On 1/9/23 10:19 PM, olcott wrote:
The set of (analytical) expressions of (formal or natural language) have
a complete semantic connection to their truth maker axioms otherwise
they are simply untrue. Copyright 2022 PL Olcott
Right, but such a connection can be based on an INFINTE number of
Previously philosophers were trying to define truth maker for analytical
truth and empirical truth at the same time and in the same way.
Nope. Just shows you are not understanding how logic works.
This much is agreed: “x makes it true that p” is a construction that
signifies, if it signifies anything at all, a relation borne to a truth-
bearer by something else, a truth-maker. But it isn’t generally agreed
what that something else might be, or what truth-bearers are, or what
the character might be of the relationship that holds, if it does,
between them, or even whether such a relationship ever does hold.
https://plato.stanford.edu/entries/truthmakers/
But note, that the statement x -> y is NOT a assertion that x MAKES y
true, but that the Truth of x proves that Y is true.
It is NOT a statement about "Causation", in fact, it is more a statement about sub-sets of models that might exist.
The statement x -> y means that the set of possible conditions of truth values of all statements where x is true, is a subset of all the
possible conditions of truth values of all statements where y is true.
Thus, if we are in a condition where x is true, we know that y must also
be true.
*This means that Wittgenstein is correct*
'True in Russell's system' means, as was said: proved in Russell's
system; and 'false in Russell's system' means:the opposite has been
proved in Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf]
Nope, not unless you are redefinig "Proof" to include an infinite set of connections, at which point you are in a totally new language.
Nope.
If P is unprovable in Russell's system then P is simply untrue in
Russell's system.
If you hold to this, as has been pointed out, this means, unless you
system is very small, that you can't talk about statements you haven't already proven as you don't know if they are Truth Beares.
The Provablity of statements is not a Truth Bearr until you have proven
that it is.
On 1/9/2023 10:01 PM, Richard Damon wrote:
On 1/9/23 10:19 PM, olcott wrote:
The set of (analytical) expressions of (formal or natural language) have >>> a complete semantic connection to their truth maker axioms otherwise
they are simply untrue. Copyright 2022 PL Olcott
Right, but such a connection can be based on an INFINTE number of
Mathematicians and logicians make sure to ignore the philosophical
foundation of these things. or we would never get this:
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
Previously philosophers were trying to define truth maker for analytical >>> truth and empirical truth at the same time and in the same way.
Nope. Just shows you are not understanding how logic works.
This much is agreed: “x makes it true that p” is a construction that >>> signifies, if it signifies anything at all, a relation borne to a truth- >>> bearer by something else, a truth-maker. But it isn’t generally agreed >>> what that something else might be, or what truth-bearers are, or what
the character might be of the relationship that holds, if it does,
between them, or even whether such a relationship ever does hold.
https://plato.stanford.edu/entries/truthmakers/
But note, that the statement x -> y is NOT a assertion that x MAKES y
true, but that the Truth of x proves that Y is true.
x ⊨ y Aristotle's syllogism required a semantic connection based on semantic categories.
https://en.wikipedia.org/wiki/Syllogism#Basic_structure
It is NOT a statement about "Causation", in fact, it is more a
statement about sub-sets of models that might exist.
https://en.wikipedia.org/wiki/Logical_consequence#Semantic_consequence
In other words there is a semantic connection form an expression of
language to its truth maker axioms.
The statement x -> y means that the set of possible conditions of
truth values of all statements where x is true, is a subset of all the
possible conditions of truth values of all statements where y is true.
Thus, if we are in a condition where x is true, we know that y must
also be true.
*This means that Wittgenstein is correct*
'True in Russell's system' means, as was said: proved in Russell's
system; and 'false in Russell's system' means:the opposite has been
proved in Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf]
Nope, not unless you are redefinig "Proof" to include an infinite set
of connections, at which point you are in a totally new language.
Formal systems require finite proofs to truth or the expression is
untrue in the formal system. Unless and until there is a connection to
the axioms of the system the expression remains untrue in the system.
Curry calls these axioms: "elementary theorems of::T"
A theory (over F) is defined as a conceptual class of these elementary statements. Let::T be such a theory. Then the elementary statements
which belong to ::T we shall call the elementary theorems of::T; we also
say that these elementary statements are true for::T. Thus, given ::T,
an elementary theorem is an elementary statement which is true. A theory
is thus a way of picking out from the statements of F a certain subclass
of true statements. https://www.liarparadox.org/Haskell_Curry_45.pdf
Nope.
If P is unprovable in Russell's system then P is simply untrue in
Russell's system.
Yup. That mathematicians and logicians do not bother to pay attention to
the philosophical foundations of these things leads them astray. They
simply follow their learned-by-rote never realizing (or even caring)
that they are incoherent.
If you hold to this, as has been pointed out, this means, unless you
system is very small, that you can't talk about statements you haven't
already proven as you don't know if they are Truth Beares.
The Provablity of statements is not a Truth Bearr until you have
proven that it is.
