely in PArCOs axioms and inference rules, not in external
models. Any statement whose meaning requires meta-theoretic
interpretation or non-well-founded self-reference is rejected
as outside the domain of PA. This yields a coherent, internal
notion of truth in arithmetic for the first time.
The only reason that anyone ever suggested an external measure of
truth as a proxy for actual truth <in> PA is because PA did not have
its own truth predicate. I fixed that anchored in PA's own axioms. Now
we can see that an external measure of true <in> PA was never actually
true <in> PA at all. It was true about PA one level of indirect
reference away from true in PA. It was incorrectly conflated with true
in PA because no one saw any other alternatives.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~TruthBearer(PA, x) rei (~True(PA, x) reo (~False(PA, x))
PA doesn't have a truth predicate, because it CAN'T.
Tarski prove this, any actual truth predicate added to PA makes it inconsistant.
The problem is your proof-theoretic statements need truth-conditional interpreation to be evaluated, as it is typically impossible to prove
that no proof exists (except by proving the negation of the statement)
This means that your logic is just based on not-well-founded ideas.
I already just said that the proof and refutation of
Goldbach are outside the scope of PA axioms.
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
or a type theory to give True(S, Goldbach) no content when Goldbach is
out of scope, or keep it explicit with an InScope(S, P) family of propositions. Of course, the type theory approach is often easier to use
with pencil and paper.
Is there a conventional alternative to implication for an explicit alternative of type theory?
Unsatisfying: WhenInScope(S, P) -> (True(S, P) & Foo(P))
More satisfying: WhenInScope(S,P,Q in (True(S,Q) & Foo(Q)))
Hey, I see that in prolog often. Q is an indeterminate (unbound variable
in prolog) bound by WhenInScope(S,P,Q in ...) within "...".
or a lambda expression alternative:
WhenInScope(S,P,++Q.True(S,Q) & Foo(Q))
I prefer that over an implicit, semi-ad-hoc type theory.
Are there conventional names for these ideas and an author and excellent exposition textbook?
On 1/19/2026 11:29 PM, Richard Damon wrote:
On 1/18/26 10:17 PM, olcott wrote:
On 1/18/2026 6:28 PM, Richard Damon wrote:
On 1/18/26 6:41 PM, olcott wrote:
On 1/18/2026 5:28 PM, Richard Damon wrote:
On 1/18/26 4:49 PM, olcott wrote:
On 1/18/2026 2:55 PM, Richard Damon wrote:
On 1/18/26 1:38 PM, olcott wrote:
On 1/18/2026 11:37 AM, Richard Damon wrote:
On 1/17/26 11:38 PM, olcott wrote:
On 1/17/2026 10:13 PM, Richard Damon wrote:
On 1/17/26 10:59 PM, olcott wrote:
On 1/17/2026 9:20 PM, Richard Damon wrote:
On 1/17/26 8:59 PM, olcott wrote:
On 1/17/2026 7:46 PM, Richard Damon wrote:No, it uses the innate properties of the Natural Nubmers. >>>>>>>>>>>>>>
On 1/17/26 8:30 PM, olcott wrote:
On 1/17/2026 7:20 PM, Richard Damon wrote:
On 1/17/26 7:49 PM, olcott wrote:
On 1/17/2026 6:14 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>> On 1/17/26 5:50 PM, olcott wrote:
On 1/17/2026 3:54 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>> On 1/17/26 4:08 PM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>> For nearly a century, discussions of arithmetic >>>>>>>>>>>>>>>>>>>>>>> have quietly
relied on a fundamental conflation: the idea that >>>>>>>>>>>>>>>>>>>>>>> rCLtrue in arithmeticrCY meant rCLtrue in the standard >>>>>>>>>>>>>>>>>>>>>>> model of rao.rCY
But PA itself has no truth predicate, no internal >>>>>>>>>>>>>>>>>>>>>>> semantics,
and no mechanism for assigning truth values. So >>>>>>>>>>>>>>>>>>>>>>> what was
called rCLtrue in arithmeticrCY was always meta- >>>>>>>>>>>>>>>>>>>>>>> theoretic truth
about arithmetic, imported from an external model >>>>>>>>>>>>>>>>>>>>>>> and never
grounded inside PA.
Nope, just shows you don't understand what TRUTH >>>>>>>>>>>>>>>>>>>>>> means.
IrCOm distinguishing internal truth from external truth. >>>>>>>>>>>>>>>>>>>>> PA has no internal truth predicate, so it cannot >>>>>>>>>>>>>>>>>>>>> express
or evaluate truth internally.
The only notion of truth available for PA is the >>>>>>>>>>>>>>>>>>>>> external,
modelrCatheoretic one rCo which is metarCatheoretic by >>>>>>>>>>>>>>>>>>>>> definition.
But Truth *IS* Truth, or you are just misdefining it. >>>>>>>>>>>>>>>>>>>>
The fact that a system can't tell you the truth >>>>>>>>>>>>>>>>>>>> value of a statement doesn't mean the statement >>>>>>>>>>>>>>>>>>>> doesn't have a truth value.
And, the problem is that, as was shown, systems with >>>>>>>>>>>>>>>>>>>> a truth predicate CAN'T support PA or they are >>>>>>>>>>>>>>>>>>>> inconsistant.
I guess systems that lie aren't a problem to you >>>>>>>>>>>>>>>>>>>> since you think lying is valid logic.
WHich Godel proves exsits.
This conflation was rarely acknowledged, and it >>>>>>>>>>>>>>>>>>>>>>> shaped the
interpretation of G||delrCOs incompleteness >>>>>>>>>>>>>>>>>>>>>>> theorems, independence
results like Goodstein and ParisrCoHarrington, and >>>>>>>>>>>>>>>>>>>>>>> the entire
discourse around rCLtrue but unprovablerCY statements. >>>>>>>>>>>>>>>>>>>>>>
My work begins by correcting this foundational >>>>>>>>>>>>>>>>>>>>>>> error.
