On 1/17/2026 3:53 AM, Mikko wrote:
On 16/01/2026 17:38, olcott wrote:
On 1/16/2026 3:32 AM, Mikko wrote:
On 15/01/2026 22:30, olcott wrote:
On 1/15/2026 3:34 AM, Mikko wrote:
On 14/01/2026 21:32, olcott wrote:
On 1/14/2026 3:01 AM, Mikko wrote:
On 13/01/2026 16:31, olcott wrote:
On 1/13/2026 3:13 AM, Mikko wrote:
On 12/01/2026 16:32, olcott wrote:
On 1/12/2026 4:47 AM, Mikko wrote:
On 11/01/2026 16:24, Tristan Wibberley wrote:
On 11/01/2026 10:13, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
No, that does not follow. If a required result cannot be >>>>>>>>>>>>>>>> derived by
appying a finite string transformation then the it it is >>>>>>>>>>>>>>>> uncomputable.
Right. Outside the scope of computation. Requiring anything >>>>>>>>>>>>>>> outside the scope of computation is an incorrect >>>>>>>>>>>>>>> requirement.
You can't determine whether the required result is >>>>>>>>>>>>>> computable before
you have the requirement.
Right, it is /in/ scope for computer science... for the / >>>>>>>>>>>>> ology/. Olcott
here uses "computation" to refer to the practice. You give the >>>>>>>>>>>>> requirement to the /ologist/ who correctly decides that it >>>>>>>>>>>>> is not for
computation because it is not computable.
You two so often violently agree; I find it warming to the >>>>>>>>>>>>> heart.
For pracitcal programming it is useful to know what is known >>>>>>>>>>>> to be
uncomputable in order to avoid wasting time in attemlpts to >>>>>>>>>>>> do the
impossible.
It f-cking nuts that after more than 2000 years
people still don't understand that self-contradictory
expressions: "This sentence is not true" have no
truth value. A smart high school student should have
figured this out 2000 years ago.
Irrelevant. For practical programming that question needn't be >>>>>>>>>> answered.
The halting problem counter-example input is anchored
in the Liar Paradox. Proof Theoretic Semantics rejects
those two and G||del's incompleteness and a bunch more
as merely non-well-founded inputs.
For every Turing machine the halting problem counter-example
provably
exists.
Not when using Proof Theoretic Semantics grounded
in the specification language. In this case the
pathological input is simply rejected as ungrounded.
Then your "Proof Theoretic Semantics" is not useful for discussion of >>>>>> Turing machines. For every Turing machine a counter example exists. >>>>>> And so exists a Turing machine that writes the counter example when >>>>>> given a Turing machine as input.
It is "not useful" in the same way that ZFC was
"not useful" for addressing Russell's Paradox.
ZF or ZFC is to some extent useful for addressing Russell's paradox.
It is an example of a set theory where Russell's paradox is avoided.
If your "Proof Theretic Semantics" cannot handle the existence of
a counter example for every Turing decider then it is not usefule
for those who work on practical problems of program correctness.
Proof theoretic semantics addresses G||del Incompleteness
for PA in a way similar to the way that ZFC addresses
Russell's Paradox in set theory.
Not really the same way. Your "Proof theoretic semantics" redefines
truth and replaces the logic. ZFC is another theory using ordinary
logic. The problem with the naive set theory is that it is not
sound for any semantics.
ZFC redefines set theory such that Russell's Paradox cannot arise.
Proof theoretic semantics redefines formal systems such that
Incompleteness cannot arise. G||del did not do this himself because
Proof theoretic semantics did not exist at the time.
On 17/01/2026 16:47, olcott wrote:
On 1/17/2026 3:53 AM, Mikko wrote:
On 16/01/2026 17:38, olcott wrote:
On 1/16/2026 3:32 AM, Mikko wrote:
On 15/01/2026 22:30, olcott wrote:
On 1/15/2026 3:34 AM, Mikko wrote:
On 14/01/2026 21:32, olcott wrote:
On 1/14/2026 3:01 AM, Mikko wrote:
On 13/01/2026 16:31, olcott wrote:
On 1/13/2026 3:13 AM, Mikko wrote:
On 12/01/2026 16:32, olcott wrote:
On 1/12/2026 4:47 AM, Mikko wrote:
On 11/01/2026 16:24, Tristan Wibberley wrote:
On 11/01/2026 10:13, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
No, that does not follow. If a required result cannot >>>>>>>>>>>>>>>>> be derived by
appying a finite string transformation then the it it >>>>>>>>>>>>>>>>> is uncomputable.
Right. Outside the scope of computation. Requiring anything >>>>>>>>>>>>>>>> outside the scope of computation is an incorrect >>>>>>>>>>>>>>>> requirement.
You can't determine whether the required result is >>>>>>>>>>>>>>> computable before
you have the requirement.
Right, it is /in/ scope for computer science... for the / >>>>>>>>>>>>>> ology/. Olcott
here uses "computation" to refer to the practice. You give >>>>>>>>>>>>>> the
requirement to the /ologist/ who correctly decides that it >>>>>>>>>>>>>> is not for
computation because it is not computable.
You two so often violently agree; I find it warming to the >>>>>>>>>>>>>> heart.
For pracitcal programming it is useful to know what is >>>>>>>>>>>>> known to be
uncomputable in order to avoid wasting time in attemlpts to >>>>>>>>>>>>> do the
impossible.
It f-cking nuts that after more than 2000 years
people still don't understand that self-contradictory
expressions: "This sentence is not true" have no
truth value. A smart high school student should have
figured this out 2000 years ago.
Irrelevant. For practical programming that question needn't >>>>>>>>>>> be answered.
The halting problem counter-example input is anchored
in the Liar Paradox. Proof Theoretic Semantics rejects
those two and G||del's incompleteness and a bunch more
as merely non-well-founded inputs.
For every Turing machine the halting problem counter-example >>>>>>>>> provably
exists.
Not when using Proof Theoretic Semantics grounded
in the specification language. In this case the
pathological input is simply rejected as ungrounded.
Then your "Proof Theoretic Semantics" is not useful for
discussion of
Turing machines. For every Turing machine a counter example exists. >>>>>>> And so exists a Turing machine that writes the counter example when >>>>>>> given a Turing machine as input.
It is "not useful" in the same way that ZFC was
"not useful" for addressing Russell's Paradox.
ZF or ZFC is to some extent useful for addressing Russell's paradox. >>>>> It is an example of a set theory where Russell's paradox is avoided. >>>>> If your "Proof Theretic Semantics" cannot handle the existence of
a counter example for every Turing decider then it is not usefule
for those who work on practical problems of program correctness.
Proof theoretic semantics addresses G||del Incompleteness
for PA in a way similar to the way that ZFC addresses
Russell's Paradox in set theory.
Not really the same way. Your "Proof theoretic semantics" redefines
truth and replaces the logic. ZFC is another theory using ordinary
logic. The problem with the naive set theory is that it is not
sound for any semantics.
ZFC redefines set theory such that Russell's Paradox cannot arise.
No, it does not. It is just another exammle of the generic concept
of set theory. Essentially the same as ZF but has one additional
postulate.
Proof theoretic semantics redefines formal systems such that
Incompleteness cannot arise. G||del did not do this himself because
Proof theoretic semantics did not exist at the time.
G||del did not do that because his topic was Peano arithmetic and its extensions, and more generally ordinary logic.
Can you can you prove anyting analogous to G||del's completeness
theorem for your "Proof theoretic semantics"?
On 1/18/2026 5:27 AM, Mikko wrote:
On 17/01/2026 16:47, olcott wrote:
On 1/17/2026 3:53 AM, Mikko wrote:
On 16/01/2026 17:38, olcott wrote:
On 1/16/2026 3:32 AM, Mikko wrote:
On 15/01/2026 22:30, olcott wrote:
On 1/15/2026 3:34 AM, Mikko wrote:
On 14/01/2026 21:32, olcott wrote:
On 1/14/2026 3:01 AM, Mikko wrote:
On 13/01/2026 16:31, olcott wrote:
On 1/13/2026 3:13 AM, Mikko wrote:
On 12/01/2026 16:32, olcott wrote:
On 1/12/2026 4:47 AM, Mikko wrote:
On 11/01/2026 16:24, Tristan Wibberley wrote:
On 11/01/2026 10:13, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
No, that does not follow. If a required result cannot >>>>>>>>>>>>>>>>>> be derived by
appying a finite string transformation then the it it >>>>>>>>>>>>>>>>>> is uncomputable.
Right. Outside the scope of computation. Requiring >>>>>>>>>>>>>>>>> anything
outside the scope of computation is an incorrect >>>>>>>>>>>>>>>>> requirement.
You can't determine whether the required result is >>>>>>>>>>>>>>>> computable before
you have the requirement.
Right, it is /in/ scope for computer science... for the / >>>>>>>>>>>>>>> ology/. Olcott
here uses "computation" to refer to the practice. You >>>>>>>>>>>>>>> give the
requirement to the /ologist/ who correctly decides that >>>>>>>>>>>>>>> it is not for
computation because it is not computable.
You two so often violently agree; I find it warming to >>>>>>>>>>>>>>> the heart.
For pracitcal programming it is useful to know what is >>>>>>>>>>>>>> known to be
uncomputable in order to avoid wasting time in attemlpts >>>>>>>>>>>>>> to do the
impossible.
It f-cking nuts that after more than 2000 years
people still don't understand that self-contradictory >>>>>>>>>>>>> expressions: "This sentence is not true" have no
truth value. A smart high school student should have >>>>>>>>>>>>> figured this out 2000 years ago.
Irrelevant. For practical programming that question needn't >>>>>>>>>>>> be answered.
The halting problem counter-example input is anchored
in the Liar Paradox. Proof Theoretic Semantics rejects
those two and G||del's incompleteness and a bunch more
as merely non-well-founded inputs.
For every Turing machine the halting problem counter-example >>>>>>>>>> provably
exists.
Not when using Proof Theoretic Semantics grounded
in the specification language. In this case the
pathological input is simply rejected as ungrounded.
Then your "Proof Theoretic Semantics" is not useful for
discussion of
Turing machines. For every Turing machine a counter example exists. >>>>>>>> And so exists a Turing machine that writes the counter example when >>>>>>>> given a Turing machine as input.
It is "not useful" in the same way that ZFC was
"not useful" for addressing Russell's Paradox.
ZF or ZFC is to some extent useful for addressing Russell's paradox. >>>>>> It is an example of a set theory where Russell's paradox is avoided. >>>>>> If your "Proof Theretic Semantics" cannot handle the existence of
a counter example for every Turing decider then it is not usefule
for those who work on practical problems of program correctness.
Proof theoretic semantics addresses G||del Incompleteness
for PA in a way similar to the way that ZFC addresses
Russell's Paradox in set theory.
Not really the same way. Your "Proof theoretic semantics" redefines
truth and replaces the logic. ZFC is another theory using ordinary
logic. The problem with the naive set theory is that it is not
sound for any semantics.
ZFC redefines set theory such that Russell's Paradox cannot arise.
No, it does not. It is just another exammle of the generic concept
of set theory. Essentially the same as ZF but has one additional
postulate.
ZFC redefines set theory such that Russell's Paradox cannot arise
and the original set theory is now referred to as naive set theory.
Proof theoretic semantics redefines formal systems such that
Incompleteness cannot arise. G||del did not do this himself because
Proof theoretic semantics did not exist at the time.
G||del did not do that because his topic was Peano arithmetic and its
extensions, and more generally ordinary logic.
Can you can you prove anyting analogous to G||del's completeness
theorem for your "Proof theoretic semantics"?
G||delrCOs incompleteness arises only because
rCLtrue in PArCY was never an internal notion
of PA at all, but a metarCamathematical notion
of truth about PA defined externally through
models;
Once truth is defined internallyrCoby extending
PA with a truth predicate so that rCLtrue in PArCY
simply means rCLderivable from PArCOs axiomsrCYrCo
the supposed gap between truth and provability
disappears
With that disappearance PA no longer counts as
incomplete, because the statements G||del identified
as rCLtrue but unprovablerCY were never internal truths
of PA in the first place, only truths assigned from
the outside by the metarCasystem.
On 1/18/2026 5:27 AM, Mikko wrote:
On 17/01/2026 16:47, olcott wrote:
On 1/17/2026 3:53 AM, Mikko wrote:
On 16/01/2026 17:38, olcott wrote:
On 1/16/2026 3:32 AM, Mikko wrote:
On 15/01/2026 22:30, olcott wrote:
On 1/15/2026 3:34 AM, Mikko wrote:
On 14/01/2026 21:32, olcott wrote:
On 1/14/2026 3:01 AM, Mikko wrote:
On 13/01/2026 16:31, olcott wrote:
On 1/13/2026 3:13 AM, Mikko wrote:
On 12/01/2026 16:32, olcott wrote:
On 1/12/2026 4:47 AM, Mikko wrote:
On 11/01/2026 16:24, Tristan Wibberley wrote:
On 11/01/2026 10:13, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
No, that does not follow. If a required result cannot >>>>>>>>>>>>>>>>>> be derived by
appying a finite string transformation then the it it >>>>>>>>>>>>>>>>>> is uncomputable.
Right. Outside the scope of computation. Requiring >>>>>>>>>>>>>>>>> anything
outside the scope of computation is an incorrect >>>>>>>>>>>>>>>>> requirement.
You can't determine whether the required result is >>>>>>>>>>>>>>>> computable before
you have the requirement.
Right, it is /in/ scope for computer science... for the / >>>>>>>>>>>>>>> ology/. Olcott
here uses "computation" to refer to the practice. You >>>>>>>>>>>>>>> give the
requirement to the /ologist/ who correctly decides that >>>>>>>>>>>>>>> it is not for
computation because it is not computable.
You two so often violently agree; I find it warming to >>>>>>>>>>>>>>> the heart.
For pracitcal programming it is useful to know what is >>>>>>>>>>>>>> known to be
uncomputable in order to avoid wasting time in attemlpts >>>>>>>>>>>>>> to do the
impossible.
It f-cking nuts that after more than 2000 years
people still don't understand that self-contradictory >>>>>>>>>>>>> expressions: "This sentence is not true" have no
truth value. A smart high school student should have >>>>>>>>>>>>> figured this out 2000 years ago.
Irrelevant. For practical programming that question needn't >>>>>>>>>>>> be answered.
The halting problem counter-example input is anchored
in the Liar Paradox. Proof Theoretic Semantics rejects
those two and G||del's incompleteness and a bunch more
as merely non-well-founded inputs.
For every Turing machine the halting problem counter-example >>>>>>>>>> provably
exists.
Not when using Proof Theoretic Semantics grounded
in the specification language. In this case the
pathological input is simply rejected as ungrounded.
Then your "Proof Theoretic Semantics" is not useful for
discussion of
Turing machines. For every Turing machine a counter example exists. >>>>>>>> And so exists a Turing machine that writes the counter example when >>>>>>>> given a Turing machine as input.
It is "not useful" in the same way that ZFC was
"not useful" for addressing Russell's Paradox.
ZF or ZFC is to some extent useful for addressing Russell's paradox. >>>>>> It is an example of a set theory where Russell's paradox is avoided. >>>>>> If your "Proof Theretic Semantics" cannot handle the existence of
a counter example for every Turing decider then it is not usefule
for those who work on practical problems of program correctness.
Proof theoretic semantics addresses G||del Incompleteness
for PA in a way similar to the way that ZFC addresses
Russell's Paradox in set theory.
Not really the same way. Your "Proof theoretic semantics" redefines
truth and replaces the logic. ZFC is another theory using ordinary
logic. The problem with the naive set theory is that it is not
sound for any semantics.
ZFC redefines set theory such that Russell's Paradox cannot arise.
No, it does not. It is just another exammle of the generic concept
of set theory. Essentially the same as ZF but has one additional
postulate.
ZFC redefines set theory such that Russell's Paradox cannot arise
and the original set theory is now referred to as naive set theory.
Proof theoretic semantics redefines formal systems such that
Incompleteness cannot arise. G||del did not do this himself because
Proof theoretic semantics did not exist at the time.
G||del did not do that because his topic was Peano arithmetic and its
extensions, and more generally ordinary logic.
Can you can you prove anyting analogous to G||del's completeness
theorem for your "Proof theoretic semantics"?
G||delrCOs incompleteness arises only because
rCLtrue in PArCY was never an internal notion
of PA at all, but a metarCamathematical notion
of truth about PA defined externally through
models;
Once truth is defined internallyrCoby extending
PA with a truth predicate so that rCLtrue in PArCY
simply means rCLderivable from PArCOs axiomsrCYrCo
the supposed gap between truth and provability
disappears
With that disappearance PA no longer counts as
incomplete, because the statements G||del identified
as rCLtrue but unprovablerCY were never internal truths
of PA in the first place, only truths assigned from
the outside by the metarCasystem.
On 19/01/2026 17:03, olcott wrote:
On 1/19/2026 2:19 AM, Mikko wrote:
On 18/01/2026 15:28, olcott wrote:
On 1/18/2026 5:27 AM, Mikko wrote:
On 17/01/2026 16:47, olcott wrote:
On 1/17/2026 3:53 AM, Mikko wrote:
On 16/01/2026 17:38, olcott wrote:
On 1/16/2026 3:32 AM, Mikko wrote:
On 15/01/2026 22:30, olcott wrote:Proof theoretic semantics addresses G||del Incompleteness
On 1/15/2026 3:34 AM, Mikko wrote:
On 14/01/2026 21:32, olcott wrote:
On 1/14/2026 3:01 AM, Mikko wrote:
On 13/01/2026 16:31, olcott wrote:
On 1/13/2026 3:13 AM, Mikko wrote:
On 12/01/2026 16:32, olcott wrote:
On 1/12/2026 4:47 AM, Mikko wrote:
On 11/01/2026 16:24, Tristan Wibberley wrote: >>>>>>>>>>>>>>>>>> On 11/01/2026 10:13, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
No, that does not follow. If a required result >>>>>>>>>>>>>>>>>>>>> cannot be derived by
appying a finite string transformation then the it >>>>>>>>>>>>>>>>>>>>> it is uncomputable.
Right. Outside the scope of computation. Requiring >>>>>>>>>>>>>>>>>>>> anything
outside the scope of computation is an incorrect >>>>>>>>>>>>>>>>>>>> requirement.
You can't determine whether the required result is >>>>>>>>>>>>>>>>>>> computable before
you have the requirement.
Right, it is /in/ scope for computer science... for >>>>>>>>>>>>>>>>>> the / ology/. Olcott
here uses "computation" to refer to the practice. You >>>>>>>>>>>>>>>>>> give the
requirement to the /ologist/ who correctly decides >>>>>>>>>>>>>>>>>> that it is not for
computation because it is not computable.
You two so often violently agree; I find it warming to >>>>>>>>>>>>>>>>>> the heart.
For pracitcal programming it is useful to know what is >>>>>>>>>>>>>>>>> known to be
uncomputable in order to avoid wasting time in >>>>>>>>>>>>>>>>> attemlpts to do the
impossible.
It f-cking nuts that after more than 2000 years >>>>>>>>>>>>>>>> people still don't understand that self-contradictory >>>>>>>>>>>>>>>> expressions: "This sentence is not true" have no >>>>>>>>>>>>>>>> truth value. A smart high school student should have >>>>>>>>>>>>>>>> figured this out 2000 years ago.