Curry agrees that the systems define "elementary theorems of::T" that
anchor the notion of truth in these systems, thus if there is no
connection from an expression to these anchors then it remains untrue.
On 1/9/23 11:50 PM, olcott wrote:
On 1/9/2023 10:01 PM, Richard Damon wrote:
On 1/9/23 10:19 PM, olcott wrote:
The set of (analytical) expressions of (formal or natural language)
have
a complete semantic connection to their truth maker axioms otherwise
they are simply untrue. Copyright 2022 PL Olcott
Right, but such a connection can be based on an INFINTE number of
Mathematicians and logicians make sure to ignore the philosophical
foundation of these things. or we would never get this:
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
Right, a system is incomplete if there exist a statement (which has a
Truth Value) but that statment can neither be proven or disproven in T.
Alternatively, a system is incomplete if there exists a TRUE statement
which can not be proven in T
(the disproven half of the above becoming a
seperate piece as if the statement is false (not just "untrue") then we
can form the negation of the statement and not be able to prove that one.
Previously philosophers were trying to define truth maker for
analytical
truth and empirical truth at the same time and in the same way.
Nope. Just shows you are not understanding how logic works.
This much is agreed: “x makes it true that p” is a construction that >>>> signifies, if it signifies anything at all, a relation borne to a
truth-
bearer by something else, a truth-maker. But it isn’t generally agreed >>>> what that something else might be, or what truth-bearers are, or what
the character might be of the relationship that holds, if it does,
between them, or even whether such a relationship ever does hold.
https://plato.stanford.edu/entries/truthmakers/
But note, that the statement x -> y is NOT a assertion that x MAKES y
true, but that the Truth of x proves that Y is true.
x ⊨ y Aristotle's syllogism required a semantic connection based on
semantic categories.
https://en.wikipedia.org/wiki/Syllogism#Basic_structure
And Aristotle's logic system is only for CATEGORICAL statements, and
thus only a first order logic system.
Note, your "Semantic" connection here comes out as a neccesary condition based on you being in CATEGORICAL logic.
Such a system can not express the required logic to create a full
description of the Natural Numbers.
It is NOT a statement about "Causation", in fact, it is more a
statement about sub-sets of models that might exist.
https://en.wikipedia.org/wiki/Logical_consequence#Semantic_consequence
In other words there is a semantic connection form an expression of
language to its truth maker axioms.
(quoting and changing some symbols to make typable)
A Formula A is a semantic consequence within a formal system FS of a set
of statements L, if and only if there is no model M in which all of L
are True and A is false.
On 1/10/2023 6:56 AM, Richard Damon wrote:
On 1/9/23 11:50 PM, olcott wrote:
On 1/9/2023 10:01 PM, Richard Damon wrote:
On 1/9/23 10:19 PM, olcott wrote:
The set of (analytical) expressions of (formal or natural language) >>>> have
a complete semantic connection to their truth maker axioms otherwise >>>> they are simply untrue. Copyright 2022 PL Olcott
Right, but such a connection can be based on an INFINTE number of
Mathematicians and logicians make sure to ignore the philosophical
foundation of these things. or we would never get this:
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
Right, a system is incomplete if there exist a statement (which has a Truth Value) but that statment can neither be proven or disproven in T.
Yet the above expression allows epistemological antinomies to show it is incomplete, whereas epistemological antinomies are not truth bearers
thus not members of any formal system: ∀φ ∈ T ((T ⊢ φ) ∨ (T ⊢ ¬φ))
Alternatively, a system is incomplete if there exists a TRUE statement which can not be proven in TThe set of (analytical) expressions of (formal or natural language) have
a complete semantic connection to their truth maker axioms otherwise
they are simply untrue. Copyright 2022 PL Olcott
If they are not true in a formal system because they are epistemological antinomies thus self-contradictory, thus not truth bearers in this
formal system then they are simply not members of this formal system.
(the disproven half of the above becoming a
seperate piece as if the statement is false (not just "untrue") then we can form the negation of the statement and not be able to prove that one.
Previously philosophers were trying to define truth maker for
analytical
truth and empirical truth at the same time and in the same way.
Nope. Just shows you are not understanding how logic works.
This much is agreed: “x makes it true that p” is a construction that
signifies, if it signifies anything at all, a relation borne to a
truth-
bearer by something else, a truth-maker. But it isn’t generally agreed
what that something else might be, or what truth-bearers are, or what >>>> the character might be of the relationship that holds, if it does,
between them, or even whether such a relationship ever does hold.
https://plato.stanford.edu/entries/truthmakers/
But note, that the statement x -> y is NOT a assertion that x MAKES y >>> true, but that the Truth of x proves that Y is true.
x ⊨ y Aristotle's syllogism required a semantic connection based on
semantic categories.
https://en.wikipedia.org/wiki/Syllogism#Basic_structure
And Aristotle's logic system is only for CATEGORICAL statements, and
thus only a first order logic system.
