On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:
On 07/01/2026 06:44, olcott wrote:
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
The counter-example input to requires more than
can be derived from finite string transformation
rules applied to this specific input thus the
Halting Problem requires too much.
In a sense the halting problem asks too much: the problem is >>>>>>>>> proven to
be unsolvable. In another sense it asks too little: usually we >>>>>>>>> want to
know whether a method halts on every input, not just one.
Although the halting problem is unsolvable, there are partial >>>>>>>>> solutions
to the halting problem. In particular, every counter-example to >>>>>>>>> the
full solution is correctly solved by some partial deciders.
*if undecidability is correct then truth itself is broken*
Depends on whether the word "truth" is interpeted in the standard >>>>>>> sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory
expressions are correctly rejected as semantically
incoherent thus form no undecidability or incompleteness.
The misconception is yours. No expression in the language of the first >>>>> order group theory is self-contradictory. But the first order goupr
theory is incomplete: it is impossible to prove that AB = BA is true >>>>> for every A and every B but it is also impossible to prove that AB
= BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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.
Of course, it one can prove that the required result is not computable
then that helps to avoid wasting effort to try the impossible. The
situation is worse if it is not known that the required result is not
computable.
That something is not computable does not mean that there is anyting
"incorrect" in the requirement.
Yes it certainly does. Requiring the impossible is always an error.
Requiring an answer to a yes/no question that has no correct yes/no
answer is an incorrect question that must be rejected.
In order to claim that a requirement
is incorrect one must at least prove that the requirement does not
serve its intended purpose.
Requiring the impossible cannot possibly serve any purpose
except perhaps to exemplify one's own ignorance.
Even then it is possible that the
requirement serves some other purpose. Even if a requirement serves
no purpose that need not mean that it be "incorrect", only that it
is useless.
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:
On 07/01/2026 06:44, olcott wrote:
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
The counter-example input to requires more than
can be derived from finite string transformation
rules applied to this specific input thus the
Halting Problem requires too much.
In a sense the halting problem asks too much: the problem is >>>>>>>>> proven to
be unsolvable. In another sense it asks too little: usually we >>>>>>>>> want to
know whether a method halts on every input, not just one.
Although the halting problem is unsolvable, there are partial >>>>>>>>> solutions
to the halting problem. In particular, every counter-example to >>>>>>>>> the
full solution is correctly solved by some partial deciders.
*if undecidability is correct then truth itself is broken*
Depends on whether the word "truth" is interpeted in the standard >>>>>>> sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory
expressions are correctly rejected as semantically
incoherent thus form no undecidability or incompleteness.
The misconception is yours. No expression in the language of the first >>>>> order group theory is self-contradictory. But the first order goupr
theory is incomplete: it is impossible to prove that AB = BA is true >>>>> for every A and every B but it is also impossible to prove that AB
= BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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.
Of course, it one can prove that the required result is not computable
then that helps to avoid wasting effort to try the impossible. The
situation is worse if it is not known that the required result is not
computable.
That something is not computable does not mean that there is anyting
"incorrect" in the requirement.
Yes it certainly does. Requiring the impossible is always an error.
It is a perfectly valid question to ask whther a particular reuqirement
is satisfiable.
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:
On 07/01/2026 06:44, olcott wrote:*if undecidability is correct then truth itself is broken*
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
The counter-example input to requires more than
can be derived from finite string transformation
rules applied to this specific input thus the
Halting Problem requires too much.
In a sense the halting problem asks too much: the problem is >>>>>>>>>> proven to
be unsolvable. In another sense it asks too little: usually we >>>>>>>>>> want to
know whether a method halts on every input, not just one.
Although the halting problem is unsolvable, there are partial >>>>>>>>>> solutions
to the halting problem. In particular, every counter-example >>>>>>>>>> to the
full solution is correctly solved by some partial deciders. >>>>>>>>>
Depends on whether the word "truth" is interpeted in the standard >>>>>>>> sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory
expressions are correctly rejected as semantically
incoherent thus form no undecidability or incompleteness.
The misconception is yours. No expression in the language of the
first
order group theory is self-contradictory. But the first order goupr >>>>>> theory is incomplete: it is impossible to prove that AB = BA is true >>>>>> for every A and every B but it is also impossible to prove that AB >>>>>> = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as
ill-posed with respect to computable semantics.
When the specification is constrained to properties
detectable via finite simulation and finite pattern
recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
On 1/11/2026 4:22 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:
On 07/01/2026 06:44, olcott wrote:*if undecidability is correct then truth itself is broken*
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
The counter-example input to requires more than
can be derived from finite string transformation
rules applied to this specific input thus the
Halting Problem requires too much.
In a sense the halting problem asks too much: the problem is >>>>>>>>>> proven to
be unsolvable. In another sense it asks too little: usually we >>>>>>>>>> want to
know whether a method halts on every input, not just one.
Although the halting problem is unsolvable, there are partial >>>>>>>>>> solutions
to the halting problem. In particular, every counter-example >>>>>>>>>> to the
full solution is correctly solved by some partial deciders. >>>>>>>>>
Depends on whether the word "truth" is interpeted in the standard >>>>>>>> sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory
expressions are correctly rejected as semantically
incoherent thus form no undecidability or incompleteness.
The misconception is yours. No expression in the language of the
first
order group theory is self-contradictory. But the first order goupr >>>>>> theory is incomplete: it is impossible to prove that AB = BA is true >>>>>> for every A and every B but it is also impossible to prove that AB >>>>>> = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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.
Of course, it one can prove that the required result is not computable >>>> then that helps to avoid wasting effort to try the impossible. The
situation is worse if it is not known that the required result is not
computable.
That something is not computable does not mean that there is anyting
"incorrect" in the requirement.
Yes it certainly does. Requiring the impossible is always an error.
It is a perfectly valid question to ask whther a particular reuqirement
is satisfiable.
Any yes/no question lacking a correct yes/no answer
is an incorrect question that must be rejected on
that basis.
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
On 09/01/2026 17:52, olcott wrote:Right. Outside the scope of computation. Requiring anything
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:
On 07/01/2026 06:44, olcott wrote:*if undecidability is correct then truth itself is broken*
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
The counter-example input to requires more than
can be derived from finite string transformation
rules applied to this specific input thus the
Halting Problem requires too much.
In a sense the halting problem asks too much: the problem is >>>>>>>>>>> proven to
be unsolvable. In another sense it asks too little: usually >>>>>>>>>>> we want to
know whether a method halts on every input, not just one. >>>>>>>>>>>
Although the halting problem is unsolvable, there are partial >>>>>>>>>>> solutions
to the halting problem. In particular, every counter-example >>>>>>>>>>> to the
full solution is correctly solved by some partial deciders. >>>>>>>>>>
Depends on whether the word "truth" is interpeted in the standard >>>>>>>>> sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory
expressions are correctly rejected as semantically
incoherent thus form no undecidability or incompleteness.
The misconception is yours. No expression in the language of the >>>>>>> first
order group theory is self-contradictory. But the first order goupr >>>>>>> theory is incomplete: it is impossible to prove that AB = BA is true >>>>>>> for every A and every B but it is also impossible to prove that >>>>>>> AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
No, that does not follow. If a required result cannot be derived by
appying a finite string transformation then the it it is uncomputable. >>>>
outside the scope of computation is an incorrect requirement.
You can't determine whether the required result is computable before
you have the requirement.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must determine
whether the computation presented by its input halts has already
been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as
ill-posed with respect to computable semantics.
When the specification is constrained to properties
detectable via finite simulation and finite pattern
recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
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 byRight. Outside the scope of computation. Requiring anything
appying a finite string transformation then the it it is uncomputable. >>>>
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.
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:Depends on whether the word "truth" is interpeted in the standard >>>>>>>>>> sense or in Olcott's sense.
On 07/01/2026 06:44, olcott wrote:*if undecidability is correct then truth itself is broken* >>>>>>>>>>
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
The counter-example input to requires more than
can be derived from finite string transformation
rules applied to this specific input thus the
Halting Problem requires too much.
In a sense the halting problem asks too much: the problem is >>>>>>>>>>>> proven to
be unsolvable. In another sense it asks too little: usually >>>>>>>>>>>> we want to
know whether a method halts on every input, not just one. >>>>>>>>>>>>
Although the halting problem is unsolvable, there are >>>>>>>>>>>> partial solutions
to the halting problem. In particular, every counter-example >>>>>>>>>>>> to the
full solution is correctly solved by some partial deciders. >>>>>>>>>>>
Undecidability is misconception. Self-contradictory
expressions are correctly rejected as semantically
incoherent thus form no undecidability or incompleteness.
The misconception is yours. No expression in the language of the >>>>>>>> first
order group theory is self-contradictory. But the first order goupr >>>>>>>> theory is incomplete: it is impossible to prove that AB = BA is >>>>>>>> true
for every A and every B but it is also impossible to prove that >>>>>>>> AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must determine
whether the computation presented by its input halts has already
been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as
ill-posed with respect to computable semantics.
When the specification is constrained to properties
detectable via finite simulation and finite pattern
recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is that an input that does the opposite of whatever
value the halt decider returns is non-well-founded
within proof-theoretic semantics.
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.
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:Depends on whether the word "truth" is interpeted in the standard >>>>>>>>>> sense or in Olcott's sense.
On 07/01/2026 06:44, olcott wrote:*if undecidability is correct then truth itself is broken* >>>>>>>>>>
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
The counter-example input to requires more than
can be derived from finite string transformation
rules applied to this specific input thus the
Halting Problem requires too much.
In a sense the halting problem asks too much: the problem is >>>>>>>>>>>> proven to
be unsolvable. In another sense it asks too little: usually >>>>>>>>>>>> we want to
know whether a method halts on every input, not just one. >>>>>>>>>>>>
Although the halting problem is unsolvable, there are >>>>>>>>>>>> partial solutions
to the halting problem. In particular, every counter-example >>>>>>>>>>>> to the
full solution is correctly solved by some partial deciders. >>>>>>>>>>>
Undecidability is misconception. Self-contradictory
expressions are correctly rejected as semantically
incoherent thus form no undecidability or incompleteness.
The misconception is yours. No expression in the language of the >>>>>>>> first
order group theory is self-contradictory. But the first order goupr >>>>>>>> theory is incomplete: it is impossible to prove that AB = BA is >>>>>>>> true
for every A and every B but it is also impossible to prove that >>>>>>>> AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must determine
whether the computation presented by its input halts has already
been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as
ill-posed with respect to computable semantics.
When the specification is constrained to properties
detectable via finite simulation and finite pattern
recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is that an input that does the opposite of whatever
value the halt decider returns is non-well-founded
within proof-theoretic semantics.
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.
On 12/01/2026 16:29, olcott wrote:
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:Depends on whether the word "truth" is interpeted in the >>>>>>>>>>> standard
On 07/01/2026 06:44, olcott wrote:*if undecidability is correct then truth itself is broken* >>>>>>>>>>>
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
The counter-example input to requires more than
can be derived from finite string transformation
rules applied to this specific input thus the
Halting Problem requires too much.
In a sense the halting problem asks too much: the problem >>>>>>>>>>>>> is proven to
be unsolvable. In another sense it asks too little: usually >>>>>>>>>>>>> we want to
know whether a method halts on every input, not just one. >>>>>>>>>>>>>
Although the halting problem is unsolvable, there are >>>>>>>>>>>>> partial solutions
to the halting problem. In particular, every counter- >>>>>>>>>>>>> example to the
full solution is correctly solved by some partial deciders. >>>>>>>>>>>>
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory
expressions are correctly rejected as semantically
incoherent thus form no undecidability or incompleteness.
The misconception is yours. No expression in the language of >>>>>>>>> the first
order group theory is self-contradictory. But the first order >>>>>>>>> goupr
theory is incomplete: it is impossible to prove that AB = BA is >>>>>>>>> true
for every A and every B but it is also impossible to prove that >>>>>>>>> AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must determine
whether the computation presented by its input halts has already
been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as
ill-posed with respect to computable semantics.
When the specification is constrained to properties
detectable via finite simulation and finite pattern
recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is proven that an universal halt decider does not exist.
A Turing
machine cannot determine the halting of all Turing machines and is
therefore not an universla halt decider.
An oracle machine may be
able to determine the haltinf of all Turing machines but not of all
oracle machines with the same oracle (or oracles) so it is not
universal.
