Interpreting incompleteness as a gap between mathematical truth and
proof depends on truth-conditional semantics; once this is replaced by proof-theoretic semantics a framework not yet sufficiently developed at
the time of Gödel’s proof the notion of such a gap becomes unfounded.
On 1/14/2026 3:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth and
proof depends on truth-conditional semantics; once this is replaced by
proof-theoretic semantics a framework not yet sufficiently developed
at the time of Gödel’s proof the notion of such a gap becomes unfounded. >>
Gödel and Turing incompleteness results expose the limits of
denotational and truth-conditional semantics, not limits of proof or computation per se. When meaning is grounded operationally or proof- theoretically, the problematic self-referential constructions are
rejected as semantically unfounded rather than treated as determinate
but unknowable facts.
Interpreting incompleteness as a gap between mathematical truth and
proof depends on truth-conditional semantics; once this is replaced by proof-theoretic semantics a framework not yet sufficiently developed at
the time of Gödel’s proof the notion of such a gap becomes unfounded.
On 1/14/26 5:11 PM, olcott wrote:
On 1/14/2026 3:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth and
proof depends on truth-conditional semantics; once this is replaced
by proof-theoretic semantics a framework not yet sufficiently
developed at the time of Gödel’s proof the notion of such a gap
becomes unfounded.
Gödel and Turing incompleteness results expose the limits of
denotational and truth-conditional semantics, not limits of proof or
computation per se. When meaning is grounded operationally or proof-
theoretically, the problematic self-referential constructions are
rejected as semantically unfounded rather than treated as determinate
but unknowable facts.
The problem is that "Computation" relys on truth-conditional semantics,
as the behavior of a program *IS* what it actually does, not what you
can generically prove about it.
I guess you are giving up on your idea of making "Truth Compuational",--
as by your logic you can't imbue meaning to things, and thus you can't actually write even a proof checker for a system, let alone a truth
checker.
On 1/14/26 4:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth and
proof depends on truth-conditional semantics; once this is replaced by
proof-theoretic semantics a framework not yet sufficiently developed
at the time of Gödel’s proof the notion of such a gap becomes unfounded. >>
But that isn't what Incompleteness is about, so you are just showing
your ignorance of the meaning of words.
You can't just "change" the meaning of truth in a system.
I guess your problme is you don't understand what Truth actually is.--
YOUR "gap" in understand is enormous.
On 1/14/2026 9:44 PM, Richard Damon wrote:
On 1/14/26 4:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth and
proof depends on truth-conditional semantics; once this is replaced
by proof-theoretic semantics a framework not yet sufficiently
developed at the time of Gödel’s proof the notion of such a gap
becomes unfounded.
But that isn't what Incompleteness is about, so you are just showing
your ignorance of the meaning of words.
You can't just "change" the meaning of truth in a system.
Yet that is what happens when you replace the foundational basis
from truth-conditional semantics to proof-theoretic semantics.
On 1/14/2026 9:44 PM, Richard Damon wrote:
On 1/14/26 5:11 PM, olcott wrote:
On 1/14/2026 3:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth and
proof depends on truth-conditional semantics; once this is replaced
by proof-theoretic semantics a framework not yet sufficiently
developed at the time of Gödel’s proof the notion of such a gap
becomes unfounded.
Gödel and Turing incompleteness results expose the limits of
denotational and truth-conditional semantics, not limits of proof or
computation per se. When meaning is grounded operationally or proof-
theoretically, the problematic self-referential constructions are
rejected as semantically unfounded rather than treated as determinate
but unknowable facts.
The problem is that "Computation" relys on truth-conditional
semantics, as the behavior of a program *IS* what it actually does,
not what you can generically prove about it.
Proof in terms of the behavior of DD simulated by HHH.
I guess you are giving up on your idea of making "Truth Compuational",
as by your logic you can't imbue meaning to things, and thus you can't
actually write even a proof checker for a system, let alone a truth
checker.
On 1/14/2026 9:44 PM, Richard Damon wrote:
On 1/14/26 4:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth and
proof depends on truth-conditional semantics; once this is replaced
by proof-theoretic semantics a framework not yet sufficiently
developed at the time of Gödel’s proof the notion of such a gap
becomes unfounded.
But that isn't what Incompleteness is about, so you are just showing
your ignorance of the meaning of words.