By LYING and destroying the meaninf of truth. >>>>>>>>>>>>>>>>>>>>>>
PA has no internal truth predicate, so classical >>>>>>>>>>>>>>>>>>>>>>> claims of
rCLtrue in arithmeticrCY were always meta-theoretic. >>>>>>>>>>>>>>>>>>>>>>> My system
introduces a truth predicate whose meaning is >>>>>>>>>>>>>>>>>>>>>>> anchored
entirely in PArCOs axioms and inference rules, not >>>>>>>>>>>>>>>>>>>>>>> in external
models. Any statement whose meaning requires >>>>>>>>>>>>>>>>>>>>>>> meta- theoretic
interpretation or non-well-founded self-reference >>>>>>>>>>>>>>>>>>>>>>> is rejected
as outside the domain of PA. This yields a >>>>>>>>>>>>>>>>>>>>>>> coherent, internal
notion of truth in arithmetic for the first time. >>>>>>>>>>>>>>>>>>>>>>>
Not having a "Predicate" doesn't mean not having a >>>>>>>>>>>>>>>>>>>>>> definition of truth.
A metarCatheoretic definition of truth is not the same >>>>>>>>>>>>>>>>>>>>> as an internal truth predicate. TarskirCOs definition of >>>>>>>>>>>>>>>>>>>>> truth for arithmetic is external to PA and cannot be >>>>>>>>>>>>>>>>>>>>> expressed inside PA. ThatrCOs exactly the distinction >>>>>>>>>>>>>>>>>>>>> IrCOm drawing.
No, he shows that any system that support PA and a >>>>>>>>>>>>>>>>>>>> Truth Predicate is inconstant.
It seems you just want to let your system be >>>>>>>>>>>>>>>>>>>> inconsistent, as then you can "prove" whatever you >>>>>>>>>>>>>>>>>>>> want.
PA can prove statements, but it cannot assert that >>>>>>>>>>>>>>>>>>>>> those statements are true. Those are different >>>>>>>>>>>>>>>>>>>>> notions.
Right, but statments in PA can be True even without >>>>>>>>>>>>>>>>>>>> such a predicate.
Unless PA can prove it then they never were actually >>>>>>>>>>>>>>>>>>> true in PA. They were true outside of PA in meta-math. >>>>>>>>>>>>>>>>>>>
Sure it is. Truth goes beyond knowledge.
You're assuming 'truth in arithmetic' means truth-in- >>>>>>>>>>>>>>>>> the- standard- model. But that's a meta-theoretic >>>>>>>>>>>>>>>>> constructrCo it's truth about arithmetic from outside PA, >>>>>>>>>>>>>>>>> not truth in arithmetic. PA has no internal truth >>>>>>>>>>>>>>>>> predicate and no way to access the standard model from >>>>>>>>>>>>>>>>> within.
No, PA (Peano Arithmetic) itself defines the numbers and >>>>>>>>>>>>>>>> the arithmatic.
Why do you think otherwise?
And why does it NEED to access the model from within? >>>>>>>>>>>>>>>>
G||delrCastyle incompleteness only appears when rCLtruthrCY is >>>>>>>>>>>>>>> defined using an outside model of the natural numbers. >>>>>>>>>>>>>>
meta-math is outside of math.
If you stop using modelrCatheoretic truth and rely only >>>>>>>>>>>>>>> on the meanings that come from the rules of the system >>>>>>>>>>>>>>> itself, then rCLtruerCY and rCLprovablerCY coincide rCo so the >>>>>>>>>>>>>>> incompleteness gap never arises.
That doesn't make sense. The answer to the arithmatic >>>>>>>>>>>>>> doesn't depend on anything outside the rules, as numbers >>>>>>>>>>>>>> mean themselves.
That a number statisfies the relationship derived doesn't >>>>>>>>>>>>>> depend on anything outside of that arithmatic.
meta-math is outside of math.
No, it uses just the math of PA.
The meta-system just embues some additional meaning into the >>>>>>>>>>>> numbers.
That is where it steps outside of math
But that meaning doesn't actually affect the results in the >>>>>>>>>> system, only to let us KNOW the results.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~TruthBearer(PA, x) rei (~True(PA, x) reo (~False(PA, x))
When we look at what is actually true directly in PA
and not what is true about PA in meta-math then G||del
Incompleteness cannot arise. The nearly century long
mistake was conflating true about PA in meta-math for
what is actually true in PA.
Except that none of the those statements are well-formed for all >>>>>>>> x, since we can't check ALL possible proofs (since there is an >>>>>>>> infinite number of them) to determine if a given statement is >>>>>>>> True, False, or Not a TruthBearer.
True(PA, x) rei PA reo x
does not require PA to search all proofs. It simply states:
---If PA proves x, then True(PA, x) holds.
---If PA does not prove x, then True(PA, x) does not hold.
And how can you tell if PA proves something?
Every expression such as "2 + 3 = 5" that can be verified
entirely on the basis of PA axioms is provable in PA.
You might know of the proof, but there might be one you don't know. >>>>>>
THus, you STILL need a state for Truth Value exists but is unknown. >>>>>>
You criteria only works in a system with only a finite number of >>>>>>>> possible proofs, of which PA doesn't fit.
For instance, Which is the Goldbach conjecture?
We think it is likely true, but don't have a proof YET.
There COULD be a counter example, but we haven't found it.
It might not be provable, but we don't know that either.
Thus, your system can't even classify a simple problem, because >>>>>>>> your criteria are not well-founded.
Goldbach is outside PA because PA neither proves
it nor refutes it. In a proofrCatheoretic framework,
a statement belongs to PArCOs inferential domain only
if it is derivable from PArCOs axioms. Since Goldbach
is undecidable in PA, it has no inferential grounding
there. Therefore, if a proof of Goldbach exists at
all, it must lie outside PArCOs deductive power.
DO you KNOW that PA can't prove it? or is it you just don't know
of a way to prove it in PA.
Do you KNOW that PA can't refute it? or is it you just haven't
found a refuation.
If you can actually prove one of those statement then you will be >>>>>> famous.
I already just said that the proof and refutation of
Goldbach are outside the scope of PA axioms.
But you didn't PROVE it, you just claim it based on lack of knowledge. >>>>
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.
Why do you say that?
Can you PROVE it?
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.
And how do you determine if its truth value cannot be determined in a
finite number of steps?
Your proof-theoretic definitions still require truth-conditional logic
to be used.
The bigger problem is that we have statements that can not be shown by
proof-theoretic means to be one of True, False, or Not-Well-Founded,
and in fact forcing that makes a contradiction.
PA is only a little tiny example of how my greater
system that makes the body of knowledge that is
"true on the basis of meaning expressed in language"
computable. Unknowns are outside of this domain.