Irrelevant. For practical programming that question >>>>>>>>>>>>>>> needn't be answered.
The halting problem counter-example input is anchored >>>>>>>>>>>>>> in the Liar Paradox. Proof Theoretic Semantics rejects >>>>>>>>>>>>>> those two and G||del's incompleteness and a bunch more >>>>>>>>>>>>>> as merely non-well-founded inputs.
For every Turing machine the halting problem counter- >>>>>>>>>>>>> example provably
exists.
Not when using Proof Theoretic Semantics grounded
in the specification language. In this case the
pathological input is simply rejected as ungrounded.
Then your "Proof Theoretic Semantics" is not useful for >>>>>>>>>>> discussion of
Turing machines. For every Turing machine a counter example >>>>>>>>>>> exists.
And so exists a Turing machine that writes the counter
example when
given a Turing machine as input.
It is "not useful" in the same way that ZFC was
"not useful" for addressing Russell's Paradox.
ZF or ZFC is to some extent useful for addressing Russell's >>>>>>>>> paradox.
It is an example of a set theory where Russell's paradox is >>>>>>>>> avoided.
If your "Proof Theretic Semantics" cannot handle the existence of >>>>>>>>> a counter example for every Turing decider then it is not usefule >>>>>>>>> for those who work on practical problems of program correctness. >>>>>>>>
for PA in a way similar to the way that ZFC addresses
Russell's Paradox in set theory.
Not really the same way. Your "Proof theoretic semantics" redefines >>>>>>> truth and replaces the logic. ZFC is another theory using ordinary >>>>>>> logic. The problem with the naive set theory is that it is not
sound for any semantics.
ZFC redefines set theory such that Russell's Paradox cannot arise.
No, it does not. It is just another exammle of the generic concept
of set theory. Essentially the same as ZF but has one additional
postulate.
ZFC redefines set theory such that Russell's Paradox cannot arise
and the original set theory is now referred to as naive set theory.
ZF and ZFC are not redefinitions. ZF is another theory. It can be
called a "set theory" because its structure is similar to Cnator's
original informal set theory. Cantor did not specify whther a set
must be well-founded but ZF specifies that it must. A set theory
were all sets are well-founded does not have Russell's paradox.
ZF is a redefinition in the only sense that matters:
it changes the foundational rules so that RussellrCOs
paradox cannot arise.
The only sense that matters is: to give a new meaning to an exsisting
term. That is OK when the new meaning is only used in a context where
the old one does not make sense.
What you are trying is to give a new meaning to "true" but preted that
it still means 'true'.
On 1/20/2026 3:58 AM, Mikko wrote:
On 19/01/2026 17:03, olcott wrote:
On 1/19/2026 2:19 AM, Mikko wrote:
On 18/01/2026 15:28, olcott wrote:
On 1/18/2026 5:27 AM, Mikko wrote:
On 17/01/2026 16:47, olcott wrote:
On 1/17/2026 3:53 AM, Mikko wrote:No, it does not. It is just another exammle of the generic concept >>>>>> of set theory. Essentially the same as ZF but has one additional
On 16/01/2026 17:38, olcott wrote:
On 1/16/2026 3:32 AM, Mikko wrote:
On 15/01/2026 22:30, olcott wrote:Proof theoretic semantics addresses G||del Incompleteness
On 1/15/2026 3:34 AM, Mikko wrote:
On 14/01/2026 21:32, olcott wrote:
On 1/14/2026 3:01 AM, Mikko wrote:
On 13/01/2026 16:31, olcott wrote:
On 1/13/2026 3:13 AM, Mikko wrote:
On 12/01/2026 16:32, olcott wrote:
On 1/12/2026 4:47 AM, Mikko wrote:
On 11/01/2026 16:24, Tristan Wibberley wrote: >>>>>>>>>>>>>>>>>>> On 11/01/2026 10:13, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
No, that does not follow. If a required result >>>>>>>>>>>>>>>>>>>>>> cannot be derived by
appying a finite string transformation then the it >>>>>>>>>>>>>>>>>>>>>> it is uncomputable.
Right. Outside the scope of computation. Requiring >>>>>>>>>>>>>>>>>>>>> anything
outside the scope of computation is an incorrect >>>>>>>>>>>>>>>>>>>>> requirement.
You can't determine whether the required result is >>>>>>>>>>>>>>>>>>>> computable before
you have the requirement.
Right, it is /in/ scope for computer science... for >>>>>>>>>>>>>>>>>>> the / ology/. Olcott
here uses "computation" to refer to the practice. You >>>>>>>>>>>>>>>>>>> give the
requirement to the /ologist/ who correctly decides >>>>>>>>>>>>>>>>>>> that it is not for
computation because it is not computable. >>>>>>>>>>>>>>>>>>>
You two so often violently agree; I find it warming >>>>>>>>>>>>>>>>>>> to the heart.
For pracitcal programming it is useful to know what is >>>>>>>>>>>>>>>>>> known to be
uncomputable in order to avoid wasting time in >>>>>>>>>>>>>>>>>> attemlpts to do the
impossible.
It f-cking nuts that after more than 2000 years >>>>>>>>>>>>>>>>> people still don't understand that self-contradictory >>>>>>>>>>>>>>>>> expressions: "This sentence is not true" have no >>>>>>>>>>>>>>>>> truth value. A smart high school student should have >>>>>>>>>>>>>>>>> figured this out 2000 years ago.
Irrelevant. For practical programming that question >>>>>>>>>>>>>>>> needn't be answered.
The halting problem counter-example input is anchored >>>>>>>>>>>>>>> in the Liar Paradox. Proof Theoretic Semantics rejects >>>>>>>>>>>>>>> those two and G||del's incompleteness and a bunch more >>>>>>>>>>>>>>> as merely non-well-founded inputs.
For every Turing machine the halting problem counter- >>>>>>>>>>>>>> example provably
exists.
Not when using Proof Theoretic Semantics grounded
in the specification language. In this case the
pathological input is simply rejected as ungrounded.
Then your "Proof Theoretic Semantics" is not useful for >>>>>>>>>>>> discussion of
Turing machines. For every Turing machine a counter example >>>>>>>>>>>> exists.
And so exists a Turing machine that writes the counter >>>>>>>>>>>> example when
given a Turing machine as input.
It is "not useful" in the same way that ZFC was
"not useful" for addressing Russell's Paradox.
ZF or ZFC is to some extent useful for addressing Russell's >>>>>>>>>> paradox.
It is an example of a set theory where Russell's paradox is >>>>>>>>>> avoided.
If your "Proof Theretic Semantics" cannot handle the existence of >>>>>>>>>> a counter example for every Turing decider then it is not usefule >>>>>>>>>> for those who work on practical problems of program correctness. >>>>>>>>>
for PA in a way similar to the way that ZFC addresses
Russell's Paradox in set theory.
Not really the same way. Your "Proof theoretic semantics" redefines >>>>>>>> truth and replaces the logic. ZFC is another theory using ordinary >>>>>>>> logic. The problem with the naive set theory is that it is not >>>>>>>> sound for any semantics.
ZFC redefines set theory such that Russell's Paradox cannot arise. >>>>>>
postulate.
ZFC redefines set theory such that Russell's Paradox cannot arise
and the original set theory is now referred to as naive set theory.
ZF and ZFC are not redefinitions. ZF is another theory. It can be
called a "set theory" because its structure is similar to Cnator's
original informal set theory. Cantor did not specify whther a set
must be well-founded but ZF specifies that it must. A set theory
were all sets are well-founded does not have Russell's paradox.
ZF is a redefinition in the only sense that matters:
it changes the foundational rules so that RussellrCOs
paradox cannot arise.
The only sense that matters is: to give a new meaning to an exsisting
term. That is OK when the new meaning is only used in a context where
the old one does not make sense.
What you are trying is to give a new meaning to "true" but preted that
it still means 'true'.
True in the standard model of arithmetic using meta-math
has always been misconstrued as true <in> arithmetic
only because back then proof theoretic semantics did
not exist.
On 20/01/2026 20:35, olcott wrote:
On 1/20/2026 3:58 AM, Mikko wrote:
On 19/01/2026 17:03, olcott wrote:
On 1/19/2026 2:19 AM, Mikko wrote:
On 18/01/2026 15:28, olcott wrote:
On 1/18/2026 5:27 AM, Mikko wrote:ZF and ZFC are not redefinitions. ZF is another theory. It can be
On 17/01/2026 16:47, olcott wrote:
On 1/17/2026 3:53 AM, Mikko wrote:No, it does not. It is just another exammle of the generic concept >>>>>>> of set theory. Essentially the same as ZF but has one additional >>>>>>> postulate.
On 16/01/2026 17:38, olcott wrote:
On 1/16/2026 3:32 AM, Mikko wrote:
On 15/01/2026 22:30, olcott wrote:Proof theoretic semantics addresses G||del Incompleteness
On 1/15/2026 3:34 AM, Mikko wrote:
On 14/01/2026 21:32, olcott wrote:
On 1/14/2026 3:01 AM, Mikko wrote:Then your "Proof Theoretic Semantics" is not useful for >>>>>>>>>>>>> discussion of
On 13/01/2026 16:31, olcott wrote:
On 1/13/2026 3:13 AM, Mikko wrote:
On 12/01/2026 16:32, olcott wrote:
On 1/12/2026 4:47 AM, Mikko wrote:
On 11/01/2026 16:24, Tristan Wibberley wrote: >>>>>>>>>>>>>>>>>>>> On 11/01/2026 10:13, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
No, that does not follow. If a required result >>>>>>>>>>>>>>>>>>>>>>> cannot be derived by
appying a finite string transformation then the >>>>>>>>>>>>>>>>>>>>>>> it it is uncomputable.
Right. Outside the scope of computation. Requiring >>>>>>>>>>>>>>>>>>>>>> anything
outside the scope of computation is an incorrect >>>>>>>>>>>>>>>>>>>>>> requirement.
You can't determine whether the required result is >>>>>>>>>>>>>>>>>>>>> computable before
you have the requirement.
Right, it is /in/ scope for computer science... for >>>>>>>>>>>>>>>>>>>> the / ology/. Olcott
here uses "computation" to refer to the practice. >>>>>>>>>>>>>>>>>>>> You give the
requirement to the /ologist/ who correctly decides >>>>>>>>>>>>>>>>>>>> that it is not for
computation because it is not computable. >>>>>>>>>>>>>>>>>>>>
You two so often violently agree; I find it warming >>>>>>>>>>>>>>>>>>>> to the heart.
For pracitcal programming it is useful to know what >>>>>>>>>>>>>>>>>>> is known to be
uncomputable in order to avoid wasting time in >>>>>>>>>>>>>>>>>>> attemlpts to do the
impossible.
It f-cking nuts that after more than 2000 years >>>>>>>>>>>>>>>>>> people still don't understand that self-contradictory >>>>>>>>>>>>>>>>>> expressions: "This sentence is not true" have no >>>>>>>>>>>>>>>>>> truth value. A smart high school student should have >>>>>>>>>>>>>>>>>> figured this out 2000 years ago.
Irrelevant. For practical programming that question >>>>>>>>>>>>>>>>> needn't be answered.
The halting problem counter-example input is anchored >>>>>>>>>>>>>>>> in the Liar Paradox. Proof Theoretic Semantics rejects >>>>>>>>>>>>>>>> those two and G||del's incompleteness and a bunch more >>>>>>>>>>>>>>>> as merely non-well-founded inputs.
For every Turing machine the halting problem counter- >>>>>>>>>>>>>>> example provably
exists.
Not when using Proof Theoretic Semantics grounded
in the specification language. In this case the
pathological input is simply rejected as ungrounded. >>>>>>>>>>>>>
Turing machines. For every Turing machine a counter example >>>>>>>>>>>>> exists.
And so exists a Turing machine that writes the counter >>>>>>>>>>>>> example when
given a Turing machine as input.
It is "not useful" in the same way that ZFC was
"not useful" for addressing Russell's Paradox.
ZF or ZFC is to some extent useful for addressing Russell's >>>>>>>>>>> paradox.
It is an example of a set theory where Russell's paradox is >>>>>>>>>>> avoided.
If your "Proof Theretic Semantics" cannot handle the
existence of
a counter example for every Turing decider then it is not >>>>>>>>>>> usefule
for those who work on practical problems of program correctness. >>>>>>>>>>
for PA in a way similar to the way that ZFC addresses
Russell's Paradox in set theory.
Not really the same way. Your "Proof theoretic semantics"
redefines
truth and replaces the logic. ZFC is another theory using ordinary >>>>>>>>> logic. The problem with the naive set theory is that it is not >>>>>>>>> sound for any semantics.
ZFC redefines set theory such that Russell's Paradox cannot arise. >>>>>>>
ZFC redefines set theory such that Russell's Paradox cannot arise
and the original set theory is now referred to as naive set theory. >>>>>
called a "set theory" because its structure is similar to Cnator's
original informal set theory. Cantor did not specify whther a set
must be well-founded but ZF specifies that it must. A set theory
were all sets are well-founded does not have Russell's paradox.
ZF is a redefinition in the only sense that matters:
it changes the foundational rules so that RussellrCOs
paradox cannot arise.
The only sense that matters is: to give a new meaning to an exsisting
term. That is OK when the new meaning is only used in a context where
the old one does not make sense.
What you are trying is to give a new meaning to "true" but preted that
it still means 'true'.
True in the standard model of arithmetic using meta-math
has always been misconstrued as true <in> arithmetic
No, it hasn't. In the way theories are usually discussed nothing is
"ture in arithmetic". Every sentence of a first order theory that
can be proven in the theory is true in every model theory. Every
sentence of a theory that cannot be proven in the theory is false
in some model of the theory.
only because back then proof theoretic semantics did
not exist.
Every interpretation of the theory is a definition of semantics.
On 1/21/2026 3:03 AM, Mikko wrote:
On 20/01/2026 20:35, olcott wrote:
On 1/20/2026 3:58 AM, Mikko wrote:
On 19/01/2026 17:03, olcott wrote:
On 1/19/2026 2:19 AM, Mikko wrote:
On 18/01/2026 15:28, olcott wrote:
On 1/18/2026 5:27 AM, Mikko wrote:ZF and ZFC are not redefinitions. ZF is another theory. It can be
On 17/01/2026 16:47, olcott wrote:
On 1/17/2026 3:53 AM, Mikko wrote:No, it does not. It is just another exammle of the generic concept >>>>>>>> of set theory. Essentially the same as ZF but has one additional >>>>>>>> postulate.
On 16/01/2026 17:38, olcott wrote:
On 1/16/2026 3:32 AM, Mikko wrote:
On 15/01/2026 22:30, olcott wrote:
On 1/15/2026 3:34 AM, Mikko wrote:
On 14/01/2026 21:32, olcott wrote:
On 1/14/2026 3:01 AM, Mikko wrote:Then your "Proof Theoretic Semantics" is not useful for >>>>>>>>>>>>>> discussion of
On 13/01/2026 16:31, olcott wrote:
On 1/13/2026 3:13 AM, Mikko wrote:
On 12/01/2026 16:32, olcott wrote:
On 1/12/2026 4:47 AM, Mikko wrote:
On 11/01/2026 16:24, Tristan Wibberley wrote: >>>>>>>>>>>>>>>>>>>>> On 11/01/2026 10:13, Mikko wrote:
On 10/01/2026 17:47, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>> On 1/10/2026 2:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>
No, that does not follow. If a required result >>>>>>>>>>>>>>>>>>>>>>>> cannot be derived by
appying a finite string transformation then the >>>>>>>>>>>>>>>>>>>>>>>> it it is uncomputable.
Right. Outside the scope of computation. >>>>>>>>>>>>>>>>>>>>>>> Requiring anything
outside the scope of computation is an incorrect >>>>>>>>>>>>>>>>>>>>>>> requirement.
You can't determine whether the required result is >>>>>>>>>>>>>>>>>>>>>> computable before
you have the requirement.
Right, it is /in/ scope for computer science... for >>>>>>>>>>>>>>>>>>>>> the / ology/. Olcott
here uses "computation" to refer to the practice. >>>>>>>>>>>>>>>>>>>>> You give the
requirement to the /ologist/ who correctly decides >>>>>>>>>>>>>>>>>>>>> that it is not for
computation because it is not computable. >>>>>>>>>>>>>>>>>>>>>
You two so often violently agree; I find it warming >>>>>>>>>>>>>>>>>>>>> to the heart.
For pracitcal programming it is useful to know what >>>>>>>>>>>>>>>>>>>> is known to be
uncomputable in order to avoid wasting time in >>>>>>>>>>>>>>>>>>>> attemlpts to do the
impossible.
It f-cking nuts that after more than 2000 years >>>>>>>>>>>>>>>>>>> people still don't understand that self-contradictory >>>>>>>>>>>>>>>>>>> expressions: "This sentence is not true" have no >>>>>>>>>>>>>>>>>>> truth value. A smart high school student should have >>>>>>>>>>>>>>>>>>> figured this out 2000 years ago.
Irrelevant. For practical programming that question >>>>>>>>>>>>>>>>>> needn't be answered.
The halting problem counter-example input is anchored >>>>>>>>>>>>>>>>> in the Liar Paradox. Proof Theoretic Semantics rejects >>>>>>>>>>>>>>>>> those two and G||del's incompleteness and a bunch more >>>>>>>>>>>>>>>>> as merely non-well-founded inputs.
For every Turing machine the halting problem counter- >>>>>>>>>>>>>>>> example provably
exists.
Not when using Proof Theoretic Semantics grounded >>>>>>>>>>>>>>> in the specification language. In this case the
pathological input is simply rejected as ungrounded. >>>>>>>>>>>>>>
Turing machines. For every Turing machine a counter >>>>>>>>>>>>>> example exists.
And so exists a Turing machine that writes the counter >>>>>>>>>>>>>> example when
given a Turing machine as input.
It is "not useful" in the same way that ZFC was
"not useful" for addressing Russell's Paradox.
ZF or ZFC is to some extent useful for addressing Russell's >>>>>>>>>>>> paradox.
It is an example of a set theory where Russell's paradox is >>>>>>>>>>>> avoided.
If your "Proof Theretic Semantics" cannot handle the
existence of
a counter example for every Turing decider then it is not >>>>>>>>>>>> usefule
for those who work on practical problems of program
correctness.
Proof theoretic semantics addresses G||del Incompleteness >>>>>>>>>>> for PA in a way similar to the way that ZFC addresses
Russell's Paradox in set theory.
Not really the same way. Your "Proof theoretic semantics" >>>>>>>>>> redefines
truth and replaces the logic. ZFC is another theory using >>>>>>>>>> ordinary
logic. The problem with the naive set theory is that it is not >>>>>>>>>> sound for any semantics.
ZFC redefines set theory such that Russell's Paradox cannot arise. >>>>>>>>
ZFC redefines set theory such that Russell's Paradox cannot arise >>>>>>> and the original set theory is now referred to as naive set theory. >>>>>>
called a "set theory" because its structure is similar to Cnator's >>>>>> original informal set theory. Cantor did not specify whther a set
must be well-founded but ZF specifies that it must. A set theory
were all sets are well-founded does not have Russell's paradox.
ZF is a redefinition in the only sense that matters:
it changes the foundational rules so that RussellrCOs
paradox cannot arise.