Note, your "Semantic" connection here comes out as a neccesary condition based on you being in CATEGORICAL logic.
Such a system can not express the required logic to create a full description of the Natural Numbers.
It is NOT a statement about "Causation", in fact, it is more a
statement about sub-sets of models that might exist.
https://en.wikipedia.org/wiki/Logical_consequence#Semantic_consequence
In other words there is a semantic connection form an expression of
language to its truth maker axioms.
(quoting and changing some symbols to make typable)
A Formula A is a semantic consequence within a formal system FS of a set of statements L, if and only if there is no model M in which all of L
are True and A is false.
*I am referring to the Haskell Curry notion of true in the system*
A theory (over 𝓕) is defined as a conceptual class of these elementary statements. Let 𝓣 be such a theory. Then the elementary statements
which belong to 𝓣 we shall call the elementary theorems of 𝓣; we also say that these elementary statements are true for 𝓣. Thus, given 𝓣,
an elementary theorem is an elementary statement which is true. A
theory is thus a way of picking out from the statements of 𝓕 a certain subclass of true statements. https://www.liarparadox.org/Haskell_Curry_45.pdf
In this case we only need a syntactic connection from the expression to
its truth maker axioms, {AKA elementary theorems of 𝓣} otherwise the expression is simply untrue in 𝓣.
--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 1/10/2023 6:56 AM, Richard Damon wrote:
On 1/9/23 11:50 PM, olcott wrote:
On 1/9/2023 10:01 PM, Richard Damon wrote:
On 1/9/23 10:19 PM, olcott wrote:
The set of (analytical) expressions of (formal or natural language)
have
a complete semantic connection to their truth maker axioms otherwise >>>>> they are simply untrue. Copyright 2022 PL Olcott
Right, but such a connection can be based on an INFINTE number of
Mathematicians and logicians make sure to ignore the philosophical
foundation of these things. or we would never get this:
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
Right, a system is incomplete if there exist a statement (which has a
Truth Value) but that statment can neither be proven or disproven in T.
Yet the above expression allows epistemological antinomies to show it is incomplete, whereas epistemological antinomies are not truth bearers
thus not members of any formal system: ∀φ ∈ T ((T ⊢ φ) ∨ (T ⊢ ¬φ))
Alternatively, a system is incomplete if there exists a TRUE statement
which can not be proven in T
The set of (analytical) expressions of (formal or natural language) have
a complete semantic connection to their truth maker axioms otherwise
they are simply untrue. Copyright 2022 PL Olcott
If they are not true in a formal system because they are epistemological antinomies thus self-contradictory, thus not truth bearers in this
formal system then they are simply not members of this formal system.
(the disproven half of the above becoming a seperate piece as if the
statement is false (not just "untrue") then we can form the negation
of the statement and not be able to prove that one.
Previously philosophers were trying to define truth maker for
analytical
truth and empirical truth at the same time and in the same way.
Nope. Just shows you are not understanding how logic works.
This much is agreed: “x makes it true that p” is a construction that >>>>> signifies, if it signifies anything at all, a relation borne to a
truth-
bearer by something else, a truth-maker. But it isn’t generally agreed >>>>> what that something else might be, or what truth-bearers are, or what >>>>> the character might be of the relationship that holds, if it does,
between them, or even whether such a relationship ever does hold.
https://plato.stanford.edu/entries/truthmakers/
But note, that the statement x -> y is NOT a assertion that x MAKES
y true, but that the Truth of x proves that Y is true.
x ⊨ y Aristotle's syllogism required a semantic connection based on
semantic categories.
https://en.wikipedia.org/wiki/Syllogism#Basic_structure
And Aristotle's logic system is only for CATEGORICAL statements, and
thus only a first order logic system.
Note, your "Semantic" connection here comes out as a neccesary
condition based on you being in CATEGORICAL logic.
Such a system can not express the required logic to create a full
description of the Natural Numbers.
It is NOT a statement about "Causation", in fact, it is more a
statement about sub-sets of models that might exist.
https://en.wikipedia.org/wiki/Logical_consequence#Semantic_consequence
In other words there is a semantic connection form an expression of
language to its truth maker axioms.
(quoting and changing some symbols to make typable)
A Formula A is a semantic consequence within a formal system FS of a
set of statements L, if and only if there is no model M in which all
of L are True and A is false.
*I am referring to the Haskell Curry notion of true in the system*
A theory (over 𝓕) is defined as a conceptual class of these elementary statements. Let 𝓣 be such a theory. Then the elementary statements
which belong to 𝓣 we shall call the elementary theorems of 𝓣; we also say that these elementary statements are true for 𝓣. Thus, given 𝓣,
an elementary theorem is an elementary statement which is true. A
theory is thus a way of picking out from the statements of 𝓕 a certain subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
In this case we only need a syntactic connection from the expression to
its truth maker axioms, {AKA elementary theorems of 𝓣} otherwise the expression is simply untrue in 𝓣.
