On 1/25/26 3:07 PM, olcott wrote:
On 1/25/2026 1:54 PM, Richard Damon wrote:
On 1/25/26 2:09 PM, olcott wrote:
On 1/25/2026 12:36 PM, Richard Damon wrote:
On 1/24/26 8:44 PM, olcott wrote:
On 1/24/2026 6:52 PM, Richard Damon wrote:
On 1/24/26 5:31 PM, olcott wrote:
On 1/24/2026 4:25 PM, Richard Damon wrote:
On 1/24/26 3:38 PM, olcott wrote:
On 1/24/2026 1:52 PM, Richard Damon wrote:
On 1/24/26 2:25 PM, olcott wrote:
On 1/24/2026 1:23 PM, Richard Damon wrote:
On 1/24/26 12:54 PM, olcott wrote:G||delrCOs incompleteness theorem only rCLworksrCY if
On 1/24/2026 11:10 AM, Richard Damon wrote:
On 1/24/26 10:44 AM, olcott wrote:
Sure it is. At least it is a FACT that no natural number >>>>>>>>>>>>>>> will statisfy that relationship, and there is no proof in >>>>>>>>>>>>>>> PA of that fact.
The statement that G is true and unprovable in PA has >>>>>>>>>>>>>>>> always been counter-factual. It has never actually been >>>>>>>>>>>>>>>> true <in> PA and that is why it is unprovable in PA. >>>>>>>>>>>>>>>
Have you ever heard of: "true in the standard model of >>>>>>>>>>>>>> arithmetic"?
Sure, but they are not in Peano Arithmatic, but are >>>>>>>>>>>>> (generally) 1st order variations of the Peano Axioms which >>>>>>>>>>>>> lead to alternate number systems.
Godel's proof is statd to be in a system with at least the >>>>>>>>>>>>> properties of Peano Arithmatic, having the ability to show >>>>>>>>>>>>> the properties of the "Natural Numbers"
one smuggles in an external notion of truth
(truth in rao) and then pretends it is an
internal notion of truth (truth in PA).
If we refuse to make that identification,
incompleteness evaporates.
But Truth in N is part of Peano Arithmatic, as Peano
Arithmatic is a axiomiation to create the Natural Numbers. >>>>>>>>>>>
You have that backwards. Truth in rao requires PA
as part of it, and PA itself has no notion of
Truth in rao. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in rao.
Which means you don't understand what N actually is.
Nothing can be "True in N" unless that truth comes from the >>>>>>>>> Axioms of PA, as N is the result of PA.
combined with the meta-math external model.
Nope. N is just a set of object built in the Formal System
defined by PA. 0 comes from Axiom 1 which states there is a 0.
If G is true and not provable then you have
the wrong kind of true. I have known that
the entire body of knowledge is a semantic
tautology for 28 years.
No, YOU do. The problem is Truth in the real world isn't based on
being about to prove the fact, and most things are not actually
provable, just well approximatable.
That is why this insight was so important:
"true on the basis of meaning expressed in language"
I broke through the 75 year logjam of the analytic/synthetic
distinction.
In other words, you don't accept the Pythgorean Theorem as "True",
since its Tru-ness doesn't come out of the meaning of its words.
[meaning of its words]
My sentence is not restricted to words and
does include mathematical expressions.
Then it accepts Godel's G as a valid statement
and Goldbach's
conjecture, even if improbably true, is a truth bearer.
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
You are just admitting to your own equivocation of meaning.--
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can deny >>>>>> the proof but you cannot perform what is meta-provably impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can deny >>>>>>> the proof but you cannot perform what is meta-provably impossible. >>>>>The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can >>>>>>>> denyThe meta-proof does not exist in the axioms of PA
the proof but you cannot perform what is meta-provably impossible. >>>>>>
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also >>>>>> a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
On 1/25/2026 2:44 PM, Richard Damon wrote:
On 1/25/26 3:07 PM, olcott wrote:
On 1/25/2026 1:54 PM, Richard Damon wrote:
On 1/25/26 2:09 PM, olcott wrote:
On 1/25/2026 12:36 PM, Richard Damon wrote:
On 1/24/26 8:44 PM, olcott wrote:
On 1/24/2026 6:52 PM, Richard Damon wrote:
On 1/24/26 5:31 PM, olcott wrote:
On 1/24/2026 4:25 PM, Richard Damon wrote:
On 1/24/26 3:38 PM, olcott wrote:
On 1/24/2026 1:52 PM, Richard Damon wrote:
On 1/24/26 2:25 PM, olcott wrote:
On 1/24/2026 1:23 PM, Richard Damon wrote:
On 1/24/26 12:54 PM, olcott wrote:G||delrCOs incompleteness theorem only rCLworksrCY if >>>>>>>>>>>>> one smuggles in an external notion of truth
On 1/24/2026 11:10 AM, Richard Damon wrote:
On 1/24/26 10:44 AM, olcott wrote:
Sure it is. At least it is a FACT that no natural number >>>>>>>>>>>>>>>> will statisfy that relationship, and there is no proof >>>>>>>>>>>>>>>> in PA of that fact.