The only sense that matters is: to give a new meaning to an exsisting
term. That is OK when the new meaning is only used in a context where
the old one does not make sense.
What you are trying is to give a new meaning to "true" but preted that >>>> it still means 'true'.
True in the standard model of arithmetic using meta-math
has always been misconstrued as true <in> arithmetic
No, it hasn't. In the way theories are usually discussed nothing is
"ture in arithmetic". Every sentence of a first order theory that
can be proven in the theory is true in every model theory. Every
sentence of a theory that cannot be proven in the theory is false
in some model of the theory.
only because back then proof theoretic semantics did
not exist.
Every interpretation of the theory is a definition of semantics.
MetarCamath relations about numbers donrCOt exist in PA
because PA only contains arithmetical relationsrCoaddition,
multiplication, ordering, primitiverCarecursive predicates
about numbers themselvesrCowhile relations that talk about
PArCOs own proofs, syntax, or truth conditions live entirely
in the metarCatheory;
so when someone appeals to a G||delrCastyle relation like
rCLn encodes a proof of this very sentence,rCY theyrCOre
invoking a metarCamathematical predicate that PA cannot
internalize, which is exactly why your framework draws
a clean boundary between internal proofrCatheoretic truth
and external modelrCatheoretic truth.
On 21/01/2026 17:22, olcott wrote:
On 1/21/2026 3:03 AM, Mikko wrote:
No, it hasn't. In the way theories are usually discussed nothing is
"ture in arithmetic". Every sentence of a first order theory that
can be proven in the theory is true in every model theory. Every
sentence of a theory that cannot be proven in the theory is false
in some model of the theory.
only because back then proof theoretic semantics did
not exist.
Every interpretation of the theory is a definition of semantics.
MetarCamath relations about numbers donrCOt exist in PA
because PA only contains arithmetical relationsrCoaddition,
multiplication, ordering, primitiverCarecursive predicates
about numbers themselvesrCowhile relations that talk about
PArCOs own proofs, syntax, or truth conditions live entirely
in the metarCatheory;
Methamathematics does not need any other relations between numbers
than what PA has. But relations that map other things to numbers
can be useful for methamathematical purposes.
so when someone appeals to a G||delrCastyle relation like
rCLn encodes a proof of this very sentence,rCY theyrCOre
invoking a metarCamathematical predicate that PA cannot
internalize, which is exactly why your framework draws
a clean boundary between internal proofrCatheoretic truth
and external modelrCatheoretic truth.
Anyway, what can be provven that way is true aboout PA. You can deny
the proof but you cannot perform what is meta-provably impossible.
Anyway, what can be provven that way is true aboout PA. You can deny
the proof but you cannot perform what is meta-provably impossible.
On 1/22/2026 2:21 AM, Mikko wrote:
On 21/01/2026 17:22, olcott wrote:
On 1/21/2026 3:03 AM, Mikko wrote:
No, it hasn't. In the way theories are usually discussed nothing is
"ture in arithmetic". Every sentence of a first order theory that
can be proven in the theory is true in every model theory. Every
sentence of a theory that cannot be proven in the theory is false
in some model of the theory.
only because back then proof theoretic semantics did
not exist.
Every interpretation of the theory is a definition of semantics.
MetarCamath relations about numbers donrCOt exist in PA
because PA only contains arithmetical relationsrCoaddition,
multiplication, ordering, primitiverCarecursive predicates
about numbers themselvesrCowhile relations that talk about
PArCOs own proofs, syntax, or truth conditions live entirely
in the metarCatheory;
Methamathematics does not need any other relations between numbers
than what PA has. But relations that map other things to numbers
can be useful for methamathematical purposes.
so when someone appeals to a G||delrCastyle relation like
rCLn encodes a proof of this very sentence,rCY theyrCOre
invoking a metarCamathematical predicate that PA cannot
internalize, which is exactly why your framework draws
a clean boundary between internal proofrCatheoretic truth
and external modelrCatheoretic truth.
Anyway, what can be provven that way is true aboout PA. You can deny
the proof but you cannot perform what is meta-provably impossible.
G||delrCOs sentence is not rCLtrue in arithmetic.rCY
It is true only in the metarCatheory, under an
external interpretation of PA (typically the
standard model rao). Inside PA itself, the sentence
is not a truthrCabearer at all.
On 22/01/2026 18:40, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
On 21/01/2026 17:22, olcott wrote:
On 1/21/2026 3:03 AM, Mikko wrote:
No, it hasn't. In the way theories are usually discussed nothing is
"ture in arithmetic". Every sentence of a first order theory that
can be proven in the theory is true in every model theory. Every
sentence of a theory that cannot be proven in the theory is false
in some model of the theory.
only because back then proof theoretic semantics did
not exist.
Every interpretation of the theory is a definition of semantics.
MetarCamath relations about numbers donrCOt exist in PA
because PA only contains arithmetical relationsrCoaddition,
multiplication, ordering, primitiverCarecursive predicates
about numbers themselvesrCowhile relations that talk about
PArCOs own proofs, syntax, or truth conditions live entirely
in the metarCatheory;
Methamathematics does not need any other relations between numbers
than what PA has. But relations that map other things to numbers
can be useful for methamathematical purposes.
so when someone appeals to a G||delrCastyle relation like
rCLn encodes a proof of this very sentence,rCY theyrCOre
invoking a metarCamathematical predicate that PA cannot
internalize, which is exactly why your framework draws
a clean boundary between internal proofrCatheoretic truth
and external modelrCatheoretic truth.
Anyway, what can be provven that way is true aboout PA. You can deny
the proof but you cannot perform what is meta-provably impossible.
G||delrCOs sentence is not rCLtrue in arithmetic.rCY
It is true only in the metarCatheory, under an
external interpretation of PA (typically the
standard model rao). Inside PA itself, the sentence
is not a truthrCabearer at all.
There is no concept of "truth-bearer" in an uninterpreted theory because there is not concept of "truth". The relevant concept is "sell-formed- formula" and G||dels sentence is one. It may be true or false in an interpretation.
G|ndel's metatheory contains PA. In G||del's interpretation PA is
interpreted in the same way as the PA part of the metatho|-ory.
G||del proves that G of PA as interpreted in the metatheory is
true but cannot be proven in PA.
On 1/23/2026 3:13 AM, Mikko wrote:
On 22/01/2026 18:40, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
On 21/01/2026 17:22, olcott wrote:
On 1/21/2026 3:03 AM, Mikko wrote:
No, it hasn't. In the way theories are usually discussed nothing is >>>>>> "ture in arithmetic". Every sentence of a first order theory that
can be proven in the theory is true in every model theory. Every
sentence of a theory that cannot be proven in the theory is false
in some model of the theory.
only because back then proof theoretic semantics did
not exist.
Every interpretation of the theory is a definition of semantics.
MetarCamath relations about numbers donrCOt exist in PA
because PA only contains arithmetical relationsrCoaddition,
multiplication, ordering, primitiverCarecursive predicates
about numbers themselvesrCowhile relations that talk about
PArCOs own proofs, syntax, or truth conditions live entirely
in the metarCatheory;
Methamathematics does not need any other relations between numbers
than what PA has. But relations that map other things to numbers
can be useful for methamathematical purposes.
so when someone appeals to a G||delrCastyle relation like
rCLn encodes a proof of this very sentence,rCY theyrCOre
invoking a metarCamathematical predicate that PA cannot
internalize, which is exactly why your framework draws
a clean boundary between internal proofrCatheoretic truth
and external modelrCatheoretic truth.
Anyway, what can be provven that way is true aboout PA. You can deny
the proof but you cannot perform what is meta-provably impossible.
G||delrCOs sentence is not rCLtrue in arithmetic.rCY
It is true only in the metarCatheory, under an
external interpretation of PA (typically the
standard model rao). Inside PA itself, the sentence
is not a truthrCabearer at all.
There is no concept of "truth-bearer" in an uninterpreted theory because
there is not concept of "truth". The relevant concept is "sell-formed-
formula" and G||dels sentence is one. It may be true or false in an
interpretation.
There is a
"true on the basis of meaning expressed in language"
and I figured out how to make it computable over the
body of knowledge.
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can deny
the proof but you cannot perform what is meta-provably impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
On 23/01/2026 12:22, olcott wrote:
On 1/23/2026 3:13 AM, Mikko wrote:
On 22/01/2026 18:40, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
On 21/01/2026 17:22, olcott wrote:
On 1/21/2026 3:03 AM, Mikko wrote:
No, it hasn't. In the way theories are usually discussed nothing is >>>>>>> "ture in arithmetic". Every sentence of a first order theory that >>>>>>> can be proven in the theory is true in every model theory. Every >>>>>>> sentence of a theory that cannot be proven in the theory is false >>>>>>> in some model of the theory.
only because back then proof theoretic semantics did
not exist.
Every interpretation of the theory is a definition of semantics. >>>>>>>
MetarCamath relations about numbers donrCOt exist in PA
because PA only contains arithmetical relationsrCoaddition,
multiplication, ordering, primitiverCarecursive predicates
about numbers themselvesrCowhile relations that talk about
PArCOs own proofs, syntax, or truth conditions live entirely
in the metarCatheory;
Methamathematics does not need any other relations between numbers
than what PA has. But relations that map other things to numbers
can be useful for methamathematical purposes.
so when someone appeals to a G||delrCastyle relation like
rCLn encodes a proof of this very sentence,rCY theyrCOre
invoking a metarCamathematical predicate that PA cannot
internalize, which is exactly why your framework draws
a clean boundary between internal proofrCatheoretic truth
and external modelrCatheoretic truth.
Anyway, what can be provven that way is true aboout PA. You can deny >>>>> the proof but you cannot perform what is meta-provably impossible.
G||delrCOs sentence is not rCLtrue in arithmetic.rCY
It is true only in the metarCatheory, under an
external interpretation of PA (typically the
standard model rao). Inside PA itself, the sentence
is not a truthrCabearer at all.
There is no concept of "truth-bearer" in an uninterpreted theory because >>> there is not concept of "truth". The relevant concept is "sell-formed-
formula" and G||dels sentence is one. It may be true or false in an
interpretation.
There is a
"true on the basis of meaning expressed in language"
and I figured out how to make it computable over the
body of knowledge.
Except that "true on the basis of meaning expressed in language" is
nmt computable and does not cover all of the body of knowldge.
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can deny
the proof but you cannot perform what is meta-provably impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which.
But every proof in PA is also
a proof in G||del's metatheory.
On 1/20/2026 3:58 AM, Mikko wrote:
On 19/01/2026 17:03, olcott wrote:
On 1/19/2026 2:19 AM, Mikko wrote:
On 18/01/2026 15:28, olcott wrote:
On 1/18/2026 5:27 AM, Mikko wrote:
On 17/01/2026 16:47, olcott wrote:
On 1/17/2026 3:53 AM, Mikko wrote:No, it does not. It is just another exammle of the generic concept >>>>>> of set theory. Essentially the same as ZF but has one additional
On 16/01/2026 17:38, olcott wrote:
On 1/16/2026 3:32 AM, Mikko wrote:
On 15/01/2026 22:30, olcott wrote:Proof theoretic semantics addresses G||del Incompleteness
On 1/15/2026 3:34 AM, Mikko wrote:
On 14/01/2026 21:32, olcott wrote:
On 1/14/2026 3:01 AM, Mikko wrote:
On 13/01/2026 16:31, olcott wrote:
On 1/13/2026 3:13 AM, Mikko wrote:
On 12/01/2026 16:32, olcott wrote:
On 1/12/2026 4:47 AM, Mikko wrote:
On 11/01/2026 16:24, Tristan Wibberley wrote: >>>>>>>>>>>>>>>>>>> On 11/01/2026 10:13, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
No, that does not follow. If a required result >>>>>>>>>>>>>>>>>>>>>> cannot be derived by
appying a finite string transformation then the it >>>>>>>>>>>>>>>>>>>>>> it is uncomputable.
Right. Outside the scope of computation. Requiring >>>>>>>>>>>>>>>>>>>>> anything
outside the scope of computation is an incorrect >>>>>>>>>>>>>>>>>>>>> requirement.
You can't determine whether the required result is >>>>>>>>>>>>>>>>>>>> computable before
you have the requirement.
Right, it is /in/ scope for computer science... for >>>>>>>>>>>>>>>>>>> the / ology/. Olcott
here uses "computation" to refer to the practice. You >>>>>>>>>>>>>>>>>>> give the
requirement to the /ologist/ who correctly decides >>>>>>>>>>>>>>>>>>> that it is not for
computation because it is not computable. >>>>>>>>>>>>>>>>>>>
You two so often violently agree; I find it warming >>>>>>>>>>>>>>>>>>> to the heart.
For pracitcal programming it is useful to know what is >>>>>>>>>>>>>>>>>> known to be
uncomputable in order to avoid wasting time in >>>>>>>>>>>>>>>>>> attemlpts to do the
impossible.
It f-cking nuts that after more than 2000 years >>>>>>>>>>>>>>>>> people still don't understand that self-contradictory >>>>>>>>>>>>>>>>> expressions: "This sentence is not true" have no >>>>>>>>>>>>>>>>> truth value. A smart high school student should have >>>>>>>>>>>>>>>>> figured this out 2000 years ago.
Irrelevant. For practical programming that question >>>>>>>>>>>>>>>> needn't be answered.
The halting problem counter-example input is anchored >>>>>>>>>>>>>>> in the Liar Paradox. Proof Theoretic Semantics rejects >>>>>>>>>>>>>>> those two and G||del's incompleteness and a bunch more >>>>>>>>>>>>>>> as merely non-well-founded inputs.
For every Turing machine the halting problem counter- >>>>>>>>>>>>>> example provably
exists.
Not when using Proof Theoretic Semantics grounded
in the specification language. In this case the
pathological input is simply rejected as ungrounded.
Then your "Proof Theoretic Semantics" is not useful for >>>>>>>>>>>> discussion of
Turing machines. For every Turing machine a counter example >>>>>>>>>>>> exists.
And so exists a Turing machine that writes the counter >>>>>>>>>>>> example when
given a Turing machine as input.
It is "not useful" in the same way that ZFC was
"not useful" for addressing Russell's Paradox.
ZF or ZFC is to some extent useful for addressing Russell's >>>>>>>>>> paradox.
It is an example of a set theory where Russell's paradox is >>>>>>>>>> avoided.
If your "Proof Theretic Semantics" cannot handle the existence of >>>>>>>>>> a counter example for every Turing decider then it is not usefule >>>>>>>>>> for those who work on practical problems of program correctness. >>>>>>>>>
for PA in a way similar to the way that ZFC addresses
Russell's Paradox in set theory.
Not really the same way. Your "Proof theoretic semantics" redefines >>>>>>>> truth and replaces the logic. ZFC is another theory using ordinary >>>>>>>> logic. The problem with the naive set theory is that it is not >>>>>>>> sound for any semantics.
ZFC redefines set theory such that Russell's Paradox cannot arise. >>>>>>
postulate.
ZFC redefines set theory such that Russell's Paradox cannot arise
and the original set theory is now referred to as naive set theory.
ZF and ZFC are not redefinitions. ZF is another theory. It can be
called a "set theory" because its structure is similar to Cnator's
original informal set theory. Cantor did not specify whther a set
must be well-founded but ZF specifies that it must. A set theory
were all sets are well-founded does not have Russell's paradox.
ZF is a redefinition in the only sense that matters:
it changes the foundational rules so that RussellrCOs
paradox cannot arise.
The only sense that matters is: to give a new meaning to an exsisting
term. That is OK when the new meaning is only used in a context where
the old one does not make sense.
What you are trying is to give a new meaning to "true" but preted that
it still means 'true'.
True in the standard model of arithmetic using meta-math
has always been misconstrued as true <in> arithmetic
only because back then proof theoretic semantics did
not exist.
No one ever understood how a truth predicate could be
directly added to PA. Now with Proof theoretic semantics
and the Haskell Curry notion of true in the system it
is easy to directly define a truth predicate <is> PA.
Truth in the standard model is metarCamathematical.
Truth in PA is proofrCatheoretic. These were historically
conflated only because proofrCatheoretic semantics did not
exist. With CurryrCOs notion of internal truth, PArCOs truth
predicate is simply:
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
On 1/20/26 1:35 PM, olcott wrote:
On 1/20/2026 3:58 AM, Mikko wrote:
On 19/01/2026 17:03, olcott wrote:
On 1/19/2026 2:19 AM, Mikko wrote:
On 18/01/2026 15:28, olcott wrote:
On 1/18/2026 5:27 AM, Mikko wrote:ZF and ZFC are not redefinitions. ZF is another theory. It can be
On 17/01/2026 16:47, olcott wrote:
On 1/17/2026 3:53 AM, Mikko wrote:No, it does not. It is just another exammle of the generic concept >>>>>>> of set theory. Essentially the same as ZF but has one additional >>>>>>> postulate.
On 16/01/2026 17:38, olcott wrote:
On 1/16/2026 3:32 AM, Mikko wrote:
On 15/01/2026 22:30, olcott wrote:Proof theoretic semantics addresses G||del Incompleteness
On 1/15/2026 3:34 AM, Mikko wrote:
On 14/01/2026 21:32, olcott wrote:
On 1/14/2026 3:01 AM, Mikko wrote:Then your "Proof Theoretic Semantics" is not useful for >>>>>>>>>>>>> discussion of
On 13/01/2026 16:31, olcott wrote:
On 1/13/2026 3:13 AM, Mikko wrote:
On 12/01/2026 16:32, olcott wrote:
On 1/12/2026 4:47 AM, Mikko wrote:
On 11/01/2026 16:24, Tristan Wibberley wrote: >>>>>>>>>>>>>>>>>>>> On 11/01/2026 10:13, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
No, that does not follow. If a required result >>>>>>>>>>>>>>>>>>>>>>> cannot be derived by
appying a finite string transformation then the >>>>>>>>>>>>>>>>>>>>>>> it it is uncomputable.
Right. Outside the scope of computation. Requiring >>>>>>>>>>>>>>>>>>>>>> anything
outside the scope of computation is an incorrect >>>>>>>>>>>>>>>>>>>>>> requirement.
You can't determine whether the required result is >>>>>>>>>>>>>>>>>>>>> computable before
you have the requirement.
Right, it is /in/ scope for computer science... for >>>>>>>>>>>>>>>>>>>> the / ology/. Olcott
here uses "computation" to refer to the practice. >>>>>>>>>>>>>>>>>>>> You give the
requirement to the /ologist/ who correctly decides >>>>>>>>>>>>>>>>>>>> that it is not for
computation because it is not computable. >>>>>>>>>>>>>>>>>>>>
You two so often violently agree; I find it warming >>>>>>>>>>>>>>>>>>>> to the heart.
For pracitcal programming it is useful to know what >>>>>>>>>>>>>>>>>>> is known to be
uncomputable in order to avoid wasting time in >>>>>>>>>>>>>>>>>>> attemlpts to do the
impossible.
It f-cking nuts that after more than 2000 years >>>>>>>>>>>>>>>>>> people still don't understand that self-contradictory >>>>>>>>>>>>>>>>>> expressions: "This sentence is not true" have no >>>>>>>>>>>>>>>>>> truth value. A smart high school student should have >>>>>>>>>>>>>>>>>> figured this out 2000 years ago.
Irrelevant. For practical programming that question >>>>>>>>>>>>>>>>> needn't be answered.