For instance, look at Godel's G, which states that there is no natural
number g that satisfies a given computable relationship, which is a
pure mathematical operation, so thus totally determined.
You know that it was never a pure mathematical
operation it is all performed in meta-math.
This statement can NOT be proven to be not-well-founded, as to do so
means we can prove that its converse isn't true. which means we can
prove that no number g can exist that meets the requirement, (as if it
could, we couldn't prove that the statement can't be false), and thus
we now HAVE a proof that it is true, as that condition is EXACTLY what
the statement claims.
The problem comes because some problem are inhenently following the
laws of the excluded middle, and thus MUST be True or False. But while
they must be true or false, there doesn't need to be a finite proof
that makes that true.
Two levels of the law of the excluded middle are required
Is X a truth-bearer if yes then is x true
Godel's statement is an example of this, as mathementics, because of
it correlation to programming, is able to create "computations" that
embue meaning into the numbers, a meaning that can't be seen in the
base number system, but is "understood" by the program/relationship
that was created with it. This means that while PA doesn't understand
the meaning, the determinism of mathematics brings the results of that
meaning into the system.
The relationship turns out to be a proof checker, in particular, a
proof checker for the statement of G. a number represents a
"statement" (or seires of statements) in PA, and a number that
satisfies it will represent a valid proof of G.
Thus, if a number existed, then the statements it represented exist in
PA, and those statements become a proof that no such number could exist.
Since this is logically impossible it can not be, thus the statement
must be true.
But, if a proof existed of this fact, then we could compute in the
meta system the number that proof represents, and that number would by
the construciton of the relationship statisfy the relationship,
makeing the statement false.
So, unless you think that it is possible for there to be a proof that
a false statement is true, the statement MUST be true but unprovable.
That, or you get crasyness like mathematics is inconsistant, that some
basic mathematical operation of two natural numbers can give different
results at different times. As in while we THINK that 1 + 2 = 3, it
might be that sometimes 1 + 2 = 4.
That, or you think that it is impossible to create a program that
given a proof in a specified system, checks that the proof is valid
with 100% certainty in that system.
Sorry, your problem is that your concept just can't work in PA and
similar systems.
This is one way to interprete Godel's Incompleteness proof.
G||delrCOs G is not a truthrCabearer inside PA.
It is only interpretable as true in an external
semantic model.
The classical argument that G is rCLtrue but unprovablerCY
relies entirely on metarCamathematical assumptions
about rao, satisfaction, and semantic bivalence.
Once truth is internalized, the argument no longer
applies. G is simply outside the domain of PArCOs
internal truth predicate. Therefore the classical
G||del conclusion is not a fact about PA, but a
fact about how the metarCatheory interprets PA.
On 1/19/2026 11:29 PM, Richard Damon wrote:
My system is not supposed to decide in advance whether
Goldbach is wellrCafounded. A formula becomes a truthrCabearer
only when PA can classify it in finitely many steps.
Goldbach may or may not be classifiable; thatrCOs an open
computational fact, not a semantic requirement. This has
no effect on G||del, because G||delrCOs sentence is structurally
nonrCatruthrCabearing, 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.
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.
On 1/19/2026 11:29 PM, Richard Damon wrote:
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 |a 00:41, olcott a |-crit :
..
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" ? ?-a 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.
But,
This has no effect on my claim that I got rid of
G||del Incompleteness.
Sure it does, because it shows your system is not well founded.
Not at all. At you own repeated insistence the
domain of all of my systems is the set of knowledge
"true on the basis of meaning expressed in language"
When we change the foundation of formal systems
to proof theoretic semantics and add my truth
predicates then G||del's claim of applying to
every formal system that can do a little bit of
arithmetic becomes simply false.
But proof-theoretic semantics are not-well-founded when applied to
systems like PA, as they need to use truth-conditional logic to
determine their proof-theoretic fvalues.
rCLYourCOre assuming proofrCatheoretic semantics must be grounded
in truthrCaconditional semantics. That assumption is false.
In proofrCatheoretic semantics, meaning is given by inferential
rules, not external truthrCaconditions.
So the internal truth predicate for PA is perfectly wellrCafounded,
and G||delrCOs semantic argument no longer applies.rCY
Every attempt at showing incompleteness <in> PA
has never actually been <in> PA.
Sure they were in PA. PA as a system defines the basics of
mathematics. It DEFINES a version of the Natural Numbers with a set of
properties.
These properties can not all be resloved with the finite proofs that
proof theoretic semantics allows.
In particular, you often can't determine that no proof exists (except
by finding the proof of the negation of the statement) as there are an
infinte number of possible proofs to rule out.
This means that actually PROVING that a statement is not-well-founded
can't be done in a proof-theoretic manner.
The only reason anyone ever treated an external, modelrCatheoretic
notion of truth as a proxy for truth in PA is that PA originally
lacked its own internal truth predicate.
Once you anchor a truth predicate directly in PArCOs axioms, it
becomes clear that the sorCacalled rCytruth in PArCO used by G||del and Tarski was never truth in PA at all.
It was truth about PA rCo one level of metarCamathematical reference removed. The two were conflated only because no one had a viable
alternative at the time.rCY
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.
No, proof-theoretic semantics are just not well founded in PA.
As you can't determine a proof-theoretic truth value for some statements.
it isn't that the value is unknown, as that just means that further
search can find the answer, but that literally there is NO valid
proof- theoretic truth value by your definition.
There is no finite proof that it is true.
There is no finite proof that it is false.
There is no finite proof of the above two statements.
Thus, there is no proof-theoretic "truth value" for the statement, not
even not-well-founded, so the definition creates a system that is not
well founded.
In my system of PA non-well founded x can always be
detected one of two ways within the body of knowledge
that can be expressed as language.
There is no finite back chained inference from
x or ~x to the axioms of PA. The inference that
does exist has a cycle in the directed graph of
its evaluation sequence.
On 1/19/2026 11:29 PM, Richard Damon wrote:
On 1/19/26 9:39 PM, olcott wrote:
On 1/17/2026 3:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY >>>> But PA itself has no truth predicate, no internal semantics,
and no mechanism for assigning truth values. So what was
called rCLtrue in arithmeticrCY was always meta-theoretic truth
about arithmetic, imported from an external model and never
grounded inside PA.