The statement that G is true and unprovable in PA has >>>>>>>>>>>>>>>>> always been counter-factual. It has never actually been >>>>>>>>>>>>>>>>> true <in> PA and that is why it is unprovable in PA. >>>>>>>>>>>>>>>>
Have you ever heard of: "true in the standard model of >>>>>>>>>>>>>>> arithmetic"?
Sure, but they are not in Peano Arithmatic, but are >>>>>>>>>>>>>> (generally) 1st order variations of the Peano Axioms which >>>>>>>>>>>>>> lead to alternate number systems.
Godel's proof is statd to be in a system with at least the >>>>>>>>>>>>>> properties of Peano Arithmatic, having the ability to show >>>>>>>>>>>>>> the properties of the "Natural Numbers"
(truth in rao) and then pretends it is an
internal notion of truth (truth in PA).
If we refuse to make that identification,
incompleteness evaporates.
But Truth in N is part of Peano Arithmatic, as Peano
Arithmatic is a axiomiation to create the Natural Numbers. >>>>>>>>>>>>
You have that backwards. Truth in rao requires PA
as part of it, and PA itself has no notion of
Truth in rao. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in rao.
Which means you don't understand what N actually is.
Nothing can be "True in N" unless that truth comes from the >>>>>>>>>> Axioms of PA, as N is the result of PA.
combined with the meta-math external model.
Nope. N is just a set of object built in the Formal System
defined by PA. 0 comes from Axiom 1 which states there is a 0. >>>>>>>>
If G is true and not provable then you have
the wrong kind of true. I have known that
the entire body of knowledge is a semantic
tautology for 28 years.
No, YOU do. The problem is Truth in the real world isn't based on >>>>>> being about to prove the fact, and most things are not actually
provable, just well approximatable.
That is why this insight was so important:
"true on the basis of meaning expressed in language"
I broke through the 75 year logjam of the analytic/synthetic
distinction.
In other words, you don't accept the Pythgorean Theorem as "True",
since its Tru-ness doesn't come out of the meaning of its words.
[meaning of its words]
My sentence is not restricted to words and
does include mathematical expressions.
Then it accepts Godel's G as a valid statement
That has no truth value in PA.
and Goldbach's conjecture, even if improbably true, is a truth bearer.
As a truth bearer with a currently unknown truth value.
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
I never said anything about words.
It took me 25 years to derive that exact phrase.
You are just admitting to your own equivocation of meaning.
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can >>>>>>>>> denyThe meta-proof does not exist in the axioms of PA
the proof but you cannot perform what is meta-provably impossible. >>>>>>>
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also >>>>>>> a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what is true
in PA?
I thought you said that PA had to be able to determine the truth itself?
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You >>>>>>>>>> can deny
the proof but you cannot perform what is meta-provably
impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also >>>>>>>> a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what is
true in PA?
I thought you said that PA had to be able to determine the truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
On 1/25/26 9:31 PM, olcott wrote:
On 1/25/2026 2:44 PM, Richard Damon wrote:
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
Which can't handle math.
As you have effective admitted by not answering about my example with
the Pythgorean Theorem.
I never said anything about words.
It took me 25 years to derive that exact phrase.
What is language, but meaning expressed in "words".
I think your problem is a fundamental failure to understand what you are talking about as you accept your own double-speak.
You are just admitting to your own equivocation of meaning.
On 1/26/26 11:58 AM, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You >>>>>>>>>>> can deny
the proof but you cannot perform what is meta-provably
impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is >>>>>>>>> also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a >>>>>>> language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what is
true in PA?
I thought you said that PA had to be able to determine the truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
But Peano Arithmatic *IS* a standard model of arithmetic.
The Induction Axiom makes it so.
The other models tend to come from making a variant of PA by changing--
that 2nd order Induction Axiom to various first order versions to
"simulate" its power using the rest of the Peano Axioms, and then adding something more to complete it.
On 1/26/2026 10:49 AM, Richard Damon wrote:
On 1/25/26 9:31 PM, olcott wrote:All math and logic has a language of math and logic.
On 1/25/2026 2:44 PM, Richard Damon wrote:
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
Which can't handle math.
I literally spent 25 years coming up with that.
All of the LLM systems understand that
"true on the basis of meaning expressed in language"
breaks the logjam established in:
"Two Dogmas of Empiricism" Willard Van Orman Quine https://www.theologie.uzh.ch/dam/jcr:ffffffff- fbd6-1538-0000-000070cf64bc/Quine51.pdf
regarding the fundamental nature of truth itself
previously called the analytic/synthetic distinction
now renamed to the analytic/empirical distinction.
These LLM systems do not yet understand that
succinctly. It takes them some back and forth
to understand that.
As you have effective admitted by not answering about my example with
the Pythgorean Theorem.
I never said anything about words.
It took me 25 years to derive that exact phrase.
What is language, but meaning expressed in "words".
I think your problem is a fundamental failure to understand what you
are talking about as you accept your own double-speak.