On 1/10/23 12:43 PM, olcott wrote:
On 1/10/2023 6:56 AM, Richard Damon wrote:
On 1/9/23 11:50 PM, olcott wrote:
On 1/9/2023 10:01 PM, Richard Damon wrote:
On 1/9/23 10:19 PM, olcott wrote:
The set of (analytical) expressions of (formal or natural
language) have
a complete semantic connection to their truth maker axioms otherwise >>>>>> they are simply untrue. Copyright 2022 PL Olcott
Right, but such a connection can be based on an INFINTE number of
Mathematicians and logicians make sure to ignore the philosophical
foundation of these things. or we would never get this:
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
Right, a system is incomplete if there exist a statement (which has a
Truth Value) but that statment can neither be proven or disproven in T.
Yet the above expression allows epistemological antinomies to show it is
incomplete, whereas epistemological antinomies are not truth bearers
thus not members of any formal system: ∀φ ∈ T ((T ⊢ φ) ∨ (T ⊢ ¬φ))
No, because an epistemological antinomie is not a "∀φ ∈ T", which my verbal statement makes clear.
The elements of T are only those statements with a Truth Value in T.
Alternatively, a system is incomplete if there exists a TRUE
statement which can not be proven in T
The set of (analytical) expressions of (formal or natural language)
have a complete semantic connection to their truth maker axioms
otherwise they are simply untrue. Copyright 2022 PL Olcott
Right, and that semantic connection can b infinite in length, and thus
not a proof.
If they are not true in a formal system because they are epistemological
antinomies thus self-contradictory, thus not truth bearers in this
formal system then they are simply not members of this formal system.
Right, even you have agreed that a statement asking about the existance
of a proof of a statement WILL be a Truth Bearer, (as such a proof
either does or does not exist) a thus G, even in the meta-theory is a
Truth Bearer.
(the disproven half of the above becoming a seperate piece as if the
statement is false (not just "untrue") then we can form the negation
of the statement and not be able to prove that one.
Previously philosophers were trying to define truth maker for
analytical
truth and empirical truth at the same time and in the same way.
Nope. Just shows you are not understanding how logic works.
This much is agreed: “x makes it true that p” is a construction that >>>>>> signifies, if it signifies anything at all, a relation borne to a
truth-
bearer by something else, a truth-maker. But it isn’t generally
agreed
what that something else might be, or what truth-bearers are, or what >>>>>> the character might be of the relationship that holds, if it does, >>>>>> between them, or even whether such a relationship ever does hold.
https://plato.stanford.edu/entries/truthmakers/
But note, that the statement x -> y is NOT a assertion that x MAKES
y true, but that the Truth of x proves that Y is true.
x ⊨ y Aristotle's syllogism required a semantic connection based on
semantic categories.
https://en.wikipedia.org/wiki/Syllogism#Basic_structure
And Aristotle's logic system is only for CATEGORICAL statements, and
thus only a first order logic system.
Note, your "Semantic" connection here comes out as a neccesary
condition based on you being in CATEGORICAL logic.
Such a system can not express the required logic to create a full
description of the Natural Numbers.
It is NOT a statement about "Causation", in fact, it is more a
statement about sub-sets of models that might exist.
https://en.wikipedia.org/wiki/Logical_consequence#Semantic_consequence >>>> In other words there is a semantic connection form an expression of
language to its truth maker axioms.
(quoting and changing some symbols to make typable)
A Formula A is a semantic consequence within a formal system FS of a
set of statements L, if and only if there is no model M in which all
of L are True and A is false.
*I am referring to the Haskell Curry notion of true in the system*
A theory (over 𝓕) is defined as a conceptual class of these elementary
statements. Let 𝓣 be such a theory. Then the elementary statements
which belong to 𝓣 we shall call the elementary theorems of 𝓣; we also >> say that these elementary statements are true for 𝓣. Thus, given 𝓣,
an elementary theorem is an elementary statement which is true. A
theory is thus a way of picking out from the statements of 𝓕 a certain
subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
In this case we only need a syntactic connection from the expression to
its truth maker axioms, {AKA elementary theorems of 𝓣} otherwise the
expression is simply untrue in 𝓣.
Right, and True in the system can come from an infinite set of
connections, and thus not a proof.
You confuse True in the system with KNOWN in the system.
Note, When he says these statements are "True for 𝓣", he is NOT saying that ONLY these statements are True for 𝓣, so this doesn't actually
define what True for 𝓣 actually means.
On 1/10/2023 5:38 PM, Richard Damon wrote:
On 1/10/23 12:43 PM, olcott wrote:
On 1/10/2023 6:56 AM, Richard Damon wrote:
On 1/9/23 11:50 PM, olcott wrote:
On 1/9/2023 10:01 PM, Richard Damon wrote:
On 1/9/23 10:19 PM, olcott wrote:
The set of (analytical) expressions of (formal or natural
language) have
a complete semantic connection to their truth maker axioms otherwise >>>>>>> they are simply untrue. Copyright 2022 PL Olcott
Right, but such a connection can be based on an INFINTE number of
Mathematicians and logicians make sure to ignore the philosophical
foundation of these things. or we would never get this:
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
Right, a system is incomplete if there exist a statement (which has
a Truth Value) but that statment can neither be proven or disproven
in T.