This conflation was rarely acknowledged, and it shaped the
interpretation of G||delrCOs incompleteness theorems, independence
results like Goodstein and ParisrCoHarrington, and the entire
discourse around rCLtrue but unprovablerCY statements.
My work begins by correcting this foundational error.
PA has no internal truth predicate, so classical claims of
rCLtrue in arithmeticrCY were always meta-theoretic. My system
introduces a truth predicate whose meaning is anchored
entirely in PArCOs axioms and inference rules, not in external
models. Any statement whose meaning requires meta-theoretic
interpretation or non-well-founded self-reference is rejected
as outside the domain of PA. This yields a coherent, internal
notion of truth in arithmetic for the first time.
The only reason that anyone ever suggested an external measure of
truth as a proxy for actual truth <in> PA is because PA did not have
its own truth predicate. I fixed that anchored in PA's own axioms.
Now we can see that an external measure of true <in> PA was never
actually true <in> PA at all. It was true about PA one level of
indirect reference away from true in PA. It was incorrectly conflated
with true in PA because no one saw any other alternatives.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~TruthBearer(PA, x) rei (~True(PA, x) reo (~False(PA, x))
PA doesn't have a truth predicate, because it CAN'T.
Tarski prove this, any actual truth predicate added to PA makes it
inconsistant.
The problem is your proof-theoretic statements need truth-conditional
interpreation to be evaluated, as it is typically impossible to prove
that no proof exists (except by proving the negation of the statement)
This means that your logic is just based on not-well-founded ideas.
a metarCalevel system is required to stand above PA and
filter expressions before PA ever evaluates them. The
metarCasystem performs the structural work PA cannot do:
it detects cycles, blocks diagonalization, rejects
nonrCatruthrCabearers, and prevents PA from entering
infinite loops. Once the metarCasystem has screened out
nonrCawellrCafounded formulas, PA can safely apply its
internal truth predicaterCodefined purely as provabilityrCoto
the remaining expressions.
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 wellrCafounded. A formula becomes a truthrCabearer
only when PA can classify it in finitely many steps.
Goldbach may or may not be classifiable; thatrCOs an open
computational fact, not a semantic requirement. This has
no effect on G||del, because G||delrCOs sentence is structurally
nonrCatruthrCabearing, 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).
You only can find a cycle when you accept the interpretations in the meta-math.--
So, do you accept that interpreation (and thus the proof) or do you
reject it, and thus have no grounds to deny the effect of the proof.
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 wellrCafounded. A formula becomes a truthrCabearer
only when PA can classify it in finitely many steps.
Goldbach may or may not be classifiable; thatrCOs an open
computational fact, not a semantic requirement. This has
no effect on G||del, because G||delrCOs sentence is structurally
nonrCatruthrCabearing, 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
You only can find a cycle when you accept the interpretations in the
meta-math.
So, do you accept that interpreation (and thus the proof) or do you
reject it, and thus have no grounds to deny the effect of the proof.
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 wellrCafounded. A formula becomes a truthrCabearer
only when PA can classify it in finitely many steps.
Goldbach may or may not be classifiable; thatrCOs an open
computational fact, not a semantic requirement. This has
no effect on G||del, because G||delrCOs sentence is structurally
nonrCatruthrCabearing, 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.
You only can find a cycle when you accept the interpretations in the
meta-math.
So, do you accept that interpreation (and thus the proof) or do you
reject it, and thus have no grounds to deny the effect of the proof.
Le 19/01/2026 |a 00:41, olcott a |-crit :
I already just said that the proof [or] 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 happen[ed] not to be the case.
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 wellrCafounded. A formula becomes a truthrCabearer
only when PA can classify it in finitely many steps.
Goldbach may or may not be classifiable; thatrCOs an open
computational fact, not a semantic requirement. This has
no effect on G||del, because G||delrCOs sentence is structurally >>>>>>> nonrCatruthrCabearing, 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 rao -4Prove_F(G||del_Number(G_F)) contains a semantic
dependency loop, because evaluating G_F requires
evaluating Prove_F on the G||del number of G_F, which
in turn requires evaluating G_F again;
this cycle in the directed graph of its evaluation
sequence makes the formula nonrCawellrCafounded at the
metarCamathematical level, and under a wellrCafounded
proofrCatheoretic semantics such expressions are
filtered out before interpretation, so the diagonal
sentence never enters PA at all.
In that framework G||delrCOs incompleteness construction
never gets off the groundrConot because G||del erred, but
because the wellrCafoundedness criterion he didnrCOt have
in 1931 blocks the selfrCareferential step that his proof
relies on.
You only can find a cycle when you accept the interpretations in the
meta-math.
So, do you accept that interpreation (and thus the proof) or do you
reject it, and thus have no grounds to deny the effect of the proof.
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 wellrCafounded. A formula becomes a truthrCabearer >>>>>>>> only when PA can classify it in finitely many steps.
Goldbach may or may not be classifiable; thatrCOs an open
computational fact, not a semantic requirement. This has
no effect on G||del, because G||delrCOs sentence is structurally >>>>>>>> nonrCatruthrCabearing, 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 rao -4Prove_F(G||del_Number(G_F)) contains a semantic
dependency loop, because evaluating G_F requires
evaluating Prove_F on the G||del 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.
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 wellrCafounded. A formula becomes a truthrCabearer >>>>>>>>> only when PA can classify it in finitely many steps.
Goldbach may or may not be classifiable; thatrCOs an open
computational fact, not a semantic requirement. This has
no effect on G||del, because G||delrCOs sentence is structurally >>>>>>>>> nonrCatruthrCabearing, 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 rao -4Prove_F(G||del_Number(G_F)) contains a semantic
dependency loop, because evaluating G_F requires
evaluating Prove_F on the G||del 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
Le 22/01/2026 |a 04:54, olcott a |-crit :
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 wellrCafounded. A formula becomes a truthrCabearer >>>>>>>>>> only when PA can classify it in finitely many steps.
Goldbach may or may not be classifiable; thatrCOs an open
computational fact, not a semantic requirement. This has
no effect on G||del, because G||delrCOs sentence is structurally >>>>>>>>>> nonrCatruthrCabearing, 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 rao -4Prove_F(G||del_Number(G_F)) contains a semantic
dependency loop, because evaluating G_F requires
evaluating Prove_F on the G||del 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.
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 wellrCafounded. A formula becomes a truthrCabearer >>>>>>>>> only when PA can classify it in finitely many steps.