You can't just "change" the meaning of truth in a system.
Yet that is what happens when you replace the foundational basis
from truth-conditional semantics to proof-theoretic semantics.
I guess your problme is you don't understand what Truth actually is.
YOUR "gap" in understand is enormous.
On 15/01/2026 07:30, olcott wrote:
On 1/14/2026 9:44 PM, Richard Damon wrote:
On 1/14/26 4:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth and
proof depends on truth-conditional semantics; once this is replaced
by proof-theoretic semantics a framework not yet sufficiently
developed at the time of Gödel’s proof the notion of such a gap
becomes unfounded.
But that isn't what Incompleteness is about, so you are just showing
your ignorance of the meaning of words.
You can't just "change" the meaning of truth in a system.
Yet that is what happens when you replace the foundational basis
from truth-conditional semantics to proof-theoretic semantics.
Gödel constructed a sentence that is correct by the rules of first
order Peano arithmetic
but neither a theorem nor the negaition of
a theorem of Peano artihmetic. Being a theorem does not depend on
semantics, only on the existence of a syntatically valid proof.
Gödel's sentence can be interpreted as a perfiectly valid game.
That at the start of the game neither player can choose a strategy
that ensures the winning does not invalidate the game.
On 1/15/26 12:27 AM, olcott wrote:Proof-theoretic semantics proves that I have been
On 1/14/2026 9:44 PM, Richard Damon wrote:
On 1/14/26 5:11 PM, olcott wrote:
On 1/14/2026 3:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth and >>>>> proof depends on truth-conditional semantics; once this is replaced >>>>> by proof-theoretic semantics a framework not yet sufficiently
developed at the time of Gödel’s proof the notion of such a gap
becomes unfounded.
Gödel and Turing incompleteness results expose the limits of
denotational and truth-conditional semantics, not limits of proof or
computation per se. When meaning is grounded operationally or proof-
theoretically, the problematic self-referential constructions are
rejected as semantically unfounded rather than treated as
determinate but unknowable facts.
The problem is that "Computation" relys on truth-conditional
semantics, as the behavior of a program *IS* what it actually does,
not what you can generically prove about it.
Proof in terms of the behavior of DD simulated by HHH.
Since your HHH doesn't correctly simulate DD,
On 1/15/26 12:30 AM, olcott wrote:
On 1/14/2026 9:44 PM, Richard Damon wrote:
On 1/14/26 4:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth and
proof depends on truth-conditional semantics; once this is replaced
by proof-theoretic semantics a framework not yet sufficiently
developed at the time of Gödel’s proof the notion of such a gap
becomes unfounded.
But that isn't what Incompleteness is about, so you are just showing
your ignorance of the meaning of words.
You can't just "change" the meaning of truth in a system.
Yet that is what happens when you replace the foundational basis
from truth-conditional semantics to proof-theoretic semantics.
Which isn't allowed.
You don't seem to understand that "changing the basis" means you have a different system.
On 1/15/2026 5:50 AM, Richard Damon wrote:
On 1/15/26 12:27 AM, olcott wrote:Proof-theoretic semantics proves that I have been
On 1/14/2026 9:44 PM, Richard Damon wrote:
On 1/14/26 5:11 PM, olcott wrote:
On 1/14/2026 3:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth
and proof depends on truth-conditional semantics; once this is
replaced by proof-theoretic semantics a framework not yet
sufficiently developed at the time of Gödel’s proof the notion of >>>>>> such a gap becomes unfounded.
Gödel and Turing incompleteness results expose the limits of
denotational and truth-conditional semantics, not limits of proof
or computation per se. When meaning is grounded operationally or
proof- theoretically, the problematic self-referential
constructions are rejected as semantically unfounded rather than
treated as determinate but unknowable facts.
The problem is that "Computation" relys on truth-conditional
semantics, as the behavior of a program *IS* what it actually does,
not what you can generically prove about it.
Proof in terms of the behavior of DD simulated by HHH.
Since your HHH doesn't correctly simulate DD,
correct all along.
On 1/15/2026 5:50 AM, Richard Damon wrote:
On 1/15/26 12:30 AM, olcott wrote:
On 1/14/2026 9:44 PM, Richard Damon wrote:
On 1/14/26 4:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth and >>>>> proof depends on truth-conditional semantics; once this is replaced >>>>> by proof-theoretic semantics a framework not yet sufficiently
developed at the time of Gödel’s proof the notion of such a gap
becomes unfounded.