Yet the above expression allows epistemological antinomies to show it is >>> incomplete, whereas epistemological antinomies are not truth bearers
thus not members of any formal system: ∀φ ∈ T ((T ⊢ φ) ∨ (T ⊢ ¬φ))
No, because an epistemological antinomie is not a "∀φ ∈ T", which my
verbal statement makes clear.
The elements of T are only those statements with a Truth Value in T.
Yes thus negating: Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
The definition of incompleteness.
Alternatively, a system is incomplete if there exists a TRUE
statement which can not be proven in T
The set of (analytical) expressions of (formal or natural language)
have a complete semantic connection to their truth maker axioms
otherwise they are simply untrue. Copyright 2022 PL Olcott
Right, and that semantic connection can b infinite in length, and thus
not a proof.
It is not allowed to be infinite length within formal systems (and you
know this) no proof in T means untrue in T.
If they are not true in a formal system because they are epistemological >>> antinomies thus self-contradictory, thus not truth bearers in this
formal system then they are simply not members of this formal system.
Right, even you have agreed that a statement asking about the
existance of a proof of a statement WILL be a Truth Bearer, (as such a
proof either does or does not exist) a thus G, even in the meta-theory
is a Truth Bearer.
A self-contradictory epistemological antinomy in one formal system can
be resolved in any formal system where it is not self-contradictory.
(the disproven half of the above becoming a seperate piece as if the
statement is false (not just "untrue") then we can form the negation
of the statement and not be able to prove that one.
Previously philosophers were trying to define truth maker for
analytical
truth and empirical truth at the same time and in the same way.
Nope. Just shows you are not understanding how logic works.
This much is agreed: “x makes it true that p” is a construction that
signifies, if it signifies anything at all, a relation borne to a >>>>>>> truth-
bearer by something else, a truth-maker. But it isn’t generally >>>>>>> agreed
what that something else might be, or what truth-bearers are, or >>>>>>> what
the character might be of the relationship that holds, if it does, >>>>>>> between them, or even whether such a relationship ever does hold. >>>>>>> https://plato.stanford.edu/entries/truthmakers/
But note, that the statement x -> y is NOT a assertion that x
MAKES y true, but that the Truth of x proves that Y is true.
x ⊨ y Aristotle's syllogism required a semantic connection based on >>>>> semantic categories.
https://en.wikipedia.org/wiki/Syllogism#Basic_structure
And Aristotle's logic system is only for CATEGORICAL statements, and
thus only a first order logic system.
Note, your "Semantic" connection here comes out as a neccesary
condition based on you being in CATEGORICAL logic.
Such a system can not express the required logic to create a full
description of the Natural Numbers.
It is NOT a statement about "Causation", in fact, it is more a
statement about sub-sets of models that might exist.
https://en.wikipedia.org/wiki/Logical_consequence#Semantic_consequence >>>>> In other words there is a semantic connection form an expression of
language to its truth maker axioms.
(quoting and changing some symbols to make typable)
A Formula A is a semantic consequence within a formal system FS of a
set of statements L, if and only if there is no model M in which all
of L are True and A is false.
*I am referring to the Haskell Curry notion of true in the system*
A theory (over 𝓕) is defined as a conceptual class of these elementary >>> statements. Let 𝓣 be such a theory. Then the elementary statements
which belong to 𝓣 we shall call the elementary theorems of 𝓣; we also >>> say that these elementary statements are true for 𝓣. Thus, given 𝓣, >>> an elementary theorem is an elementary statement which is true. A
theory is thus a way of picking out from the statements of 𝓕 a certain >>> subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
In this case we only need a syntactic connection from the expression to
its truth maker axioms, {AKA elementary theorems of 𝓣} otherwise the
expression is simply untrue in 𝓣.
Right, and True in the system can come from an infinite set of
connections, and thus not a proof.
True within the system requires provable in the system.
True outside the system does not require provable within the system.
Gödel did not even attempt to show that G is true in F.
Gödel showed that G is true outside of F.
For this reason, the sentence GF is often said to be "true but
unprovable." (Raatikainen 2015). However, since the Gödel sentence
cannot itself formally specify its intended interpretation, the truth of
the sentence GF may only be arrived at via a meta-analysis from outside
the system.
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
You confuse True in the system with KNOWN in the system.
Note, When he says these statements are "True for 𝓣", he is NOT saying
that ONLY these statements are True for 𝓣, so this doesn't actually
define what True for 𝓣 actually means.