It is that an input that does the opposite of whatever
value the halt decider returns is non-well-founded
within proof-theoretic semantics.
Yes, it is. What the "halt decider" returns is determinable: just run
it and see what it returns. From that the rest can be proven with a
well founded proof. In particular, there is a well-founded proof that
the "halt decider" is not a halt decider.
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.
On 13/01/2026 09:11, Mikko wrote:
An oracle machine may be
able to determine the haltinf of all Turing machines but not of all
oracle machines with the same oracle (or oracles) so it is not
universal.
What's the formal definition of "an oracle machine" ?
I would have thought an oracle always halts because it's an oracle it
answers every question that has an answer with either "HasAnswer answer"
or "HasNoAnswer".
On 1/13/2026 8:23 AM, Tristan Wibberley wrote:
On 13/01/2026 09:11, Mikko wrote:
An oracle machine may be
able to determine the haltinf of all Turing machines but not of all
oracle machines with the same oracle (or oracles) so it is not
universal.
What's the formal definition of "an oracle machine" ?
I would have thought an oracle always halts because it's an oracle it
answers every question that has an answer with either "HasAnswer answer"
or "HasNoAnswer".
It seems outside of computer science and into fantasy. https://en.wikipedia.org/wiki/Oracle_machine
On 13/01/2026 14:34, olcott wrote:
On 1/13/2026 8:23 AM, Tristan Wibberley wrote:
On 13/01/2026 09:11, Mikko wrote:
An oracle machine may be
able to determine the haltinf of all Turing machines but not of all
oracle machines with the same oracle (or oracles) so it is not
universal.
What's the formal definition of "an oracle machine" ?
I would have thought an oracle always halts because it's an oracle it
answers every question that has an answer with either "HasAnswer answer" >>> or "HasNoAnswer".
It seems outside of computer science and into fantasy.
https://en.wikipedia.org/wiki/Oracle_machine
Perhaps a halting oracle is real computer science, if it's own actions
are nondeterministic (ie, use bits of entropy from the environment via /dev/random to guide its search through confluent paths) then it could
always find whether a deterministic program halts because no
deterministic program has the oracle as a subprogram.
Then we have a new but different problem of making sure no two oracles receive the same sequence of entropy bits so an oracle can report on a program that contains it.
On 1/13/2026 3:11 AM, Mikko wrote:
On 12/01/2026 16:29, olcott wrote:
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:The misconception is yours. No expression in the language of >>>>>>>>>> the first
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:Depends on whether the word "truth" is interpeted in the >>>>>>>>>>>> standard
On 07/01/2026 06:44, olcott wrote:*if undecidability is correct then truth itself is broken* >>>>>>>>>>>>
All deciders essentially: Transform finite string >>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>> {Accept, Reject} values.
The counter-example input to requires more than
can be derived from finite string transformation >>>>>>>>>>>>>>> rules applied to this specific input thus the
Halting Problem requires too much.
In a sense the halting problem asks too much: the problem >>>>>>>>>>>>>> is proven to
be unsolvable. In another sense it asks too little: >>>>>>>>>>>>>> usually we want to
know whether a method halts on every input, not just one. >>>>>>>>>>>>>>
Although the halting problem is unsolvable, there are >>>>>>>>>>>>>> partial solutions
to the halting problem. In particular, every counter- >>>>>>>>>>>>>> example to the
full solution is correctly solved by some partial deciders. >>>>>>>>>>>>>
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory
expressions are correctly rejected as semantically
incoherent thus form no undecidability or incompleteness. >>>>>>>>>>
order group theory is self-contradictory. But the first order >>>>>>>>>> goupr
theory is incomplete: it is impossible to prove that AB = BA >>>>>>>>>> is true
for every A and every B but it is also impossible to prove >>>>>>>>>> that AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must determine
whether the computation presented by its input halts has already
been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as
ill-posed with respect to computable semantics.
When the specification is constrained to properties
detectable via finite simulation and finite pattern
recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is proven that an universal halt decider does not exist.
“The system adopts Proof-Theoretic Semantics: meaning is determined by inferential role, and truth is internal to the theory. A theory T is
defined by a finite set of stipulated atomic statements together with
all expressions derivable from them under the inference rules. The statements belonging to T constitute its theorems, and these are exactly
the statements that are true-in-T.”
Under a system like the above rough draft all inputs
having pathological self reference such as the halting
problem counter-example input are simply rejected as
non-well-founded. Tarski Undefinability, Gödel's
incompleteness and the halting problem cease to exist.
A Turing
machine cannot determine the halting of all Turing machines and is
therefore not an universla halt decider.
This is not true in Proof Theoretic Semantics. I
still have to refine my words. I may not have said
that exactly correctly. The result is that in Proof
Theoretic Semantics the counter-example is rejected
as non-well-founded.
On 1/13/2026 8:23 AM, Tristan Wibberley wrote:
On 13/01/2026 09:11, Mikko wrote:
An oracle machine may be
able to determine the haltinf of all Turing machines but not of all
oracle machines with the same oracle (or oracles) so it is not
universal.
What's the formal definition of "an oracle machine" ?
I would have thought an oracle always halts because it's an oracle it
answers every question that has an answer with either "HasAnswer answer"
or "HasNoAnswer".
It seems outside of computer science and into fantasy. https://en.wikipedia.org/wiki/Oracle_machine
On 13/01/2026 14:34, olcott wrote:
On 1/13/2026 8:23 AM, Tristan Wibberley wrote:
On 13/01/2026 09:11, Mikko wrote:
An oracle machine may be
able to determine the haltinf of all Turing machines but not of all
oracle machines with the same oracle (or oracles) so it is not
universal.
What's the formal definition of "an oracle machine" ?
I would have thought an oracle always halts because it's an oracle it
answers every question that has an answer with either "HasAnswer answer" >>> or "HasNoAnswer".
It seems outside of computer science and into fantasy.
https://en.wikipedia.org/wiki/Oracle_machine
Perhaps a halting oracle is real computer science, if it's own actions
are nondeterministic (ie, use bits of entropy from the environment via /dev/random to guide its search through confluent paths) then it could
always find whether a deterministic program halts because no
deterministic program has the oracle as a subprogram.
Then we have a new but different problem of making sure no two oracles receive the same sequence of entropy bits so an oracle can report on a program that contains it.
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 Gödel's incompleteness and a bunch more
as merely non-well-founded inputs.
Definition: An abstract machine with access to an "oracle"—a black box
that provides immediate answers to complex, even undecidable, problems
(like the Halting Problem). AKA a majick genie.
For a non-deterministic machine there are three possibilities: it may
halt always, sometimes, or never. THere is no oracle that can find the
right answer about every meachne that contains the same oracle.
On 13/01/2026 18:50, olcott wrote:
Definition: An abstract machine with access to an "oracle"—a black box
that provides immediate answers to complex, even undecidable, problems
(like the Halting Problem). AKA a majick genie.
What's it called when its almost an oracle but is arbitrarily slow?
On 13/01/2026 16:27, olcott wrote:
On 1/13/2026 3:11 AM, Mikko wrote:
On 12/01/2026 16:29, olcott wrote:
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:The misconception is yours. No expression in the language of >>>>>>>>>>> the first
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:Depends on whether the word "truth" is interpeted in the >>>>>>>>>>>>> standard
On 07/01/2026 06:44, olcott wrote:*if undecidability is correct then truth itself is broken* >>>>>>>>>>>>>
All deciders essentially: Transform finite string >>>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>>> {Accept, Reject} values.
The counter-example input to requires more than >>>>>>>>>>>>>>>> can be derived from finite string transformation >>>>>>>>>>>>>>>> rules applied to this specific input thus the
Halting Problem requires too much.
In a sense the halting problem asks too much: the problem >>>>>>>>>>>>>>> is proven to
be unsolvable. In another sense it asks too little: >>>>>>>>>>>>>>> usually we want to
know whether a method halts on every input, not just one. >>>>>>>>>>>>>>>
Although the halting problem is unsolvable, there are >>>>>>>>>>>>>>> partial solutions
to the halting problem. In particular, every counter- >>>>>>>>>>>>>>> example to the
full solution is correctly solved by some partial deciders. >>>>>>>>>>>>>>
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory
expressions are correctly rejected as semantically
incoherent thus form no undecidability or incompleteness. >>>>>>>>>>>
order group theory is self-contradictory. But the first order >>>>>>>>>>> goupr
theory is incomplete: it is impossible to prove that AB = BA >>>>>>>>>>> is true
for every A and every B but it is also impossible to prove >>>>>>>>>>> that AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must determine
whether the computation presented by its input halts has already
been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as
ill-posed with respect to computable semantics.
When the specification is constrained to properties
detectable via finite simulation and finite pattern
recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is proven that an universal halt decider does not exist.
“The system adopts Proof-Theoretic Semantics: meaning is determined by
inferential role, and truth is internal to the theory. A theory T is
defined by a finite set of stipulated atomic statements together with
all expressions derivable from them under the inference rules. The
statements belonging to T constitute its theorems, and these are
exactly the statements that are true-in-T.”
Under a system like the above rough draft all inputs
having pathological self reference such as the halting
problem counter-example input are simply rejected as
non-well-founded. Tarski Undefinability, Gödel's
incompleteness and the halting problem cease to exist.
A Turing
machine cannot determine the halting of all Turing machines and is
therefore not an universla halt decider.
This is not true in Proof Theoretic Semantics. I
still have to refine my words. I may not have said
that exactly correctly. The result is that in Proof
Theoretic Semantics the counter-example is rejected
as non-well-founded.
That no Turing machine is a halt decider is a proven theorem and a
truth about Turing machines. If your "Proof Thoeretic Semnatics"
does not regard it as true then your "Proof Theoretic Semantics"
is incomplete.
On 13/01/2026 16:17, olcott wrote:
On 1/13/2026 2:46 AM, Mikko wrote:Yes, it is. How to handle questions that lack a yes/no answer is
On 12/01/2026 16:43, olcott wrote:
On 1/12/2026 4:51 AM, Mikko wrote:
On 11/01/2026 16:23, olcott wrote:
On 1/11/2026 4:22 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:It is a perfectly valid question to ask whther a particular
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:The misconception is yours. No expression in the language of >>>>>>>>>>> the first
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:Depends on whether the word "truth" is interpeted in the >>>>>>>>>>>>> standard
On 07/01/2026 06:44, olcott wrote:*if undecidability is correct then truth itself is broken* >>>>>>>>>>>>>
All deciders essentially: Transform finite string >>>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>>> {Accept, Reject} values.
The counter-example input to requires more than >>>>>>>>>>>>>>>> can be derived from finite string transformation >>>>>>>>>>>>>>>> rules applied to this specific input thus the
Halting Problem requires too much.
In a sense the halting problem asks too much: the problem >>>>>>>>>>>>>>> is proven to
be unsolvable. In another sense it asks too little: >>>>>>>>>>>>>>> usually we want to
know whether a method halts on every input, not just one. >>>>>>>>>>>>>>>
Although the halting problem is unsolvable, there are >>>>>>>>>>>>>>> partial solutions
to the halting problem. In particular, every counter- >>>>>>>>>>>>>>> example to the
full solution is correctly solved by some partial deciders. >>>>>>>>>>>>>>
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory
expressions are correctly rejected as semantically
incoherent thus form no undecidability or incompleteness. >>>>>>>>>>>
order group theory is self-contradictory. But the first order >>>>>>>>>>> goupr
theory is incomplete: it is impossible to prove that AB = BA >>>>>>>>>>> is true
for every A and every B but it is also impossible to prove >>>>>>>>>>> that AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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.
Of course, it one can prove that the required result is not >>>>>>>>> computable
then that helps to avoid wasting effort to try the impossible. The >>>>>>>>> situation is worse if it is not known that the required result >>>>>>>>> is not
computable.
That something is not computable does not mean that there is >>>>>>>>> anyting
"incorrect" in the requirement.
Yes it certainly does. Requiring the impossible is always an error. >>>>>>>
reuqirement
is satisfiable.
Any yes/no question lacking a correct yes/no answer
is an incorrect question that must be rejected on
that basis.
Irrelevant. The question whether a particular requirement is
satisfiable
does have an answer that is either "yes" or "no". In some ases it is >>>>> not known whether it is "yes" or "no" and there may be no known way to >>>>> find out be even then either "yes" or "no" is the correct answer.