But that isn't what Incompleteness is about, so you are just showing
your ignorance of the meaning of words.
You can't just "change" the meaning of truth in a system.
Yet that is what happens when you replace the foundational basis
from truth-conditional semantics to proof-theoretic semantics.
Which isn't allowed.
You don't seem to understand that "changing the basis" means you have
a different system.
Yes just like you always said that I needed.
A formal system such that
"true on the basis of meaning expressed in language"
is always reliably computable.
On 1/15/26 6:43 PM, olcott wrote:
On 1/15/2026 5:50 AM, Richard Damon wrote:
On 1/15/26 12:27 AM, olcott wrote:Proof-theoretic semantics proves that I have been
On 1/14/2026 9:44 PM, Richard Damon wrote:
On 1/14/26 5:11 PM, olcott wrote:
On 1/14/2026 3:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth >>>>>>> and proof depends on truth-conditional semantics; once this is
replaced by proof-theoretic semantics a framework not yet
sufficiently developed at the time of Gödel’s proof the notion of >>>>>>> such a gap becomes unfounded.
Gödel and Turing incompleteness results expose the limits of
denotational and truth-conditional semantics, not limits of proof >>>>>> or computation per se. When meaning is grounded operationally or
proof- theoretically, the problematic self-referential
constructions are rejected as semantically unfounded rather than
treated as determinate but unknowable facts.
The problem is that "Computation" relys on truth-conditional
semantics, as the behavior of a program *IS* what it actually does, >>>>> not what you can generically prove about it.
Proof in terms of the behavior of DD simulated by HHH.
Since your HHH doesn't correctly simulate DD,
correct all along.
Nope, only in a system that USES proof-theoretic semantics, which don't
meet the requirements for the systems that Godel uses.
They don't even meet the requirements for your goal, as such systems can
not encode human knowledge, as everything we know about the real world
just violates the requrements of needing to be derived from the axioms
of the system, as real-world knowledge isn't based on an axiomatic system.
On 1/15/26 6:45 PM, olcott wrote:
On 1/15/2026 5:50 AM, Richard Damon wrote:
On 1/15/26 12:30 AM, olcott wrote:
On 1/14/2026 9:44 PM, Richard Damon wrote:
On 1/14/26 4:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth
and proof depends on truth-conditional semantics; once this is
replaced by proof-theoretic semantics a framework not yet
sufficiently developed at the time of Gödel’s proof the notion of >>>>>> such a gap becomes unfounded.
But that isn't what Incompleteness is about, so you are just
showing your ignorance of the meaning of words.
You can't just "change" the meaning of truth in a system.
Yet that is what happens when you replace the foundational basis
from truth-conditional semantics to proof-theoretic semantics.
Which isn't allowed.
You don't seem to understand that "changing the basis" means you have
a different system.
Yes just like you always said that I needed.
A formal system such that
"true on the basis of meaning expressed in language"
is always reliably computable.
So, go ahead and try to do it.
The problem is your system now can't encode "all human knowledge", as
that doesn't work in the system.
The problem is that the knowledge is based on logic that isn't
compatible with you logic system.
All you could do is try to encode the human knowledge that is still true under the restrictions, which can't include details about the real
world, as that is inherently truth-conditional, as we don't know the "axioms" the world works on.
Thus, your "all human knowledge" just fails to be available.
On 1/15/2026 9:28 PM, Richard Damon wrote:
On 1/15/26 6:43 PM, olcott wrote:
On 1/15/2026 5:50 AM, Richard Damon wrote:
On 1/15/26 12:27 AM, olcott wrote:Proof-theoretic semantics proves that I have been
On 1/14/2026 9:44 PM, Richard Damon wrote:
On 1/14/26 5:11 PM, olcott wrote:
On 1/14/2026 3:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth >>>>>>>> and proof depends on truth-conditional semantics; once this is >>>>>>>> replaced by proof-theoretic semantics a framework not yet
sufficiently developed at the time of Gödel’s proof the notion >>>>>>>> of such a gap becomes unfounded.