The halting problem counter-example input is anchored >>>>>>>>>>>>>>>> in the Liar Paradox. Proof Theoretic Semantics rejects >>>>>>>>>>>>>>>> those two and G||del's incompleteness and a bunch more >>>>>>>>>>>>>>>> as merely non-well-founded inputs.
For every Turing machine the halting problem counter- >>>>>>>>>>>>>>> example provably
exists.
Not when using Proof Theoretic Semantics grounded
in the specification language. In this case the
pathological input is simply rejected as ungrounded. >>>>>>>>>>>>>
Turing machines. For every Turing machine a counter example >>>>>>>>>>>>> exists.
And so exists a Turing machine that writes the counter >>>>>>>>>>>>> example when
given a Turing machine as input.
It is "not useful" in the same way that ZFC was
"not useful" for addressing Russell's Paradox.
ZF or ZFC is to some extent useful for addressing Russell's >>>>>>>>>>> paradox.
It is an example of a set theory where Russell's paradox is >>>>>>>>>>> avoided.
If your "Proof Theretic Semantics" cannot handle the
existence of
a counter example for every Turing decider then it is not >>>>>>>>>>> usefule
for those who work on practical problems of program correctness. >>>>>>>>>>
for PA in a way similar to the way that ZFC addresses
Russell's Paradox in set theory.
Not really the same way. Your "Proof theoretic semantics"
redefines
truth and replaces the logic. ZFC is another theory using ordinary >>>>>>>>> logic. The problem with the naive set theory is that it is not >>>>>>>>> sound for any semantics.
ZFC redefines set theory such that Russell's Paradox cannot arise. >>>>>>>
ZFC redefines set theory such that Russell's Paradox cannot arise
and the original set theory is now referred to as naive set theory. >>>>>
called a "set theory" because its structure is similar to Cnator's
original informal set theory. Cantor did not specify whther a set
must be well-founded but ZF specifies that it must. A set theory
were all sets are well-founded does not have Russell's paradox.
ZF is a redefinition in the only sense that matters:
it changes the foundational rules so that RussellrCOs
paradox cannot arise.
The only sense that matters is: to give a new meaning to an exsisting
term. That is OK when the new meaning is only used in a context where
the old one does not make sense.
What you are trying is to give a new meaning to "true" but preted that
it still means 'true'.
True in the standard model of arithmetic using meta-math
has always been misconstrued as true <in> arithmetic
only because back then proof theoretic semantics did
not exist.
But no one says that the information in the meta-math CHANGED the
behavior of the arithmatic, in fact, it specificially doesn't.
No one ever understood how a truth predicate could be
directly added to PA. Now with Proof theoretic semantics
and the Haskell Curry notion of true in the system it
is easy to directly define a truth predicate <is> PA.
No, Tarski showed what happens if you add a presumed working Truth
Predicate to PA, it breaks the system.
Truth in the standard model is metarCamathematical.
Nope, but then, you don't understand what Truth actually is.
Truth in PA is proofrCatheoretic. These were historically
conflated only because proofrCatheoretic semantics did not
exist. With CurryrCOs notion of internal truth, PArCOs truth
predicate is simply:
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
Which isn't a predicate as it doesn't give a value for all possible x's.
As there exist x's that are neither provable or refutable in PA.--
Perhaps your problem is you don't understand what a PREDICATE is.
And then you have the problem that "PA reo x" can't always be determined
by purely Proof-Theoretic analysis, so we also end up with statements
that might be true, or might be false, or might not have a truth value,
or maybe even can't be classified into one of those by proof-theoretic semantics.
On 1/24/2026 8:51 AM, Richard Damon wrote:
On 1/20/26 1:35 PM, olcott wrote:
On 1/20/2026 3:58 AM, Mikko wrote:
On 19/01/2026 17:03, olcott wrote:
On 1/19/2026 2:19 AM, Mikko wrote:
On 18/01/2026 15:28, olcott wrote:
On 1/18/2026 5:27 AM, Mikko wrote:ZF and ZFC are not redefinitions. ZF is another theory. It can be
On 17/01/2026 16:47, olcott wrote:
On 1/17/2026 3:53 AM, Mikko wrote:No, it does not. It is just another exammle of the generic concept >>>>>>>> of set theory. Essentially the same as ZF but has one additional >>>>>>>> postulate.
On 16/01/2026 17:38, olcott wrote:
On 1/16/2026 3:32 AM, Mikko wrote:
On 15/01/2026 22:30, olcott wrote:
On 1/15/2026 3:34 AM, Mikko wrote:
On 14/01/2026 21:32, olcott wrote:
On 1/14/2026 3:01 AM, Mikko wrote:Then your "Proof Theoretic Semantics" is not useful for >>>>>>>>>>>>>> discussion of
On 13/01/2026 16:31, olcott wrote:
On 1/13/2026 3:13 AM, Mikko wrote:
On 12/01/2026 16:32, olcott wrote:
On 1/12/2026 4:47 AM, Mikko wrote:
On 11/01/2026 16:24, Tristan Wibberley wrote: >>>>>>>>>>>>>>>>>>>>> On 11/01/2026 10:13, Mikko wrote:
On 10/01/2026 17:47, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>> On 1/10/2026 2:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>
No, that does not follow. If a required result >>>>>>>>>>>>>>>>>>>>>>>> cannot be derived by
appying a finite string transformation then the >>>>>>>>>>>>>>>>>>>>>>>> it it is uncomputable.
Right. Outside the scope of computation. >>>>>>>>>>>>>>>>>>>>>>> Requiring anything
outside the scope of computation is an incorrect >>>>>>>>>>>>>>>>>>>>>>> requirement.
You can't determine whether the required result is >>>>>>>>>>>>>>>>>>>>>> computable before
you have the requirement.
Right, it is /in/ scope for computer science... for >>>>>>>>>>>>>>>>>>>>> the / ology/. Olcott
here uses "computation" to refer to the practice. >>>>>>>>>>>>>>>>>>>>> You give the
requirement to the /ologist/ who correctly decides >>>>>>>>>>>>>>>>>>>>> that it is not for
computation because it is not computable. >>>>>>>>>>>>>>>>>>>>>
You two so often violently agree; I find it warming >>>>>>>>>>>>>>>>>>>>> to the heart.
For pracitcal programming it is useful to know what >>>>>>>>>>>>>>>>>>>> is known to be
uncomputable in order to avoid wasting time in >>>>>>>>>>>>>>>>>>>> attemlpts to do the
impossible.
It f-cking nuts that after more than 2000 years >>>>>>>>>>>>>>>>>>> people still don't understand that self-contradictory >>>>>>>>>>>>>>>>>>> expressions: "This sentence is not true" have no >>>>>>>>>>>>>>>>>>> truth value. A smart high school student should have >>>>>>>>>>>>>>>>>>> figured this out 2000 years ago.
Irrelevant. For practical programming that question >>>>>>>>>>>>>>>>>> needn't be answered.
The halting problem counter-example input is anchored >>>>>>>>>>>>>>>>> in the Liar Paradox. Proof Theoretic Semantics rejects >>>>>>>>>>>>>>>>> those two and G||del's incompleteness and a bunch more >>>>>>>>>>>>>>>>> as merely non-well-founded inputs.
For every Turing machine the halting problem counter- >>>>>>>>>>>>>>>> example provably
exists.
Not when using Proof Theoretic Semantics grounded >>>>>>>>>>>>>>> in the specification language. In this case the
pathological input is simply rejected as ungrounded. >>>>>>>>>>>>>>
Turing machines. For every Turing machine a counter >>>>>>>>>>>>>> example exists.
And so exists a Turing machine that writes the counter >>>>>>>>>>>>>> example when
given a Turing machine as input.
It is "not useful" in the same way that ZFC was
"not useful" for addressing Russell's Paradox.
ZF or ZFC is to some extent useful for addressing Russell's >>>>>>>>>>>> paradox.
It is an example of a set theory where Russell's paradox is >>>>>>>>>>>> avoided.
If your "Proof Theretic Semantics" cannot handle the
existence of
a counter example for every Turing decider then it is not >>>>>>>>>>>> usefule
for those who work on practical problems of program
correctness.
Proof theoretic semantics addresses G||del Incompleteness >>>>>>>>>>> for PA in a way similar to the way that ZFC addresses
Russell's Paradox in set theory.
Not really the same way. Your "Proof theoretic semantics" >>>>>>>>>> redefines
truth and replaces the logic. ZFC is another theory using >>>>>>>>>> ordinary
logic. The problem with the naive set theory is that it is not >>>>>>>>>> sound for any semantics.
ZFC redefines set theory such that Russell's Paradox cannot arise. >>>>>>>>
ZFC redefines set theory such that Russell's Paradox cannot arise >>>>>>> and the original set theory is now referred to as naive set theory. >>>>>>
called a "set theory" because its structure is similar to Cnator's >>>>>> original informal set theory. Cantor did not specify whther a set
must be well-founded but ZF specifies that it must. A set theory
were all sets are well-founded does not have Russell's paradox.
ZF is a redefinition in the only sense that matters:
it changes the foundational rules so that RussellrCOs
paradox cannot arise.
The only sense that matters is: to give a new meaning to an exsisting
term. That is OK when the new meaning is only used in a context where
the old one does not make sense.
What you are trying is to give a new meaning to "true" but preted that >>>> it still means 'true'.
True in the standard model of arithmetic using meta-math
has always been misconstrued as true <in> arithmetic
only because back then proof theoretic semantics did
not exist.
But no one says that the information in the meta-math CHANGED the
behavior of the arithmatic, in fact, it specificially doesn't.
The statement that G is true and unprovable in PA has
always been counter-factual. It has never actually been
true <in> PA and that is why it is unprovable in PA.
No one ever understood how a truth predicate could be
directly added to PA. Now with Proof theoretic semantics
and the Haskell Curry notion of true in the system it
is easy to directly define a truth predicate <is> PA.
No, Tarski showed what happens if you add a presumed working Truth
Predicate to PA, it breaks the system.
Only when we use model theory
Swap the foundation to proof theory
and the problem goes away.
Truth in the standard model is metarCamathematical.
Nope, but then, you don't understand what Truth actually is.
Truth in PA is proofrCatheoretic. These were historically
conflated only because proofrCatheoretic semantics did not
exist. With CurryrCOs notion of internal truth, PArCOs truth
predicate is simply:
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
Which isn't a predicate as it doesn't give a value for all possible x's.
Is this sentence true or false: "What time is it?"
A truth predicate can be defined over the domain
of meaningful truth-apt expressions.
As there exist x's that are neither provable or refutable in PA.
Perhaps your problem is you don't understand what a PREDICATE is.
And then you have the problem that "PA reo x" can't always be determined
by purely Proof-Theoretic analysis, so we also end up with statements
that might be true, or might be false, or might not have a truth
value, or maybe even can't be classified into one of those by proof-
theoretic semantics.
On 1/24/26 10:44 AM, olcott wrote:
On 1/24/2026 8:51 AM, Richard Damon wrote:
On 1/20/26 1:35 PM, olcott wrote:
On 1/20/2026 3:58 AM, Mikko wrote:
On 19/01/2026 17:03, olcott wrote:
On 1/19/2026 2:19 AM, Mikko wrote:
On 18/01/2026 15:28, olcott wrote:
On 1/18/2026 5:27 AM, Mikko wrote:ZF and ZFC are not redefinitions. ZF is another theory. It can be >>>>>>> called a "set theory" because its structure is similar to Cnator's >>>>>>> original informal set theory. Cantor did not specify whther a set >>>>>>> must be well-founded but ZF specifies that it must. A set theory >>>>>>> were all sets are well-founded does not have Russell's paradox.
On 17/01/2026 16:47, olcott wrote:
On 1/17/2026 3:53 AM, Mikko wrote:
On 16/01/2026 17:38, olcott wrote:
On 1/16/2026 3:32 AM, Mikko wrote:
On 15/01/2026 22:30, olcott wrote:
On 1/15/2026 3:34 AM, Mikko wrote:
On 14/01/2026 21:32, olcott wrote:
On 1/14/2026 3:01 AM, Mikko wrote:Then your "Proof Theoretic Semantics" is not useful for >>>>>>>>>>>>>>> discussion of
On 13/01/2026 16:31, olcott wrote:
On 1/13/2026 3:13 AM, Mikko wrote:
On 12/01/2026 16:32, olcott wrote:
On 1/12/2026 4:47 AM, Mikko wrote:
On 11/01/2026 16:24, Tristan Wibberley wrote: >>>>>>>>>>>>>>>>>>>>>> On 11/01/2026 10:13, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 10/01/2026 17:47, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 1/10/2026 2:23 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>
No, that does not follow. If a required result >>>>>>>>>>>>>>>>>>>>>>>>> cannot be derived by
appying a finite string transformation then the >>>>>>>>>>>>>>>>>>>>>>>>> it it is uncomputable.
Right. Outside the scope of computation. >>>>>>>>>>>>>>>>>>>>>>>> Requiring anything
outside the scope of computation is an incorrect >>>>>>>>>>>>>>>>>>>>>>>> requirement.
You can't determine whether the required result >>>>>>>>>>>>>>>>>>>>>>> is computable before
you have the requirement.
Right, it is /in/ scope for computer science... >>>>>>>>>>>>>>>>>>>>>> for the / ology/. Olcott
here uses "computation" to refer to the practice. >>>>>>>>>>>>>>>>>>>>>> You give the
requirement to the /ologist/ who correctly decides >>>>>>>>>>>>>>>>>>>>>> that it is not for
computation because it is not computable. >>>>>>>>>>>>>>>>>>>>>>
You two so often violently agree; I find it >>>>>>>>>>>>>>>>>>>>>> warming to the heart.
For pracitcal programming it is useful to know what >>>>>>>>>>>>>>>>>>>>> is known to be
uncomputable in order to avoid wasting time in >>>>>>>>>>>>>>>>>>>>> attemlpts to do the
impossible.
It f-cking nuts that after more than 2000 years >>>>>>>>>>>>>>>>>>>> people still don't understand that self-contradictory >>>>>>>>>>>>>>>>>>>> expressions: "This sentence is not true" have no >>>>>>>>>>>>>>>>>>>> truth value. A smart high school student should have >>>>>>>>>>>>>>>>>>>> figured this out 2000 years ago.
Irrelevant. For practical programming that question >>>>>>>>>>>>>>>>>>> needn't be answered.
The halting problem counter-example input is anchored >>>>>>>>>>>>>>>>>> in the Liar Paradox. Proof Theoretic Semantics rejects >>>>>>>>>>>>>>>>>> those two and G||del's incompleteness and a bunch more >>>>>>>>>>>>>>>>>> as merely non-well-founded inputs.
For every Turing machine the halting problem counter- >>>>>>>>>>>>>>>>> example provably
exists.
Not when using Proof Theoretic Semantics grounded >>>>>>>>>>>>>>>> in the specification language. In this case the >>>>>>>>>>>>>>>> pathological input is simply rejected as ungrounded. >>>>>>>>>>>>>>>
Turing machines. For every Turing machine a counter >>>>>>>>>>>>>>> example exists.
And so exists a Turing machine that writes the counter >>>>>>>>>>>>>>> example when
given a Turing machine as input.
It is "not useful" in the same way that ZFC was
"not useful" for addressing Russell's Paradox.
ZF or ZFC is to some extent useful for addressing Russell's >>>>>>>>>>>>> paradox.
It is an example of a set theory where Russell's paradox is >>>>>>>>>>>>> avoided.
If your "Proof Theretic Semantics" cannot handle the >>>>>>>>>>>>> existence of
a counter example for every Turing decider then it is not >>>>>>>>>>>>> usefule
for those who work on practical problems of program >>>>>>>>>>>>> correctness.
Proof theoretic semantics addresses G||del Incompleteness >>>>>>>>>>>> for PA in a way similar to the way that ZFC addresses
Russell's Paradox in set theory.
Not really the same way. Your "Proof theoretic semantics" >>>>>>>>>>> redefines
truth and replaces the logic. ZFC is another theory using >>>>>>>>>>> ordinary
logic. The problem with the naive set theory is that it is not >>>>>>>>>>> sound for any semantics.
ZFC redefines set theory such that Russell's Paradox cannot >>>>>>>>>> arise.
No, it does not. It is just another exammle of the generic concept >>>>>>>>> of set theory. Essentially the same as ZF but has one additional >>>>>>>>> postulate.
ZFC redefines set theory such that Russell's Paradox cannot arise >>>>>>>> and the original set theory is now referred to as naive set theory. >>>>>>>
ZF is a redefinition in the only sense that matters:
it changes the foundational rules so that RussellrCOs
paradox cannot arise.
The only sense that matters is: to give a new meaning to an exsisting >>>>> term. That is OK when the new meaning is only used in a context where >>>>> the old one does not make sense.
What you are trying is to give a new meaning to "true" but preted that >>>>> it still means 'true'.
True in the standard model of arithmetic using meta-math
has always been misconstrued as true <in> arithmetic
only because back then proof theoretic semantics did
not exist.
But no one says that the information in the meta-math CHANGED the
behavior of the arithmatic, in fact, it specificially doesn't.
The statement that G is true and unprovable in PA has
always been counter-factual. It has never actually been
true <in> PA and that is why it is unprovable in PA.
Sure it is. At least it is a FACT that no natural number will statisfy
that relationship, and there is no proof in PA of that fact.
IF you want to define that the statement isn't called true because you
can't prove it, then your definition of truth just ends up being problematical as you can't say any of:
It is true (as you can't prove it)
It is false (since you can't prove that either)
It is not-well-founded, since you can't prove that statement either, as proving that you can't prove it false ends up being a proof that it is
true, which gives us a number that makes it false.
Thus, in a pure Proof-Theoretic Semantics framework, all you can say is
you don't know the truth category of the statement (True, False, Non- Well-Founded), or even if there IS a truth category of the statement. It turns out it is just a statement that Proof-Theretics Semantics can't
talk about, and shows that such a framework can't even decide if it can
talk about a given statement until its actual answer is known.
This fundamentally breaks the system from being usable.
No one ever understood how a truth predicate could be
directly added to PA. Now with Proof theoretic semantics
and the Haskell Curry notion of true in the system it
is easy to directly define a truth predicate <is> PA.
No, Tarski showed what happens if you add a presumed working Truth
Predicate to PA, it breaks the system.
Only when we use model theory
Swap the foundation to proof theory
and the problem goes away.
Nope. The basics of mathematics itself, which *IS* what PA has defined, breaks it
Your problem is you don't understand what PA actually entails.
Truth in the standard model is metarCamathematical.
Nope, but then, you don't understand what Truth actually is.
Truth in PA is proofrCatheoretic. These were historically
conflated only because proofrCatheoretic semantics did not
exist. With CurryrCOs notion of internal truth, PArCOs truth
predicate is simply:
reCx ree PA ((True(PA, x)-a rei (PA reo x))
reCx ree PA ((False(PA, x) rei (PA reo ~x))
Which isn't a predicate as it doesn't give a value for all possible x's. >>>
Is this sentence true or false: "What time is it?"
Fallacy of Proof by example.
A truth predicate can be defined over the domain
of meaningful truth-apt expressions.
Nope.
What value does it give to G, the statement that no number exists that satisifies the specified computatable relationship that was developed in Godels proof?