Goldbach may or may not be classifiable; thatrCOs an open
computational fact, not a semantic requirement. This has
no effect on G||del, because G||delrCOs sentence is structurally >>>>>>>>> nonrCatruthrCabearing, 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 rao -4Prove_F(G||del_Number(G_F)) contains a semantic
dependency loop, because evaluating G_F requires
evaluating Prove_F on the G||del 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 exists only in metarCamathematics, which
people often misconstrue as arithmetic itself.
Satisfaction is not a notion available within
arithmeticrCoonly within models of arithmetic.
On 1/21/2026 10:59 PM, Python wrote:
Le 22/01/2026 |a 04:54, olcott a |-crit :
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 wellrCafounded. A formula becomes a truthrCabearer >>>>>>>>>>> only when PA can classify it in finitely many steps.
Goldbach may or may not be classifiable; thatrCOs an open >>>>>>>>>>> computational fact, not a semantic requirement. This has >>>>>>>>>>> no effect on G||del, because G||delrCOs sentence is structurally >>>>>>>>>>> nonrCatruthrCabearing, 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 rao -4Prove_F(G||del_Number(G_F)) contains a semantic
dependency loop, because evaluating G_F requires
evaluating Prove_F on the G||del 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.
When PA is actually given its own truth predicate
anchored only in its own axioms then for the first
time one see that meta-math truth in the standard
model of arithmetic never was actually true in PA
itself at all.
On 1/20/2026 5:08 PM, 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
or a type theory to give True(S, Goldbach) no content when Goldbach is
out of scope, or keep it explicit with an InScope(S, P) family of
propositions. Of course, the type theory approach is often easier to use
with pencil and paper.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
In PA itself this requires pathological self-reference to
be computed in meta-math by detecting a cycle in the
directed graph of the evaluation sequence of the expression.
This seems to block all of the undecidability that would
otherwise be construed as incompleteness.
reCx ree PA (~WellFounded(PA, x) rei (~True(PA, x) reo (~False(PA, x))
It is also best that an outer knowledge level resolve
Goldbach as outside of the domain of knowledge. If not
then PA might try brute force ans get stuck in a loop.
So within the domain of knowledge Goldbach is ~WellFounded.
Is there a conventional alternative to implication for an explicit
alternative of type theory?
Unsatisfying: WhenInScope(S, P) -> (True(S, P) & Foo(P))
More satisfying: WhenInScope(S,P,Q in (True(S,Q) & Foo(Q)))
Hey, I see that in prolog often. Q is an indeterminate (unbound variable
in prolog) bound by WhenInScope(S,P,Q in ...) within "...".
or a lambda expression alternative:
WhenInScope(S,P,++Q.True(S,Q) & Foo(Q))
I prefer that over an implicit, semi-ad-hoc type theory.
Are there conventional names for these ideas and an author and excellent
exposition textbook?
Well-founded proof theoretic semantics.
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
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.
In addition to equality, it requires negation, either forall or exists, either conjunction or disjunction... but my memory says not all of those
are available in PA in sufficient generality! Oh if only my brain worked
as well as it once did I could work this through in a sitting, instead I
get mentally disorganised.
On 1/22/26 12:18 AM, olcott wrote:
On 1/21/2026 10:59 PM, Python wrote:
Le 22/01/2026 |a 04:54, olcott a |-crit :
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:
PA augmented with its own True(PA,x) and False(PA,x)My system is not supposed to decide in advance whether >>>>>>>>>>>> Goldbach is wellrCafounded. A formula becomes a truthrCabearer >>>>>>>>>>>> only when PA can classify it in finitely many steps.
Goldbach may or may not be classifiable; thatrCOs an open >>>>>>>>>>>> computational fact, not a semantic requirement. This has >>>>>>>>>>>> no effect on G||del, because G||delrCOs sentence is structurally >>>>>>>>>>>> nonrCatruthrCabearing, 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. >>>>>>>>>>
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 rao -4Prove_F(G||del_Number(G_F)) contains a semantic
dependency loop, because evaluating G_F requires
evaluating Prove_F on the G||del 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.
When PA is actually given its own truth predicate
anchored only in its own axioms then for the first
time one see that meta-math truth in the standard
model of arithmetic never was actually true in PA
itself at all.
But PA can't be given such a truth predicate and reamin consistant.
Your provblem is you are too stupid to understand the problem.--
I guess you claim is that if the meta arithmatic uses the fact that 2 *
3 = 6, then maybe in the base arithmatic 2 * 3 might now be 8.
On 1/20/26 6:33 PM, olcott wrote:
On 1/20/2026 5:08 PM, 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
or a type theory to give True(S, Goldbach) no content when Goldbach is
out of scope, or keep it explicit with an InScope(S, P) family of
propositions. Of course, the type theory approach is often easier to use >>> with pencil and paper.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
In PA itself this requires pathological self-reference to
be computed in meta-math by detecting a cycle in the
directed graph of the evaluation sequence of the expression.
This seems to block all of the undecidability that would
otherwise be construed as incompleteness.
Which just shows that you are a liar, as the relaitionship in PA doesn't refer to itself.
You think you are allowed to equivocate on your use of looking into the meta-system.
reCx ree PA (~WellFounded(PA, x) rei (~True(PA, x) reo (~False(PA, x))
For which there is no Proof-Theoretic resolution of, and thus no
handling in Proof-Theoretic semantics.
It is also best that an outer knowledge level resolve
Goldbach as outside of the domain of knowledge. If not
then PA might try brute force ans get stuck in a loop.
So within the domain of knowledge Goldbach is ~WellFounded.
But that isn't true, there is no knowledge of that fact, ax we don't
know if it can be proven or refuted.
In fact, it CAN'T be proved to be not-well-founded in PA, as proving
that we can't disprove it ends up proving it.
The problem is that some statements are inherently logically truth-
beares, as there can be no middle to try to exclude, but they also might
not be provable.
Your Proof-Theoretic Semantics concept can't handle systems where that
is a fact.
Is there a conventional alternative to implication for an explicit
alternative of type theory?
Unsatisfying: WhenInScope(S, P) -> (True(S, P) & Foo(P))
More satisfying: WhenInScope(S,P,Q in (True(S,Q) & Foo(Q)))
Hey, I see that in prolog often. Q is an indeterminate (unbound variable >>> in prolog) bound by WhenInScope(S,P,Q in ...) within "...".
or a lambda expression alternative:
WhenInScope(S,P,++Q.True(S,Q) & Foo(Q))
I prefer that over an implicit, semi-ad-hoc type theory.