On 1/10/23 7:05 PM, olcott wrote:
On 1/10/2023 5:38 PM, Richard Damon wrote:
On 1/10/23 12:43 PM, olcott wrote:
On 1/10/2023 6:56 AM, Richard Damon wrote:
On 1/9/23 11:50 PM, olcott wrote:
On 1/9/2023 10:01 PM, Richard Damon wrote:
On 1/9/23 10:19 PM, olcott wrote:
The set of (analytical) expressions of (formal or natural
language) have
a complete semantic connection to their truth maker axioms
otherwise
they are simply untrue. Copyright 2022 PL Olcott
Right, but such a connection can be based on an INFINTE number of >>>>>>>
Mathematicians and logicians make sure to ignore the philosophical >>>>>> foundation of these things. or we would never get this:
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
Right, a system is incomplete if there exist a statement (which has
a Truth Value) but that statment can neither be proven or disproven
in T.
Yet the above expression allows epistemological antinomies to show
it is
incomplete, whereas epistemological antinomies are not truth bearers
thus not members of any formal system: ∀φ ∈ T ((T ⊢ φ) ∨ (T ⊢ ¬φ))
No, because an epistemological antinomie is not a "∀φ ∈ T", which my >>> verbal statement makes clear.
The elements of T are only those statements with a Truth Value in T.
Yes thus negating: Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
The definition of incompleteness.
Except that the definiton of ⊬ is "Does not PROVE", not is not true (or
you are quoting the wrong definition of incompleteness"
This seems to be your standard problem.
Alternatively, a system is incomplete if there exists a TRUE
statement which can not be proven in T
The set of (analytical) expressions of (formal or natural language)
have a complete semantic connection to their truth maker axioms
otherwise they are simply untrue. Copyright 2022 PL Olcott
Right, and that semantic connection can b infinite in length, and
thus not a proof.
It is not allowed to be infinite length within formal systems (and you
know this) no proof in T means untrue in T.
Source of that claim?
For TRUTH (not proof)
Until you can document that, it isn't true, and you are just making
yourself to be a liar.
If they are not true in a formal system because they are
epistemological
antinomies thus self-contradictory, thus not truth bearers in this
formal system then they are simply not members of this formal system.
Right, even you have agreed that a statement asking about the
existance of a proof of a statement WILL be a Truth Bearer, (as such
a proof either does or does not exist) a thus G, even in the
meta-theory is a Truth Bearer.
A self-contradictory epistemological antinomy in one formal system can
be resolved in any formal system where it is not self-contradictory.
So? in F, G is just a statement about the existance of a number.
in Meta-F, G is that same staement, but from it you can prove that G
being true implies that G can not be proven
So, where is the "Self-Contradictory Statement"?
True within the system requires provable in the system.
Source!!!
That is the lie that you will DIE on
True outside the system does not require provable within the system.
SOURCE.
Gödel did not even attempt to show that G is true in F.
Gödel showed that G is true outside of F.
No, In his proof he shows that due to the things that can be proved in Meta-F, G must be true in F.
Try to actually READ his proof.
My guess is you are looking at the Cliff notes version because it is
beyond you, and yo aren't even understand those Cliff Notes.
For this reason, the sentence GF is often said to be "true but
unprovable." (Raatikainen 2015). However, since the Gödel sentence
cannot itself formally specify its intended interpretation, the truth
of the sentence GF may only be arrived at via a meta-analysis from
outside the system.
No, the Godel sentence if F EXACTLY specifies its direct meaning in F.
That no number exists that meets a certain criteria. PERIOD.
The Truth of that statement is proven in Meta F.
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
You confuse True in the system with KNOWN in the system.
Note, When he says these statements are "True for 𝓣", he is NOT
saying that ONLY these statements are True for 𝓣, so this doesn't
actually define what True for 𝓣 actually means.
On 1/10/2023 6:27 PM, Richard Damon wrote:
On 1/10/23 7:05 PM, olcott wrote:
On 1/10/2023 5:38 PM, Richard Damon wrote:
On 1/10/23 12:43 PM, olcott wrote:
On 1/10/2023 6:56 AM, Richard Damon wrote:
On 1/9/23 11:50 PM, olcott wrote:
On 1/9/2023 10:01 PM, Richard Damon wrote:
On 1/9/23 10:19 PM, olcott wrote:
The set of (analytical) expressions of (formal or natural
language) have
a complete semantic connection to their truth maker axioms
otherwise
they are simply untrue. Copyright 2022 PL Olcott
Right, but such a connection can be based on an INFINTE number of >>>>>>>>
Mathematicians and logicians make sure to ignore the philosophical >>>>>>> foundation of these things. or we would never get this:
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
Right, a system is incomplete if there exist a statement (which
has a Truth Value) but that statment can neither be proven or
disproven in T.
Yet the above expression allows epistemological antinomies to show
it is
incomplete, whereas epistemological antinomies are not truth bearers >>>>> thus not members of any formal system: ∀φ ∈ T ((T ⊢ φ) ∨ (T ⊢ ¬φ))
No, because an epistemological antinomie is not a "∀φ ∈ T", which my >>>> verbal statement makes clear.