You are just admitting to your own equivocation of meaning.
On 1/26/26 12:23 PM, olcott wrote:
On 1/26/2026 10:49 AM, Richard Damon wrote:
On 1/25/26 9:31 PM, olcott wrote:All math and logic has a language of math and logic.
On 1/25/2026 2:44 PM, Richard Damon wrote:
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
Which can't handle math.
I literally spent 25 years coming up with that.
And is build of "words" which is the symbolism of that langauge.
On 1/26/2026 12:24 PM, Richard Damon wrote:
On 1/26/26 12:23 PM, olcott wrote:All of those copies of what I said ARE VERY SPECIFICALLY
On 1/26/2026 10:49 AM, Richard Damon wrote:
On 1/25/26 9:31 PM, olcott wrote:All math and logic has a language of math and logic.
On 1/25/2026 2:44 PM, Richard Damon wrote:
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
Which can't handle math.
I literally spent 25 years coming up with that.
And is build of "words" which is the symbolism of that langauge.
AND VERY INTENTIONALLY NOT LIMITED TO WORDS.
Do I need to say that 10,000 times to get
you to notice that I said it at least once?
I build Minimal Type Theory entirely on the
basis of the YACC grammar specification of the
language of FOL.
https://www.researchgate.net/ publication/331859461_Minimal_Type_Theory_YACC_BNF
On 1/26/26 1:43 PM, olcott wrote:
On 1/26/2026 12:24 PM, Richard Damon wrote:
On 1/26/26 12:23 PM, olcott wrote:All of those copies of what I said ARE VERY SPECIFICALLY
On 1/26/2026 10:49 AM, Richard Damon wrote:
On 1/25/26 9:31 PM, olcott wrote:All math and logic has a language of math and logic.
On 1/25/2026 2:44 PM, Richard Damon wrote:
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
Which can't handle math.
I literally spent 25 years coming up with that.
And is build of "words" which is the symbolism of that langauge.
AND VERY INTENTIONALLY NOT LIMITED TO WORDS.
Do I need to say that 10,000 times to get
you to notice that I said it at least once?
No just answer the question.
But, I guess since you don't actually knoew what you mean, you can't do that,
I build Minimal Type Theory entirely on the
basis of the YACC grammar specification of the
language of FOL.
So?
https://www.researchgate.net/
publication/331859461_Minimal_Type_Theory_YACC_BNF
So, what is a "language" built on if not what it considers its "words"?
And, how does you system handle the truth of something like the
Pythagorean Theorem?
Your repeated failure just proves that you CAN'T answer as you know your system is broken but need to continue clinging to its lie.--
On 1/26/2026 3:58 PM, Richard Damon wrote:
On 1/26/26 1:43 PM, olcott wrote:
On 1/26/2026 12:24 PM, Richard Damon wrote:
On 1/26/26 12:23 PM, olcott wrote:All of those copies of what I said ARE VERY SPECIFICALLY
On 1/26/2026 10:49 AM, Richard Damon wrote:
On 1/25/26 9:31 PM, olcott wrote:All math and logic has a language of math and logic.
On 1/25/2026 2:44 PM, Richard Damon wrote:
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
Which can't handle math.
I literally spent 25 years coming up with that.
And is build of "words" which is the symbolism of that langauge.
AND VERY INTENTIONALLY NOT LIMITED TO WORDS.
Do I need to say that 10,000 times to get
you to notice that I said it at least once?
No just answer the question.
But, I guess since you don't actually knoew what you mean, you can't
do that,
I build Minimal Type Theory entirely on the
basis of the YACC grammar specification of the
language of FOL.
So?
https://www.researchgate.net/
publication/331859461_Minimal_Type_Theory_YACC_BNF
So, what is a "language" built on if not what it considers its "words"?
All of the logic, math and computation languages
are not grounded in words deep ship.
And, how does you system handle the truth of something like the
Pythagorean Theorem?
written in PA syntax as:
reCa reCb reCc (a-+a + b-+b = c-+c)
Your repeated failure just proves that you CAN'T answer as you know
your system is broken but need to continue clinging to its lie.
On 1/26/26 5:08 PM, olcott wrote:
On 1/26/2026 3:58 PM, Richard Damon wrote:
On 1/26/26 1:43 PM, olcott wrote:
On 1/26/2026 12:24 PM, Richard Damon wrote:
On 1/26/26 12:23 PM, olcott wrote:All of those copies of what I said ARE VERY SPECIFICALLY
On 1/26/2026 10:49 AM, Richard Damon wrote:
On 1/25/26 9:31 PM, olcott wrote:All math and logic has a language of math and logic.
On 1/25/2026 2:44 PM, Richard Damon wrote:
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
Which can't handle math.
I literally spent 25 years coming up with that.
And is build of "words" which is the symbolism of that langauge.
AND VERY INTENTIONALLY NOT LIMITED TO WORDS.
Do I need to say that 10,000 times to get
you to notice that I said it at least once?
No just answer the question.
But, I guess since you don't actually knoew what you mean, you can't
do that,
I build Minimal Type Theory entirely on the
basis of the YACC grammar specification of the
language of FOL.