Are there conventional names for these ideas and an author and excellent >>> exposition textbook?
Well-founded proof theoretic semantics.
So, have a reference that uses it the way you do, or are you admitting
that it is you own cockamamie perversion of a system that works for the things it works for but not this sort of system.
On 1/20/26 6:33 PM, olcott wrote:
On 1/20/2026 5:08 PM, 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
or a type theory to give True(S, Goldbach) no content when Goldbach is
out of scope, or keep it explicit with an InScope(S, P) family of
propositions. Of course, the type theory approach is often easier to use >>> with pencil and paper.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
In PA itself this requires pathological self-reference to
be computed in meta-math by detecting a cycle in the
directed graph of the evaluation sequence of the expression.
This seems to block all of the undecidability that would
otherwise be construed as incompleteness.
Which just shows that you are a liar, as the relaitionship in PA doesn't refer to itself.
You think you are allowed to equivocate on your use of looking into the meta-system.
reCx ree PA (~WellFounded(PA, x) rei (~True(PA, x) reo (~False(PA, x))
For which there is no Proof-Theoretic resolution of, and thus no
handling in Proof-Theoretic semantics.
It is also best that an outer knowledge level resolve
Goldbach as outside of the domain of knowledge. If not
then PA might try brute force ans get stuck in a loop.
So within the domain of knowledge Goldbach is ~WellFounded.
But that isn't true, there is no knowledge of that fact, ax we don't
know if it can be proven or refuted.
In fact, it CAN'T be proved to be not-well-founded in PA, as proving
that we can't disprove it ends up proving it.
The problem is that some statements are inherently logically truth-
beares, as there can be no middle to try to exclude, but they also might
not be provable.
Your Proof-Theoretic Semantics concept can't handle systems where that
is a fact.
Is there a conventional alternative to implication for an explicit
alternative of type theory?
Unsatisfying: WhenInScope(S, P) -> (True(S, P) & Foo(P))
More satisfying: WhenInScope(S,P,Q in (True(S,Q) & Foo(Q)))
Hey, I see that in prolog often. Q is an indeterminate (unbound variable >>> in prolog) bound by WhenInScope(S,P,Q in ...) within "...".
or a lambda expression alternative:
WhenInScope(S,P,++Q.True(S,Q) & Foo(Q))
I prefer that over an implicit, semi-ad-hoc type theory.
Are there conventional names for these ideas and an author and excellent >>> exposition textbook?
Well-founded proof theoretic semantics.
So, have a reference that uses it the way you do, or are you admitting
that it is you own cockamamie perversion of a system that works for the things it works for but not this sort of system.
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.
On 1/20/26 6:33 PM, olcott wrote:
On 1/20/2026 5:08 PM, 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
or a type theory to give True(S, Goldbach) no content when Goldbach is
out of scope, or keep it explicit with an InScope(S, P) family of
propositions. Of course, the type theory approach is often easier to use >>> with pencil and paper.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
In PA itself this requires pathological self-reference to
be computed in meta-math by detecting a cycle in the
directed graph of the evaluation sequence of the expression.
This seems to block all of the undecidability that would
otherwise be construed as incompleteness.
Which just shows that you are a liar, as the relaitionship in PA doesn't refer to itself.
You think you are allowed to equivocate on your use of looking into the meta-system.
reCx ree PA (~WellFounded(PA, x) rei (~True(PA, x) reo (~False(PA, x))
For which there is no Proof-Theoretic resolution of, and thus no
handling in Proof-Theoretic semantics.
It is also best that an outer knowledge level resolve
Goldbach as outside of the domain of knowledge. If not
then PA might try brute force ans get stuck in a loop.
So within the domain of knowledge Goldbach is ~WellFounded.
But that isn't true, there is no knowledge of that fact, ax we don't
know if it can be proven or refuted.
In fact, it CAN'T be proved to be not-well-founded in PA, as proving
that we can't disprove it ends up proving it.
The problem is that some statements are inherently logically truth-
beares, as there can be no middle to try to exclude, but they also might
not be provable.
Your Proof-Theoretic Semantics concept can't handle systems where that
is a fact.
Is there a conventional alternative to implication for an explicit
alternative of type theory?
Unsatisfying: WhenInScope(S, P) -> (True(S, P) & Foo(P))
More satisfying: WhenInScope(S,P,Q in (True(S,Q) & Foo(Q)))
Hey, I see that in prolog often. Q is an indeterminate (unbound variable >>> in prolog) bound by WhenInScope(S,P,Q in ...) within "...".
or a lambda expression alternative:
WhenInScope(S,P,++Q.True(S,Q) & Foo(Q))
I prefer that over an implicit, semi-ad-hoc type theory.
Are there conventional names for these ideas and an author and excellent >>> exposition textbook?
Well-founded proof theoretic semantics.
So, have a reference that uses it the way you do, or are you admitting
that it is you own cockamamie perversion of a system that works for the things it works for but not this sort of system.
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.
On 1/22/2026 6:23 PM, Richard Damon wrote:
On 1/20/26 6:33 PM, olcott wrote:
On 1/20/2026 5:08 PM, 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
or a type theory to give True(S, Goldbach) no content when Goldbach is >>>> out of scope, or keep it explicit with an InScope(S, P) family of
propositions. Of course, the type theory approach is often easier to
use
with pencil and paper.
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
In PA itself this requires pathological self-reference to
be computed in meta-math by detecting a cycle in the
directed graph of the evaluation sequence of the expression.
This seems to block all of the undecidability that would
otherwise be construed as incompleteness.
Which just shows that you are a liar, as the relaitionship in PA
doesn't refer to itself.
You think you are allowed to equivocate on your use of looking into
the meta-system.
reCx ree PA (~WellFounded(PA, x) rei (~True(PA, x) reo (~False(PA, x))
For which there is no Proof-Theoretic resolution of, and thus no
handling in Proof-Theoretic semantics.
It is also best that an outer knowledge level resolve
Goldbach as outside of the domain of knowledge. If not
then PA might try brute force ans get stuck in a loop.
So within the domain of knowledge Goldbach is ~WellFounded.