The elements of T are only those statements with a Truth Value in T.
Yes thus negating: Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)) >>> The definition of incompleteness.
Except that the definiton of ⊬ is "Does not PROVE", not is not true
(or you are quoting the wrong definition of incompleteness"
This seems to be your standard problem.
Alternatively, a system is incomplete if there exists a TRUE
statement which can not be proven in T
The set of (analytical) expressions of (formal or natural language)
have a complete semantic connection to their truth maker axioms
otherwise they are simply untrue. Copyright 2022 PL Olcott
Right, and that semantic connection can b infinite in length, and
thus not a proof.
It is not allowed to be infinite length within formal systems (and
you know this) no proof in T means untrue in T.
Source of that claim?
For TRUTH (not proof)
Until you can document that, it isn't true, and you are just making
yourself to be a liar.
You said that infinite proofs are not allowed are you changing your
mind? Did you forget that you said this?
Right, even you have agreed that a statement asking about the
If they are not true in a formal system because they are
epistemological
antinomies thus self-contradictory, thus not truth bearers in this
formal system then they are simply not members of this formal system. >>>>
existance of a proof of a statement WILL be a Truth Bearer, (as such
a proof either does or does not exist) a thus G, even in the
meta-theory is a Truth Bearer.
A self-contradictory epistemological antinomy in one formal system
can be resolved in any formal system where it is not self-contradictory. >>>
So? in F, G is just a statement about the existance of a number.
G is not true in F. An expression is true in a formal system iff it is provable from the axioms of this formal system.
in Meta-F, G is that same staement, but from it you can prove that G
being true implies that G can not be proven
So, where is the "Self-Contradictory Statement"?
The G says of itself that it is unprovable in F is self-connradictory in F.
True within the system requires provable in the system.
Source!!!
That is the lie that you will DIE on
True outside the system does not require provable within the system.
SOURCE.
Gödel did not even attempt to show that G is true in F.
Gödel showed that G is true outside of F.
No, In his proof he shows that due to the things that can be proved in
Meta-F, G must be true in F.
Try to actually READ his proof.
My guess is you are looking at the Cliff notes version because it is
beyond you, and yo aren't even understand those Cliff Notes.
For this reason, the sentence GF is often said to be "true but
unprovable." (Raatikainen 2015). However, since the Gödel sentence
cannot itself formally specify its intended interpretation, the truth
of the sentence GF may only be arrived at via a meta-analysis from
outside the system.
No, the Godel sentence if F EXACTLY specifies its direct meaning in F.
That no number exists that meets a certain criteria. PERIOD.
The Truth of that statement is proven in Meta F.
"This sentence is not true" is proven true in meta F.
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
You confuse True in the system with KNOWN in the system.
Note, When he says these statements are "True for 𝓣", he is NOT
saying that ONLY these statements are True for 𝓣, so this doesn't
actually define what True for 𝓣 actually means.
On 1/10/23 9:03 PM, olcott wrote:
On 1/10/2023 6:27 PM, Richard Damon wrote:
On 1/10/23 7:05 PM, olcott wrote:
On 1/10/2023 5:38 PM, Richard Damon wrote:
On 1/10/23 12:43 PM, olcott wrote:
On 1/10/2023 6:56 AM, Richard Damon wrote:
On 1/9/23 11:50 PM, olcott wrote:
On 1/9/2023 10:01 PM, Richard Damon wrote:
On 1/9/23 10:19 PM, olcott wrote:
The set of (analytical) expressions of (formal or natural
language) have
a complete semantic connection to their truth maker axioms >>>>>>>>>> otherwise
they are simply untrue. Copyright 2022 PL Olcott
Right, but such a connection can be based on an INFINTE number of >>>>>>>>>
Mathematicians and logicians make sure to ignore the philosophical >>>>>>>> foundation of these things. or we would never get this:
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
Right, a system is incomplete if there exist a statement (which
has a Truth Value) but that statment can neither be proven or
disproven in T.
Yet the above expression allows epistemological antinomies to show >>>>>> it is
incomplete, whereas epistemological antinomies are not truth bearers >>>>>> thus not members of any formal system: ∀φ ∈ T ((T ⊢ φ) ∨ (T ⊢ ¬φ))
No, because an epistemological antinomie is not a "∀φ ∈ T", which >>>>> my verbal statement makes clear.
The elements of T are only those statements with a Truth Value in T.
Yes thus negating: Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)) >>>> The definition of incompleteness.
Except that the definiton of ⊬ is "Does not PROVE", not is not true
(or you are quoting the wrong definition of incompleteness"
This seems to be your standard problem.
Alternatively, a system is incomplete if there exists a TRUE
statement which can not be proven in T
The set of (analytical) expressions of (formal or natural
language) have a complete semantic connection to their truth maker >>>>>> axioms otherwise they are simply untrue. Copyright 2022 PL Olcott
Right, and that semantic connection can b infinite in length, and
thus not a proof.