So?
https://www.researchgate.net/
publication/331859461_Minimal_Type_Theory_YACC_BNF
So, what is a "language" built on if not what it considers its "words"?
All of the logic, math and computation languages
are not grounded in words deep ship.
sure they are, when you consider a "word" to include the symbols and
number they use.
And, how does you system handle the truth of something like the
Pythagorean Theorem?
written in PA syntax as:
reCa reCb reCc (a-+a + b-+b = c-+c)
So, why is that true for EVERY a and b that are sides of a right triangle?
Note, the Pythagorean Theorem isn't part of PA, but Plain Geometry.
I guess you just belive in truth conditional logic.
Your problem is you just don't know that truth or proof means because of your ignorance.
Your repeated failure just proves that you CAN'T answer as you know
your system is broken but need to continue clinging to its lie.
On 1/26/2026 4:36 PM, Richard Damon wrote:
On 1/26/26 5:08 PM, olcott wrote:
On 1/26/2026 3:58 PM, Richard Damon wrote:
On 1/26/26 1:43 PM, olcott wrote:
On 1/26/2026 12:24 PM, Richard Damon wrote:
On 1/26/26 12:23 PM, olcott wrote:All of those copies of what I said ARE VERY SPECIFICALLY
On 1/26/2026 10:49 AM, Richard Damon wrote:
On 1/25/26 9:31 PM, olcott wrote:All math and logic has a language of math and logic.
On 1/25/2026 2:44 PM, Richard Damon wrote:
If not, you don't know what you words mean,
And how is "Meaning of its words" not about the WORDS?
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
"true on the basis of meaning expressed in language"
Which can't handle math.
I literally spent 25 years coming up with that.
And is build of "words" which is the symbolism of that langauge.
AND VERY INTENTIONALLY NOT LIMITED TO WORDS.
Do I need to say that 10,000 times to get
you to notice that I said it at least once?
No just answer the question.
But, I guess since you don't actually knoew what you mean, you can't
do that,
I build Minimal Type Theory entirely on the
basis of the YACC grammar specification of the
language of FOL.
So?
https://www.researchgate.net/
publication/331859461_Minimal_Type_Theory_YACC_BNF
So, what is a "language" built on if not what it considers its "words"? >>>>
All of the logic, math and computation languages
are not grounded in words deep ship.
sure they are, when you consider a "word" to include the symbols and
number they use.
And, how does you system handle the truth of something like the
Pythagorean Theorem?
written in PA syntax as:
reCa reCb reCc (a-+a + b-+b = c-+c)
So, why is that true for EVERY a and b that are sides of a right
triangle?
Note, the Pythagorean Theorem isn't part of PA, but Plain Geometry.
I guess you just belive in truth conditional logic.
"true on the basis of meaning expressed in language"
Inherently includes every element of the entire body
of knowledge that can be expressed in any formal
mathematical or natural language.
Your problem is you just don't know that truth or proof means because
of your ignorance.
Your repeated failure just proves that you CAN'T answer as you know
your system is broken but need to continue clinging to its lie.
On 1/26/26 5:44 PM, olcott wrote:>>
"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.
ONLY if meaning means the (possibly infinite) operation of the logical operations of the system to its axioms.
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can >>>>>>>> denyThe meta-proof does not exist in the axioms of PA
the proof but you cannot perform what is meta-provably impossible. >>>>>>
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also >>>>>> a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You >>>>>>>>>> can deny
the proof but you cannot perform what is meta-provably
impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also >>>>>>>> a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what is
true in PA?
I thought you said that PA had to be able to determine the truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can >>>>>>>>> denyThe meta-proof does not exist in the axioms of PA
the proof but you cannot perform what is meta-provably impossible. >>>>>>>
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also >>>>>>> a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what is true
in PA?
I thought you said that PA had to be able to determine the truth itself?
On 26/01/2026 17:22, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You can >>>>>>>>> denyThe meta-proof does not exist in the axioms of PA
the proof but you cannot perform what is meta-provably impossible. >>>>>>>
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also >>>>>>> a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
The above is not a notational convention. The symbols may be defined
in some context but they are undefined elsewhere.
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You >>>>>>>>>>> can deny
the proof but you cannot perform what is meta-provably
impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is >>>>>>>>> also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a >>>>>>> language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what is
true in PA?
I thought you said that PA had to be able to determine the truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
On 1/27/2026 2:05 AM, Mikko wrote:
On 26/01/2026 17:22, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You >>>>>>>>>> can deny
the proof but you cannot perform what is meta-provably
impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is also >>>>>>>> a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
The above is not a notational convention. The symbols may be defined
in some context but they are undefined elsewhere.
Mendelson simply uses reo EYAR to indicate that EYAR is a theorem.
reCx (True(x) rei reo EYAR)
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You >>>>>>>>>>>> can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA is >>>>>>>>>> also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a >>>>>>>> language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and
WellFounded with two arguments. And you did not specify in which
context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what is
true in PA?