Now that I finally have the standard terminology:
Proof-theoretic semantics has always been the correct
formal system to handle decision problems.
When it is asked a yes/no question lacking a correct
yes/no answer it correctly determines non-well-founded.
I have been correct all along and merely lacked the
standard terminology.
Irrelevant, as already noted above.
irrelevant when discussing questions that do have a yes/no asnwer.
Whether a particular requirement is satisriable always has a yes/no
answer, so it is irrelevat how to handle questions that don't.
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 Gödel's incompleteness and a bunch more
as merely non-well-founded inputs.
For every Turing machine the halting problem counter-example provably
exists.
From the existence of the counter-example it is provable that
the first Turing machine is not a halting decider. With universal quationfication follows that no Turing machine is a halting decider.
Besides, there are other ways to prove that halting is not Turing
decidable.
On 1/12/26 9:29 AM, olcott wrote:
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:Depends on whether the word "truth" is interpeted in the >>>>>>>>>>> standard
On 07/01/2026 06:44, olcott wrote:*if undecidability is correct then truth itself is broken* >>>>>>>>>>>
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
The counter-example input to requires more than
can be derived from finite string transformation
rules applied to this specific input thus the
Halting Problem requires too much.
In a sense the halting problem asks too much: the problem >>>>>>>>>>>>> is proven to
be unsolvable. In another sense it asks too little: usually >>>>>>>>>>>>> we want to
know whether a method halts on every input, not just one. >>>>>>>>>>>>>
Although the halting problem is unsolvable, there are >>>>>>>>>>>>> partial solutions
to the halting problem. In particular, every counter- >>>>>>>>>>>>> example to the
full solution is correctly solved by some partial deciders. >>>>>>>>>>>>
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory
expressions are correctly rejected as semantically
incoherent thus form no undecidability or incompleteness.
The misconception is yours. No expression in the language of >>>>>>>>> the first
order group theory is self-contradictory. But the first order >>>>>>>>> goupr
theory is incomplete: it is impossible to prove that AB = BA is >>>>>>>>> true
for every A and every B but it is also impossible to prove that >>>>>>>>> AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must determine
whether the computation presented by its input halts has already
been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as
ill-posed with respect to computable semantics.
When the specification is constrained to properties
detectable via finite simulation and finite pattern
recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is that an input that does the opposite of whatever
value the halt decider returns is non-well-founded
within proof-theoretic semantics.
But the problem is that Computation is not a proof-theoretic semantic system, and thus those rules don't apply.
On 1/12/2026 9:19 PM, Richard Damon wrote:
On 1/12/26 9:29 AM, olcott wrote:
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:The misconception is yours. No expression in the language of >>>>>>>>>> the first
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:Depends on whether the word "truth" is interpeted in the >>>>>>>>>>>> standard
On 07/01/2026 06:44, olcott wrote:*if undecidability is correct then truth itself is broken* >>>>>>>>>>>>
All deciders essentially: Transform finite string >>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>> {Accept, Reject} values.
The counter-example input to requires more than
can be derived from finite string transformation >>>>>>>>>>>>>>> rules applied to this specific input thus the
Halting Problem requires too much.
In a sense the halting problem asks too much: the problem >>>>>>>>>>>>>> is proven to
be unsolvable. In another sense it asks too little: >>>>>>>>>>>>>> usually we want to
know whether a method halts on every input, not just one. >>>>>>>>>>>>>>
Although the halting problem is unsolvable, there are >>>>>>>>>>>>>> partial solutions
to the halting problem. In particular, every counter- >>>>>>>>>>>>>> example to the
full solution is correctly solved by some partial deciders. >>>>>>>>>>>>>
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory
expressions are correctly rejected as semantically
incoherent thus form no undecidability or incompleteness. >>>>>>>>>>
order group theory is self-contradictory. But the first order >>>>>>>>>> goupr
theory is incomplete: it is impossible to prove that AB = BA >>>>>>>>>> is true
for every A and every B but it is also impossible to prove >>>>>>>>>> that AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must determine
whether the computation presented by its input halts has already
been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as
ill-posed with respect to computable semantics.
When the specification is constrained to properties
detectable via finite simulation and finite pattern
recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is that an input that does the opposite of whatever
value the halt decider returns is non-well-founded
within proof-theoretic semantics.
But the problem is that Computation is not a proof-theoretic semantic
system, and thus those rules don't apply.
The dumbed down version is that the halting problem asks
a question outside of the scope of finite string transformations.
The halting problem proof does not fail because finite computation
is too weak. It fails because it asks finite computation to
decide a judgment that is not finitely grounded under operational
semantics.
By operational semantics I mean the standard proof‑theoretic
account of program meaning, where execution judgments are
given by inference rules and termination corresponds to the
existence of a finite derivation.
By proof‑theoretic semantics I mean the approach in which the
meaning of a statement is determined by its rules of proof
rather than by truth conditions in an external model.
Operational semantics fits this pattern: programs have meaning
through their execution rules, not through abstract denotations.
By denotational semantics I mean any semantics that assigns
mathematical objects—functions, truth values, domains-to programs independently of how they are executed or proved. This contrasts
with operational or proof‑theoretic semantics, where meaning is
grounded in the structure of derivations rather than in an abstract mathematical object.
I use “denotational semantics” simply to refer to any framework
that assigns meanings independently of operational behavior.
On 14/01/2026 08:53, Mikko wrote:
For a non-deterministic machine there are three possibilities: it may
halt always, sometimes, or never. THere is no oracle that can find the
right answer about every meachne that contains the same oracle.
We well into Turing c-machine territory here aren't we?
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: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.
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
You can't determine whether the required result is computable >>>>>>>> beforeNo, 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 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. >>>>>>
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 Gödel'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.
On 1/14/2026 1:58 AM, Mikko wrote:
On 13/01/2026 16:17, olcott wrote:
On 1/13/2026 2:46 AM, Mikko wrote:Yes, it is. How to handle questions that lack a yes/no answer is
On 12/01/2026 16:43, olcott wrote:
On 1/12/2026 4:51 AM, Mikko wrote:
On 11/01/2026 16:23, olcott wrote:
On 1/11/2026 4:22 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:The misconception is yours. No expression in the language of >>>>>>>>>>>> the first
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:Depends on whether the word "truth" is interpeted in the >>>>>>>>>>>>>> standard
On 07/01/2026 06:44, olcott wrote:*if undecidability is correct then truth itself is broken* >>>>>>>>>>>>>>
All deciders essentially: Transform finite string >>>>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>>>> {Accept, Reject} values.
The counter-example input to requires more than >>>>>>>>>>>>>>>>> can be derived from finite string transformation >>>>>>>>>>>>>>>>> rules applied to this specific input thus the >>>>>>>>>>>>>>>>> Halting Problem requires too much.
In a sense the halting problem asks too much: the >>>>>>>>>>>>>>>> problem is proven to
be unsolvable. In another sense it asks too little: >>>>>>>>>>>>>>>> usually we want to
know whether a method halts on every input, not just one. >>>>>>>>>>>>>>>>
Although the halting problem is unsolvable, there are >>>>>>>>>>>>>>>> partial solutions
to the halting problem. In particular, every counter- >>>>>>>>>>>>>>>> example to the
full solution is correctly solved by some partial deciders. >>>>>>>>>>>>>>>
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory
expressions are correctly rejected as semantically
incoherent thus form no undecidability or incompleteness. >>>>>>>>>>>>
order group theory is self-contradictory. But the first >>>>>>>>>>>> order goupr
theory is incomplete: it is impossible to prove that AB = BA >>>>>>>>>>>> is true
for every A and every B but it is also impossible to prove >>>>>>>>>>>> that AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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. >>>>>>>>>
Of course, it one can prove that the required result is not >>>>>>>>>> computable
then that helps to avoid wasting effort to try the impossible. >>>>>>>>>> The
situation is worse if it is not known that the required result >>>>>>>>>> is not
computable.
That something is not computable does not mean that there is >>>>>>>>>> anyting
"incorrect" in the requirement.
Yes it certainly does. Requiring the impossible is always an >>>>>>>>> error.
It is a perfectly valid question to ask whther a particular
reuqirement
is satisfiable.
Any yes/no question lacking a correct yes/no answer
is an incorrect question that must be rejected on
that basis.
Irrelevant. The question whether a particular requirement is
satisfiable
does have an answer that is either "yes" or "no". In some ases it is >>>>>> not known whether it is "yes" or "no" and there may be no known
way to
find out be even then either "yes" or "no" is the correct answer.
Now that I finally have the standard terminology:
Proof-theoretic semantics has always been the correct
formal system to handle decision problems.
When it is asked a yes/no question lacking a correct
yes/no answer it correctly determines non-well-founded.
I have been correct all along and merely lacked the
standard terminology.
Irrelevant, as already noted above.
irrelevant when discussing questions that do have a yes/no asnwer.
Whether a particular requirement is satisriable always has a yes/no
answer, so it is irrelevat how to handle questions that don't.
The classical diagonal argument for the Halting Problem asks a halt
decider H to evaluate a program D whose behavior depends on H’s own output.
On 1/14/2026 1:40 AM, Mikko wrote:
On 13/01/2026 16:27, olcott wrote:
On 1/13/2026 3:11 AM, Mikko wrote:
On 12/01/2026 16:29, olcott wrote:
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:You can't determine whether the required result is computable >>>>>>>> before
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:The misconception is yours. No expression in the language of >>>>>>>>>>>> the first
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:Depends on whether the word "truth" is interpeted in the >>>>>>>>>>>>>> standard
On 07/01/2026 06:44, olcott wrote:*if undecidability is correct then truth itself is broken* >>>>>>>>>>>>>>
All deciders essentially: Transform finite string >>>>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>>>> {Accept, Reject} values.
The counter-example input to requires more than >>>>>>>>>>>>>>>>> can be derived from finite string transformation >>>>>>>>>>>>>>>>> rules applied to this specific input thus the >>>>>>>>>>>>>>>>> Halting Problem requires too much.
In a sense the halting problem asks too much: the >>>>>>>>>>>>>>>> problem is proven to
be unsolvable. In another sense it asks too little: >>>>>>>>>>>>>>>> usually we want to
know whether a method halts on every input, not just one. >>>>>>>>>>>>>>>>
Although the halting problem is unsolvable, there are >>>>>>>>>>>>>>>> partial solutions
to the halting problem. In particular, every counter- >>>>>>>>>>>>>>>> example to the
full solution is correctly solved by some partial deciders. >>>>>>>>>>>>>>>
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory
expressions are correctly rejected as semantically
incoherent thus form no undecidability or incompleteness. >>>>>>>>>>>>
order group theory is self-contradictory. But the first >>>>>>>>>>>> order goupr
theory is incomplete: it is impossible to prove that AB = BA >>>>>>>>>>>> is true
for every A and every B but it is also impossible to prove >>>>>>>>>>>> that AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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 have the requirement.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must determine >>>>>> whether the computation presented by its input halts has already
been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as
ill-posed with respect to computable semantics.
When the specification is constrained to properties
detectable via finite simulation and finite pattern
recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is proven that an universal halt decider does not exist.
“The system adopts Proof-Theoretic Semantics: meaning is determined
by inferential role, and truth is internal to the theory. A theory T
is defined by a finite set of stipulated atomic statements together
with all expressions derivable from them under the inference rules.
The statements belonging to T constitute its theorems, and these are
exactly the statements that are true-in-T.”
Under a system like the above rough draft all inputs
having pathological self reference such as the halting
problem counter-example input are simply rejected as
non-well-founded. Tarski Undefinability, Gödel's
incompleteness and the halting problem cease to exist.
A Turing
machine cannot determine the halting of all Turing machines and is
therefore not an universla halt decider.
This is not true in Proof Theoretic Semantics. I
still have to refine my words. I may not have said
that exactly correctly. The result is that in Proof
Theoretic Semantics the counter-example is rejected
as non-well-founded.
That no Turing machine is a halt decider is a proven theorem and a
truth about Turing machines. If your "Proof Thoeretic Semnatics"
does not regard it as true then your "Proof Theoretic Semantics"
is incomplete.
My long‑term goal is to make ‘true on the basis of meaning’ computable.