What value CAN it give to it? (That might be correct)
Remember, Proof-Theoretic only asserts what it can prove, so to assert
that it is not-well-founded means it can prove that it can't be proven false, and since a simple proof of falsehood is showing a specific
number g exists that satisfies it, but since proving that no such number
g exists that satisfies it is proving the statement of G itself, so you won't be able to prove that no such proof exists, since you have one.
As there exist x's that are neither provable or refutable in PA.
Perhaps your problem is you don't understand what a PREDICATE is.
And then you have the problem that "PA reo x" can't always be
determined by purely Proof-Theoretic analysis, so we also end up with
statements that might be true, or might be false, or might not have a
truth value, or maybe even can't be classified into one of those by
proof- theoretic semantics.
On 1/24/2026 1:52 PM, Richard Damon wrote:
On 1/24/26 2:25 PM, olcott wrote:
On 1/24/2026 1:23 PM, Richard Damon wrote:
On 1/24/26 12:54 PM, olcott wrote:G||delrCOs incompleteness theorem only rCLworksrCY if
On 1/24/2026 11:10 AM, Richard Damon wrote:
On 1/24/26 10:44 AM, olcott wrote:
The statement that G is true and unprovable in PA has
always been counter-factual. It has never actually been
true <in> PA and that is why it is unprovable in PA.
Sure it is. At least it is a FACT that no natural number will
statisfy that relationship, and there is no proof in PA of that fact. >>>>>>
Have you ever heard of: "true in the standard model of arithmetic"?
Sure, but they are not in Peano Arithmatic, but are (generally) 1st
order variations of the Peano Axioms which lead to alternate number
systems.
Godel's proof is statd to be in a system with at least the
properties of Peano Arithmatic, having the ability to show the
properties of the "Natural Numbers"
one smuggles in an external notion of truth
(truth in rao) and then pretends it is an
internal notion of truth (truth in PA).
If we refuse to make that identification,
incompleteness evaporates.
But Truth in N is part of Peano Arithmatic, as Peano Arithmatic is a
axiomiation to create the Natural Numbers.
You have that backwards. Truth in rao requires PA
as part of it, and PA itself has no notion of
Truth in rao. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in rao.
On 1/24/26 3:38 PM, olcott wrote:
On 1/24/2026 1:52 PM, Richard Damon wrote:
On 1/24/26 2:25 PM, olcott wrote:
On 1/24/2026 1:23 PM, Richard Damon wrote:
On 1/24/26 12:54 PM, olcott wrote:G||delrCOs incompleteness theorem only rCLworksrCY if
On 1/24/2026 11:10 AM, Richard Damon wrote:
On 1/24/26 10:44 AM, olcott wrote:
The statement that G is true and unprovable in PA has
always been counter-factual. It has never actually been
true <in> PA and that is why it is unprovable in PA.
Sure it is. At least it is a FACT that no natural number will
statisfy that relationship, and there is no proof in PA of that >>>>>>> fact.
Have you ever heard of: "true in the standard model of arithmetic"? >>>>>
Sure, but they are not in Peano Arithmatic, but are (generally) 1st >>>>> order variations of the Peano Axioms which lead to alternate number >>>>> systems.
Godel's proof is statd to be in a system with at least the
properties of Peano Arithmatic, having the ability to show the
properties of the "Natural Numbers"
one smuggles in an external notion of truth
(truth in rao) and then pretends it is an
internal notion of truth (truth in PA).
If we refuse to make that identification,
incompleteness evaporates.
But Truth in N is part of Peano Arithmatic, as Peano Arithmatic is a
axiomiation to create the Natural Numbers.
You have that backwards. Truth in rao requires PA
as part of it, and PA itself has no notion of
Truth in rao. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in rao.
Which means you don't understand what N actually is.
Nothing can be "True in N" unless that truth comes from the Axioms of
PA, as N is the result of PA.
But then, your claim of not knowing what is true in the world you are creating somes on point for you.--
On 1/24/2026 4:25 PM, Richard Damon wrote:
On 1/24/26 3:38 PM, olcott wrote:
On 1/24/2026 1:52 PM, Richard Damon wrote:
On 1/24/26 2:25 PM, olcott wrote:
On 1/24/2026 1:23 PM, Richard Damon wrote:
On 1/24/26 12:54 PM, olcott wrote:G||delrCOs incompleteness theorem only rCLworksrCY if
On 1/24/2026 11:10 AM, Richard Damon wrote:
On 1/24/26 10:44 AM, olcott wrote:
The statement that G is true and unprovable in PA has
always been counter-factual. It has never actually been
true <in> PA and that is why it is unprovable in PA.
Sure it is. At least it is a FACT that no natural number will >>>>>>>> statisfy that relationship, and there is no proof in PA of that >>>>>>>> fact.
Have you ever heard of: "true in the standard model of arithmetic"? >>>>>>
Sure, but they are not in Peano Arithmatic, but are (generally)
1st order variations of the Peano Axioms which lead to alternate
number systems.
Godel's proof is statd to be in a system with at least the
properties of Peano Arithmatic, having the ability to show the
properties of the "Natural Numbers"
one smuggles in an external notion of truth
(truth in rao) and then pretends it is an
internal notion of truth (truth in PA).
If we refuse to make that identification,
incompleteness evaporates.
But Truth in N is part of Peano Arithmatic, as Peano Arithmatic is a
axiomiation to create the Natural Numbers.
You have that backwards. Truth in rao requires PA
as part of it, and PA itself has no notion of
Truth in rao. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in rao.
Which means you don't understand what N actually is.
Nothing can be "True in N" unless that truth comes from the Axioms of
PA, as N is the result of PA.
combined with the meta-math external model.
But then, your claim of not knowing what is true in the world you are
creating somes on point for you.
On 1/24/26 5:31 PM, olcott wrote:
On 1/24/2026 4:25 PM, Richard Damon wrote:
On 1/24/26 3:38 PM, olcott wrote:
On 1/24/2026 1:52 PM, Richard Damon wrote:
On 1/24/26 2:25 PM, olcott wrote:
On 1/24/2026 1:23 PM, Richard Damon wrote:
On 1/24/26 12:54 PM, olcott wrote:G||delrCOs incompleteness theorem only rCLworksrCY if
On 1/24/2026 11:10 AM, Richard Damon wrote:
On 1/24/26 10:44 AM, olcott wrote:
The statement that G is true and unprovable in PA has
always been counter-factual. It has never actually been
true <in> PA and that is why it is unprovable in PA.
Sure it is. At least it is a FACT that no natural number will >>>>>>>>> statisfy that relationship, and there is no proof in PA of that >>>>>>>>> fact.
Have you ever heard of: "true in the standard model of arithmetic"? >>>>>>>
Sure, but they are not in Peano Arithmatic, but are (generally) >>>>>>> 1st order variations of the Peano Axioms which lead to alternate >>>>>>> number systems.
Godel's proof is statd to be in a system with at least the
properties of Peano Arithmatic, having the ability to show the
properties of the "Natural Numbers"
one smuggles in an external notion of truth
(truth in rao) and then pretends it is an
internal notion of truth (truth in PA).
If we refuse to make that identification,
incompleteness evaporates.
But Truth in N is part of Peano Arithmatic, as Peano Arithmatic is
a axiomiation to create the Natural Numbers.
You have that backwards. Truth in rao requires PA
as part of it, and PA itself has no notion of
Truth in rao. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in rao.
Which means you don't understand what N actually is.
Nothing can be "True in N" unless that truth comes from the Axioms of
PA, as N is the result of PA.
combined with the meta-math external model.
Nope. N is just a set of object built in the Formal System defined by
PA. 0 comes from Axiom 1 which states there is a 0.
The rest come from the successor function where n+1 = S(n)
And the induction property makes sure we get to the full set.
No "meta-math" needed.
You are just smoking your category errors and believing your own lies.
But then, your claim of not knowing what is true in the world you are
creating somes on point for you.
On 1/24/26 5:31 PM, olcott wrote:
On 1/24/2026 4:25 PM, Richard Damon wrote:
On 1/24/26 3:38 PM, olcott wrote:
On 1/24/2026 1:52 PM, Richard Damon wrote:
On 1/24/26 2:25 PM, olcott wrote:
On 1/24/2026 1:23 PM, Richard Damon wrote:
On 1/24/26 12:54 PM, olcott wrote:G||delrCOs incompleteness theorem only rCLworksrCY if
On 1/24/2026 11:10 AM, Richard Damon wrote:
On 1/24/26 10:44 AM, olcott wrote:
The statement that G is true and unprovable in PA has
always been counter-factual. It has never actually been
true <in> PA and that is why it is unprovable in PA.
Sure it is. At least it is a FACT that no natural number will >>>>>>>>> statisfy that relationship, and there is no proof in PA of that >>>>>>>>> fact.
Have you ever heard of: "true in the standard model of arithmetic"? >>>>>>>
Sure, but they are not in Peano Arithmatic, but are (generally) >>>>>>> 1st order variations of the Peano Axioms which lead to alternate >>>>>>> number systems.
Godel's proof is statd to be in a system with at least the
properties of Peano Arithmatic, having the ability to show the
properties of the "Natural Numbers"
one smuggles in an external notion of truth
(truth in rao) and then pretends it is an
internal notion of truth (truth in PA).
If we refuse to make that identification,
incompleteness evaporates.
But Truth in N is part of Peano Arithmatic, as Peano Arithmatic is
a axiomiation to create the Natural Numbers.
You have that backwards. Truth in rao requires PA
as part of it, and PA itself has no notion of
Truth in rao. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in rao.
Which means you don't understand what N actually is.
Nothing can be "True in N" unless that truth comes from the Axioms of
PA, as N is the result of PA.
combined with the meta-math external model.
Nope. N is just a set of object built in the Formal System defined by
PA. 0 comes from Axiom 1 which states there is a 0.
The rest come from the successor function where n+1 = S(n)
And the induction property makes sure we get to the full set.
No "meta-math" needed.
You are just smoking your category errors and believing your own lies.
But then, your claim of not knowing what is true in the world you are
creating somes on point for you.
On 1/24/2026 2:20 AM, Mikko wrote:
On 23/01/2026 12:22, olcott wrote:
On 1/23/2026 3:13 AM, Mikko wrote:
On 22/01/2026 18:40, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
On 21/01/2026 17:22, olcott wrote:
On 1/21/2026 3:03 AM, Mikko wrote:
No, it hasn't. In the way theories are usually discussed nothing is >>>>>>>> "ture in arithmetic". Every sentence of a first order theory that >>>>>>>> can be proven in the theory is true in every model theory. Every >>>>>>>> sentence of a theory that cannot be proven in the theory is false >>>>>>>> in some model of the theory.
only because back then proof theoretic semantics did
not exist.
Every interpretation of the theory is a definition of semantics. >>>>>>>>
MetarCamath relations about numbers donrCOt exist in PA
because PA only contains arithmetical relationsrCoaddition,
multiplication, ordering, primitiverCarecursive predicates
about numbers themselvesrCowhile relations that talk about
PArCOs own proofs, syntax, or truth conditions live entirely
in the metarCatheory;
Methamathematics does not need any other relations between numbers >>>>>> than what PA has. But relations that map other things to numbers
can be useful for methamathematical purposes.
so when someone appeals to a G||delrCastyle relation like
rCLn encodes a proof of this very sentence,rCY theyrCOre
invoking a metarCamathematical predicate that PA cannot
internalize, which is exactly why your framework draws
a clean boundary between internal proofrCatheoretic truth
and external modelrCatheoretic truth.
Anyway, what can be provven that way is true aboout PA. You can deny >>>>>> the proof but you cannot perform what is meta-provably impossible.
G||delrCOs sentence is not rCLtrue in arithmetic.rCY
It is true only in the metarCatheory, under an
external interpretation of PA (typically the
standard model rao). Inside PA itself, the sentence
is not a truthrCabearer at all.
There is no concept of "truth-bearer" in an uninterpreted theory
because
there is not concept of "truth". The relevant concept is "sell-formed- >>>> formula" and G||dels sentence is one. It may be true or false in an
interpretation.
There is a
"true on the basis of meaning expressed in language"
and I figured out how to make it computable over the
body of knowledge.
Except that "true on the basis of meaning expressed in language" is
nmt computable and does not cover all of the body of knowldge.
When the basis of "true" is proof theoretic semantics
internal to the formal system relative to its own axioms
and not truth conditional in a separate model outside
of the system undecidability ceases to exist.
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can deny
the proof but you cannot perform what is meta-provably impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
On 24/01/2026 16:01, olcott wrote:
On 1/24/2026 2:20 AM, Mikko wrote:
On 23/01/2026 12:22, olcott wrote:
On 1/23/2026 3:13 AM, Mikko wrote:
On 22/01/2026 18:40, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
On 21/01/2026 17:22, olcott wrote:G||delrCOs sentence is not rCLtrue in arithmetic.rCY
On 1/21/2026 3:03 AM, Mikko wrote:
No, it hasn't. In the way theories are usually discussed
nothing is
"ture in arithmetic". Every sentence of a first order theory that >>>>>>>>> can be proven in the theory is true in every model theory. Every >>>>>>>>> sentence of a theory that cannot be proven in the theory is false >>>>>>>>> in some model of the theory.
only because back then proof theoretic semantics did
not exist.
Every interpretation of the theory is a definition of semantics. >>>>>>>>>
MetarCamath relations about numbers donrCOt exist in PA
because PA only contains arithmetical relationsrCoaddition,
multiplication, ordering, primitiverCarecursive predicates
about numbers themselvesrCowhile relations that talk about
PArCOs own proofs, syntax, or truth conditions live entirely
in the metarCatheory;
Methamathematics does not need any other relations between numbers >>>>>>> than what PA has. But relations that map other things to numbers >>>>>>> can be useful for methamathematical purposes.
so when someone appeals to a G||delrCastyle relation like
rCLn encodes a proof of this very sentence,rCY theyrCOre
invoking a metarCamathematical predicate that PA cannot
internalize, which is exactly why your framework draws
a clean boundary between internal proofrCatheoretic truth
and external modelrCatheoretic truth.
Anyway, what can be provven that way is true aboout PA. You can deny >>>>>>> the proof but you cannot perform what is meta-provably impossible. >>>>>>
It is true only in the metarCatheory, under an
external interpretation of PA (typically the
standard model rao). Inside PA itself, the sentence
is not a truthrCabearer at all.
There is no concept of "truth-bearer" in an uninterpreted theory
because
there is not concept of "truth". The relevant concept is "sell-formed- >>>>> formula" and G||dels sentence is one. It may be true or false in an
interpretation.
There is a
"true on the basis of meaning expressed in language"
and I figured out how to make it computable over the
body of knowledge.
Except that "true on the basis of meaning expressed in language" is
nmt computable and does not cover all of the body of knowldge.
When the basis of "true" is proof theoretic semantics
internal to the formal system relative to its own axioms
and not truth conditional in a separate model outside
of the system undecidability ceases to exist.
No, it does not. It does not matter what you call it, a sentence
that cannot be neither proven nor disproven is undecidable because
that is what the word means. An example is G||del's sentence in
Peano arithmetics.
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can deny >>>>> the proof but you cannot perform what is meta-provably impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
On 1/25/2026 12:36 PM, Richard Damon wrote:
On 1/24/26 8:44 PM, olcott wrote:
On 1/24/2026 6:52 PM, Richard Damon wrote:
On 1/24/26 5:31 PM, olcott wrote:
On 1/24/2026 4:25 PM, Richard Damon wrote:
On 1/24/26 3:38 PM, olcott wrote:
On 1/24/2026 1:52 PM, Richard Damon wrote:
On 1/24/26 2:25 PM, olcott wrote:
On 1/24/2026 1:23 PM, Richard Damon wrote:
On 1/24/26 12:54 PM, olcott wrote:G||delrCOs incompleteness theorem only rCLworksrCY if
On 1/24/2026 11:10 AM, Richard Damon wrote:
On 1/24/26 10:44 AM, olcott wrote:
The statement that G is true and unprovable in PA has >>>>>>>>>>>>> always been counter-factual. It has never actually been >>>>>>>>>>>>> true <in> PA and that is why it is unprovable in PA.
Sure it is. At least it is a FACT that no natural number >>>>>>>>>>>> will statisfy that relationship, and there is no proof in PA >>>>>>>>>>>> of that fact.
Have you ever heard of: "true in the standard model of
arithmetic"?
Sure, but they are not in Peano Arithmatic, but are
(generally) 1st order variations of the Peano Axioms which >>>>>>>>>> lead to alternate number systems.
Godel's proof is statd to be in a system with at least the >>>>>>>>>> properties of Peano Arithmatic, having the ability to show the >>>>>>>>>> properties of the "Natural Numbers"
one smuggles in an external notion of truth
(truth in rao) and then pretends it is an
internal notion of truth (truth in PA).
If we refuse to make that identification,
incompleteness evaporates.
But Truth in N is part of Peano Arithmatic, as Peano Arithmatic >>>>>>>> is a axiomiation to create the Natural Numbers.
You have that backwards. Truth in rao requires PA
as part of it, and PA itself has no notion of
Truth in rao. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in rao.
Which means you don't understand what N actually is.
Nothing can be "True in N" unless that truth comes from the Axioms >>>>>> of PA, as N is the result of PA.
combined with the meta-math external model.
Nope. N is just a set of object built in the Formal System defined
by PA. 0 comes from Axiom 1 which states there is a 0.
If G is true and not provable then you have
the wrong kind of true. I have known that
the entire body of knowledge is a semantic
tautology for 28 years.
No, YOU do. The problem is Truth in the real world isn't based on
being about to prove the fact, and most things are not actually
provable, just well approximatable.
That is why this insight was so important:
"true on the basis of meaning expressed in language"
I broke through the 75 year logjam of the analytic/synthetic
distinction.
And your ideas just prove your stupidity and being a pathological liar.
That the sum of the squares of the length of the two sides of a right
triangle is equal to the square of the length of the hypotenuse is NOT
"true by the meaning of words" or a Tautology, but is part of the body
of Knowledge.
Now I have all of the details to prove how
any sort of "true and unprovable" has always
been complete nonsense:
Proof theoretic semantics anchored in the values
of true / false / non-well-founded derived from
axioms where non-well-founded are expressions that
are not truth-apt.
Which just can't handle systems like PA.
But then, it is clear those are beyond your ability to understand, so
it doesn't bother you.
The rest come from the successor function where n+1 = S(n)
And the induction property makes sure we get to the full set.
No "meta-math" needed.
You are just smoking your category errors and believing your own lies. >>>>
But then, your claim of not knowing what is true in the world you >>>>>> are creating somes on point for you.
On 1/25/2026 12:40 PM, Richard Damon wrote:
On 1/25/26 1:33 PM, olcott wrote:
On 1/25/2026 12:27 PM, Richard Damon wrote:
On 1/25/26 8:24 AM, olcott wrote:
On 1/25/2026 5:19 AM, Mikko wrote:
On 24/01/2026 16:01, olcott wrote:
On 1/24/2026 2:20 AM, Mikko wrote:
On 23/01/2026 12:22, olcott wrote:
On 1/23/2026 3:13 AM, Mikko wrote:
On 22/01/2026 18:40, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
On 21/01/2026 17:22, olcott wrote:
On 1/21/2026 3:03 AM, Mikko wrote:
No, it hasn't. In the way theories are usually discussed >>>>>>>>>>>>>> nothing is
"ture in arithmetic". Every sentence of a first order >>>>>>>>>>>>>> theory that
can be proven in the theory is true in every model theory. >>>>>>>>>>>>>> Every
sentence of a theory that cannot be proven in the theory >>>>>>>>>>>>>> is false
in some model of the theory.
only because back then proof theoretic semantics did >>>>>>>>>>>>>>> not exist.