I thought you said that PA had to be able to determine the truth
itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
All of the expressions where True(L, x) is not computable
x is semantically incoherent or outside of the domain of knowledge.
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. You >>>>>>>>>>>>> can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA.
All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA >>>>>>>>>>> is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a >>>>>>>>> language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and >>>>>>> WellFounded with two arguments. And you did not specify in which >>>>>>> context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what is
true in PA?
I thought you said that PA had to be able to determine the truth
itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results
are related and there are much similarity in the proofs. Tarski did
not use Turing machines in the proof but a computability proof must
use that.
All of the expressions where True(L, x) is not computable
x is semantically incoherent or outside of the domain of knowledge.
Computability does not depend on semantics or knowledge.
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. >>>>>>>>>>>>>> You can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA. >>>>>>>>>>>>> All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA >>>>>>>>>>>> is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x)))
Those sentences don't mean anything without specificantions of a >>>>>>>>>> language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms
of PA.
There are no notational convention that defines True, False, and >>>>>>>> WellFounded with two arguments. And you did not specify in which >>>>>>>> context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what is >>>>>> true in PA?
I thought you said that PA had to be able to determine the truth
itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results
are related and there are much similarity in the proofs. Tarski did
not use Turing machines in the proof but a computability proof must
use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
On 28/01/2026 15:49, olcott wrote:
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:Those sentences don't mean anything without specificantions of a >>>>>>>>>>> language and a theory that gives them some meaning.
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. >>>>>>>>>>>>>>> You can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA
and that is the reason why an external truth in
an external model cannot be proved internally in PA. >>>>>>>>>>>>>> All of these years it was only a mere conflation
error.
It is perfectly clear which is which. But every proof in PA >>>>>>>>>>>>> is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x))) >>>>>>>>>>>
In other word you do not understand standard notational
conventions that define True for PA as provable from the
axioms of PA and False for PA as refutable from the axioms >>>>>>>>>> of PA.
There are no notational convention that defines True, False, and >>>>>>>>> WellFounded with two arguments. And you did not specify in which >>>>>>>>> context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what >>>>>>> is true in PA?
I thought you said that PA had to be able to determine the truth >>>>>>> itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results
are related and there are much similarity in the proofs. Tarski did
not use Turing machines in the proof but a computability proof must
use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
Irrelevant. There is no "occurs_check" in Tarski's proof.
On 1/29/2026 3:12 AM, Mikko wrote:
On 28/01/2026 15:49, olcott wrote:
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:Those sentences don't mean anything without specificantions >>>>>>>>>>>> of a
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. >>>>>>>>>>>>>>>> You can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA >>>>>>>>>>>>>>> and that is the reason why an external truth in
an external model cannot be proved internally in PA. >>>>>>>>>>>>>>> All of these years it was only a mere conflation >>>>>>>>>>>>>>> error.
It is perfectly clear which is which. But every proof in >>>>>>>>>>>>>> PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x))) >>>>>>>>>>>>
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the >>>>>>>>>>> axioms of PA and False for PA as refutable from the axioms >>>>>>>>>>> of PA.
There are no notational convention that defines True, False, and >>>>>>>>>> WellFounded with two arguments. And you did not specify in which >>>>>>>>>> context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what >>>>>>>> is true in PA?
I thought you said that PA had to be able to determine the truth >>>>>>>> itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results
are related and there are much similarity in the proofs. Tarski did
not use Turing machines in the proof but a computability proof must
use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
Irrelevant. There is no "occurs_check" in Tarski's proof.
On 29/01/2026 15:57, olcott wrote:
On 1/29/2026 3:12 AM, Mikko wrote:
On 28/01/2026 15:49, olcott wrote:
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:Those sentences don't mean anything without specificantions >>>>>>>>>>>>> of a
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. >>>>>>>>>>>>>>>>> You can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA >>>>>>>>>>>>>>>> and that is the reason why an external truth in >>>>>>>>>>>>>>>> an external model cannot be proved internally in PA. >>>>>>>>>>>>>>>> All of these years it was only a mere conflation >>>>>>>>>>>>>>>> error.
It is perfectly clear which is which. But every proof in >>>>>>>>>>>>>>> PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x))) >>>>>>>>>>>>>
language and a theory that gives them some meaning.
In other word you do not understand standard notational >>>>>>>>>>>> conventions that define True for PA as provable from the >>>>>>>>>>>> axioms of PA and False for PA as refutable from the axioms >>>>>>>>>>>> of PA.
There are no notational convention that defines True, False, and >>>>>>>>>>> WellFounded with two arguments. And you did not specify in which >>>>>>>>>>> context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what >>>>>>>>> is true in PA?
I thought you said that PA had to be able to determine the
truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results
are related and there are much similarity in the proofs. Tarski did
not use Turing machines in the proof but a computability proof must
use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
Irrelevant. There is no "occurs_check" in Tarski's proof.
That would have no effet. Even if the metalanguage had an occcurs_check
it would not be necessary to use it in a proof.