On 1/14/26 8:25 PM, olcott wrote:
By operational semantics I mean the standard proof‑theoretic
account of program meaning, where execution judgments are
given by inference rules and termination corresponds to the
existence of a finite derivation.
Which is just incorrect. Since infinite derivation has meaning in the
field.
On 1/14/26 8:25 PM, olcott wrote:
On 1/12/2026 9:19 PM, Richard Damon wrote:
On 1/12/26 9:29 AM, olcott wrote:
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:The misconception is yours. No expression in the language of >>>>>>>>>>> the first
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:Depends on whether the word "truth" is interpeted in the >>>>>>>>>>>>> standard
On 07/01/2026 06:44, olcott wrote:*if undecidability is correct then truth itself is broken* >>>>>>>>>>>>>
All deciders essentially: Transform finite string >>>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>>> {Accept, Reject} values.
The counter-example input to requires more than >>>>>>>>>>>>>>>> can be derived from finite string transformation >>>>>>>>>>>>>>>> rules applied to this specific input thus the
Halting Problem requires too much.
In a sense the halting problem asks too much: the problem >>>>>>>>>>>>>>> is proven to
be unsolvable. In another sense it asks too little: >>>>>>>>>>>>>>> usually we want to
know whether a method halts on every input, not just one. >>>>>>>>>>>>>>>
Although the halting problem is unsolvable, there are >>>>>>>>>>>>>>> partial solutions
to the halting problem. In particular, every counter- >>>>>>>>>>>>>>> example to the
full solution is correctly solved by some partial deciders. >>>>>>>>>>>>>>
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory
expressions are correctly rejected as semantically
incoherent thus form no undecidability or incompleteness. >>>>>>>>>>>
order group theory is self-contradictory. But the first order >>>>>>>>>>> goupr
theory is incomplete: it is impossible to prove that AB = BA >>>>>>>>>>> is true
for every A and every B but it is also impossible to prove >>>>>>>>>>> that AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must determine
whether the computation presented by its input halts has already
been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as
ill-posed with respect to computable semantics.
When the specification is constrained to properties
detectable via finite simulation and finite pattern
recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is that an input that does the opposite of whatever
value the halt decider returns is non-well-founded
within proof-theoretic semantics.
But the problem is that Computation is not a proof-theoretic semantic
system, and thus those rules don't apply.
The dumbed down version is that the halting problem asks
a question outside of the scope of finite string transformations.
But it doesn't, not unless you think that programs can't be represented
as finite strings.
The halting problem proof does not fail because finite computation
is too weak. It fails because it asks finite computation to
decide a judgment that is not finitely grounded under operational
semantics.
But that is the issue, Operational Semantics for Programs are not
actually finitely based, since programs can be non-halting.
Just shows you don't know what your words actually mean.
By operational semantics I mean the standard proof‑theoretic
account of program meaning, where execution judgments are
given by inference rules and termination corresponds to the
existence of a finite derivation.
Which is just incorrect. Since infinite derivation has meaning in the
field.
On 1/14/26 8:25 PM, olcott wrote:
On 1/12/2026 9:19 PM, Richard Damon wrote:
On 1/12/26 9:29 AM, olcott wrote:
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:The misconception is yours. No expression in the language of >>>>>>>>>>> the first
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:Depends on whether the word "truth" is interpeted in the >>>>>>>>>>>>> standard
On 07/01/2026 06:44, olcott wrote:*if undecidability is correct then truth itself is broken* >>>>>>>>>>>>>
All deciders essentially: Transform finite string >>>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>>> {Accept, Reject} values.
The counter-example input to requires more than >>>>>>>>>>>>>>>> can be derived from finite string transformation >>>>>>>>>>>>>>>> rules applied to this specific input thus the
Halting Problem requires too much.
In a sense the halting problem asks too much: the problem >>>>>>>>>>>>>>> is proven to
be unsolvable. In another sense it asks too little: >>>>>>>>>>>>>>> usually we want to
know whether a method halts on every input, not just one. >>>>>>>>>>>>>>>
Although the halting problem is unsolvable, there are >>>>>>>>>>>>>>> partial solutions
to the halting problem. In particular, every counter- >>>>>>>>>>>>>>> example to the
full solution is correctly solved by some partial deciders. >>>>>>>>>>>>>>
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory
expressions are correctly rejected as semantically
incoherent thus form no undecidability or incompleteness. >>>>>>>>>>>
order group theory is self-contradictory. But the first order >>>>>>>>>>> goupr
theory is incomplete: it is impossible to prove that AB = BA >>>>>>>>>>> is true
for every A and every B but it is also impossible to prove >>>>>>>>>>> that AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must determine
whether the computation presented by its input halts has already
been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as
ill-posed with respect to computable semantics.
When the specification is constrained to properties
detectable via finite simulation and finite pattern
recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is that an input that does the opposite of whatever
value the halt decider returns is non-well-founded
within proof-theoretic semantics.
But the problem is that Computation is not a proof-theoretic semantic
system, and thus those rules don't apply.
The dumbed down version is that the halting problem asks
a question outside of the scope of finite string transformations.
But it doesn't, not unless you think that programs can't be represented
as finite strings.
The halting problem proof does not fail because finite computation
is too weak. It fails because it asks finite computation to
decide a judgment that is not finitely grounded under operational
semantics.
But that is the issue, Operational Semantics for Programs are not
actually finitely based, since programs can be non-halting.
Just shows you don't know what your words actually mean.
By operational semantics I mean the standard proof‑theoretic
account of program meaning, where execution judgments are
given by inference rules and termination corresponds to the
existence of a finite derivation.
Which is just incorrect. Since infinite derivation has meaning in the
field.
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: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.
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
You can't determine whether the required result is computable >>>>>>>>> beforeNo, 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 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. >>>>>>>
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 Gödel'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.
On 14/01/2026 19:28, olcott wrote:
On 1/14/2026 1:40 AM, Mikko wrote:
On 13/01/2026 16:27, olcott wrote:
On 1/13/2026 3:11 AM, Mikko wrote:
On 12/01/2026 16:29, olcott wrote:
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:You can't determine whether the required result is computable >>>>>>>>> before
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:The misconception is yours. No expression in the language >>>>>>>>>>>>> of the first
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:Depends on whether the word "truth" is interpeted in the >>>>>>>>>>>>>>> standard
On 07/01/2026 06:44, olcott wrote:
All deciders essentially: Transform finite string >>>>>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>>>>> {Accept, Reject} values.
The counter-example input to requires more than >>>>>>>>>>>>>>>>>> can be derived from finite string transformation >>>>>>>>>>>>>>>>>> rules applied to this specific input thus the >>>>>>>>>>>>>>>>>> Halting Problem requires too much.
In a sense the halting problem asks too much: the >>>>>>>>>>>>>>>>> problem is proven to
be unsolvable. In another sense it asks too little: >>>>>>>>>>>>>>>>> usually we want to
know whether a method halts on every input, not just one. >>>>>>>>>>>>>>>>>
Although the halting problem is unsolvable, there are >>>>>>>>>>>>>>>>> partial solutions
to the halting problem. In particular, every counter- >>>>>>>>>>>>>>>>> example to the
full solution is correctly solved by some partial >>>>>>>>>>>>>>>>> deciders.
*if undecidability is correct then truth itself is broken* >>>>>>>>>>>>>>>
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory >>>>>>>>>>>>>> expressions are correctly rejected as semantically >>>>>>>>>>>>>> incoherent thus form no undecidability or incompleteness. >>>>>>>>>>>>>
order group theory is self-contradictory. But the first >>>>>>>>>>>>> order goupr
theory is incomplete: it is impossible to prove that AB = >>>>>>>>>>>>> BA is true
for every A and every B but it is also impossible to prove >>>>>>>>>>>>> that AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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 have the requirement.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must determine >>>>>>> whether the computation presented by its input halts has already >>>>>>> been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as
ill-posed with respect to computable semantics.
When the specification is constrained to properties
detectable via finite simulation and finite pattern
recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is proven that an universal halt decider does not exist.
“The system adopts Proof-Theoretic Semantics: meaning is determined >>>> by inferential role, and truth is internal to the theory. A theory T
is defined by a finite set of stipulated atomic statements together
with all expressions derivable from them under the inference rules.
The statements belonging to T constitute its theorems, and these are
exactly the statements that are true-in-T.”
Under a system like the above rough draft all inputs
having pathological self reference such as the halting
problem counter-example input are simply rejected as
non-well-founded. Tarski Undefinability, Gödel's
incompleteness and the halting problem cease to exist.
A Turing
machine cannot determine the halting of all Turing machines and is
therefore not an universla halt decider.
This is not true in Proof Theoretic Semantics. I
still have to refine my words. I may not have said
that exactly correctly. The result is that in Proof
Theoretic Semantics the counter-example is rejected
as non-well-founded.
That no Turing machine is a halt decider is a proven theorem and a
truth about Turing machines. If your "Proof Thoeretic Semnatics"
does not regard it as true then your "Proof Theoretic Semantics"
is incomplete.
My long‑term goal is to make ‘true on the basis of meaning’ computable.
As meaning is not computable, how can "true on the balsis of meaning"
be commputable?
On 1/14/2026 9:51 PM, Richard Damon wrote:
On 1/14/26 8:25 PM, olcott wrote:
On 1/12/2026 9:19 PM, Richard Damon wrote:
On 1/12/26 9:29 AM, olcott wrote:
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:You can't determine whether the required result is computable >>>>>>>> before
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:The misconception is yours. No expression in the language of >>>>>>>>>>>> the first
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:Depends on whether the word "truth" is interpeted in the >>>>>>>>>>>>>> standard
On 07/01/2026 06:44, olcott wrote:*if undecidability is correct then truth itself is broken* >>>>>>>>>>>>>>
All deciders essentially: Transform finite string >>>>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>>>> {Accept, Reject} values.
The counter-example input to requires more than >>>>>>>>>>>>>>>>> can be derived from finite string transformation >>>>>>>>>>>>>>>>> rules applied to this specific input thus the >>>>>>>>>>>>>>>>> Halting Problem requires too much.
In a sense the halting problem asks too much: the >>>>>>>>>>>>>>>> problem is proven to
be unsolvable. In another sense it asks too little: >>>>>>>>>>>>>>>> usually we want to
know whether a method halts on every input, not just one. >>>>>>>>>>>>>>>>
Although the halting problem is unsolvable, there are >>>>>>>>>>>>>>>> partial solutions
to the halting problem. In particular, every counter- >>>>>>>>>>>>>>>> example to the
full solution is correctly solved by some partial deciders. >>>>>>>>>>>>>>>
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory
expressions are correctly rejected as semantically
incoherent thus form no undecidability or incompleteness. >>>>>>>>>>>>
order group theory is self-contradictory. But the first >>>>>>>>>>>> order goupr
theory is incomplete: it is impossible to prove that AB = BA >>>>>>>>>>>> is true
for every A and every B but it is also impossible to prove >>>>>>>>>>>> that AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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 have the requirement.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must determine >>>>>> whether the computation presented by its input halts has already
been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as
ill-posed with respect to computable semantics.
When the specification is constrained to properties
detectable via finite simulation and finite pattern
recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is that an input that does the opposite of whatever
value the halt decider returns is non-well-founded
within proof-theoretic semantics.
But the problem is that Computation is not a proof-theoretic
semantic system, and thus those rules don't apply.
The dumbed down version is that the halting problem asks
a question outside of the scope of finite string transformations.
But it doesn't, not unless you think that programs can't be
represented as finite strings.
The halting problem proof does not fail because finite computation
is too weak. It fails because it asks finite computation to
decide a judgment that is not finitely grounded under operational
semantics.
But that is the issue, Operational Semantics for Programs are not
actually finitely based, since programs can be non-halting.
Just shows you don't know what your words actually mean.
By operational semantics I mean the standard proof‑theoretic
account of program meaning, where execution judgments are
given by inference rules and termination corresponds to the
existence of a finite derivation.
Which is just incorrect. Since infinite derivation has meaning in the
field.
The halting problem is not undecidable because computation is weak, but because the classical formulation uses a denotational semantics that is
too permissive.
In operational/proof‑theoretic semantics, where meaning is grounded in finite derivations, the halting predicate is not a well‑formed judgment — just as unrestricted comprehension was not a well‑formed judgment in naïve set theory.
On 1/15/26 12:34 PM, olcott wrote:
On 1/14/2026 9:51 PM, Richard Damon wrote:
On 1/14/26 8:25 PM, olcott wrote:
On 1/12/2026 9:19 PM, Richard Damon wrote:
On 1/12/26 9:29 AM, olcott wrote:
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:You can't determine whether the required result is computable >>>>>>>>> before
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:The misconception is yours. No expression in the language >>>>>>>>>>>>> of the first
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:Depends on whether the word "truth" is interpeted in the >>>>>>>>>>>>>>> standard
On 07/01/2026 06:44, olcott wrote:
All deciders essentially: Transform finite string >>>>>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>>>>> {Accept, Reject} values.
The counter-example input to requires more than >>>>>>>>>>>>>>>>>> can be derived from finite string transformation >>>>>>>>>>>>>>>>>> rules applied to this specific input thus the >>>>>>>>>>>>>>>>>> Halting Problem requires too much.
In a sense the halting problem asks too much: the >>>>>>>>>>>>>>>>> problem is proven to
be unsolvable. In another sense it asks too little: >>>>>>>>>>>>>>>>> usually we want to
know whether a method halts on every input, not just one. >>>>>>>>>>>>>>>>>
Although the halting problem is unsolvable, there are >>>>>>>>>>>>>>>>> partial solutions
to the halting problem. In particular, every counter- >>>>>>>>>>>>>>>>> example to the
full solution is correctly solved by some partial >>>>>>>>>>>>>>>>> deciders.
*if undecidability is correct then truth itself is broken* >>>>>>>>>>>>>>>
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory >>>>>>>>>>>>>> expressions are correctly rejected as semantically >>>>>>>>>>>>>> incoherent thus form no undecidability or incompleteness. >>>>>>>>>>>>>
order group theory is self-contradictory. But the first >>>>>>>>>>>>> order goupr
theory is incomplete: it is impossible to prove that AB = >>>>>>>>>>>>> BA is true
for every A and every B but it is also impossible to prove >>>>>>>>>>>>> that AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying
finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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 have the requirement.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must determine >>>>>>> whether the computation presented by its input halts has already >>>>>>> been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as
ill-posed with respect to computable semantics.
When the specification is constrained to properties
detectable via finite simulation and finite pattern
recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is that an input that does the opposite of whatever
value the halt decider returns is non-well-founded
within proof-theoretic semantics.
But the problem is that Computation is not a proof-theoretic
semantic system, and thus those rules don't apply.
The dumbed down version is that the halting problem asks
a question outside of the scope of finite string transformations.
But it doesn't, not unless you think that programs can't be
represented as finite strings.
The halting problem proof does not fail because finite computation
is too weak. It fails because it asks finite computation to
decide a judgment that is not finitely grounded under operational
semantics.
But that is the issue, Operational Semantics for Programs are not
actually finitely based, since programs can be non-halting.
Just shows you don't know what your words actually mean.
By operational semantics I mean the standard proof‑theoretic
account of program meaning, where execution judgments are
given by inference rules and termination corresponds to the
existence of a finite derivation.
Which is just incorrect. Since infinite derivation has meaning in the
field.
The halting problem is not undecidable because computation is weak,
but because the classical formulation uses a denotational semantics
that is too permissive.
Nope.
In operational/proof‑theoretic semantics, where meaning is grounded in
finite derivations, the halting predicate is not a well‑formed
judgment — just as unrestricted comprehension was not a well‑formed
judgment in naïve set theory.
In other words, by trying to enforce your interpreation, you system
becomes unworkable, as you can't tell if you can ask a question.
The problem is that systems like this grow faster in power to generate--
than your logic grow in power to decide, and either you accept that some truths are unprovable (and thus accept the truth-conditional view) or
you need to just abandon the ability to actually work in the system as
you can't tell what questions are reasonable.
All you are doing is proving that you are just too stupid to understand
the implications of what you are talking about, because you never really understood what the words actually mean.
On 1/15/2026 3:48 AM, Mikko wrote:
On 14/01/2026 19:28, olcott wrote:
On 1/14/2026 1:40 AM, Mikko wrote:
On 13/01/2026 16:27, olcott wrote:
On 1/13/2026 3:11 AM, Mikko wrote:
On 12/01/2026 16:29, olcott wrote:
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:You can't determine whether the required result is computable >>>>>>>>>> before
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:The misconception is yours. No expression in the language >>>>>>>>>>>>>> of the first
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:Depends on whether the word "truth" is interpeted in the >>>>>>>>>>>>>>>> standard
On 07/01/2026 06:44, olcott wrote:
All deciders essentially: Transform finite string >>>>>>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>>>>>> {Accept, Reject} values.
The counter-example input to requires more than >>>>>>>>>>>>>>>>>>> can be derived from finite string transformation >>>>>>>>>>>>>>>>>>> rules applied to this specific input thus the >>>>>>>>>>>>>>>>>>> Halting Problem requires too much.
In a sense the halting problem asks too much: the >>>>>>>>>>>>>>>>>> problem is proven to
be unsolvable. In another sense it asks too little: >>>>>>>>>>>>>>>>>> usually we want to
know whether a method halts on every input, not just one. >>>>>>>>>>>>>>>>>>
Although the halting problem is unsolvable, there are >>>>>>>>>>>>>>>>>> partial solutions
to the halting problem. In particular, every counter- >>>>>>>>>>>>>>>>>> example to the
full solution is correctly solved by some partial >>>>>>>>>>>>>>>>>> deciders.
*if undecidability is correct then truth itself is broken* >>>>>>>>>>>>>>>>
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory >>>>>>>>>>>>>>> expressions are correctly rejected as semantically >>>>>>>>>>>>>>> incoherent thus form no undecidability or incompleteness. >>>>>>>>>>>>>>
order group theory is self-contradictory. But the first >>>>>>>>>>>>>> order goupr
theory is incomplete: it is impossible to prove that AB = >>>>>>>>>>>>>> BA is true
for every A and every B but it is also impossible to prove >>>>>>>>>>>>>> that AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying >>>>>>>>>>>>> finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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 have the requirement.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must determine >>>>>>>> whether the computation presented by its input halts has already >>>>>>>> been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as
ill-posed with respect to computable semantics.
When the specification is constrained to properties
detectable via finite simulation and finite pattern
recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is proven that an universal halt decider does not exist.
“The system adopts Proof-Theoretic Semantics: meaning is determined >>>>> by inferential role, and truth is internal to the theory. A theory
T is defined by a finite set of stipulated atomic statements
together with all expressions derivable from them under the
inference rules. The statements belonging to T constitute its
theorems, and these are exactly the statements that are true-in-T.” >>>>>
Under a system like the above rough draft all inputs
having pathological self reference such as the halting
problem counter-example input are simply rejected as
non-well-founded. Tarski Undefinability, Gödel's
incompleteness and the halting problem cease to exist.
A Turing
machine cannot determine the halting of all Turing machines and is >>>>>> therefore not an universla halt decider.
This is not true in Proof Theoretic Semantics. I
still have to refine my words. I may not have said
that exactly correctly. The result is that in Proof
Theoretic Semantics the counter-example is rejected
as non-well-founded.
That no Turing machine is a halt decider is a proven theorem and a
truth about Turing machines. If your "Proof Thoeretic Semnatics"
does not regard it as true then your "Proof Theoretic Semantics"
is incomplete.
My long‑term goal is to make ‘true on the basis of meaning’ computable.
As meaning is not computable, how can "true on the balsis of meaning"
be commputable?
Under *proof‑theoretic semantics*
"true on the basis of meaning expressed in language"
has always been entirely computable.
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: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.
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
You can't determine whether the required result is computable >>>>>>>>>> beforeNo, 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 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. >>>>>>>>
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 Gödel'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.
On 16/01/2026 01:38, olcott wrote:
On 1/15/2026 3:48 AM, Mikko wrote:
On 14/01/2026 19:28, olcott wrote:
On 1/14/2026 1:40 AM, Mikko wrote:
On 13/01/2026 16:27, olcott wrote:
On 1/13/2026 3:11 AM, Mikko wrote:
On 12/01/2026 16:29, olcott wrote:
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:You can't determine whether the required result is computable >>>>>>>>>>> before
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:The misconception is yours. No expression in the language >>>>>>>>>>>>>>> of the first
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:
On 07/01/2026 06:44, olcott wrote:
All deciders essentially: Transform finite string >>>>>>>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>>>>>>> {Accept, Reject} values.
The counter-example input to requires more than >>>>>>>>>>>>>>>>>>>> can be derived from finite string transformation >>>>>>>>>>>>>>>>>>>> rules applied to this specific input thus the >>>>>>>>>>>>>>>>>>>> Halting Problem requires too much.
In a sense the halting problem asks too much: the >>>>>>>>>>>>>>>>>>> problem is proven to
be unsolvable. In another sense it asks too little: >>>>>>>>>>>>>>>>>>> usually we want to
know whether a method halts on every input, not just >>>>>>>>>>>>>>>>>>> one.
Although the halting problem is unsolvable, there are >>>>>>>>>>>>>>>>>>> partial solutions
to the halting problem. In particular, every counter- >>>>>>>>>>>>>>>>>>> example to the
full solution is correctly solved by some partial >>>>>>>>>>>>>>>>>>> deciders.
*if undecidability is correct then truth itself is >>>>>>>>>>>>>>>>>> broken*
Depends on whether the word "truth" is interpeted in >>>>>>>>>>>>>>>>> the standard
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory >>>>>>>>>>>>>>>> expressions are correctly rejected as semantically >>>>>>>>>>>>>>>> incoherent thus form no undecidability or incompleteness. >>>>>>>>>>>>>>>
order group theory is self-contradictory. But the first >>>>>>>>>>>>>>> order goupr
theory is incomplete: it is impossible to prove that AB = >>>>>>>>>>>>>>> BA is true
for every A and every B but it is also impossible to >>>>>>>>>>>>>>> prove that AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying >>>>>>>>>>>>>> finite string transformation rules to actual finite >>>>>>>>>>>>>> string inputs, then the required result exceeds the >>>>>>>>>>>>>> scope of computation and must be rejected as an
incorrect requirement.
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 have the requirement.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must determine >>>>>>>>> whether the computation presented by its input halts has already >>>>>>>>> been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as >>>>>>>>>> ill-posed with respect to computable semantics.
When the specification is constrained to properties
detectable via finite simulation and finite pattern
recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is proven that an universal halt decider does not exist.
“The system adopts Proof-Theoretic Semantics: meaning is
determined by inferential role, and truth is internal to the
theory. A theory T is defined by a finite set of stipulated atomic >>>>>> statements together with all expressions derivable from them under >>>>>> the inference rules. The statements belonging to T constitute its >>>>>> theorems, and these are exactly the statements that are true-in-T.” >>>>>>
Under a system like the above rough draft all inputs
having pathological self reference such as the halting
problem counter-example input are simply rejected as
non-well-founded. Tarski Undefinability, Gödel's
incompleteness and the halting problem cease to exist.
A Turing
machine cannot determine the halting of all Turing machines and is >>>>>>> therefore not an universla halt decider.
This is not true in Proof Theoretic Semantics. I
still have to refine my words. I may not have said
that exactly correctly. The result is that in Proof
Theoretic Semantics the counter-example is rejected
as non-well-founded.
That no Turing machine is a halt decider is a proven theorem and a
truth about Turing machines. If your "Proof Thoeretic Semnatics"
does not regard it as true then your "Proof Theoretic Semantics"
is incomplete.
My long‑term goal is to make ‘true on the basis of meaning’ computable.
As meaning is not computable, how can "true on the balsis of meaning"
be commputable?
Under *proof‑theoretic semantics*
"true on the basis of meaning expressed in language"
has always been entirely computable.
Have you already put the algorithm to some web page?
On 16/01/2026 01:38, olcott wrote:
On 1/15/2026 3:48 AM, Mikko wrote:
On 14/01/2026 19:28, olcott wrote:
On 1/14/2026 1:40 AM, Mikko wrote:
On 13/01/2026 16:27, olcott wrote:
On 1/13/2026 3:11 AM, Mikko wrote:
On 12/01/2026 16:29, olcott wrote:
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:You can't determine whether the required result is computable >>>>>>>>>>> before
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:The misconception is yours. No expression in the language >>>>>>>>>>>>>>> of the first
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:
On 07/01/2026 06:44, olcott wrote:
All deciders essentially: Transform finite string >>>>>>>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>>>>>>> {Accept, Reject} values.
The counter-example input to requires more than >>>>>>>>>>>>>>>>>>>> can be derived from finite string transformation >>>>>>>>>>>>>>>>>>>> rules applied to this specific input thus the >>>>>>>>>>>>>>>>>>>> Halting Problem requires too much.
In a sense the halting problem asks too much: the >>>>>>>>>>>>>>>>>>> problem is proven to
be unsolvable. In another sense it asks too little: >>>>>>>>>>>>>>>>>>> usually we want to
know whether a method halts on every input, not just >>>>>>>>>>>>>>>>>>> one.
Although the halting problem is unsolvable, there are >>>>>>>>>>>>>>>>>>> partial solutions
to the halting problem. In particular, every counter- >>>>>>>>>>>>>>>>>>> example to the
full solution is correctly solved by some partial >>>>>>>>>>>>>>>>>>> deciders.
*if undecidability is correct then truth itself is >>>>>>>>>>>>>>>>>> broken*
Depends on whether the word "truth" is interpeted in >>>>>>>>>>>>>>>>> the standard
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory >>>>>>>>>>>>>>>> expressions are correctly rejected as semantically >>>>>>>>>>>>>>>> incoherent thus form no undecidability or incompleteness. >>>>>>>>>>>>>>>
order group theory is self-contradictory. But the first >>>>>>>>>>>>>>> order goupr
theory is incomplete: it is impossible to prove that AB = >>>>>>>>>>>>>>> BA is true
for every A and every B but it is also impossible to >>>>>>>>>>>>>>> prove that AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying >>>>>>>>>>>>>> finite string transformation rules to actual finite >>>>>>>>>>>>>> string inputs, then the required result exceeds the >>>>>>>>>>>>>> scope of computation and must be rejected as an
incorrect requirement.
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 have the requirement.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must determine >>>>>>>>> whether the computation presented by its input halts has already >>>>>>>>> been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as >>>>>>>>>> ill-posed with respect to computable semantics.
When the specification is constrained to properties
detectable via finite simulation and finite pattern
recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is proven that an universal halt decider does not exist.
“The system adopts Proof-Theoretic Semantics: meaning is
determined by inferential role, and truth is internal to the
theory. A theory T is defined by a finite set of stipulated atomic >>>>>> statements together with all expressions derivable from them under >>>>>> the inference rules. The statements belonging to T constitute its >>>>>> theorems, and these are exactly the statements that are true-in-T.” >>>>>>
Under a system like the above rough draft all inputs
having pathological self reference such as the halting
problem counter-example input are simply rejected as
non-well-founded. Tarski Undefinability, Gödel's
incompleteness and the halting problem cease to exist.
A Turing
machine cannot determine the halting of all Turing machines and is >>>>>>> therefore not an universla halt decider.
This is not true in Proof Theoretic Semantics. I
still have to refine my words. I may not have said
that exactly correctly. The result is that in Proof
Theoretic Semantics the counter-example is rejected
as non-well-founded.
That no Turing machine is a halt decider is a proven theorem and a
truth about Turing machines. If your "Proof Thoeretic Semnatics"
does not regard it as true then your "Proof Theoretic Semantics"
is incomplete.
My long‑term goal is to make ‘true on the basis of meaning’ computable.
As meaning is not computable, how can "true on the balsis of meaning"
be commputable?
Under *proof‑theoretic semantics*
"true on the basis of meaning expressed in language"
has always been entirely computable.
Have you already put the algorithm to some web page?
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: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.
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:
You can't determine whether the required result is computable >>>>>>>>>>> beforeNo, 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 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. >>>>>>>>>
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 Gödel'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.
On 1/15/2026 9:27 PM, Richard Damon wrote:
On 1/15/26 12:34 PM, olcott wrote:
On 1/14/2026 9:51 PM, Richard Damon wrote:
On 1/14/26 8:25 PM, olcott wrote:
On 1/12/2026 9:19 PM, Richard Damon wrote:
On 1/12/26 9:29 AM, olcott wrote:
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:You can't determine whether the required result is computable >>>>>>>>>> before
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:The misconception is yours. No expression in the language >>>>>>>>>>>>>> of the first
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:Depends on whether the word "truth" is interpeted in the >>>>>>>>>>>>>>>> standard
On 07/01/2026 06:44, olcott wrote:
All deciders essentially: Transform finite string >>>>>>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>>>>>> {Accept, Reject} values.
The counter-example input to requires more than >>>>>>>>>>>>>>>>>>> can be derived from finite string transformation >>>>>>>>>>>>>>>>>>> rules applied to this specific input thus the >>>>>>>>>>>>>>>>>>> Halting Problem requires too much.
In a sense the halting problem asks too much: the >>>>>>>>>>>>>>>>>> problem is proven to
be unsolvable. In another sense it asks too little: >>>>>>>>>>>>>>>>>> usually we want to
know whether a method halts on every input, not just one. >>>>>>>>>>>>>>>>>>
Although the halting problem is unsolvable, there are >>>>>>>>>>>>>>>>>> partial solutions
to the halting problem. In particular, every counter- >>>>>>>>>>>>>>>>>> example to the
full solution is correctly solved by some partial >>>>>>>>>>>>>>>>>> deciders.
*if undecidability is correct then truth itself is broken* >>>>>>>>>>>>>>>>
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory >>>>>>>>>>>>>>> expressions are correctly rejected as semantically >>>>>>>>>>>>>>> incoherent thus form no undecidability or incompleteness. >>>>>>>>>>>>>>
order group theory is self-contradictory. But the first >>>>>>>>>>>>>> order goupr
theory is incomplete: it is impossible to prove that AB = >>>>>>>>>>>>>> BA is true
for every A and every B but it is also impossible to prove >>>>>>>>>>>>>> that AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
When a required result cannot be derived by applying >>>>>>>>>>>>> finite string transformation rules to actual finite
string inputs, then the required result exceeds the
scope of computation and must be rejected as an
incorrect requirement.
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 have the requirement.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must determine >>>>>>>> whether the computation presented by its input halts has already >>>>>>>> been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as
ill-posed with respect to computable semantics.
When the specification is constrained to properties
detectable via finite simulation and finite pattern
recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is that an input that does the opposite of whatever
value the halt decider returns is non-well-founded
within proof-theoretic semantics.
But the problem is that Computation is not a proof-theoretic
semantic system, and thus those rules don't apply.
The dumbed down version is that the halting problem asks
a question outside of the scope of finite string transformations.
But it doesn't, not unless you think that programs can't be
represented as finite strings.
The halting problem proof does not fail because finite computation
is too weak. It fails because it asks finite computation to
decide a judgment that is not finitely grounded under operational
semantics.
But that is the issue, Operational Semantics for Programs are not
actually finitely based, since programs can be non-halting.
Just shows you don't know what your words actually mean.
By operational semantics I mean the standard proof‑theoretic
account of program meaning, where execution judgments are
given by inference rules and termination corresponds to the
existence of a finite derivation.
Which is just incorrect. Since infinite derivation has meaning in
the field.
The halting problem is not undecidable because computation is weak,
but because the classical formulation uses a denotational semantics
that is too permissive.
Nope.
In operational/proof‑theoretic semantics, where meaning is grounded
in finite derivations, the halting predicate is not a well‑formed
judgment — just as unrestricted comprehension was not a well‑formed >>> judgment in naïve set theory.
In other words, by trying to enforce your interpreation, you system
becomes unworkable, as you can't tell if you can ask a question.
It is the same ∀x ∈ T ((True(T, x) ≡ (T ⊢ x))
that I have been talking about for years except that
it is now grounded in well-founded proof‑theoretic
semantics.
The problem is that systems like this grow faster in power to generate
than your logic grow in power to decide, and either you accept that
some truths are unprovable (and thus accept the truth-conditional
view) or you need to just abandon the ability to actually work in the
system as you can't tell what questions are reasonable.
All you are doing is proving that you are just too stupid to
understand the implications of what you are talking about, because you
never really understood what the words actually mean.
On 1/16/2026 3:17 AM, Mikko wrote:
On 16/01/2026 01:38, olcott wrote:
On 1/15/2026 3:48 AM, Mikko wrote:
On 14/01/2026 19:28, olcott wrote:
On 1/14/2026 1:40 AM, Mikko wrote:
On 13/01/2026 16:27, olcott wrote:
On 1/13/2026 3:11 AM, Mikko wrote:
On 12/01/2026 16:29, olcott wrote:
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:You can't determine whether the required result is
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:The misconception is yours. No expression in the >>>>>>>>>>>>>>>> language of the first
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:
On 07/01/2026 06:44, olcott wrote:
All deciders essentially: Transform finite string >>>>>>>>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>>>>>>>> {Accept, Reject} values.
The counter-example input to requires more than >>>>>>>>>>>>>>>>>>>>> can be derived from finite string transformation >>>>>>>>>>>>>>>>>>>>> rules applied to this specific input thus the >>>>>>>>>>>>>>>>>>>>> Halting Problem requires too much.
In a sense the halting problem asks too much: the >>>>>>>>>>>>>>>>>>>> problem is proven to
be unsolvable. In another sense it asks too little: >>>>>>>>>>>>>>>>>>>> usually we want to
know whether a method halts on every input, not just >>>>>>>>>>>>>>>>>>>> one.
Although the halting problem is unsolvable, there >>>>>>>>>>>>>>>>>>>> are partial solutions
to the halting problem. In particular, every >>>>>>>>>>>>>>>>>>>> counter- example to the
full solution is correctly solved by some partial >>>>>>>>>>>>>>>>>>>> deciders.
*if undecidability is correct then truth itself is >>>>>>>>>>>>>>>>>>> broken*
Depends on whether the word "truth" is interpeted in >>>>>>>>>>>>>>>>>> the standard
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory >>>>>>>>>>>>>>>>> expressions are correctly rejected as semantically >>>>>>>>>>>>>>>>> incoherent thus form no undecidability or incompleteness. >>>>>>>>>>>>>>>>
order group theory is self-contradictory. But the first >>>>>>>>>>>>>>>> order goupr
theory is incomplete: it is impossible to prove that AB >>>>>>>>>>>>>>>> = BA is true
for every A and every B but it is also impossible to >>>>>>>>>>>>>>>> prove that AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string >>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>> {Accept, Reject} values.
When a required result cannot be derived by applying >>>>>>>>>>>>>>> finite string transformation rules to actual finite >>>>>>>>>>>>>>> string inputs, then the required result exceeds the >>>>>>>>>>>>>>> scope of computation and must be rejected as an
incorrect requirement.
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. >>>>>>>>>>>>
computable before
you have the requirement.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must
determine
whether the computation presented by its input halts has already >>>>>>>>>> been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as >>>>>>>>>>> ill-posed with respect to computable semantics.
When the specification is constrained to properties >>>>>>>>>>> detectable via finite simulation and finite pattern >>>>>>>>>>> recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is proven that an universal halt decider does not exist.
“The system adopts Proof-Theoretic Semantics: meaning is
determined by inferential role, and truth is internal to the
theory. A theory T is defined by a finite set of stipulated
atomic statements together with all expressions derivable from
them under the inference rules. The statements belonging to T
constitute its theorems, and these are exactly the statements
that are true-in-T.”
Under a system like the above rough draft all inputs
having pathological self reference such as the halting
problem counter-example input are simply rejected as
non-well-founded. Tarski Undefinability, Gödel's
incompleteness and the halting problem cease to exist.
A Turing
machine cannot determine the halting of all Turing machines and is >>>>>>>> therefore not an universla halt decider.
This is not true in Proof Theoretic Semantics. I
still have to refine my words. I may not have said
that exactly correctly. The result is that in Proof
Theoretic Semantics the counter-example is rejected
as non-well-founded.
That no Turing machine is a halt decider is a proven theorem and a >>>>>> truth about Turing machines. If your "Proof Thoeretic Semnatics"
does not regard it as true then your "Proof Theoretic Semantics"
is incomplete.
My long‑term goal is to make ‘true on the basis of meaning’
computable.
As meaning is not computable, how can "true on the balsis of meaning"
be commputable?
Under *proof‑theoretic semantics*
"true on the basis of meaning expressed in language"
has always been entirely computable.
Have you already put the algorithm to some web page?
I am still working on refining the presentation.
On 1/16/2026 3:17 AM, Mikko wrote:
On 16/01/2026 01:38, olcott wrote:
On 1/15/2026 3:48 AM, Mikko wrote:
On 14/01/2026 19:28, olcott wrote:
On 1/14/2026 1:40 AM, Mikko wrote:
On 13/01/2026 16:27, olcott wrote:
On 1/13/2026 3:11 AM, Mikko wrote:
On 12/01/2026 16:29, olcott wrote:
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:You can't determine whether the required result is
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:The misconception is yours. No expression in the >>>>>>>>>>>>>>>> language of the first
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:
On 07/01/2026 06:44, olcott wrote:
All deciders essentially: Transform finite string >>>>>>>>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>>>>>>>> {Accept, Reject} values.
The counter-example input to requires more than >>>>>>>>>>>>>>>>>>>>> can be derived from finite string transformation >>>>>>>>>>>>>>>>>>>>> rules applied to this specific input thus the >>>>>>>>>>>>>>>>>>>>> Halting Problem requires too much.
In a sense the halting problem asks too much: the >>>>>>>>>>>>>>>>>>>> problem is proven to
be unsolvable. In another sense it asks too little: >>>>>>>>>>>>>>>>>>>> usually we want to
know whether a method halts on every input, not just >>>>>>>>>>>>>>>>>>>> one.
Although the halting problem is unsolvable, there >>>>>>>>>>>>>>>>>>>> are partial solutions
to the halting problem. In particular, every >>>>>>>>>>>>>>>>>>>> counter- example to the
full solution is correctly solved by some partial >>>>>>>>>>>>>>>>>>>> deciders.
*if undecidability is correct then truth itself is >>>>>>>>>>>>>>>>>>> broken*
Depends on whether the word "truth" is interpeted in >>>>>>>>>>>>>>>>>> the standard
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory >>>>>>>>>>>>>>>>> expressions are correctly rejected as semantically >>>>>>>>>>>>>>>>> incoherent thus form no undecidability or incompleteness. >>>>>>>>>>>>>>>>
order group theory is self-contradictory. But the first >>>>>>>>>>>>>>>> order goupr
theory is incomplete: it is impossible to prove that AB >>>>>>>>>>>>>>>> = BA is true
for every A and every B but it is also impossible to >>>>>>>>>>>>>>>> prove that AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string >>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>> {Accept, Reject} values.
When a required result cannot be derived by applying >>>>>>>>>>>>>>> finite string transformation rules to actual finite >>>>>>>>>>>>>>> string inputs, then the required result exceeds the >>>>>>>>>>>>>>> scope of computation and must be rejected as an
incorrect requirement.
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. >>>>>>>>>>>>
computable before
you have the requirement.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must
determine
whether the computation presented by its input halts has already >>>>>>>>>> been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as >>>>>>>>>>> ill-posed with respect to computable semantics.
When the specification is constrained to properties >>>>>>>>>>> detectable via finite simulation and finite pattern >>>>>>>>>>> recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is proven that an universal halt decider does not exist.
“The system adopts Proof-Theoretic Semantics: meaning is
determined by inferential role, and truth is internal to the
theory. A theory T is defined by a finite set of stipulated
atomic statements together with all expressions derivable from
them under the inference rules. The statements belonging to T
constitute its theorems, and these are exactly the statements
that are true-in-T.”
Under a system like the above rough draft all inputs
having pathological self reference such as the halting
problem counter-example input are simply rejected as
non-well-founded. Tarski Undefinability, Gödel's
incompleteness and the halting problem cease to exist.
A Turing
machine cannot determine the halting of all Turing machines and is >>>>>>>> therefore not an universla halt decider.
This is not true in Proof Theoretic Semantics. I
still have to refine my words. I may not have said
that exactly correctly. The result is that in Proof
Theoretic Semantics the counter-example is rejected
as non-well-founded.
That no Turing machine is a halt decider is a proven theorem and a >>>>>> truth about Turing machines. If your "Proof Thoeretic Semnatics"
does not regard it as true then your "Proof Theoretic Semantics"
is incomplete.
My long‑term goal is to make ‘true on the basis of meaning’
computable.
As meaning is not computable, how can "true on the balsis of meaning"
be commputable?
Under *proof‑theoretic semantics*
"true on the basis of meaning expressed in language"
has always been entirely computable.
Have you already put the algorithm to some web page?
Proof Theoretic Semantics Blocks Pathological Self-Reference https://philpapers.org/rec/OLCPTS
On 1/16/2026 3:17 AM, Mikko wrote:
On 16/01/2026 01:38, olcott wrote:
On 1/15/2026 3:48 AM, Mikko wrote:
On 14/01/2026 19:28, olcott wrote:
On 1/14/2026 1:40 AM, Mikko wrote:
On 13/01/2026 16:27, olcott wrote:
On 1/13/2026 3:11 AM, Mikko wrote:
On 12/01/2026 16:29, olcott wrote:
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:You can't determine whether the required result is
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:The misconception is yours. No expression in the >>>>>>>>>>>>>>>> language of the first
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:
On 07/01/2026 06:44, olcott wrote:
All deciders essentially: Transform finite string >>>>>>>>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>>>>>>>> {Accept, Reject} values.
The counter-example input to requires more than >>>>>>>>>>>>>>>>>>>>> can be derived from finite string transformation >>>>>>>>>>>>>>>>>>>>> rules applied to this specific input thus the >>>>>>>>>>>>>>>>>>>>> Halting Problem requires too much.
In a sense the halting problem asks too much: the >>>>>>>>>>>>>>>>>>>> problem is proven to
be unsolvable. In another sense it asks too little: >>>>>>>>>>>>>>>>>>>> usually we want to
know whether a method halts on every input, not just >>>>>>>>>>>>>>>>>>>> one.
Although the halting problem is unsolvable, there >>>>>>>>>>>>>>>>>>>> are partial solutions
to the halting problem. In particular, every >>>>>>>>>>>>>>>>>>>> counter- example to the
full solution is correctly solved by some partial >>>>>>>>>>>>>>>>>>>> deciders.
*if undecidability is correct then truth itself is >>>>>>>>>>>>>>>>>>> broken*
Depends on whether the word "truth" is interpeted in >>>>>>>>>>>>>>>>>> the standard
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory >>>>>>>>>>>>>>>>> expressions are correctly rejected as semantically >>>>>>>>>>>>>>>>> incoherent thus form no undecidability or incompleteness. >>>>>>>>>>>>>>>>
order group theory is self-contradictory. But the first >>>>>>>>>>>>>>>> order goupr
theory is incomplete: it is impossible to prove that AB >>>>>>>>>>>>>>>> = BA is true
for every A and every B but it is also impossible to >>>>>>>>>>>>>>>> prove that AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string >>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>> {Accept, Reject} values.
When a required result cannot be derived by applying >>>>>>>>>>>>>>> finite string transformation rules to actual finite >>>>>>>>>>>>>>> string inputs, then the required result exceeds the >>>>>>>>>>>>>>> scope of computation and must be rejected as an
incorrect requirement.
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. >>>>>>>>>>>>
computable before
you have the requirement.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must
determine
whether the computation presented by its input halts has already >>>>>>>>>> been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as >>>>>>>>>>> ill-posed with respect to computable semantics.
When the specification is constrained to properties >>>>>>>>>>> detectable via finite simulation and finite pattern >>>>>>>>>>> recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is proven that an universal halt decider does not exist.
“The system adopts Proof-Theoretic Semantics: meaning is
determined by inferential role, and truth is internal to the
theory. A theory T is defined by a finite set of stipulated
atomic statements together with all expressions derivable from
them under the inference rules. The statements belonging to T
constitute its theorems, and these are exactly the statements
that are true-in-T.”
Under a system like the above rough draft all inputs
having pathological self reference such as the halting
problem counter-example input are simply rejected as
non-well-founded. Tarski Undefinability, Gödel's
incompleteness and the halting problem cease to exist.
A Turing
machine cannot determine the halting of all Turing machines and is >>>>>>>> therefore not an universla halt decider.
This is not true in Proof Theoretic Semantics. I
still have to refine my words. I may not have said
that exactly correctly. The result is that in Proof
Theoretic Semantics the counter-example is rejected
as non-well-founded.
That no Turing machine is a halt decider is a proven theorem and a >>>>>> truth about Turing machines. If your "Proof Thoeretic Semnatics"
does not regard it as true then your "Proof Theoretic Semantics"
is incomplete.
My long‑term goal is to make ‘true on the basis of meaning’
computable.
As meaning is not computable, how can "true on the balsis of meaning"
be commputable?
Under *proof‑theoretic semantics*
"true on the basis of meaning expressed in language"
has always been entirely computable.
Have you already put the algorithm to some web page?
Proof Theoretic Semantics Blocks Pathological Self-Reference https://philpapers.org/rec/OLCPTS
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:
You can't determine whether the required result isNo, 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. >>>>>>>>>>>>
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 Gödel'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 Gödel Incompleteness
for PA in a way similar to the way that ZFC addresses
Russell's Paradox in set theory.
On 1/16/2026 3:17 AM, Mikko wrote:
On 16/01/2026 01:38, olcott wrote:
On 1/15/2026 3:48 AM, Mikko wrote:
On 14/01/2026 19:28, olcott wrote:
On 1/14/2026 1:40 AM, Mikko wrote:
On 13/01/2026 16:27, olcott wrote:
On 1/13/2026 3:11 AM, Mikko wrote:
On 12/01/2026 16:29, olcott wrote:
On 1/12/2026 4:44 AM, Mikko wrote:
On 11/01/2026 16:18, olcott wrote:
On 1/11/2026 4:13 AM, Mikko wrote:
On 10/01/2026 17:47, olcott wrote:
On 1/10/2026 2:23 AM, Mikko wrote:You can't determine whether the required result is
On 09/01/2026 17:52, olcott wrote:
On 1/9/2026 3:59 AM, Mikko wrote:
On 08/01/2026 16:22, olcott wrote:
On 1/8/2026 4:22 AM, Mikko wrote:The misconception is yours. No expression in the >>>>>>>>>>>>>>>> language of the first
On 07/01/2026 13:54, olcott wrote:
On 1/7/2026 5:49 AM, Mikko wrote:
On 07/01/2026 06:44, olcott wrote:
All deciders essentially: Transform finite string >>>>>>>>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>>>>>>>> {Accept, Reject} values.
The counter-example input to requires more than >>>>>>>>>>>>>>>>>>>>> can be derived from finite string transformation >>>>>>>>>>>>>>>>>>>>> rules applied to this specific input thus the >>>>>>>>>>>>>>>>>>>>> Halting Problem requires too much.
In a sense the halting problem asks too much: the >>>>>>>>>>>>>>>>>>>> problem is proven to
be unsolvable. In another sense it asks too little: >>>>>>>>>>>>>>>>>>>> usually we want to
know whether a method halts on every input, not just >>>>>>>>>>>>>>>>>>>> one.
Although the halting problem is unsolvable, there >>>>>>>>>>>>>>>>>>>> are partial solutions
to the halting problem. In particular, every >>>>>>>>>>>>>>>>>>>> counter- example to the
full solution is correctly solved by some partial >>>>>>>>>>>>>>>>>>>> deciders.
*if undecidability is correct then truth itself is >>>>>>>>>>>>>>>>>>> broken*
Depends on whether the word "truth" is interpeted in >>>>>>>>>>>>>>>>>> the standard
sense or in Olcott's sense.
Undecidability is misconception. Self-contradictory >>>>>>>>>>>>>>>>> expressions are correctly rejected as semantically >>>>>>>>>>>>>>>>> incoherent thus form no undecidability or incompleteness. >>>>>>>>>>>>>>>>
order group theory is self-contradictory. But the first >>>>>>>>>>>>>>>> order goupr
theory is incomplete: it is impossible to prove that AB >>>>>>>>>>>>>>>> = BA is true
for every A and every B but it is also impossible to >>>>>>>>>>>>>>>> prove that AB = BA
is false for some A and some B.
All deciders essentially: Transform finite string >>>>>>>>>>>>>>> inputs by finite string transformation rules into >>>>>>>>>>>>>>> {Accept, Reject} values.