Every interpretation of the theory is a definition of >>>>>>>>>>>>>> semantics.
MetarCamath relations about numbers donrCOt exist in PA >>>>>>>>>>>>> because PA only contains arithmetical relationsrCoaddition, >>>>>>>>>>>>> multiplication, ordering, primitiverCarecursive predicates >>>>>>>>>>>>> about numbers themselvesrCowhile relations that talk about >>>>>>>>>>>>> PArCOs own proofs, syntax, or truth conditions live entirely >>>>>>>>>>>>> in the metarCatheory;
Methamathematics does not need any other relations between >>>>>>>>>>>> numbers
than what PA has. But relations that map other things to >>>>>>>>>>>> numbers
can be useful for methamathematical purposes.
so when someone appeals to a G||delrCastyle relation like >>>>>>>>>>>>> rCLn encodes a proof of this very sentence,rCY theyrCOre >>>>>>>>>>>>> invoking a metarCamathematical predicate that PA cannot >>>>>>>>>>>>> internalize, which is exactly why your framework draws >>>>>>>>>>>>> a clean boundary between internal proofrCatheoretic truth >>>>>>>>>>>>> and external modelrCatheoretic truth.
Anyway, what can be provven that way is true aboout PA. You >>>>>>>>>>>> can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>> impossible.
G||delrCOs sentence is not rCLtrue in arithmetic.rCY
It is true only in the metarCatheory, under an
external interpretation of PA (typically the
standard model rao). Inside PA itself, the sentence
is not a truthrCabearer at all.
There is no concept of "truth-bearer" in an uninterpreted >>>>>>>>>> theory because
there is not concept of "truth". The relevant concept is
"sell- formed-
formula" and G||dels sentence is one. It may be true or false >>>>>>>>>> in an
interpretation.
There is a
"true on the basis of meaning expressed in language"
and I figured out how to make it computable over the
body of knowledge.
Except that "true on the basis of meaning expressed in language" is >>>>>>>> nmt computable and does not cover all of the body of knowldge.
When the basis of "true" is proof theoretic semantics
internal to the formal system relative to its own axioms
and not truth conditional in a separate model outside
of the system undecidability ceases to exist.
No, it does not. It does not matter what you call it, a sentence
that cannot be neither proven nor disproven is undecidable because >>>>>> that is what the word means. An example is G||del's sentence in
Peano arithmetics.
When a truth predicate gets the input "What time is?"
this input is rejected as not truth-apt.
That fine.
When PA gets an expression that cannot be proven or
refuted using its own axioms then this expression is
not within its domain.
Then most of Natural Number mathematics is isn't in its domain,
It is what it is.
But PA was CREATED to allow us to define the Natural Numbers in an
axiomatic way.
Yet only within the actual axioms of PA.
PA doesn't even know PA until you add a truth predicate.
When you do add a truth predicate then PA knows PA. If
you want more than that then meta-math can know "about" PA.
This is one level of indirect reference away from knowing PA.
In other words, you world is just inconsistant because it can't handle
itself.
You just build your logic on equivocations and lies.
But since PA doesn't have a truth predicate, you can't add it.
What PA has, if you actually understand it, is that it was built on a
definition of logic that defines truth based on what flows out of the
possible infinite application of its axioms.
When you try to build with a lessor logic, you don't get a PA that can
do what it needs to, and thus isn't actually an arithmatic.
And, you can't KNOW if somehting is a valid question to ask until
you know the answer.
This makes a fairly worthless domain to learn things in.
By your definition, a question like can every even number, greater
than 2, be the sum of two prime numbers MIGHT not be within its
domain, even though it is purely a question about the capability of
numbers.
On 1/25/2026 12:31 PM, Richard Damon wrote:
On 1/25/26 8:30 AM, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can >>>>>>>> denyThe meta-proof does not exist in the axioms of PA
the proof but you cannot perform what is meta-provably impossible. >>>>>>
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also >>>>>> a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
And you don't understand that those definitions aren't defined in a
proof theoretic semantics.
PA reo-a x
can't be evaluated itself in proof theoretic semantics and always get
a value, as you can't PROVE that statement.
I have carefully researched Proof theoretic semantics
from its original papers and will be able to tutor you
on this basis pretty soon.
On 1/25/26 2:09 PM, olcott wrote:
On 1/25/2026 12:36 PM, Richard Damon wrote:
On 1/24/26 8:44 PM, olcott wrote:
On 1/24/2026 6:52 PM, Richard Damon wrote:
On 1/24/26 5:31 PM, olcott wrote:
On 1/24/2026 4:25 PM, Richard Damon wrote:
On 1/24/26 3:38 PM, olcott wrote:
On 1/24/2026 1:52 PM, Richard Damon wrote:
On 1/24/26 2:25 PM, olcott wrote:
On 1/24/2026 1:23 PM, Richard Damon wrote:
On 1/24/26 12:54 PM, olcott wrote:G||delrCOs incompleteness theorem only rCLworksrCY if
On 1/24/2026 11:10 AM, Richard Damon wrote:
On 1/24/26 10:44 AM, olcott wrote:
Sure it is. At least it is a FACT that no natural number >>>>>>>>>>>>> will statisfy that relationship, and there is no proof in >>>>>>>>>>>>> PA of that fact.
The statement that G is true and unprovable in PA has >>>>>>>>>>>>>> always been counter-factual. It has never actually been >>>>>>>>>>>>>> true <in> PA and that is why it is unprovable in PA. >>>>>>>>>>>>>
Have you ever heard of: "true in the standard model of >>>>>>>>>>>> arithmetic"?
Sure, but they are not in Peano Arithmatic, but are
(generally) 1st order variations of the Peano Axioms which >>>>>>>>>>> lead to alternate number systems.
Godel's proof is statd to be in a system with at least the >>>>>>>>>>> properties of Peano Arithmatic, having the ability to show >>>>>>>>>>> the properties of the "Natural Numbers"
one smuggles in an external notion of truth
(truth in rao) and then pretends it is an
internal notion of truth (truth in PA).
If we refuse to make that identification,
incompleteness evaporates.
But Truth in N is part of Peano Arithmatic, as Peano Arithmatic >>>>>>>>> is a axiomiation to create the Natural Numbers.
You have that backwards. Truth in rao requires PA
as part of it, and PA itself has no notion of
Truth in rao. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in rao.
Which means you don't understand what N actually is.
Nothing can be "True in N" unless that truth comes from the
Axioms of PA, as N is the result of PA.
combined with the meta-math external model.
Nope. N is just a set of object built in the Formal System defined
by PA. 0 comes from Axiom 1 which states there is a 0.
If G is true and not provable then you have
the wrong kind of true. I have known that
the entire body of knowledge is a semantic
tautology for 28 years.
No, YOU do. The problem is Truth in the real world isn't based on
being about to prove the fact, and most things are not actually
provable, just well approximatable.
That is why this insight was so important:
"true on the basis of meaning expressed in language"
I broke through the 75 year logjam of the analytic/synthetic
distinction.
In other words, you don't accept the Pythgorean Theorem as "True", since
its Tru-ness doesn't come out of the meaning of its words.
On 1/25/26 2:10 PM, olcott wrote:
On 1/25/2026 12:40 PM, Richard Damon wrote:
On 1/25/26 1:33 PM, olcott wrote:
On 1/25/2026 12:27 PM, Richard Damon wrote:
On 1/25/26 8:24 AM, olcott wrote:
On 1/25/2026 5:19 AM, Mikko wrote:
On 24/01/2026 16:01, olcott wrote:
On 1/24/2026 2:20 AM, Mikko wrote:
On 23/01/2026 12:22, olcott wrote:When the basis of "true" is proof theoretic semantics
On 1/23/2026 3:13 AM, Mikko wrote:
On 22/01/2026 18:40, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
On 21/01/2026 17:22, olcott wrote:
On 1/21/2026 3:03 AM, Mikko wrote:
No, it hasn't. In the way theories are usually discussed >>>>>>>>>>>>>>> nothing is
"ture in arithmetic". Every sentence of a first order >>>>>>>>>>>>>>> theory that
can be proven in the theory is true in every model >>>>>>>>>>>>>>> theory. Every
sentence of a theory that cannot be proven in the theory >>>>>>>>>>>>>>> is false
in some model of the theory.
only because back then proof theoretic semantics did >>>>>>>>>>>>>>>> not exist.
Every interpretation of the theory is a definition of >>>>>>>>>>>>>>> semantics.
MetarCamath relations about numbers donrCOt exist in PA >>>>>>>>>>>>>> because PA only contains arithmetical relationsrCoaddition, >>>>>>>>>>>>>> multiplication, ordering, primitiverCarecursive predicates >>>>>>>>>>>>>> about numbers themselvesrCowhile relations that talk about >>>>>>>>>>>>>> PArCOs own proofs, syntax, or truth conditions live entirely >>>>>>>>>>>>>> in the metarCatheory;
Methamathematics does not need any other relations between >>>>>>>>>>>>> numbers
than what PA has. But relations that map other things to >>>>>>>>>>>>> numbers
can be useful for methamathematical purposes.
so when someone appeals to a G||delrCastyle relation like >>>>>>>>>>>>>> rCLn encodes a proof of this very sentence,rCY theyrCOre >>>>>>>>>>>>>> invoking a metarCamathematical predicate that PA cannot >>>>>>>>>>>>>> internalize, which is exactly why your framework draws >>>>>>>>>>>>>> a clean boundary between internal proofrCatheoretic truth >>>>>>>>>>>>>> and external modelrCatheoretic truth.
Anyway, what can be provven that way is true aboout PA. You >>>>>>>>>>>>> can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>> impossible.
G||delrCOs sentence is not rCLtrue in arithmetic.rCY
It is true only in the metarCatheory, under an
external interpretation of PA (typically the
standard model rao). Inside PA itself, the sentence
is not a truthrCabearer at all.
There is no concept of "truth-bearer" in an uninterpreted >>>>>>>>>>> theory because
there is not concept of "truth". The relevant concept is >>>>>>>>>>> "sell- formed-
formula" and G||dels sentence is one. It may be true or false >>>>>>>>>>> in an
interpretation.
There is a
"true on the basis of meaning expressed in language"
and I figured out how to make it computable over the
body of knowledge.
Except that "true on the basis of meaning expressed in
language" is
nmt computable and does not cover all of the body of knowldge. >>>>>>>>
internal to the formal system relative to its own axioms
and not truth conditional in a separate model outside
of the system undecidability ceases to exist.
No, it does not. It does not matter what you call it, a sentence >>>>>>> that cannot be neither proven nor disproven is undecidable because >>>>>>> that is what the word means. An example is G||del's sentence in
Peano arithmetics.
When a truth predicate gets the input "What time is?"
this input is rejected as not truth-apt.
That fine.
When PA gets an expression that cannot be proven or
refuted using its own axioms then this expression is
not within its domain.
Then most of Natural Number mathematics is isn't in its domain,
It is what it is.
But PA was CREATED to allow us to define the Natural Numbers in an
axiomatic way.
Yet only within the actual axioms of PA.
Yes, the Natural Numbers are object created within the formal system of Peano Arithmetic (as one way to define them) and in that system there
are a lot of properties of them that are True (or False).
If there is a property of them that PA Created that it can't talk about, that sounds very much like PA is just incomplete in its understanding of what it does, just by the basic normal definition of incomplete.
On 1/25/26 2:05 PM, olcott wrote:
On 1/25/2026 12:31 PM, Richard Damon wrote:
On 1/25/26 8:30 AM, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can >>>>>>>>> denyThe meta-proof does not exist in the axioms of PA
the proof but you cannot perform what is meta-provably impossible. >>>>>>>
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also >>>>>>> a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
And you don't understand that those definitions aren't defined in a
proof theoretic semantics.
PA reo-a x
can't be evaluated itself in proof theoretic semantics and always get
a value, as you can't PROVE that statement.
I have carefully researched Proof theoretic semantics
from its original papers and will be able to tutor you
on this basis pretty soon.
I doubt it, as you never had the logical framework to acutally
understand what you are reading.
Since, you have been spouting for years that your ideas must be true, we
can see that you are going to likely misread things to twist them to
your ideas, just like you try to do with everything else.
On 1/25/2026 1:54 PM, Richard Damon wrote:
On 1/25/26 2:09 PM, olcott wrote:
On 1/25/2026 12:36 PM, Richard Damon wrote:
On 1/24/26 8:44 PM, olcott wrote:
On 1/24/2026 6:52 PM, Richard Damon wrote:
On 1/24/26 5:31 PM, olcott wrote:
On 1/24/2026 4:25 PM, Richard Damon wrote:
On 1/24/26 3:38 PM, olcott wrote:
On 1/24/2026 1:52 PM, Richard Damon wrote:
On 1/24/26 2:25 PM, olcott wrote:
On 1/24/2026 1:23 PM, Richard Damon wrote:
On 1/24/26 12:54 PM, olcott wrote:G||delrCOs incompleteness theorem only rCLworksrCY if
On 1/24/2026 11:10 AM, Richard Damon wrote:
On 1/24/26 10:44 AM, olcott wrote:
Sure it is. At least it is a FACT that no natural number >>>>>>>>>>>>>> will statisfy that relationship, and there is no proof in >>>>>>>>>>>>>> PA of that fact.
The statement that G is true and unprovable in PA has >>>>>>>>>>>>>>> always been counter-factual. It has never actually been >>>>>>>>>>>>>>> true <in> PA and that is why it is unprovable in PA. >>>>>>>>>>>>>>
Have you ever heard of: "true in the standard model of >>>>>>>>>>>>> arithmetic"?
Sure, but they are not in Peano Arithmatic, but are
(generally) 1st order variations of the Peano Axioms which >>>>>>>>>>>> lead to alternate number systems.
Godel's proof is statd to be in a system with at least the >>>>>>>>>>>> properties of Peano Arithmatic, having the ability to show >>>>>>>>>>>> the properties of the "Natural Numbers"
one smuggles in an external notion of truth
(truth in rao) and then pretends it is an
internal notion of truth (truth in PA).
If we refuse to make that identification,
incompleteness evaporates.
But Truth in N is part of Peano Arithmatic, as Peano
Arithmatic is a axiomiation to create the Natural Numbers. >>>>>>>>>>
You have that backwards. Truth in rao requires PA
as part of it, and PA itself has no notion of
Truth in rao. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in rao.
Which means you don't understand what N actually is.
Nothing can be "True in N" unless that truth comes from the
Axioms of PA, as N is the result of PA.
combined with the meta-math external model.
Nope. N is just a set of object built in the Formal System defined >>>>>> by PA. 0 comes from Axiom 1 which states there is a 0.
If G is true and not provable then you have
the wrong kind of true. I have known that
the entire body of knowledge is a semantic
tautology for 28 years.
No, YOU do. The problem is Truth in the real world isn't based on
being about to prove the fact, and most things are not actually
provable, just well approximatable.
That is why this insight was so important:
"true on the basis of meaning expressed in language"
I broke through the 75 year logjam of the analytic/synthetic
distinction.
In other words, you don't accept the Pythgorean Theorem as "True",
since its Tru-ness doesn't come out of the meaning of its words.
[meaning of its words]
My sentence is not restricted to words and
does include mathematical expressions.
On 1/25/2026 1:57 PM, Richard Damon wrote:
On 1/25/26 2:10 PM, olcott wrote:
On 1/25/2026 12:40 PM, Richard Damon wrote:
On 1/25/26 1:33 PM, olcott wrote:
On 1/25/2026 12:27 PM, Richard Damon wrote:
On 1/25/26 8:24 AM, olcott wrote:
On 1/25/2026 5:19 AM, Mikko wrote:
On 24/01/2026 16:01, olcott wrote:
On 1/24/2026 2:20 AM, Mikko wrote:
On 23/01/2026 12:22, olcott wrote:When the basis of "true" is proof theoretic semantics
On 1/23/2026 3:13 AM, Mikko wrote:
On 22/01/2026 18:40, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
On 21/01/2026 17:22, olcott wrote:
On 1/21/2026 3:03 AM, Mikko wrote:
No, it hasn't. In the way theories are usually discussed >>>>>>>>>>>>>>>> nothing is
"ture in arithmetic". Every sentence of a first order >>>>>>>>>>>>>>>> theory that
can be proven in the theory is true in every model >>>>>>>>>>>>>>>> theory. Every
sentence of a theory that cannot be proven in the theory >>>>>>>>>>>>>>>> is false
in some model of the theory.
only because back then proof theoretic semantics did >>>>>>>>>>>>>>>>> not exist.
Every interpretation of the theory is a definition of >>>>>>>>>>>>>>>> semantics.
MetarCamath relations about numbers donrCOt exist in PA >>>>>>>>>>>>>>> because PA only contains arithmetical relationsrCoaddition, >>>>>>>>>>>>>>> multiplication, ordering, primitiverCarecursive predicates >>>>>>>>>>>>>>> about numbers themselvesrCowhile relations that talk about >>>>>>>>>>>>>>> PArCOs own proofs, syntax, or truth conditions live entirely >>>>>>>>>>>>>>> in the metarCatheory;
Methamathematics does not need any other relations between >>>>>>>>>>>>>> numbers
than what PA has. But relations that map other things to >>>>>>>>>>>>>> numbers
can be useful for methamathematical purposes.
so when someone appeals to a G||delrCastyle relation like >>>>>>>>>>>>>>> rCLn encodes a proof of this very sentence,rCY theyrCOre >>>>>>>>>>>>>>> invoking a metarCamathematical predicate that PA cannot >>>>>>>>>>>>>>> internalize, which is exactly why your framework draws >>>>>>>>>>>>>>> a clean boundary between internal proofrCatheoretic truth >>>>>>>>>>>>>>> and external modelrCatheoretic truth.
Anyway, what can be provven that way is true aboout PA. >>>>>>>>>>>>>> You can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>>> impossible.
G||delrCOs sentence is not rCLtrue in arithmetic.rCY >>>>>>>>>>>>> It is true only in the metarCatheory, under an
external interpretation of PA (typically the
standard model rao). Inside PA itself, the sentence
is not a truthrCabearer at all.
There is no concept of "truth-bearer" in an uninterpreted >>>>>>>>>>>> theory because
there is not concept of "truth". The relevant concept is >>>>>>>>>>>> "sell- formed-
formula" and G||dels sentence is one. It may be true or false >>>>>>>>>>>> in an
interpretation.
There is a
"true on the basis of meaning expressed in language"
and I figured out how to make it computable over the
body of knowledge.
Except that "true on the basis of meaning expressed in
language" is
nmt computable and does not cover all of the body of knowldge. >>>>>>>>>
internal to the formal system relative to its own axioms
and not truth conditional in a separate model outside
of the system undecidability ceases to exist.
No, it does not. It does not matter what you call it, a sentence >>>>>>>> that cannot be neither proven nor disproven is undecidable because >>>>>>>> that is what the word means. An example is G||del's sentence in >>>>>>>> Peano arithmetics.
When a truth predicate gets the input "What time is?"
this input is rejected as not truth-apt.
That fine.