Gödel and Turing incompleteness results expose the limits of
denotational and truth-conditional semantics, not limits of proof >>>>>>> or computation per se. When meaning is grounded operationally or >>>>>>> proof- theoretically, the problematic self-referential
constructions are rejected as semantically unfounded rather than >>>>>>> treated as determinate but unknowable facts.
The problem is that "Computation" relys on truth-conditional
semantics, as the behavior of a program *IS* what it actually
does, not what you can generically prove about it.
Proof in terms of the behavior of DD simulated by HHH.
Since your HHH doesn't correctly simulate DD,
correct all along.
Nope, only in a system that USES proof-theoretic semantics, which
don't meet the requirements for the systems that Godel uses.
It is the same ∀x ∈ T ((True(T, x) ≡ (T ⊢ x))
that I have been talking about for years except that
it is now grounded in well-founded proof‑theoretic
semantics.
*That exactly maps to this*
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
Proof‑theoretic semantics didn't exist when Gödel
wrote his paper.
They don't even meet the requirements for your goal, as such systems
can not encode human knowledge, as everything we know about the real
world just violates the requrements of needing to be derived from the
axioms of the system, as real-world knowledge isn't based on an
axiomatic system.
everything we know about the real world
is encoded as a finite set of atomic facts
that ARE the Haskell Curry Axioms:
Thus, given {T}, an elementary theorem
is an elementary statement which is true.
On 1/15/2026 9:28 PM, Richard Damon wrote:
On 1/15/26 6:45 PM, olcott wrote:
On 1/15/2026 5:50 AM, Richard Damon wrote:
On 1/15/26 12:30 AM, olcott wrote:
On 1/14/2026 9:44 PM, Richard Damon wrote:
On 1/14/26 4:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth >>>>>>> and proof depends on truth-conditional semantics; once this is
replaced by proof-theoretic semantics a framework not yet
sufficiently developed at the time of Gödel’s proof the notion of >>>>>>> such a gap becomes unfounded.
But that isn't what Incompleteness is about, so you are just
showing your ignorance of the meaning of words.
You can't just "change" the meaning of truth in a system.
Yet that is what happens when you replace the foundational basis
from truth-conditional semantics to proof-theoretic semantics.
Which isn't allowed.
You don't seem to understand that "changing the basis" means you
have a different system.
Yes just like you always said that I needed.
A formal system such that
"true on the basis of meaning expressed in language"
is always reliably computable.
So, go ahead and try to do it.
The problem is your system now can't encode "all human knowledge", as
that doesn't work in the system.
We start with the smaller goal a defining PA without
Gödel Incompleteness.
The problem is that the knowledge is based on logic that isn't
compatible with you logic system.
All you could do is try to encode the human knowledge that is still
true under the restrictions, which can't include details about the
real world, as that is inherently truth-conditional, as we don't know
the "axioms" the world works on.
Thus, your "all human knowledge" just fails to be available.
Actual "true on the basis of meaning expressed in language"
has always been proof-theoretic semantics.
On 1/15/26 11:20 PM, olcott wrote:
On 1/15/2026 9:28 PM, Richard Damon wrote:
On 1/15/26 6:43 PM, olcott wrote:
On 1/15/2026 5:50 AM, Richard Damon wrote:
On 1/15/26 12:27 AM, olcott wrote:Proof-theoretic semantics proves that I have been
On 1/14/2026 9:44 PM, Richard Damon wrote:
On 1/14/26 5:11 PM, olcott wrote:
On 1/14/2026 3:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth >>>>>>>> and proof depends on truth-conditional semantics; once this is >>>>>>>> replaced by proof-theoretic semantics a framework not yet
sufficiently developed at the time of Gödel’s proof the notion >>>>>>>> of such a gap becomes unfounded.
Gödel and Turing incompleteness results expose the limits of >>>>>>> denotational and truth-conditional semantics, not limits of proof >>>>>>> or computation per se. When meaning is grounded operationally or >>>>>>> proof- theoretically, the problematic self-referential
constructions are rejected as semantically unfounded rather than >>>>>>> treated as determinate but unknowable facts.
The problem is that "Computation" relys on truth-conditional
semantics, as the behavior of a program *IS* what it actually
does, not what you can generically prove about it.
Proof in terms of the behavior of DD simulated by HHH.
Since your HHH doesn't correctly simulate DD,
correct all along.