On 16/01/2026 04:03, olcott wrote:
It is the same reCx ree T ((True(T, x) rei (T reo x))
I still think you're asking for confusion with that use of the turnstile.
But it does make it very obvious that we should expect negation to be restricted in your system which might overcome a psychological hurdle.
How is negation restricted in your system?
On 1/30/2026 3:34 AM, Mikko wrote:
On 29/01/2026 15:57, olcott wrote:
On 1/29/2026 3:12 AM, Mikko wrote:
On 28/01/2026 15:49, olcott wrote:
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:In other word you do not understand standard notational >>>>>>>>>>>>> conventions that define True for PA as provable from the >>>>>>>>>>>>> axioms of PA and False for PA as refutable from the axioms >>>>>>>>>>>>> of PA.
On 1/24/2026 2:23 AM, Mikko wrote:Those sentences don't mean anything without
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout >>>>>>>>>>>>>>>>>> PA. You can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA >>>>>>>>>>>>>>>>> and that is the reason why an external truth in >>>>>>>>>>>>>>>>> an external model cannot be proved internally in PA. >>>>>>>>>>>>>>>>> All of these years it was only a mere conflation >>>>>>>>>>>>>>>>> error.
It is perfectly clear which is which. But every proof in >>>>>>>>>>>>>>>> PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x))) >>>>>>>>>>>>>>
specificantions of a
language and a theory that gives them some meaning. >>>>>>>>>>>>>
There are no notational convention that defines True, False, >>>>>>>>>>>> and
WellFounded with two arguments. And you did not specify in >>>>>>>>>>>> which
context your sentences are true or otherwise relevant. >>>>>>>>>>>>
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define >>>>>>>>>> what is true in PA?
I thought you said that PA had to be able to determine the >>>>>>>>>> truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results >>>>>> are related and there are much similarity in the proofs. Tarski did >>>>>> not use Turing machines in the proof but a computability proof must >>>>>> use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
Irrelevant. There is no "occurs_check" in Tarski's proof.
That would have no effet. Even if the metalanguage had an occcurs_check
it would not be necessary to use it in a proof.
It would only seem to have no effect because you
never bothered to understand what an occurs_check is.
Truth is computable because rCLmeaningful sentencerCY
is defined as rCLsentence with a well-founded
justification tree,rCY and evaluating any well-founded
tree always terminates. Anything else isnrCOt truth-apt.
On 1/29/2026 3:12 AM, Mikko wrote:
On 28/01/2026 15:49, olcott wrote:
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:Those sentences don't mean anything without specificantions >>>>>>>>>>>> of a
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. >>>>>>>>>>>>>>>> You can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA >>>>>>>>>>>>>>> and that is the reason why an external truth in
an external model cannot be proved internally in PA. >>>>>>>>>>>>>>> All of these years it was only a mere conflation >>>>>>>>>>>>>>> error.
It is perfectly clear which is which. But every proof in >>>>>>>>>>>>>> PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x))) >>>>>>>>>>>>
language and a theory that gives them some meaning.
In other word you do not understand standard notational
conventions that define True for PA as provable from the >>>>>>>>>>> axioms of PA and False for PA as refutable from the axioms >>>>>>>>>>> of PA.
There are no notational convention that defines True, False, and >>>>>>>>>> WellFounded with two arguments. And you did not specify in which >>>>>>>>>> context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what >>>>>>>> is true in PA?
I thought you said that PA had to be able to determine the truth >>>>>>>> itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results
are related and there are much similarity in the proofs. Tarski did
not use Turing machines in the proof but a computability proof must
use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
Irrelevant. There is no "occurs_check" in Tarski's proof.
If there was then there never would be a Tarski proof. https://liarparadox.org/Tarski_247_248.pdf
On 30/01/2026 16:35, olcott wrote:
On 1/30/2026 3:34 AM, Mikko wrote:
On 29/01/2026 15:57, olcott wrote:
On 1/29/2026 3:12 AM, Mikko wrote:
On 28/01/2026 15:49, olcott wrote:
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:In other word you do not understand standard notational >>>>>>>>>>>>>> conventions that define True for PA as provable from the >>>>>>>>>>>>>> axioms of PA and False for PA as refutable from the axioms >>>>>>>>>>>>>> of PA.
On 1/24/2026 2:23 AM, Mikko wrote:Those sentences don't mean anything without
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout >>>>>>>>>>>>>>>>>>> PA. You can deny
the proof but you cannot perform what is meta- >>>>>>>>>>>>>>>>>>> provably impossible.
The meta-proof does not exist in the axioms of PA >>>>>>>>>>>>>>>>>> and that is the reason why an external truth in >>>>>>>>>>>>>>>>>> an external model cannot be proved internally in PA. >>>>>>>>>>>>>>>>>> All of these years it was only a mere conflation >>>>>>>>>>>>>>>>>> error.
It is perfectly clear which is which. But every proof >>>>>>>>>>>>>>>>> in PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x))) >>>>>>>>>>>>>>>
specificantions of a
language and a theory that gives them some meaning. >>>>>>>>>>>>>>
There are no notational convention that defines True, >>>>>>>>>>>>> False, and
WellFounded with two arguments. And you did not specify in >>>>>>>>>>>>> which
context your sentences are true or otherwise relevant. >>>>>>>>>>>>>
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define >>>>>>>>>>> what is true in PA?
I thought you said that PA had to be able to determine the >>>>>>>>>>> truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results >>>>>>> are related and there are much similarity in the proofs. Tarski did >>>>>>> not use Turing machines in the proof but a computability proof must >>>>>>> use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
Irrelevant. There is no "occurs_check" in Tarski's proof.
That would have no effet. Even if the metalanguage had an occcurs_check
it would not be necessary to use it in a proof.
It would only seem to have no effect because you
never bothered to understand what an occurs_check is.
That assumption is false.
Truth is computable because rCLmeaningful sentencerCY
is defined as rCLsentence with a well-founded
justification tree,rCY and evaluating any well-founded
tree always terminates. Anything else isnrCOt truth-apt.
That "bcause" is wrong. Whether a sentence has a well-founded
justifiation tree is not computable, especially for arithmetic
sentences.
But that does not alter the fact that an existence or non-existence
of a metalanguage feature that is not present in the justification
tree is irrelevant.
On 29/01/2026 15:57, olcott wrote:
On 1/29/2026 3:12 AM, Mikko wrote:
On 28/01/2026 15:49, olcott wrote:
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:
On 1/24/2026 2:23 AM, Mikko wrote:Those sentences don't mean anything without specificantions >>>>>>>>>>>>> of a
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout PA. >>>>>>>>>>>>>>>>> You can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA >>>>>>>>>>>>>>>> and that is the reason why an external truth in >>>>>>>>>>>>>>>> an external model cannot be proved internally in PA. >>>>>>>>>>>>>>>> All of these years it was only a mere conflation >>>>>>>>>>>>>>>> error.
It is perfectly clear which is which. But every proof in >>>>>>>>>>>>>>> PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x))) >>>>>>>>>>>>>
language and a theory that gives them some meaning.
In other word you do not understand standard notational >>>>>>>>>>>> conventions that define True for PA as provable from the >>>>>>>>>>>> axioms of PA and False for PA as refutable from the axioms >>>>>>>>>>>> of PA.
There are no notational convention that defines True, False, and >>>>>>>>>>> WellFounded with two arguments. And you did not specify in which >>>>>>>>>>> context your sentences are true or otherwise relevant.
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define what >>>>>>>>> is true in PA?
I thought you said that PA had to be able to determine the
truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results
are related and there are much similarity in the proofs. Tarski did
not use Turing machines in the proof but a computability proof must
use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
Irrelevant. There is no "occurs_check" in Tarski's proof.
If there was then there never would be a Tarski proof.
https://liarparadox.org/Tarski_247_248.pdf
Irrelevant. Tarski's proof is what it is and there is no "occurs_check" there.
On 1/31/2026 2:56 AM, Mikko wrote:
On 29/01/2026 15:57, olcott wrote:
On 1/29/2026 3:12 AM, Mikko wrote:
On 28/01/2026 15:49, olcott wrote:
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:In other word you do not understand standard notational >>>>>>>>>>>>> conventions that define True for PA as provable from the >>>>>>>>>>>>> axioms of PA and False for PA as refutable from the axioms >>>>>>>>>>>>> of PA.
On 1/24/2026 2:23 AM, Mikko wrote:Those sentences don't mean anything without
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout >>>>>>>>>>>>>>>>>> PA. You can deny
the proof but you cannot perform what is meta-provably >>>>>>>>>>>>>>>>>> impossible.
The meta-proof does not exist in the axioms of PA >>>>>>>>>>>>>>>>> and that is the reason why an external truth in >>>>>>>>>>>>>>>>> an external model cannot be proved internally in PA. >>>>>>>>>>>>>>>>> All of these years it was only a mere conflation >>>>>>>>>>>>>>>>> error.
It is perfectly clear which is which. But every proof in >>>>>>>>>>>>>>>> PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x )
reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x))) >>>>>>>>>>>>>>
specificantions of a
language and a theory that gives them some meaning. >>>>>>>>>>>>>
There are no notational convention that defines True, False, >>>>>>>>>>>> and
WellFounded with two arguments. And you did not specify in >>>>>>>>>>>> which
context your sentences are true or otherwise relevant. >>>>>>>>>>>>
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define >>>>>>>>>> what is true in PA?
I thought you said that PA had to be able to determine the >>>>>>>>>> truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results >>>>>> are related and there are much similarity in the proofs. Tarski did >>>>>> not use Turing machines in the proof but a computability proof must >>>>>> use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
Irrelevant. There is no "occurs_check" in Tarski's proof.
If there was then there never would be a Tarski proof.
https://liarparadox.org/Tarski_247_248.pdf
Irrelevant. Tarski's proof is what it is and there is no "occurs_check"
there.
Sure and a car that has a missing engine will always
be a car that will not run.