When PA gets an expression that cannot be proven or
refuted using its own axioms then this expression is
not within its domain.
Then most of Natural Number mathematics is isn't in its domain,
It is what it is.
But PA was CREATED to allow us to define the Natural Numbers in an
axiomatic way.
Yet only within the actual axioms of PA.
Yes, the Natural Numbers are object created within the formal system
of Peano Arithmetic (as one way to define them) and in that system
there are a lot of properties of them that are True (or False).
If there is a property of them that PA Created that it can't talk
about, that sounds very much like PA is just incomplete in its
understanding of what it does, just by the basic normal definition of
incomplete.
G||delrCOs sentence is not rCLtrue in arithmetic.rCY
It is true only in the metarCatheory, under an
external interpretation of PA (typically the
standard model rao). Inside PA itself, the sentence
is not a truthrCabearer at all.
On 1/25/2026 1:59 PM, Richard Damon wrote:
On 1/25/26 2:05 PM, olcott wrote:
On 1/25/2026 12:31 PM, Richard Damon wrote:
On 1/25/26 8:30 AM, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You >>>>>>>>>> can deny
the proof but you cannot perform what is meta-provably
impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also >>>>>>>> a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
And you don't understand that those definitions aren't defined in a
proof theoretic semantics.
PA reo-a x
can't be evaluated itself in proof theoretic semantics and always
get a value, as you can't PROVE that statement.
I have carefully researched Proof theoretic semantics
from its original papers and will be able to tutor you
on this basis pretty soon.
I doubt it, as you never had the logical framework to acutally
understand what you are reading.
I always had a logical framework. What I lacked was
the conventional terms of the art that defined this
framework. Now I have those too.
This article was written by one of the leading authors in the field. https://plato.stanford.edu/entries/proof-theoretic-semantics/
That and two or three other papers will link together all of my
ideas as bullet points.
Since, you have been spouting for years that your ideas must be true,
we can see that you are going to likely misread things to twist them
to your ideas, just like you try to do with everything else.
It was only the power of LLMs that actually knew about all
of the philosophical foundations of math, logic, computer
science and linguistics as well as the technical details
of these fields that could understand how I linked them
all together.
On 1/25/26 3:07 PM, olcott wrote:
On 1/25/2026 1:54 PM, Richard Damon wrote:
On 1/25/26 2:09 PM, olcott wrote:
On 1/25/2026 12:36 PM, Richard Damon wrote:
On 1/24/26 8:44 PM, olcott wrote:
On 1/24/2026 6:52 PM, Richard Damon wrote:
On 1/24/26 5:31 PM, olcott wrote:
On 1/24/2026 4:25 PM, Richard Damon wrote:
On 1/24/26 3:38 PM, olcott wrote:
On 1/24/2026 1:52 PM, Richard Damon wrote:
On 1/24/26 2:25 PM, olcott wrote:
On 1/24/2026 1:23 PM, Richard Damon wrote:
On 1/24/26 12:54 PM, olcott wrote:G||delrCOs incompleteness theorem only rCLworksrCY if
On 1/24/2026 11:10 AM, Richard Damon wrote:
On 1/24/26 10:44 AM, olcott wrote:
Sure it is. At least it is a FACT that no natural number >>>>>>>>>>>>>>> will statisfy that relationship, and there is no proof in >>>>>>>>>>>>>>> PA of that fact.
The statement that G is true and unprovable in PA has >>>>>>>>>>>>>>>> always been counter-factual. It has never actually been >>>>>>>>>>>>>>>> true <in> PA and that is why it is unprovable in PA. >>>>>>>>>>>>>>>
Have you ever heard of: "true in the standard model of >>>>>>>>>>>>>> arithmetic"?
Sure, but they are not in Peano Arithmatic, but are >>>>>>>>>>>>> (generally) 1st order variations of the Peano Axioms which >>>>>>>>>>>>> lead to alternate number systems.
Godel's proof is statd to be in a system with at least the >>>>>>>>>>>>> properties of Peano Arithmatic, having the ability to show >>>>>>>>>>>>> the properties of the "Natural Numbers"
one smuggles in an external notion of truth
(truth in rao) and then pretends it is an
internal notion of truth (truth in PA).
If we refuse to make that identification,
incompleteness evaporates.
But Truth in N is part of Peano Arithmatic, as Peano
Arithmatic is a axiomiation to create the Natural Numbers. >>>>>>>>>>>
You have that backwards. Truth in rao requires PA
as part of it, and PA itself has no notion of
Truth in rao. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in rao.
Which means you don't understand what N actually is.
Nothing can be "True in N" unless that truth comes from the >>>>>>>>> Axioms of PA, as N is the result of PA.
combined with the meta-math external model.
Nope. N is just a set of object built in the Formal System
defined by PA. 0 comes from Axiom 1 which states there is a 0.
If G is true and not provable then you have
the wrong kind of true. I have known that
the entire body of knowledge is a semantic
tautology for 28 years.
No, YOU do. The problem is Truth in the real world isn't based on
being about to prove the fact, and most things are not actually
provable, just well approximatable.
That is why this insight was so important:
"true on the basis of meaning expressed in language"
I broke through the 75 year logjam of the analytic/synthetic
distinction.
In other words, you don't accept the Pythgorean Theorem as "True",
since its Tru-ness doesn't come out of the meaning of its words.
[meaning of its words]
My sentence is not restricted to words and
does include mathematical expressions.
Then it accepts Godel's G as a valid statement
and Goldbach's
conjecture, even if improbably true, is a truth bearer.
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
You are just admitting to your own equivocation of meaning.--
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can deny >>>>>> the proof but you cannot perform what is meta-provably impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can deny >>>>>>> the proof but you cannot perform what is meta-provably impossible. >>>>>The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can >>>>>>>> denyThe meta-proof does not exist in the axioms of PA
the proof but you cannot perform what is meta-provably impossible. >>>>>>
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also >>>>>> a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
On 1/25/2026 2:44 PM, Richard Damon wrote:
On 1/25/26 3:07 PM, olcott wrote:
On 1/25/2026 1:54 PM, Richard Damon wrote:
On 1/25/26 2:09 PM, olcott wrote:
On 1/25/2026 12:36 PM, Richard Damon wrote:
On 1/24/26 8:44 PM, olcott wrote:
On 1/24/2026 6:52 PM, Richard Damon wrote:
On 1/24/26 5:31 PM, olcott wrote:
On 1/24/2026 4:25 PM, Richard Damon wrote:
On 1/24/26 3:38 PM, olcott wrote:
On 1/24/2026 1:52 PM, Richard Damon wrote:
On 1/24/26 2:25 PM, olcott wrote:
On 1/24/2026 1:23 PM, Richard Damon wrote:
On 1/24/26 12:54 PM, olcott wrote:G||delrCOs incompleteness theorem only rCLworksrCY if >>>>>>>>>>>>> one smuggles in an external notion of truth
On 1/24/2026 11:10 AM, Richard Damon wrote:
On 1/24/26 10:44 AM, olcott wrote:
Sure it is. At least it is a FACT that no natural number >>>>>>>>>>>>>>>> will statisfy that relationship, and there is no proof >>>>>>>>>>>>>>>> in PA of that fact.
The statement that G is true and unprovable in PA has >>>>>>>>>>>>>>>>> always been counter-factual. It has never actually been >>>>>>>>>>>>>>>>> true <in> PA and that is why it is unprovable in PA. >>>>>>>>>>>>>>>>
Have you ever heard of: "true in the standard model of >>>>>>>>>>>>>>> arithmetic"?
Sure, but they are not in Peano Arithmatic, but are >>>>>>>>>>>>>> (generally) 1st order variations of the Peano Axioms which >>>>>>>>>>>>>> lead to alternate number systems.
Godel's proof is statd to be in a system with at least the >>>>>>>>>>>>>> properties of Peano Arithmatic, having the ability to show >>>>>>>>>>>>>> the properties of the "Natural Numbers"
(truth in rao) and then pretends it is an
internal notion of truth (truth in PA).
If we refuse to make that identification,
incompleteness evaporates.
But Truth in N is part of Peano Arithmatic, as Peano
Arithmatic is a axiomiation to create the Natural Numbers. >>>>>>>>>>>>
You have that backwards. Truth in rao requires PA
as part of it, and PA itself has no notion of
Truth in rao. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in rao.
Which means you don't understand what N actually is.
Nothing can be "True in N" unless that truth comes from the >>>>>>>>>> Axioms of PA, as N is the result of PA.
combined with the meta-math external model.
Nope. N is just a set of object built in the Formal System
defined by PA. 0 comes from Axiom 1 which states there is a 0. >>>>>>>>
If G is true and not provable then you have
the wrong kind of true. I have known that
the entire body of knowledge is a semantic
tautology for 28 years.
No, YOU do. The problem is Truth in the real world isn't based on >>>>>> being about to prove the fact, and most things are not actually
provable, just well approximatable.
That is why this insight was so important:
"true on the basis of meaning expressed in language"
I broke through the 75 year logjam of the analytic/synthetic
distinction.
In other words, you don't accept the Pythgorean Theorem as "True",
since its Tru-ness doesn't come out of the meaning of its words.
[meaning of its words]
My sentence is not restricted to words and
does include mathematical expressions.
Then it accepts Godel's G as a valid statement
That has no truth value in PA.
and Goldbach's conjecture, even if improbably true, is a truth bearer.
As a truth bearer with a currently unknown truth value.
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
I never said anything about words.
It took me 25 years to derive that exact phrase.
You are just admitting to your own equivocation of meaning.
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can >>>>>>>>> denyThe meta-proof does not exist in the axioms of PA
the proof but you cannot perform what is meta-provably impossible. >>>>>>>
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also >>>>>>> a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what is true
in PA?
I thought you said that PA had to be able to determine the truth itself?
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You >>>>>>>>>> can deny
the proof but you cannot perform what is meta-provably
impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also >>>>>>>> a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what is
true in PA?
I thought you said that PA had to be able to determine the truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
On 1/25/26 9:31 PM, olcott wrote:
On 1/25/2026 2:44 PM, Richard Damon wrote:
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
Which can't handle math.
As you have effective admitted by not answering about my example with
the Pythgorean Theorem.
I never said anything about words.
It took me 25 years to derive that exact phrase.
What is language, but meaning expressed in "words".
I think your problem is a fundamental failure to understand what you are talking about as you accept your own double-speak.
You are just admitting to your own equivocation of meaning.
On 1/26/26 11:58 AM, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You >>>>>>>>>>> can deny
the proof but you cannot perform what is meta-provably
impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is >>>>>>>>> also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a >>>>>>> language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what is
true in PA?
I thought you said that PA had to be able to determine the truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
But Peano Arithmatic *IS* a standard model of arithmetic.
The Induction Axiom makes it so.
The other models tend to come from making a variant of PA by changing--
that 2nd order Induction Axiom to various first order versions to
"simulate" its power using the rest of the Peano Axioms, and then adding something more to complete it.
On 1/26/2026 10:49 AM, Richard Damon wrote:
On 1/25/26 9:31 PM, olcott wrote:All math and logic has a language of math and logic.
On 1/25/2026 2:44 PM, Richard Damon wrote:
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
Which can't handle math.
I literally spent 25 years coming up with that.
All of the LLM systems understand that
"true on the basis of meaning expressed in language"
breaks the logjam established in:
"Two Dogmas of Empiricism" Willard Van Orman Quine https://www.theologie.uzh.ch/dam/jcr:ffffffff- fbd6-1538-0000-000070cf64bc/Quine51.pdf
regarding the fundamental nature of truth itself
previously called the analytic/synthetic distinction
now renamed to the analytic/empirical distinction.
These LLM systems do not yet understand that
succinctly. It takes them some back and forth
to understand that.
As you have effective admitted by not answering about my example with
the Pythgorean Theorem.
I never said anything about words.
It took me 25 years to derive that exact phrase.
What is language, but meaning expressed in "words".
I think your problem is a fundamental failure to understand what you
are talking about as you accept your own double-speak.
You are just admitting to your own equivocation of meaning.
On 1/26/26 12:23 PM, olcott wrote:
On 1/26/2026 10:49 AM, Richard Damon wrote:
On 1/25/26 9:31 PM, olcott wrote:All math and logic has a language of math and logic.
On 1/25/2026 2:44 PM, Richard Damon wrote:
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
Which can't handle math.
I literally spent 25 years coming up with that.
And is build of "words" which is the symbolism of that langauge.
On 1/26/2026 12:24 PM, Richard Damon wrote:
On 1/26/26 12:23 PM, olcott wrote:All of those copies of what I said ARE VERY SPECIFICALLY
On 1/26/2026 10:49 AM, Richard Damon wrote:
On 1/25/26 9:31 PM, olcott wrote:All math and logic has a language of math and logic.
On 1/25/2026 2:44 PM, Richard Damon wrote:
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
Which can't handle math.
I literally spent 25 years coming up with that.
And is build of "words" which is the symbolism of that langauge.
AND VERY INTENTIONALLY NOT LIMITED TO WORDS.
Do I need to say that 10,000 times to get
you to notice that I said it at least once?
I build Minimal Type Theory entirely on the
basis of the YACC grammar specification of the
language of FOL.
https://www.researchgate.net/ publication/331859461_Minimal_Type_Theory_YACC_BNF
On 1/26/26 1:43 PM, olcott wrote:
On 1/26/2026 12:24 PM, Richard Damon wrote:
On 1/26/26 12:23 PM, olcott wrote:All of those copies of what I said ARE VERY SPECIFICALLY
On 1/26/2026 10:49 AM, Richard Damon wrote:
On 1/25/26 9:31 PM, olcott wrote:All math and logic has a language of math and logic.
On 1/25/2026 2:44 PM, Richard Damon wrote:
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
Which can't handle math.
I literally spent 25 years coming up with that.
And is build of "words" which is the symbolism of that langauge.
AND VERY INTENTIONALLY NOT LIMITED TO WORDS.
Do I need to say that 10,000 times to get
you to notice that I said it at least once?
No just answer the question.
But, I guess since you don't actually knoew what you mean, you can't do that,
I build Minimal Type Theory entirely on the
basis of the YACC grammar specification of the
language of FOL.
So?
https://www.researchgate.net/
publication/331859461_Minimal_Type_Theory_YACC_BNF
So, what is a "language" built on if not what it considers its "words"?
And, how does you system handle the truth of something like the
Pythagorean Theorem?
Your repeated failure just proves that you CAN'T answer as you know your system is broken but need to continue clinging to its lie.--
On 1/26/2026 3:58 PM, Richard Damon wrote:
On 1/26/26 1:43 PM, olcott wrote:
On 1/26/2026 12:24 PM, Richard Damon wrote:
On 1/26/26 12:23 PM, olcott wrote:All of those copies of what I said ARE VERY SPECIFICALLY
On 1/26/2026 10:49 AM, Richard Damon wrote:
On 1/25/26 9:31 PM, olcott wrote:All math and logic has a language of math and logic.
On 1/25/2026 2:44 PM, Richard Damon wrote:
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
Which can't handle math.
I literally spent 25 years coming up with that.
And is build of "words" which is the symbolism of that langauge.
AND VERY INTENTIONALLY NOT LIMITED TO WORDS.
Do I need to say that 10,000 times to get
you to notice that I said it at least once?
No just answer the question.
But, I guess since you don't actually knoew what you mean, you can't
do that,
I build Minimal Type Theory entirely on the
basis of the YACC grammar specification of the
language of FOL.
So?
https://www.researchgate.net/
publication/331859461_Minimal_Type_Theory_YACC_BNF
So, what is a "language" built on if not what it considers its "words"?
All of the logic, math and computation languages
are not grounded in words deep ship.
And, how does you system handle the truth of something like the
Pythagorean Theorem?
written in PA syntax as:
reCa reCb reCc (a-+a + b-+b = c-+c)
Your repeated failure just proves that you CAN'T answer as you know
your system is broken but need to continue clinging to its lie.
On 1/26/26 5:08 PM, olcott wrote:
On 1/26/2026 3:58 PM, Richard Damon wrote:
On 1/26/26 1:43 PM, olcott wrote:
On 1/26/2026 12:24 PM, Richard Damon wrote:
On 1/26/26 12:23 PM, olcott wrote:All of those copies of what I said ARE VERY SPECIFICALLY
On 1/26/2026 10:49 AM, Richard Damon wrote:
On 1/25/26 9:31 PM, olcott wrote:All math and logic has a language of math and logic.
On 1/25/2026 2:44 PM, Richard Damon wrote:
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
Which can't handle math.
I literally spent 25 years coming up with that.
And is build of "words" which is the symbolism of that langauge.
AND VERY INTENTIONALLY NOT LIMITED TO WORDS.
Do I need to say that 10,000 times to get
you to notice that I said it at least once?
No just answer the question.
But, I guess since you don't actually knoew what you mean, you can't
do that,
I build Minimal Type Theory entirely on the
basis of the YACC grammar specification of the
language of FOL.
So?
https://www.researchgate.net/
publication/331859461_Minimal_Type_Theory_YACC_BNF
So, what is a "language" built on if not what it considers its "words"?
All of the logic, math and computation languages
are not grounded in words deep ship.
sure they are, when you consider a "word" to include the symbols and
number they use.
And, how does you system handle the truth of something like the
Pythagorean Theorem?
written in PA syntax as:
reCa reCb reCc (a-+a + b-+b = c-+c)
So, why is that true for EVERY a and b that are sides of a right triangle?
Note, the Pythagorean Theorem isn't part of PA, but Plain Geometry.
I guess you just belive in truth conditional logic.
Your problem is you just don't know that truth or proof means because of your ignorance.
Your repeated failure just proves that you CAN'T answer as you know
your system is broken but need to continue clinging to its lie.
On 1/26/2026 4:36 PM, Richard Damon wrote:
On 1/26/26 5:08 PM, olcott wrote:
On 1/26/2026 3:58 PM, Richard Damon wrote:
On 1/26/26 1:43 PM, olcott wrote:
On 1/26/2026 12:24 PM, Richard Damon wrote:
On 1/26/26 12:23 PM, olcott wrote:All of those copies of what I said ARE VERY SPECIFICALLY
On 1/26/2026 10:49 AM, Richard Damon wrote:
On 1/25/26 9:31 PM, olcott wrote:All math and logic has a language of math and logic.
On 1/25/2026 2:44 PM, Richard Damon wrote:
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
Which can't handle math.
I literally spent 25 years coming up with that.
And is build of "words" which is the symbolism of that langauge.
AND VERY INTENTIONALLY NOT LIMITED TO WORDS.
Do I need to say that 10,000 times to get
you to notice that I said it at least once?
No just answer the question.
But, I guess since you don't actually knoew what you mean, you can't
do that,
I build Minimal Type Theory entirely on the
basis of the YACC grammar specification of the
language of FOL.
So?
https://www.researchgate.net/
publication/331859461_Minimal_Type_Theory_YACC_BNF
So, what is a "language" built on if not what it considers its "words"? >>>>
All of the logic, math and computation languages
are not grounded in words deep ship.
sure they are, when you consider a "word" to include the symbols and
number they use.
And, how does you system handle the truth of something like the
Pythagorean Theorem?
written in PA syntax as:
reCa reCb reCc (a-+a + b-+b = c-+c)
So, why is that true for EVERY a and b that are sides of a right
triangle?