Nope, only in a system that USES proof-theoretic semantics, which
don't meet the requirements for the systems that Godel uses.
It is the same ∀x ∈ T ((True(T, x) ≡ (T ⊢ x))
that I have been talking about for years except that
it is now grounded in well-founded proof‑theoretic
semantics.
Right, which just doesn't work in the system.
*That exactly maps to this*
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
Nope. The problem is you don't understand what a "Finite String Transformation rule" is.
Proof‑theoretic semantics didn't exist when Gödel
wrote his paper.
So, that means they don't apply to the basic system.
Remember, the context of the system, which includes its "semantics" are DEFINED by the system. IT COULDN'T have meant something that wasn't
defined yet.
They don't even meet the requirements for your goal, as such systems
can not encode human knowledge, as everything we know about the real
world just violates the requrements of needing to be derived from the
axioms of the system, as real-world knowledge isn't based on an
axiomatic system.
everything we know about the real world
is encoded as a finite set of atomic facts
that ARE the Haskell Curry Axioms:
Thus, given {T}, an elementary theorem
is an elementary statement which is true.
But we don't know actual "facts" about the real world, just have a set
of observations and most likely explanations.
In other words, your system is based on the ASSUMPTION that we are 100% correct in our imperical deductions.
I could ask, WHICH set of "facts" are you going to take?
The ones that show the earth is the center of the universe? Those WERE
the best conclusions at one point.
All you are doing is showing you fundamentally don't understand what
"truth" is.
Or what "Formal Logic" is.
Richard Damon <Richard@Damon-Family.org> posted:
On 1/15/26 11:20 PM, olcott wrote:
On 1/15/2026 9:28 PM, Richard Damon wrote:
On 1/15/26 6:43 PM, olcott wrote:
On 1/15/2026 5:50 AM, Richard Damon wrote:
On 1/15/26 12:27 AM, olcott wrote:Proof-theoretic semantics proves that I have been
On 1/14/2026 9:44 PM, Richard Damon wrote:
On 1/14/26 5:11 PM, olcott wrote:
On 1/14/2026 3:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth >>>>>>>>>> and proof depends on truth-conditional semantics; once this is >>>>>>>>>> replaced by proof-theoretic semantics a framework not yet
sufficiently developed at the time of Gödel’s proof the notion >>>>>>>>>> of such a gap becomes unfounded.
Gödel and Turing incompleteness results expose the limits of >>>>>>>>> denotational and truth-conditional semantics, not limits of proof >>>>>>>>> or computation per se. When meaning is grounded operationally or >>>>>>>>> proof- theoretically, the problematic self-referential
constructions are rejected as semantically unfounded rather than >>>>>>>>> treated as determinate but unknowable facts.
The problem is that "Computation" relys on truth-conditional
semantics, as the behavior of a program *IS* what it actually
does, not what you can generically prove about it.
Proof in terms of the behavior of DD simulated by HHH.
Since your HHH doesn't correctly simulate DD,
correct all along.
Nope, only in a system that USES proof-theoretic semantics, which
don't meet the requirements for the systems that Godel uses.
It is the same ∀x ∈ T ((True(T, x) ≡ (T ⊢ x))
that I have been talking about for years except that
it is now grounded in well-founded proof‑theoretic
semantics.
Right, which just doesn't work in the system.
*That exactly maps to this*
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
Nope. The problem is you don't understand what a "Finite String
Transformation rule" is.
Proof‑theoretic semantics didn't exist when Gödel
wrote his paper.
So, that means they don't apply to the basic system.
Remember, the context of the system, which includes its "semantics" are
DEFINED by the system. IT COULDN'T have meant something that wasn't
defined yet.
They don't even meet the requirements for your goal, as such systems
can not encode human knowledge, as everything we know about the real
world just violates the requrements of needing to be derived from the
axioms of the system, as real-world knowledge isn't based on an
axiomatic system.
everything we know about the real world
is encoded as a finite set of atomic facts
that ARE the Haskell Curry Axioms:
Thus, given {T}, an elementary theorem
is an elementary statement which is true.
But we don't know actual "facts" about the real world, just have a set
of observations and most likely explanations.
In other words, your system is based on the ASSUMPTION that we are 100%
correct in our imperical deductions.
I could ask, WHICH set of "facts" are you going to take?