But that isn't true, there is no knowledge of that fact, ax we don't
know if it can be proven or refuted.
In fact, it CAN'T be proved to be not-well-founded in PA, as proving
that we can't disprove it ends up proving it.
The problem is that some statements are inherently logically truth-
beares, as there can be no middle to try to exclude, but they also
might not be provable.
Your Proof-Theoretic Semantics concept can't handle systems where that
is a fact.
Is there a conventional alternative to implication for an explicit
alternative of type theory?
Unsatisfying: WhenInScope(S, P) -> (True(S, P) & Foo(P))
More satisfying: WhenInScope(S,P,Q in (True(S,Q) & Foo(Q)))
Hey, I see that in prolog often. Q is an indeterminate (unbound
variable
in prolog) bound by WhenInScope(S,P,Q in ...) within "...".
or a lambda expression alternative:
WhenInScope(S,P,++Q.True(S,Q) & Foo(Q))
I prefer that over an implicit, semi-ad-hoc type theory.
Are there conventional names for these ideas and an author and
excellent
exposition textbook?
Well-founded proof theoretic semantics.
So, have a reference that uses it the way you do, or are you admitting
that it is you own cockamamie perversion of a system that works for
the things it works for but not this sort of system.
From rCLIntuitionistic Type TheoryrCY (1984), p. 7)
rCLTo know a proposition is true is to know a proof of it.rCY
On 1/22/2026 6:17 PM, Richard Damon wrote:
On 1/22/26 12:18 AM, olcott wrote:
On 1/21/2026 10:59 PM, Python wrote:
Le 22/01/2026 |a 04:54, olcott a |-crit :
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:
PA augmented with its own True(PA,x) and False(PA,x)My system is not supposed to decide in advance whether >>>>>>>>>>>>> Goldbach is wellrCafounded. A formula becomes a truthrCabearer >>>>>>>>>>>>> only when PA can classify it in finitely many steps. >>>>>>>>>>>>> Goldbach may or may not be classifiable; thatrCOs an open >>>>>>>>>>>>> computational fact, not a semantic requirement. This has >>>>>>>>>>>>> no effect on G||del, because G||delrCOs sentence is structurally >>>>>>>>>>>>> nonrCatruthrCabearing, 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. >>>>>>>>>>>
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 rao -4Prove_F(G||del_Number(G_F)) contains a semantic
dependency loop, because evaluating G_F requires
evaluating Prove_F on the G||del 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.
When PA is actually given its own truth predicate
anchored only in its own axioms then for the first
time one see that meta-math truth in the standard
model of arithmetic never was actually true in PA
itself at all.
But PA can't be given such a truth predicate and reamin consistant.
Unless the foundation model theory is replaced
with the foundation of proof theory and proof
theory itself is grounded in Haskell Curry's
notion of "true in the system".
Your provblem is you are too stupid to understand the problem.
I guess you claim is that if the meta arithmatic uses the fact that 2
* 3 = 6, then maybe in the base arithmatic 2 * 3 might now be 8.
On 1/22/26 7:33 PM, olcott wrote:
On 1/22/2026 6:17 PM, Richard Damon wrote:
On 1/22/26 12:18 AM, olcott wrote:
On 1/21/2026 10:59 PM, Python wrote:
Le 22/01/2026 |a 04:54, olcott a |-crit :
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:
PA augmented with its own True(PA,x) and False(PA,x)My system is not supposed to decide in advance whether >>>>>>>>>>>>>> Goldbach is wellrCafounded. A formula becomes a truthrCabearer >>>>>>>>>>>>>> only when PA can classify it in finitely many steps. >>>>>>>>>>>>>> Goldbach may or may not be classifiable; thatrCOs an open >>>>>>>>>>>>>> computational fact, not a semantic requirement. This has >>>>>>>>>>>>>> no effect on G||del, because G||delrCOs sentence is structurally >>>>>>>>>>>>>> nonrCatruthrCabearing, 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. >>>>>>>>>>>>
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 rao -4Prove_F(G||del_Number(G_F)) contains a semantic
dependency loop, because evaluating G_F requires
evaluating Prove_F on the G||del 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.
When PA is actually given its own truth predicate
anchored only in its own axioms then for the first
time one see that meta-math truth in the standard
model of arithmetic never was actually true in PA
itself at all.
But PA can't be given such a truth predicate and reamin consistant.
Unless the foundation model theory is replaced
with the foundation of proof theory and proof
theory itself is grounded in Haskell Curry's
notion of "true in the system".
Try to show that working, and HAVE a truth predicate.
Remember, a truth predicate is "True" if the input is a true expression,
and "False" if the input is something else, being either a False
statement, or a not-well-founded statement, or even just plain non-sense.
Your provblem is you are too stupid to understand the problem.
I guess you claim is that if the meta arithmatic uses the fact that 2
* 3 = 6, then maybe in the base arithmatic 2 * 3 might now be 8.
On 1/22/2026 8:51 PM, Richard Damon wrote:
On 1/22/26 7:33 PM, olcott wrote:
On 1/22/2026 6:17 PM, Richard Damon wrote:
On 1/22/26 12:18 AM, olcott wrote:
On 1/21/2026 10:59 PM, Python wrote:
Le 22/01/2026 |a 04:54, olcott a |-crit :
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:
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.My system is not supposed to decide in advance whether >>>>>>>>>>>>>>> Goldbach is wellrCafounded. A formula becomes a truthrCabearer >>>>>>>>>>>>>>> only when PA can classify it in finitely many steps. >>>>>>>>>>>>>>> Goldbach may or may not be classifiable; thatrCOs an open >>>>>>>>>>>>>>> computational fact, not a semantic requirement. This has >>>>>>>>>>>>>>> no effect on G||del, because G||delrCOs sentence is structurally
nonrCatruthrCabearing, 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. >>>>>>>>>>>>>
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 rao -4Prove_F(G||del_Number(G_F)) contains a semantic
dependency loop, because evaluating G_F requires
evaluating Prove_F on the G||del 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.
When PA is actually given its own truth predicate
anchored only in its own axioms then for the first
time one see that meta-math truth in the standard
model of arithmetic never was actually true in PA
itself at all.
But PA can't be given such a truth predicate and reamin consistant.
Unless the foundation model theory is replaced
with the foundation of proof theory and proof
theory itself is grounded in Haskell Curry's
notion of "true in the system".