Note, the Pythagorean Theorem isn't part of PA, but Plain Geometry.
I guess you just belive in truth conditional logic.
"true on the basis of meaning expressed in language"
Inherently includes every element of the entire body
of knowledge that can be expressed in any formal
mathematical or natural language.
Your problem is you just don't know that truth or proof means because
of your ignorance.
Your repeated failure just proves that you CAN'T answer as you know
your system is broken but need to continue clinging to its lie.
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can >>>>>>>> denyThe meta-proof does not exist in the axioms of PA
the proof but you cannot perform what is meta-provably impossible. >>>>>>
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also >>>>>> a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You >>>>>>>>>> can deny
the proof but you cannot perform what is meta-provably
impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also >>>>>>>> a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what is
true in PA?
I thought you said that PA had to be able to determine the truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can >>>>>>>>> denyThe meta-proof does not exist in the axioms of PA
the proof but you cannot perform what is meta-provably impossible. >>>>>>>
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also >>>>>>> a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what is true
in PA?
I thought you said that PA had to be able to determine the truth itself?
On 26/01/2026 17:22, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can >>>>>>>>> denyThe meta-proof does not exist in the axioms of PA
the proof but you cannot perform what is meta-provably impossible. >>>>>>>
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also >>>>>>> a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
The above is not a notational convention. The symbols may be defined
in some context but they are undefined elsewhere.
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You >>>>>>>>>>> can deny
the proof but you cannot perform what is meta-provably
impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is >>>>>>>>> also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a >>>>>>> language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what is
true in PA?
I thought you said that PA had to be able to determine the truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
On 1/27/2026 2:05 AM, Mikko wrote:
On 26/01/2026 17:22, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You >>>>>>>>>> can deny
the proof but you cannot perform what is meta-provably
impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also >>>>>>>> a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
The above is not a notational convention. The symbols may be defined
in some context but they are undefined elsewhere.
Mendelson simply uses reo EYAR to indicate that EYAR is a theorem.
reCx (True(x) rei reo EYAR)
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You >>>>>>>>>>>> can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is >>>>>>>>>> also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a >>>>>>>> language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what is
true in PA?
I thought you said that PA had to be able to determine the truth
itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
All of the expressions where True(L, x) is not computable
x is semantically incoherent or outside of the domain of knowledge.
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You >>>>>>>>>>>>> can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA >>>>>>>>>>> is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a >>>>>>>>> language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and >>>>>>> WellFounded with two arguments. And you did not specify in which >>>>>>> context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what is
true in PA?
I thought you said that PA had to be able to determine the truth
itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results
are related and there are much similarity in the proofs. Tarski did
not use Turing machines in the proof but a computability proof must
use that.
All of the expressions where True(L, x) is not computable
x is semantically incoherent or outside of the domain of knowledge.
Computability does not depend on semantics or knowledge.
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. >>>>>>>>>>>>>> You can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA. >>>>>>>>>>>>> All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA >>>>>>>>>>>> is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a >>>>>>>>>> language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and >>>>>>>> WellFounded with two arguments. And you did not specify in which >>>>>>>> context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what is >>>>>> true in PA?
I thought you said that PA had to be able to determine the truth
itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results
are related and there are much similarity in the proofs. Tarski did
not use Turing machines in the proof but a computability proof must
use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
On 28/01/2026 15:49, olcott wrote:
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:Those sentences don't mean anything without specificantions of a >>>>>>>>>>> language and a theory that gives them some meaning.
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. >>>>>>>>>>>>>>> You can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA. >>>>>>>>>>>>>> All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA >>>>>>>>>>>>> is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x))) >>>>>>>>>>>
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms >>>>>>>>>> of PA.
There are no notational convention that defines True, False, and >>>>>>>>> WellFounded with two arguments. And you did not specify in which >>>>>>>>> context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what >>>>>>> is true in PA?
I thought you said that PA had to be able to determine the truth >>>>>>> itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results
are related and there are much similarity in the proofs. Tarski did
not use Turing machines in the proof but a computability proof must
use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
Irrelevant. There is no "occurs_check" in Tarski's proof.
On 1/29/2026 3:12 AM, Mikko wrote:
On 28/01/2026 15:49, olcott wrote:
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:Those sentences don't mean anything without specificantions >>>>>>>>>>>> of a
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. >>>>>>>>>>>>>>>> You can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA >>>>>>>>>>>>>>> and that is the reason why an external truth in
an external model cannot be proved internally in PA. >>>>>>>>>>>>>>> All of these years it was only a mere conflation >>>>>>>>>>>>>>> error.
It is perfectly clear which is which. But every proof in >>>>>>>>>>>>>> PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x))) >>>>>>>>>>>>
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the >>>>>>>>>>> axioms of PA and False for PA as refutable from the axioms >>>>>>>>>>> of PA.
There are no notational convention that defines True, False, and >>>>>>>>>> WellFounded with two arguments. And you did not specify in which >>>>>>>>>> context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what >>>>>>>> is true in PA?
I thought you said that PA had to be able to determine the truth >>>>>>>> itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results
are related and there are much similarity in the proofs. Tarski did
not use Turing machines in the proof but a computability proof must
use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
Irrelevant. There is no "occurs_check" in Tarski's proof.
On 29/01/2026 15:57, olcott wrote:
On 1/29/2026 3:12 AM, Mikko wrote:
On 28/01/2026 15:49, olcott wrote:
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:Those sentences don't mean anything without specificantions >>>>>>>>>>>>> of a
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. >>>>>>>>>>>>>>>>> You can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA >>>>>>>>>>>>>>>> and that is the reason why an external truth in >>>>>>>>>>>>>>>> an external model cannot be proved internally in PA. >>>>>>>>>>>>>>>> All of these years it was only a mere conflation >>>>>>>>>>>>>>>> error.
It is perfectly clear which is which. But every proof in >>>>>>>>>>>>>>> PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x))) >>>>>>>>>>>>>
language and a theory that gives them some meaning.
In other word you do not understand standard notational >>>>>>>>>>>> conventions that define True for PA as provable from the >>>>>>>>>>>> axioms of PA and False for PA as refutable from the axioms >>>>>>>>>>>> of PA.
There are no notational convention that defines True, False, and >>>>>>>>>>> WellFounded with two arguments. And you did not specify in which >>>>>>>>>>> context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what >>>>>>>>> is true in PA?
I thought you said that PA had to be able to determine the
truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results
are related and there are much similarity in the proofs. Tarski did
not use Turing machines in the proof but a computability proof must
use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
Irrelevant. There is no "occurs_check" in Tarski's proof.
That would have no effet. Even if the metalanguage had an occcurs_check
it would not be necessary to use it in a proof.
On 16/01/2026 04:03, olcott wrote:
It is the same reCx ree T ((True(T, x) rei (T reo x))
I still think you're asking for confusion with that use of the turnstile.
But it does make it very obvious that we should expect negation to be restricted in your system which might overcome a psychological hurdle.
How is negation restricted in your system?
On 1/30/2026 3:34 AM, Mikko wrote:
On 29/01/2026 15:57, olcott wrote:
On 1/29/2026 3:12 AM, Mikko wrote:
On 28/01/2026 15:49, olcott wrote:
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:In other word you do not understand standard notational >>>>>>>>>>>>> conventions that define True for PA as provable from the >>>>>>>>>>>>> axioms of PA and False for PA as refutable from the axioms >>>>>>>>>>>>> of PA.
On 1/24/2026 2:23 AM, Mikko wrote:Those sentences don't mean anything without
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout >>>>>>>>>>>>>>>>>> PA. You can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA >>>>>>>>>>>>>>>>> and that is the reason why an external truth in >>>>>>>>>>>>>>>>> an external model cannot be proved internally in PA. >>>>>>>>>>>>>>>>> All of these years it was only a mere conflation >>>>>>>>>>>>>>>>> error.
It is perfectly clear which is which. But every proof in >>>>>>>>>>>>>>>> PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x))) >>>>>>>>>>>>>>
specificantions of a
language and a theory that gives them some meaning. >>>>>>>>>>>>>
There are no notational convention that defines True, False, >>>>>>>>>>>> and
WellFounded with two arguments. And you did not specify in >>>>>>>>>>>> which
context your sentences are true or otherwise relevant. >>>>>>>>>>>>
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define >>>>>>>>>> what is true in PA?
I thought you said that PA had to be able to determine the >>>>>>>>>> truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results >>>>>> are related and there are much similarity in the proofs. Tarski did >>>>>> not use Turing machines in the proof but a computability proof must >>>>>> use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
Irrelevant. There is no "occurs_check" in Tarski's proof.
That would have no effet. Even if the metalanguage had an occcurs_check
it would not be necessary to use it in a proof.
It would only seem to have no effect because you
never bothered to understand what an occurs_check is.
Truth is computable because rCLmeaningful sentencerCY
is defined as rCLsentence with a well-founded
justification tree,rCY and evaluating any well-founded
tree always terminates. Anything else isnrCOt truth-apt.
On 1/29/2026 3:12 AM, Mikko wrote:
On 28/01/2026 15:49, olcott wrote:
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:Those sentences don't mean anything without specificantions >>>>>>>>>>>> of a
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. >>>>>>>>>>>>>>>> You can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA >>>>>>>>>>>>>>> and that is the reason why an external truth in
an external model cannot be proved internally in PA. >>>>>>>>>>>>>>> All of these years it was only a mere conflation >>>>>>>>>>>>>>> error.
It is perfectly clear which is which. But every proof in >>>>>>>>>>>>>> PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x))) >>>>>>>>>>>>
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the >>>>>>>>>>> axioms of PA and False for PA as refutable from the axioms >>>>>>>>>>> of PA.
There are no notational convention that defines True, False, and >>>>>>>>>> WellFounded with two arguments. And you did not specify in which >>>>>>>>>> context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what >>>>>>>> is true in PA?
I thought you said that PA had to be able to determine the truth >>>>>>>> itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results
are related and there are much similarity in the proofs. Tarski did
not use Turing machines in the proof but a computability proof must
use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
Irrelevant. There is no "occurs_check" in Tarski's proof.
If there was then there never would be a Tarski proof. https://liarparadox.org/Tarski_247_248.pdf
On 30/01/2026 16:35, olcott wrote:
On 1/30/2026 3:34 AM, Mikko wrote:
On 29/01/2026 15:57, olcott wrote:
On 1/29/2026 3:12 AM, Mikko wrote:
On 28/01/2026 15:49, olcott wrote:
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:In other word you do not understand standard notational >>>>>>>>>>>>>> conventions that define True for PA as provable from the >>>>>>>>>>>>>> axioms of PA and False for PA as refutable from the axioms >>>>>>>>>>>>>> of PA.
On 1/24/2026 2:23 AM, Mikko wrote:Those sentences don't mean anything without
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout >>>>>>>>>>>>>>>>>>> PA. You can deny
the proof but you cannot perform what is meta- >>>>>>>>>>>>>>>>>>> provably impossible.
The meta-proof does not exist in the axioms of PA >>>>>>>>>>>>>>>>>> and that is the reason why an external truth in >>>>>>>>>>>>>>>>>> an external model cannot be proved internally in PA. >>>>>>>>>>>>>>>>>> All of these years it was only a mere conflation >>>>>>>>>>>>>>>>>> error.
It is perfectly clear which is which. But every proof >>>>>>>>>>>>>>>>> in PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x))) >>>>>>>>>>>>>>>
specificantions of a
language and a theory that gives them some meaning. >>>>>>>>>>>>>>
There are no notational convention that defines True, >>>>>>>>>>>>> False, and
WellFounded with two arguments. And you did not specify in >>>>>>>>>>>>> which
context your sentences are true or otherwise relevant. >>>>>>>>>>>>>
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define >>>>>>>>>>> what is true in PA?
I thought you said that PA had to be able to determine the >>>>>>>>>>> truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results >>>>>>> are related and there are much similarity in the proofs. Tarski did >>>>>>> not use Turing machines in the proof but a computability proof must >>>>>>> use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
Irrelevant. There is no "occurs_check" in Tarski's proof.
That would have no effet. Even if the metalanguage had an occcurs_check
it would not be necessary to use it in a proof.
It would only seem to have no effect because you
never bothered to understand what an occurs_check is.
That assumption is false.
Truth is computable because rCLmeaningful sentencerCY
is defined as rCLsentence with a well-founded
justification tree,rCY and evaluating any well-founded
tree always terminates. Anything else isnrCOt truth-apt.
That "bcause" is wrong. Whether a sentence has a well-founded
justifiation tree is not computable, especially for arithmetic
sentences.
But that does not alter the fact that an existence or non-existence
of a metalanguage feature that is not present in the justification
tree is irrelevant.
On 29/01/2026 15:57, olcott wrote:
On 1/29/2026 3:12 AM, Mikko wrote:
On 28/01/2026 15:49, olcott wrote:
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:Those sentences don't mean anything without specificantions >>>>>>>>>>>>> of a
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. >>>>>>>>>>>>>>>>> You can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA >>>>>>>>>>>>>>>> and that is the reason why an external truth in >>>>>>>>>>>>>>>> an external model cannot be proved internally in PA. >>>>>>>>>>>>>>>> All of these years it was only a mere conflation >>>>>>>>>>>>>>>> error.
It is perfectly clear which is which. But every proof in >>>>>>>>>>>>>>> PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x))) >>>>>>>>>>>>>
language and a theory that gives them some meaning.
In other word you do not understand standard notational >>>>>>>>>>>> conventions that define True for PA as provable from the >>>>>>>>>>>> axioms of PA and False for PA as refutable from the axioms >>>>>>>>>>>> of PA.
There are no notational convention that defines True, False, and >>>>>>>>>>> WellFounded with two arguments. And you did not specify in which >>>>>>>>>>> context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what >>>>>>>>> is true in PA?
I thought you said that PA had to be able to determine the
truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results
are related and there are much similarity in the proofs. Tarski did
not use Turing machines in the proof but a computability proof must
use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
Irrelevant. There is no "occurs_check" in Tarski's proof.
If there was then there never would be a Tarski proof.
https://liarparadox.org/Tarski_247_248.pdf
Irrelevant. Tarski's proof is what it is and there is no "occurs_check" there.
On 1/31/2026 2:56 AM, Mikko wrote:
On 29/01/2026 15:57, olcott wrote:
On 1/29/2026 3:12 AM, Mikko wrote:
On 28/01/2026 15:49, olcott wrote:
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:In other word you do not understand standard notational >>>>>>>>>>>>> conventions that define True for PA as provable from the >>>>>>>>>>>>> axioms of PA and False for PA as refutable from the axioms >>>>>>>>>>>>> of PA.
On 1/24/2026 2:23 AM, Mikko wrote:Those sentences don't mean anything without
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout >>>>>>>>>>>>>>>>>> PA. You can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA >>>>>>>>>>>>>>>>> and that is the reason why an external truth in >>>>>>>>>>>>>>>>> an external model cannot be proved internally in PA. >>>>>>>>>>>>>>>>> All of these years it was only a mere conflation >>>>>>>>>>>>>>>>> error.
It is perfectly clear which is which. But every proof in >>>>>>>>>>>>>>>> PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x))) >>>>>>>>>>>>>>
specificantions of a
language and a theory that gives them some meaning. >>>>>>>>>>>>>
There are no notational convention that defines True, False, >>>>>>>>>>>> and
WellFounded with two arguments. And you did not specify in >>>>>>>>>>>> which
context your sentences are true or otherwise relevant. >>>>>>>>>>>>
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define >>>>>>>>>> what is true in PA?
I thought you said that PA had to be able to determine the >>>>>>>>>> truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results >>>>>> are related and there are much similarity in the proofs. Tarski did >>>>>> not use Turing machines in the proof but a computability proof must >>>>>> use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
Irrelevant. There is no "occurs_check" in Tarski's proof.
If there was then there never would be a Tarski proof.
https://liarparadox.org/Tarski_247_248.pdf
Irrelevant. Tarski's proof is what it is and there is no "occurs_check"
there.
Sure and a car that has a missing engine will always
be a car that will not run.
On 1/31/2026 2:41 AM, Mikko wrote:
On 30/01/2026 16:35, olcott wrote:
On 1/30/2026 3:34 AM, Mikko wrote:
On 29/01/2026 15:57, olcott wrote:
On 1/29/2026 3:12 AM, Mikko wrote:
On 28/01/2026 15:49, olcott wrote:
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:In other word you do not understand standard notational >>>>>>>>>>>>>>> conventions that define True for PA as provable from the >>>>>>>>>>>>>>> axioms of PA and False for PA as refutable from the axioms >>>>>>>>>>>>>>> of PA.
On 1/24/2026 2:23 AM, Mikko wrote:Those sentences don't mean anything without
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout >>>>>>>>>>>>>>>>>>>> PA. You can deny
the proof but you cannot perform what is meta- >>>>>>>>>>>>>>>>>>>> provably impossible.
The meta-proof does not exist in the axioms of PA >>>>>>>>>>>>>>>>>>> and that is the reason why an external truth in >>>>>>>>>>>>>>>>>>> an external model cannot be proved internally in PA. >>>>>>>>>>>>>>>>>>> All of these years it was only a mere conflation >>>>>>>>>>>>>>>>>>> error.
It is perfectly clear which is which. But every proof >>>>>>>>>>>>>>>>>> in PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x ) >>>>>>>>>>>>>>>>> reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x))) >>>>>>>>>>>>>>>>
specificantions of a
language and a theory that gives them some meaning. >>>>>>>>>>>>>>>
There are no notational convention that defines True, >>>>>>>>>>>>>> False, and
WellFounded with two arguments. And you did not specify in >>>>>>>>>>>>>> which
context your sentences are true or otherwise relevant. >>>>>>>>>>>>>>
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define >>>>>>>>>>>> what is true in PA?
I thought you said that PA had to be able to determine the >>>>>>>>>>>> truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results >>>>>>>> are related and there are much similarity in the proofs. Tarski did >>>>>>>> not use Turing machines in the proof but a computability proof must >>>>>>>> use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
Irrelevant. There is no "occurs_check" in Tarski's proof.
That would have no effet. Even if the metalanguage had an occcurs_check >>>> it would not be necessary to use it in a proof.
It would only seem to have no effect because you
never bothered to understand what an occurs_check is.
That assumption is false.
So far you have conclusively proven that you
do not understand what an occurs_check is.
If you want to provide that you do know then
you must provide all of the correct details.
Merely claiming that my statement is false
is an assertion entirely bereft of supporting
reasoning thus inherently baseless.
Truth is computable because rCLmeaningful sentencerCY
is defined as rCLsentence with a well-founded
justification tree,rCY and evaluating any well-founded
tree always terminates. Anything else isnrCOt truth-apt.
That "bcause" is wrong. Whether a sentence has a well-founded
justifiation tree is not computable, especially for arithmetic
sentences.
My one half page of text explaining all of the key details
of my 28 years of work was completely validated by five
different LLM systems. proof theoretic semantics is correct
model theoretic semantics is profoundly wrong-headed.
Your ignorance of the details of well-founded proof theoretic
semantics makes your rebuttal baseless.
But that does not alter the fact that an existence or non-existence
of a metalanguage feature that is not present in the justification
tree is irrelevant.
An existence or non-existence of a metalanguage feature
is entirely anchored in a totally wrong-headed notion.
The only way that this can be seen is to become an expert
in well-founded proof theoretic semantics.
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