The ones that show the earth is the center of the universe? Those WERE
the best conclusions at one point.
All you are doing is showing you fundamentally don't understand what
"truth" is.
Or what "Formal Logic" is.
The problem I have wih Pete Olcott's posts over the past 20 years at least is that he doesn't have a clue what a human language is.
I don't mind if he wants to bore readers of comp.theory, sci.logic, sci.math, and comp.ai.philosophy with his stuff (though others may), but can he not drop
sci.lang from the list?
In any case it's rarely acceptable to send the sama message to more than three
groups (even as many as that).
On 1/16/2026 12:12 PM, athel.cb@gmail.com wrote:
Richard Damon <Richard@Damon-Family.org> posted:
On 1/15/26 11:20 PM, olcott wrote:
On 1/15/2026 9:28 PM, Richard Damon wrote:
On 1/15/26 6:43 PM, olcott wrote:
On 1/15/2026 5:50 AM, Richard Damon wrote:
On 1/15/26 12:27 AM, olcott wrote:Proof-theoretic semantics proves that I have been
On 1/14/2026 9:44 PM, Richard Damon wrote:
On 1/14/26 5:11 PM, olcott wrote:
On 1/14/2026 3:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth >>>>>>>>>>> and proof depends on truth-conditional semantics; once this is >>>>>>>>>>> replaced by proof-theoretic semantics a framework not yet >>>>>>>>>>> sufficiently developed at the time of Gödel’s proof the notion >>>>>>>>>>> of such a gap becomes unfounded.
Gödel and Turing incompleteness results expose the limits of >>>>>>>>>> denotational and truth-conditional semantics, not limits of proof >>>>>>>>>> or computation per se. When meaning is grounded operationally or >>>>>>>>>> proof- theoretically, the problematic self-referential
constructions are rejected as semantically unfounded rather than >>>>>>>>>> treated as determinate but unknowable facts.
The problem is that "Computation" relys on truth-conditional >>>>>>>>> semantics, as the behavior of a program *IS* what it actually >>>>>>>>> does, not what you can generically prove about it.
Proof in terms of the behavior of DD simulated by HHH.
Since your HHH doesn't correctly simulate DD,
correct all along.
Nope, only in a system that USES proof-theoretic semantics, which
don't meet the requirements for the systems that Godel uses.
It is the same ∀x ∈ T ((True(T, x) ≡ (T ⊢ x))
that I have been talking about for years except that
it is now grounded in well-founded proof‑theoretic
semantics.
Right, which just doesn't work in the system.
*That exactly maps to this*
All deciders essentially: Transform finite string
inputs by finite string transformation rules into
{Accept, Reject} values.
Nope. The problem is you don't understand what a "Finite String
Transformation rule" is.
Proof‑theoretic semantics didn't exist when Gödel
wrote his paper.
So, that means they don't apply to the basic system.
Remember, the context of the system, which includes its "semantics" are
DEFINED by the system. IT COULDN'T have meant something that wasn't
defined yet.
They don't even meet the requirements for your goal, as such systems >>>>> can not encode human knowledge, as everything we know about the real >>>>> world just violates the requrements of needing to be derived from the >>>>> axioms of the system, as real-world knowledge isn't based on an
axiomatic system.
everything we know about the real world
is encoded as a finite set of atomic facts
that ARE the Haskell Curry Axioms:
Thus, given {T}, an elementary theorem
is an elementary statement which is true.
But we don't know actual "facts" about the real world, just have a set
of observations and most likely explanations.
In other words, your system is based on the ASSUMPTION that we are 100%
correct in our imperical deductions.
I could ask, WHICH set of "facts" are you going to take?
The ones that show the earth is the center of the universe? Those WERE
the best conclusions at one point.
All you are doing is showing you fundamentally don't understand what
"truth" is.
Or what "Formal Logic" is.
The problem I have wih Pete Olcott's posts over the past 20 years at
least is
that he doesn't have a clue what a human language is.
I don't mind if he wants to bore readers of comp.theory, sci.logic,
sci.math,
and comp.ai.philosophy with his stuff (though others may), but can he
not drop
sci.lang from the list?
In any case it's rarely acceptable to send the sama message to more
than three
groups (even as many as that).
The actual problem on this aspect is that most
everyone in the world rejects even the idea of
Montague Semantics before having any idea what it is.
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