It is not allowed to be infinite length within formal systems (and
you know this) no proof in T means untrue in T.
Source of that claim?
For TRUTH (not proof)
Until you can document that, it isn't true, and you are just making
yourself to be a liar.
You said that infinite proofs are not allowed are you changing your
mind? Did you forget that you said this?
You seem to have a brain short between the concepts of Truth and Proof.
I said TRUTH allows an infinite connection, but PROOFS do not.
On 1/10/2023 9:26 PM, Richard Damon wrote:
On 1/10/23 9:03 PM, olcott wrote:
On 1/10/2023 6:27 PM, Richard Damon wrote:
On 1/10/23 7:05 PM, olcott wrote:
On 1/10/2023 5:38 PM, Richard Damon wrote:
On 1/10/23 12:43 PM, olcott wrote:Yes thus negating: Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)) >>>>> The definition of incompleteness.
On 1/10/2023 6:56 AM, Richard Damon wrote:
On 1/9/23 11:50 PM, olcott wrote:
On 1/9/2023 10:01 PM, Richard Damon wrote:
On 1/9/23 10:19 PM, olcott wrote:
The set of (analytical) expressions of (formal or natural >>>>>>>>>>> language) have
a complete semantic connection to their truth maker axioms >>>>>>>>>>> otherwise
they are simply untrue. Copyright 2022 PL Olcott
Right, but such a connection can be based on an INFINTE number of >>>>>>>>>>
Mathematicians and logicians make sure to ignore the philosophical >>>>>>>>> foundation of these things. or we would never get this:
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
Right, a system is incomplete if there exist a statement (which >>>>>>>> has a Truth Value) but that statment can neither be proven or
disproven in T.
Yet the above expression allows epistemological antinomies to
show it is
incomplete, whereas epistemological antinomies are not truth bearers >>>>>>> thus not members of any formal system: ∀φ ∈ T ((T ⊢ φ) ∨ (T ⊢ ¬φ))
No, because an epistemological antinomie is not a "∀φ ∈ T", which >>>>>> my verbal statement makes clear.
The elements of T are only those statements with a Truth Value in T. >>>>>
Except that the definiton of ⊬ is "Does not PROVE", not is not true
(or you are quoting the wrong definition of incompleteness"
This seems to be your standard problem.
Alternatively, a system is incomplete if there exists a TRUE
statement which can not be proven in T
The set of (analytical) expressions of (formal or natural
language) have a complete semantic connection to their truth
maker axioms otherwise they are simply untrue. Copyright 2022 PL >>>>>>> Olcott
Right, and that semantic connection can b infinite in length, and
thus not a proof.
It is not allowed to be infinite length within formal systems (and
you know this) no proof in T means untrue in T.
Source of that claim?
For TRUTH (not proof)
Until you can document that, it isn't true, and you are just making
yourself to be a liar.
You said that infinite proofs are not allowed are you changing your
mind? Did you forget that you said this?
You seem to have a brain short between the concepts of Truth and Proof.
I said TRUTH allows an infinite connection, but PROOFS do not.
If an analytic expression of language is true or false there must be a complete set of semantic connections making it true or false otherwise
it is not a truth bearer.
Because formal systems are only allowed to have finite proofs formal
systems are not allowed to have infinite connections to their semantic
truth maker. Thus an expression is only true in a formal system iff it
is provable within this system. Otherwise this expression is untrue
which may or may not include false.
On 1/12/23 8:56 PM, olcott wrote:
On 1/10/2023 9:26 PM, Richard Damon wrote:
On 1/10/23 9:03 PM, olcott wrote:
On 1/10/2023 6:27 PM, Richard Damon wrote:
On 1/10/23 7:05 PM, olcott wrote:
On 1/10/2023 5:38 PM, Richard Damon wrote:
On 1/10/23 12:43 PM, olcott wrote:Yes thus negating: Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)) >>>>>> The definition of incompleteness.
On 1/10/2023 6:56 AM, Richard Damon wrote:
On 1/9/23 11:50 PM, olcott wrote:
On 1/9/2023 10:01 PM, Richard Damon wrote:
On 1/9/23 10:19 PM, olcott wrote:
The set of (analytical) expressions of (formal or natural >>>>>>>>>>>> language) have
a complete semantic connection to their truth maker axioms >>>>>>>>>>>> otherwise
they are simply untrue. Copyright 2022 PL Olcott
Right, but such a connection can be based on an INFINTE
number of
Mathematicians and logicians make sure to ignore the
philosophical
foundation of these things. or we would never get this:
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
Right, a system is incomplete if there exist a statement (which >>>>>>>>> has a Truth Value) but that statment can neither be proven or >>>>>>>>> disproven in T.
Yet the above expression allows epistemological antinomies to
show it is
incomplete, whereas epistemological antinomies are not truth
bearers
thus not members of any formal system: ∀φ ∈ T ((T ