Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
This is best understood within the proof theoretic
notion of epistemic justification.
Truth as an Epistemic Notion --- Dag Prawitz (2011) https:// link.springer.com/article/10.1007/s11245-011-9107-6
On 9/22/2026 11:01 PM, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
This is best understood within the proof theoretic
notion of epistemic justification.
Truth as an Epistemic Notion --- Dag Prawitz (2011) https://
link.springer.com/article/10.1007/s11245-011-9107-6
Why do you restrict your proof to Pennsylvania?-a Does that mean if
you exit the state for any reason, your hypothesis becomes false?
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
On 22/09/2026 18:01, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
That is not specific to PA. That is what formal provability means
for every formal theory.
On 9/22/2026 10:32 AM, Johann 'Myrkraverk' Oskarsson wrote:
On 9/22/2026 11:01 PM, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
This is best understood within the proof theoretic
notion of epistemic justification.
Truth as an Epistemic Notion --- Dag Prawitz (2011) https://
link.springer.com/article/10.1007/s11245-011-9107-6
Why do you restrict your proof to Pennsylvania?-a Does that mean if
you exit the state for any reason, your hypothesis becomes false?
The axioms of Peano Arithmetic (PA) https://mathworld.wolfram.com/PeanosAxioms.html
On 9/23/2026 4:45 AM, Mikko wrote:
On 22/09/2026 18:01, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
That is not specific to PA. That is what formal provability means
for every formal theory.
If that was true then everyone would understand that
Wittgenstein is correct and G||del is wrong.
-a-a 'True in Russell's system' means, as was said:
-a-a proved in Russell's system; and 'false in Russell's
-a-a system' means: the opposite has been proved in
-a-a Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf
G||del's result is only achieved by going outside of Peano
Arithmetic into the standard model of Arithmetic.
Proof Theoretic Semantics does this differently:
-a-a What is the appropriate notion of truth for sentences
-a-a whose meanings are understood in epistemic terms
-a-a such as proof or ground for an assertion? It seems
-a-a that the truth of such sentences has to be identified
-a-a with the existence of proofs or grounds.
Truth as an Epistemic Notion --- Dag Prawitz (2011) https://link.springer.com/article/10.1007/s11245-011-9107-6
Provability in PA requires a sequence of syntactic
inference steps to remain totally within PA and not
leap outside of PA into any model of arithmetic.
-a-a 'True in Russell's system' means, as was said:
-a-a proved in Russell's system; and 'false in Russell's
-a-a system' means: the opposite has been proved in
-a-a Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf
G||del's result is only achieved by going outside of Peano
Arithmetic into the standard model of Arithmetic.
Proof Theoretic Semantics does this differently:
-a-a What is the appropriate notion of truth for sentences
-a-a whose meanings are understood in epistemic terms
-a-a such as proof or ground for an assertion? It seems
-a-a that the truth of such sentences has to be identified
-a-a with the existence of proofs or grounds.
Truth as an Epistemic Notion --- Dag Prawitz (2011) https://link.springer.com/article/10.1007/s11245-011-9107-6
On 23/09/2026 16:39, olcott wrote:
On 9/23/2026 4:45 AM, Mikko wrote:
On 22/09/2026 18:01, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
That is not specific to PA. That is what formal provability means
for every formal theory.
If that was true then everyone would understand that
Wittgenstein is correct and G||del is wrong.
That is clearly false. The meaning of the word "provability" (or amy
word) has no consequences beyond meanings of words.
The rest of the quoted message and my response to it are irrelevant
to my comment above.
-a-a-a 'True in Russell's system' means, as was said:
-a-a-a proved in Russell's system; and 'false in Russell's
-a-a-a system' means: the opposite has been proved in
-a-a-a Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf
With that defimition the words "true" and "false" lose a part of their
useful meanings. In order to say "proved in Russell's system" one can
say "proved in Russell's system". A statement proved in Russell's
system can also be called a "theorem of Russell's system".
G||del's result is only achieved by going outside of Peano
Arithmetic into the standard model of Arithmetic.
No, it is only necessary to go to some system where the results make
sense. G||del's most important results are not about numbers but about
formal theories. A map of Italy does not help if you have problems
with finding your way in Tokyo.
Proof Theoretic Semantics does this differently:
-a-a-a What is the appropriate notion of truth for sentences
-a-a-a whose meanings are understood in epistemic terms
-a-a-a such as proof or ground for an assertion? It seems
-a-a-a that the truth of such sentences has to be identified
-a-a-a with the existence of proofs or grounds.
Truth as an Epistemic Notion --- Dag Prawitz (2011)
https://link.springer.com/article/10.1007/s11245-011-9107-6
Truth theoretic semantics is not the kind of semantics we usually want.
Many useful things can be done with no semantics at all.
On 9/24/2026 6:24 AM, Mikko wrote:
On 23/09/2026 16:39, olcott wrote:
On 9/23/2026 4:45 AM, Mikko wrote:
On 22/09/2026 18:01, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
That is not specific to PA. That is what formal provability means
for every formal theory.
If that was true then everyone would understand that
Wittgenstein is correct and G||del is wrong.
That is clearly false. The meaning of the word "provability" (or amy
word) has no consequences beyond meanings of words.
The rest of the quoted message and my response to it are irrelevant
to my comment above.
-a-a-a 'True in Russell's system' means, as was said:
-a-a-a proved in Russell's system; and 'false in Russell's
-a-a-a system' means: the opposite has been proved in
-a-a-a Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf
With that defimition the words "true" and "false" lose a part of their
useful meanings. In order to say "proved in Russell's system" one can
say "proved in Russell's system". A statement proved in Russell's
system can also be called a "theorem of Russell's system".
G||del's result is only achieved by going outside of Peano
Arithmetic into the standard model of Arithmetic.
No, it is only necessary to go to some system where the results make
sense. G||del's most important results are not about numbers but about
formal theories. A map of Italy does not help if you have problems
with finding your way in Tokyo.
Proof Theoretic Semantics does this differently:
-a-a-a What is the appropriate notion of truth for sentences
-a-a-a whose meanings are understood in epistemic terms
-a-a-a such as proof or ground for an assertion? It seems
-a-a-a that the truth of such sentences has to be identified
-a-a-a with the existence of proofs or grounds.
Truth as an Epistemic Notion --- Dag Prawitz (2011)
https://link.springer.com/article/10.1007/s11245-011-9107-6
Truth theoretic semantics is not the kind of semantics we usually want.
Many useful things can be done with no semantics at all.
"true on the basis of meaning expressed in language" is
reliably computable for the entire body of general knowledge.
There is no coherent sense of "incompleteness" or undecidability
within such a system.
On 27/09/2026 05:59, olcott wrote:
On 9/24/2026 6:24 AM, Mikko wrote:
On 23/09/2026 16:39, olcott wrote:
On 9/23/2026 4:45 AM, Mikko wrote:
On 22/09/2026 18:01, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
That is not specific to PA. That is what formal provability means
for every formal theory.
If that was true then everyone would understand that
Wittgenstein is correct and G||del is wrong.
That is clearly false. The meaning of the word "provability" (or amy
word) has no consequences beyond meanings of words.
The rest of the quoted message and my response to it are irrelevant
to my comment above.
-a-a-a 'True in Russell's system' means, as was said:
-a-a-a proved in Russell's system; and 'false in Russell's
-a-a-a system' means: the opposite has been proved in
-a-a-a Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf
With that defimition the words "true" and "false" lose a part of their
useful meanings. In order to say "proved in Russell's system" one can
say "proved in Russell's system". A statement proved in Russell's
system can also be called a "theorem of Russell's system".
G||del's result is only achieved by going outside of Peano
Arithmetic into the standard model of Arithmetic.
No, it is only necessary to go to some system where the results make
sense. G||del's most important results are not about numbers but about
formal theories. A map of Italy does not help if you have problems
with finding your way in Tokyo.
Proof Theoretic Semantics does this differently:
-a-a-a What is the appropriate notion of truth for sentences
-a-a-a whose meanings are understood in epistemic terms
-a-a-a such as proof or ground for an assertion? It seems
-a-a-a that the truth of such sentences has to be identified
-a-a-a with the existence of proofs or grounds.
Truth as an Epistemic Notion --- Dag Prawitz (2011)
https://link.springer.com/article/10.1007/s11245-011-9107-6
Truth theoretic semantics is not the kind of semantics we usually want.
Many useful things can be done with no semantics at all.
"true on the basis of meaning expressed in language" is
reliably computable for the entire body of general knowledge.
There is no coherent sense of "incompleteness" or undecidability
within such a system.
Forst of all, that is a change of topic, apparently in order to make
people to forget what you said earlier. WHich is in fact a good idea,
there is no need to remember your errors. But it is useful to remember
that most of what you say is erroneous.
Whether "true on the basis of meaning expressed in language" is
computable depends on a language. For an uninterpreted formal
language nothing in the language is true, so the computation is
trivial.
Anyway, "true on the basis of meaning expressed in language" does
not cover the truths we usually need to know. An example is the
whether tomorrow: it is not computable. A prediction can be computed
but sometimes the truth is different.
There is no complete method to compute about a sentence is some lanugage
of natural number arithmetic whther it is true about natural numbers.
It is not quite clear whether such sentence can be said to be "true
on the basis of meaning expressed in language".
More often a truth is the basis of an expression than an expression
of a truth.
On 9/28/2026 4:31 AM, Mikko wrote:
On 27/09/2026 05:59, olcott wrote:
On 9/24/2026 6:24 AM, Mikko wrote:
On 23/09/2026 16:39, olcott wrote:
On 9/23/2026 4:45 AM, Mikko wrote:
On 22/09/2026 18:01, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
That is not specific to PA. That is what formal provability means
for every formal theory.
If that was true then everyone would understand that
Wittgenstein is correct and G||del is wrong.
That is clearly false. The meaning of the word "provability" (or amy
word) has no consequences beyond meanings of words.
The rest of the quoted message and my response to it are irrelevant
to my comment above.
-a-a-a 'True in Russell's system' means, as was said:
-a-a-a proved in Russell's system; and 'false in Russell's
-a-a-a system' means: the opposite has been proved in
-a-a-a Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf
With that defimition the words "true" and "false" lose a part of their >>>> useful meanings. In order to say "proved in Russell's system" one can
say "proved in Russell's system". A statement proved in Russell's
system can also be called a "theorem of Russell's system".
G||del's result is only achieved by going outside of Peano
Arithmetic into the standard model of Arithmetic.
No, it is only necessary to go to some system where the results make
sense. G||del's most important results are not about numbers but about >>>> formal theories. A map of Italy does not help if you have problems
with finding your way in Tokyo.
Proof Theoretic Semantics does this differently:
-a-a-a What is the appropriate notion of truth for sentences
-a-a-a whose meanings are understood in epistemic terms
-a-a-a such as proof or ground for an assertion? It seems
-a-a-a that the truth of such sentences has to be identified
-a-a-a with the existence of proofs or grounds.
Truth as an Epistemic Notion --- Dag Prawitz (2011)
https://link.springer.com/article/10.1007/s11245-011-9107-6
Truth theoretic semantics is not the kind of semantics we usually want. >>>> Many useful things can be done with no semantics at all.
"true on the basis of meaning expressed in language" is
reliably computable for the entire body of general knowledge.
There is no coherent sense of "incompleteness" or undecidability
within such a system.
Forst of all, that is a change of topic, apparently in order to make
people to forget what you said earlier. WHich is in fact a good idea,
there is no need to remember your errors. But it is useful to remember
that most of what you say is erroneous.
Whether "true on the basis of meaning expressed in language" is
computable depends on a language. For an uninterpreted formal
language nothing in the language is true, so the computation is
trivial.
The body of general knowledge expressed in formalized English
can be formalized in CycL. That "cats are animals" is one
example of "true on the basis of meaning expressed in language"
Anyway, "true on the basis of meaning expressed in language" does
not cover the truths we usually need to know. An example is the
whether tomorrow: it is not computable. A prediction can be computed
but sometimes the truth is different.
The scope of the body of general knowledge that is
"true on the basis of meaning expressed in language"
Tomorrow's weather is outside of this scope.
There is no complete method to compute about a sentence is some lanugage
of natural number arithmetic whther it is true about natural numbers.
It is not quite clear whether such sentence can be said to be "true
on the basis of meaning expressed in language".
Proof theoretic semantics is the way that "true on the
basis of meaning expressed in language" has always
worked in actual reality.
Mathematical incompleteness and undecidability has always been
artificially contrived by choosing model theory as a foundation.
When we switch> to proof theoretic semantics all that we lose is "undecidability" and "incompleteness", they were
never inherently actually there.
More often a truth is the basis of an expression than an expression
of a truth.
On 28/09/2026 15:10, olcott wrote:
On 9/28/2026 4:31 AM, Mikko wrote:
On 27/09/2026 05:59, olcott wrote:
On 9/24/2026 6:24 AM, Mikko wrote:
On 23/09/2026 16:39, olcott wrote:
On 9/23/2026 4:45 AM, Mikko wrote:
On 22/09/2026 18:01, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
That is not specific to PA. That is what formal provability means >>>>>>> for every formal theory.
If that was true then everyone would understand that
Wittgenstein is correct and G||del is wrong.
That is clearly false. The meaning of the word "provability" (or amy >>>>> word) has no consequences beyond meanings of words.
The rest of the quoted message and my response to it are irrelevant
to my comment above.
-a-a-a 'True in Russell's system' means, as was said:
-a-a-a proved in Russell's system; and 'false in Russell's
-a-a-a system' means: the opposite has been proved in
-a-a-a Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf
With that defimition the words "true" and "false" lose a part of their >>>>> useful meanings. In order to say "proved in Russell's system" one can >>>>> say "proved in Russell's system". A statement proved in Russell's
system can also be called a "theorem of Russell's system".
G||del's result is only achieved by going outside of Peano
Arithmetic into the standard model of Arithmetic.
No, it is only necessary to go to some system where the results make >>>>> sense. G||del's most important results are not about numbers but about >>>>> formal theories. A map of Italy does not help if you have problems
with finding your way in Tokyo.
Proof Theoretic Semantics does this differently:
-a-a-a What is the appropriate notion of truth for sentences
-a-a-a whose meanings are understood in epistemic terms
-a-a-a such as proof or ground for an assertion? It seems
-a-a-a that the truth of such sentences has to be identified
-a-a-a with the existence of proofs or grounds.
Truth as an Epistemic Notion --- Dag Prawitz (2011)
https://link.springer.com/article/10.1007/s11245-011-9107-6
Truth theoretic semantics is not the kind of semantics we usually
want.
Many useful things can be done with no semantics at all.
"true on the basis of meaning expressed in language" is
reliably computable for the entire body of general knowledge.
There is no coherent sense of "incompleteness" or undecidability
within such a system.
Forst of all, that is a change of topic, apparently in order to make
people to forget what you said earlier. WHich is in fact a good idea,
there is no need to remember your errors. But it is useful to remember
that most of what you say is erroneous.
Whether "true on the basis of meaning expressed in language" is
computable depends on a language. For an uninterpreted formal
language nothing in the language is true, so the computation is
trivial.
The body of general knowledge expressed in formalized English
can be formalized in CycL. That "cats are animals" is one
example of "true on the basis of meaning expressed in language"
An ordinary dictionary like https://en.wiktionary.org/wiki/cat can
tell that cats are animals. If you want to get anybody interested
you need more impressive examples.
Anyway, "true on the basis of meaning expressed in language" does
not cover the truths we usually need to know. An example is the
whether tomorrow: it is not computable. A prediction can be computed
but sometimes the truth is different.
The scope of the body of general knowledge that is
"true on the basis of meaning expressed in language"
Tomorrow's weather is outside of this scope.
And so are most of the things that people might want to find out.
There is no complete method to compute about a sentence is some lanugage >>> of natural number arithmetic whther it is true about natural numbers.
It is not quite clear whether such sentence can be said to be "true
on the basis of meaning expressed in language".
Proof theoretic semantics is the way that "true on the
basis of meaning expressed in language" has always
worked in actual reality.
It works in the sense of never giving a false answer
(as long as
applied correctly) but not in the sense of giving the answers
people most often need.
Mathematical incompleteness and undecidability has always been
artificially contrived by choosing model theory as a foundation.
That is false. A surprising discovery is not "artificially contrived".
Undecidability is not related to model theory. It is a conseqeunce of
the meaning of the words expressed in language.
The term "incomplete" has several meanings. One is that a theory is incomplete if there is an undecidable sentence. This meaning is unreated
to model theory. Another meaning is that there is a sentence that is
true is some model but cannot be proven. If no models are considered
this latter meaning is not applicable.
When we switch> to proof theoretic semantics all that we lose is
"undecidability" and "incompleteness", they were
never inherently actually there.
There still is undecidability as it is a purely proof theoretic
concept.
Undecidable sentences don't became decidable by avoiding all
consideration of decidability.
--More often a truth is the basis of an expression than an expression
of a truth.
-- Mikko
No, Pete. Calling an undecidable sentence "out-of-scope" does not make undecidability disappear.
For a formal theory T, "G is undecidable in T" simply means
-a-a T |-/- G-a-a-a and-a-a-a T |-/- ~G
That is a statement entirely about proofs. No model theory is involved.
And G is not "out-of-scope": it is a perfectly well-formed sentence of
the language of T. What is out-of-scope, apparently, is merely anything
for which your proposed semantics fails to supply an answer.
If you define your scope as--
-a-a { A : T proves A or T proves ~A }
then of course everything "in scope" is decidable.
But that does not eliminate incompleteness. It eliminates the incomplete cases from your definition of "scope".
You can make a hospital 100% successful by defining "patient" to mean "person we successfully cured", too.
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