Try to show that working, and HAVE a truth predicate.
Remember, a truth predicate is "True" if the input is a true
expression, and "False" if the input is something else, being either a
False statement, or a not-well-founded statement, or even just plain
non-sense.
I have boiled that all down so it coherently all
holds together and proves itself completely true
entirely on the basis of the meaning of its words.
I can do this now in one half page of text.
I have worked on the feverishly most every waking
moment for weeks.
Your provblem is you are too stupid to understand the problem.
I guess you claim is that if the meta arithmatic uses the fact that
2 * 3 = 6, then maybe in the base arithmatic 2 * 3 might now be 8.
On 1/19/26 9:39 PM, olcott wrote:
On 1/17/2026 3:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY
But PA itself has no truth predicate, no internal semantics,
and no mechanism for assigning truth values.
^^^reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~TruthBearer(PA, x) rei (~True(PA, x) reo (~False(PA, x))
PA doesn't have a truth predicate, because it CAN'T.
a metarCalevel system is required to stand above PA and
filter expressions before PA ever evaluates them. The
metarCasystem performs the structural work PA cannot do:
it detects cycles, blocks diagonalization, rejects
nonrCatruthrCabearers, and prevents PA from entering
infinite loops.
On 20/01/2026 05:29, Richard Damon wrote:
On 1/19/26 9:39 PM, olcott wrote:
On 1/17/2026 3:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY >>>> But PA itself has no truth predicate, no internal semantics,
and no mechanism for assigning truth values.
Don't we assume it to be (implicitly) a schematic system, where the
axioms define the deduction rules?
...
^^^reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~TruthBearer(PA, x) rei (~True(PA, x) reo (~False(PA, x))
PA doesn't have a truth predicate, because it CAN'T.
a unary truth predicate
but perhaps an operation "IsElementaryTheorem_p(system, objects...)"
for each predicate 'p' can be admitted to an extension of PA.
Perhaps importantly, I note that PA doesn't relate = with rea but both
appear in the axioms, naively avoiding the problem of "what do you mean
by 'negation'?" but leaving a problem of "what do you mean by 'contradiction'?"
What resolutions do you perceive regarding that?
On 20/01/2026 21:39, olcott wrote:
a metarCalevel system is required to stand above PA and
filter expressions before PA ever evaluates them. The
metarCasystem performs the structural work PA cannot do:
it detects cycles, blocks diagonalization, rejects
nonrCatruthrCabearers, and prevents PA from entering
infinite loops.
Then the truth predicate is a restricted truth predicate.
I think Tarski's findings don't directly apply to what Olcott is doing
as they are stated for systems with negation (of statements) carrying
the semantics of contradiction. PA doesn't seem to have that in its
axioms; then there's the matter of universal generality: is that a
predicate, connective, or an operation? Some of those take the truth "predicate" away from Elementary Theorems, some of them don't but
negation must lose its naivety as it becomes an operation.
How does one characterise PA among:
- syntactical system
- schematic system
- abstract formal system
- concrete formal system
etc...
understanding that there is some overlap.
*Please keep comp.theory because I am showing*
On 28/01/2026 18:08, olcott wrote:
*Please keep comp.theory because I am showing*
Perhaps you could post there a result derived from this discussion eventually? For the moment, we're in the nitty-gritty of logic only.
On 1/28/2026 10:21 AM, Tristan Wibberley wrote:
On 20/01/2026 05:29, Richard Damon wrote:
On 1/19/26 9:39 PM, olcott wrote:
On 1/17/2026 3:08 PM, olcott wrote:
For nearly a century, discussions of arithmetic have quietly
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY >>>>> But PA itself has no truth predicate, no internal semantics,
and no mechanism for assigning truth values.
Don't we assume it to be (implicitly) a schematic system, where the
axioms define the deduction rules?
That is the conflation error of G||del's incompleteness.
It seems to be saying what you said to casual observers.
...
-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a ^^^reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
reCx ree PA (~TruthBearer(PA, x) rei (~True(PA, x) reo (~False(PA, x)) >>>>
PA doesn't have a truth predicate, because it CAN'T.
-a-a-a-a-a-a-a-a a unary truth predicate
but perhaps an operation "IsElementaryTheorem_p(system, objects...)"
for each predicate 'p' can be admitted to an extension of PA.
You just understand these things more deeply than
anyone else here.
When we refer to Haskell Curry's notion of elementary
theorems that are true then anything derived from
them is a theorem that is also true. That is the
key foundation of proof theoretic semantics:
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
*Please keep comp.theory because I am showing*
Perhaps importantly, I note that PA doesn't relate = with rea but both
appear in the axioms, naively avoiding the problem of "what do you mean
by 'negation'?" but leaving a problem of "what do you mean by
'contradiction'?"
What resolutions do you perceive regarding that?
On 1/28/2026 10:34 AM, Tristan Wibberley wrote:
On 20/01/2026 21:39, olcott wrote:
a metarCalevel system is required to stand above PA and
filter expressions before PA ever evaluates them. The
metarCasystem performs the structural work PA cannot do:
it detects cycles, blocks diagonalization, rejects
nonrCatruthrCabearers, and prevents PA from entering
infinite loops.
Then the truth predicate is a restricted truth predicate.
It is only restricted to its domain of knowledge
expressed in language. This excludes semantic nonsense
like pathological self-reference, type mismatch errors
and unknowns such as the truth value of the Goldbach
conjecture.
I think Tarski's findings don't directly apply to what Olcott is doing
as they are stated for systems with negation (of statements) carrying
the semantics of contradiction. PA doesn't seem to have that in its
axioms; then there's the matter of universal generality: is that a
predicate, connective, or an operation? Some of those take the truth
"predicate" away from Elementary Theorems, some of them don't but
negation must lose its naivety as it becomes an operation.
How does one characterise PA among:
- syntactical system
- schematic system
- abstract formal system
- concrete formal system
etc...
understanding that there is some overlap.
What I am proposing is that PA is entirely syntactic
and when we add a truth predicate anchored entirely
in the axioms of PA that this predicate itself is
at a meta-level. When we explicitly add this predicate
then we can really see what is actually true in PA
itself and this has always been provable in PA.
Incompleteness only arose because what was true
outside of PA could not be proved inside PA. This
was a mere conflation error all along.
Your problem is you just don't understand what "semantics" actually mean.
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