When a required result cannot be derived by applying >>>>>>>>>>>>>>> finite string transformation rules to actual finite >>>>>>>>>>>>>>> string inputs, then the required result exceeds the >>>>>>>>>>>>>>> scope of computation and must be rejected as an
incorrect requirement.
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. >>>>>>>>>>>>
computable before
you have the requirement.
*Computation and Undecidability*
https://philpapers.org/go.pl?aid=OLCCAU
We know that there does not exist any finite
string transformations that H can apply to its
input P to derive the halt status of any P
that does the opposite of whatever H returns.
Which only nmakes sense when the requirement that H must
determine
whether the computation presented by its input halts has already >>>>>>>>>> been presented.
*ChatGPT explains how and why I am correct*
*Reinterpretation of undecidability*
The example of P and H demonstrates that what is
often called “undecidable” is better understood as >>>>>>>>>>> ill-posed with respect to computable semantics.
When the specification is constrained to properties >>>>>>>>>>> detectable via finite simulation and finite pattern >>>>>>>>>>> recognition, computation proceeds normally and
correctly. Undecidability only appears when the
specification overreaches that boundary.
It tries to explain but it does not prove.
Its the same thing that I have been saying for years.
It is not that a universal halt decider cannot exist.
It is proven that an universal halt decider does not exist.
“The system adopts Proof-Theoretic Semantics: meaning is
determined by inferential role, and truth is internal to the
theory. A theory T is defined by a finite set of stipulated
atomic statements together with all expressions derivable from
them under the inference rules. The statements belonging to T
constitute its theorems, and these are exactly the statements
that are true-in-T.”
Under a system like the above rough draft all inputs
having pathological self reference such as the halting
problem counter-example input are simply rejected as
non-well-founded. Tarski Undefinability, Gödel's
incompleteness and the halting problem cease to exist.
A Turing
machine cannot determine the halting of all Turing machines and is >>>>>>>> therefore not an universla halt decider.
This is not true in Proof Theoretic Semantics. I
still have to refine my words. I may not have said
that exactly correctly. The result is that in Proof
Theoretic Semantics the counter-example is rejected
as non-well-founded.
That no Turing machine is a halt decider is a proven theorem and a >>>>>> truth about Turing machines. If your "Proof Thoeretic Semnatics"
does not regard it as true then your "Proof Theoretic Semantics"
is incomplete.
My long‑term goal is to make ‘true on the basis of meaning’
computable.
As meaning is not computable, how can "true on the balsis of meaning"
be commputable?
Under *proof‑theoretic semantics*
"true on the basis of meaning expressed in language"
has always been entirely computable.
Have you already put the algorithm to some web page?
Proof Theoretic Semantics Blocks Pathological Self-Reference https://philpapers.org/rec/OLCPTS
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:
You can't determine whether the required result isNo, 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. >>>>>>>>>>>>>
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 Gödel'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 Gödel 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.
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 Gödel'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 Gödel 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. Gödel 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 Gödel'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 Gödel 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. Gödel did not do this himself because
Proof theoretic semantics did not exist at the time.
Gödel 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 Gödel'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 Gödel'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 Gödel 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. Gödel did not do this himself because
Proof theoretic semantics did not exist at the time.
Gödel 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 Gödel's completeness
theorem for your "Proof theoretic semantics"?
Gödel’s incompleteness arises only because
“true in PA” was never an internal notion
of PA at all, but a meta‑mathematical notion
of truth about PA defined externally through
models;
Once truth is defined internally—by extending
PA with a truth predicate so that “true in PA”
simply means “derivable from PA’s axioms”—
the supposed gap between truth and provability
disappears
With that disappearance PA no longer counts as
incomplete, because the statements Gödel identified
as “true but unprovable” were never internal truths
of PA in the first place, only truths assigned from
the outside by the meta‑system.
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 Gödel'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 Gödel 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. Gödel did not do this himself because
Proof theoretic semantics did not exist at the time.
Gödel 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 Gödel's completeness
theorem for your "Proof theoretic semantics"?
Gödel’s incompleteness arises only because
“true in PA” was never an internal notion
of PA at all, but a meta‑mathematical notion
of truth about PA defined externally through
models;
Once truth is defined internally—by extending
PA with a truth predicate so that “true in PA”
simply means “derivable from PA’s axioms”—
the supposed gap between truth and provability
disappears
With that disappearance PA no longer counts as
incomplete, because the statements Gödel identified
as “true but unprovable” were never internal truths
of PA in the first place, only truths assigned from
the outside by the meta‑system.
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 Gödel 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 Gödel'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 Russell’s
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 Gödel 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 Gödel'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 Russell’s
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 Gödel 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 Gödel'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 Russell’s
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 Gödel'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 Gödel 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 Russell’s
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.
Meta‑math relations about numbers don’t exist in PA
because PA only contains arithmetical relations—addition,
multiplication, ordering, primitive‑recursive predicates
about numbers themselves—while relations that talk about
PA’s own proofs, syntax, or truth conditions live entirely
in the meta‑theory;
so when someone appeals to a Gödel‑style relation like
“n encodes a proof of this very sentence,” they’re
invoking a meta‑mathematical predicate that PA cannot
internalize, which is exactly why your framework draws
a clean boundary between internal proof‑theoretic truth
and external model‑theoretic 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.
Meta‑math relations about numbers don’t exist in PA
because PA only contains arithmetical relations—addition,
multiplication, ordering, primitive‑recursive predicates
about numbers themselves—while relations that talk about
PA’s own proofs, syntax, or truth conditions live entirely
in the meta‑theory;
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 Gödel‑style relation like
“n encodes a proof of this very sentence,” they’re
invoking a meta‑mathematical predicate that PA cannot
internalize, which is exactly why your framework draws
a clean boundary between internal proof‑theoretic truth
and external model‑theoretic 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.
Meta‑math relations about numbers don’t exist in PA
because PA only contains arithmetical relations—addition,
multiplication, ordering, primitive‑recursive predicates
about numbers themselves—while relations that talk about
PA’s own proofs, syntax, or truth conditions live entirely
in the meta‑theory;
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 Gödel‑style relation like
“n encodes a proof of this very sentence,” they’re
invoking a meta‑mathematical predicate that PA cannot
internalize, which is exactly why your framework draws
a clean boundary between internal proof‑theoretic truth
and external model‑theoretic 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.
Gödel’s sentence is not “true in arithmetic.”
It is true only in the meta‑theory, under an
external interpretation of PA (typically the
standard model ℕ). Inside PA itself, the sentence
is not a truth‑bearer at all.
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 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 Gödel'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 Gödel 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 Gödel'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 Russell’s
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 meta‑mathematical.
Truth in PA is proof‑theoretic. These were historically
conflated only because proof‑theoretic semantics did not
exist. With Curry’s notion of internal truth, PA’s truth
predicate is simply:
∀x ∈ PA ((True(PA, x) ≡ (PA ⊢ x))
∀x ∈ PA ((False(PA, x) ≡ (PA ⊢ ~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 Gödel 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 Gödel'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 Russell’s
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 meta‑mathematical.
Nope, but then, you don't understand what Truth actually is.
Truth in PA is proof‑theoretic. These were historically
conflated only because proof‑theoretic semantics did not
exist. With Curry’s notion of internal truth, PA’s truth
predicate is simply:
∀x ∈ PA ((True(PA, x) ≡ (PA ⊢ x))
∀x ∈ PA ((False(PA, x) ≡ (PA ⊢ ~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 ⊢ 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 Gödel'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 Gödel 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 Russell’s
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 meta‑mathematical.
Nope, but then, you don't understand what Truth actually is.
Truth in PA is proof‑theoretic. These were historically
conflated only because proof‑theoretic semantics did not
exist. With Curry’s notion of internal truth, PA’s truth
predicate is simply:
∀x ∈ PA ((True(PA, x) ≡ (PA ⊢ x))
∀x ∈ PA ((False(PA, x) ≡ (PA ⊢ ~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 ⊢ 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 Gödel'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 Gödel 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 Russell’s
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 meta‑mathematical.
Nope, but then, you don't understand what Truth actually is.
Truth in PA is proof‑theoretic. These were historically
conflated only because proof‑theoretic semantics did not
exist. With Curry’s notion of internal truth, PA’s truth
predicate is simply:
∀x ∈ PA ((True(PA, x) ≡ (PA ⊢ x))
∀x ∈ PA ((False(PA, x) ≡ (PA ⊢ ~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 ⊢ 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 11:10 AM, Richard Damon wrote:
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:ZF is a redefinition in the only sense that matters:
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: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 Gödel'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 Gödel 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 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. >>>>>>>
it changes the foundational rules so that Russell’s
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.
Have you ever heard of: "true in the standard model of arithmetic"?
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 meta‑mathematical.
Nope, but then, you don't understand what Truth actually is.
Truth in PA is proof‑theoretic. These were historically
conflated only because proof‑theoretic semantics did not
exist. With Curry’s notion of internal truth, PA’s truth
predicate is simply:
∀x ∈ PA ((True(PA, x) ≡ (PA ⊢ x))
∀x ∈ PA ((False(PA, x) ≡ (PA ⊢ ~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 ⊢ 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 12:54 PM, olcott wrote:
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"
On 1/24/2026 1:23 PM, Richard Damon wrote:
On 1/24/26 12:54 PM, olcott wrote:Gödel’s incompleteness theorem only “works” 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 ℕ) and then pretends it is an
internal notion of truth (truth in PA).
If we refuse to make that identification,
incompleteness evaporates.
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:Gödel’s incompleteness theorem only “works” 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 ℕ) 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.
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:Gödel’s incompleteness theorem only “works” 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 ℕ) 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 ℕ requires PA
as part of it, and PA itself has no notion of
Truth in ℕ. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in ℕ.
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:Gödel’s incompleteness theorem only “works” 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 ℕ) 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 ℕ requires PA
as part of it, and PA itself has no notion of
Truth in ℕ. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in ℕ.
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:Gödel’s incompleteness theorem only “works” 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 ℕ) 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 ℕ requires PA
as part of it, and PA itself has no notion of
Truth in ℕ. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in ℕ.
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:Gödel’s incompleteness theorem only “works” 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 ℕ) 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 ℕ requires PA
as part of it, and PA itself has no notion of
Truth in ℕ. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in ℕ.
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:Gödel’s incompleteness theorem only “works” 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 ℕ) 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 ℕ requires PA
as part of it, and PA itself has no notion of
Truth in ℕ. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in ℕ.
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 2:23 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 Gödel's metatheory.
∀x ∈ PA ( True(PA, x) ≡ PA ⊢ x )
∀x ∈ PA ( False(PA, x) ≡ PA ⊢ ¬x )
∀x ∈ PA ( ¬WellFounded(PA, x) ≡
(¬True(PA, x) ∧ (¬False(PA, x)))
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:Gödel’s incompleteness theorem only “works” 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 ℕ) 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 ℕ requires PA
as part of it, and PA itself has no notion of
Truth in ℕ. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in ℕ.
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.
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.
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 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:Gödel’s incompleteness theorem only “works” 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 ℕ) 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 ℕ requires PA
as part of it, and PA itself has no notion of
Truth in ℕ. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in ℕ.
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.
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 12:36 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:Gödel’s incompleteness theorem only “works” 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 ℕ) 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 ℕ requires PA
as part of it, and PA itself has no notion of
Truth in ℕ. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in ℕ.
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 1/25/26 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:Gödel’s incompleteness theorem only “works” 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 ℕ) 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 ℕ requires PA
as part of it, and PA itself has no notion of
Truth in ℕ. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in ℕ.
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 1/25/2026 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:Gödel’s incompleteness theorem only “works” 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 ℕ) 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 ℕ requires PA
as part of it, and PA itself has no notion of
Truth in ℕ. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in ℕ.
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 1/25/26 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:Gödel’s incompleteness theorem only “works” 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 ℕ) 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 ℕ requires PA
as part of it, and PA itself has no notion of
Truth in ℕ. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in ℕ.
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 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:Gödel’s incompleteness theorem only “works” 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 ℕ) 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 ℕ requires PA
as part of it, and PA itself has no notion of
Truth in ℕ. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in ℕ.
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 1/25/26 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 1/26/2026 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:
∀a ∀b ∀c (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:
∀a ∀b ∀c (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:
∀a ∀b ∀c (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 16/01/2026 04:03, olcott wrote:
It is the same ∀x ∈ T ((True(T, x) ≡ (T ⊢ 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?
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