On 1/31/2026 2:41 AM, Mikko wrote:
On 30/01/2026 16:35, olcott wrote:
On 1/30/2026 3:34 AM, Mikko wrote:
On 29/01/2026 15:57, olcott wrote:
On 1/29/2026 3:12 AM, Mikko wrote:
On 28/01/2026 15:49, olcott wrote:
On 1/28/2026 3:54 AM, Mikko wrote:
On 27/01/2026 17:32, olcott wrote:
On 1/27/2026 2:17 AM, Mikko wrote:
On 26/01/2026 18:58, olcott wrote:
On 1/26/2026 10:45 AM, Richard Damon wrote:
On 1/26/26 10:22 AM, olcott wrote:
On 1/26/2026 6:55 AM, Mikko wrote:
On 25/01/2026 15:30, olcott wrote:
On 1/25/2026 5:24 AM, Mikko wrote:
On 24/01/2026 16:18, olcott wrote:In other word you do not understand standard notational >>>>>>>>>>>>>>> conventions that define True for PA as provable from the >>>>>>>>>>>>>>> axioms of PA and False for PA as refutable from the axioms >>>>>>>>>>>>>>> of PA.
On 1/24/2026 2:23 AM, Mikko wrote:Those sentences don't mean anything without
On 22/01/2026 18:47, olcott wrote:
On 1/22/2026 2:21 AM, Mikko wrote:
Anyway, what can be provven that way is true aboout >>>>>>>>>>>>>>>>>>>> PA. You can deny
the proof but you cannot perform what is meta- >>>>>>>>>>>>>>>>>>>> provably impossible.
The meta-proof does not exist in the axioms of PA >>>>>>>>>>>>>>>>>>> and that is the reason why an external truth in >>>>>>>>>>>>>>>>>>> an external model cannot be proved internally in PA. >>>>>>>>>>>>>>>>>>> All of these years it was only a mere conflation >>>>>>>>>>>>>>>>>>> error.
It is perfectly clear which is which. But every proof >>>>>>>>>>>>>>>>>> in PA is also
a proof in G||del's metatheory.
reCx ree PA (-a True(PA, x) rei PA reo-a x ) >>>>>>>>>>>>>>>>> reCx ree PA ( False(PA, x) rei PA reo -4x )
reCx ree PA ( -4WellFounded(PA, x) rei
-a-a-a-a-a-a-a-a-a (-4True(PA, x) reo (-4False(PA, x))) >>>>>>>>>>>>>>>>
specificantions of a
language and a theory that gives them some meaning. >>>>>>>>>>>>>>>
There are no notational convention that defines True, >>>>>>>>>>>>>> False, and
WellFounded with two arguments. And you did not specify in >>>>>>>>>>>>>> which
context your sentences are true or otherwise relevant. >>>>>>>>>>>>>>
rCLx is a single finite string representing
a PArCaformula, such as rCy2 + 3 = 5rCO.
True(PA, x), False(PA, x), and WellFounded(PA, x)
are metarCalevel unary predicates classifying
that formula by its provability in PA.rCY
In outher words, you ACCEPT that the meta level can define >>>>>>>>>>>> what is true in PA?
I thought you said that PA had to be able to determine the >>>>>>>>>>>> truth itself?
We need a meta-level truth predicate anchored
only in the axioms of PA itself and thus not
anchored in the standard model of arithmetic.
That predicate is not computable.
That was Tarski's mistake.
No, Tarski's proof is about a different problem, though the results >>>>>>>> are related and there are much similarity in the proofs. Tarski did >>>>>>>> not use Turing machines in the proof but a computability proof must >>>>>>>> use that.
Because you refuse to understand the underlying
details of what occurs_check means I cannot
explain to you how Tarski erred.
Irrelevant. There is no "occurs_check" in Tarski's proof.
That would have no effet. Even if the metalanguage had an occcurs_check >>>> it would not be necessary to use it in a proof.
It would only seem to have no effect because you
never bothered to understand what an occurs_check is.
That assumption is false.
So far you have conclusively proven that you
do not understand what an occurs_check is.
If you want to provide that you do know then
you must provide all of the correct details.
Merely claiming that my statement is false
is an assertion entirely bereft of supporting
reasoning thus inherently baseless.
Truth is computable because rCLmeaningful sentencerCY
is defined as rCLsentence with a well-founded
justification tree,rCY and evaluating any well-founded
tree always terminates. Anything else isnrCOt truth-apt.
That "bcause" is wrong. Whether a sentence has a well-founded
justifiation tree is not computable, especially for arithmetic
sentences.
My one half page of text explaining all of the key details
of my 28 years of work was completely validated by five
different LLM systems. proof theoretic semantics is correct
model theoretic semantics is profoundly wrong-headed.
Your ignorance of the details of well-founded proof theoretic
semantics makes your rebuttal baseless.
But that does not alter the fact that an existence or non-existence
of a metalanguage feature that is not present in the justification
tree is irrelevant.
An existence or non-existence of a metalanguage feature
is entirely anchored in a totally wrong-headed notion.
The only way that this can be seen is to become an expert
in well-founded proof theoretic semantics.
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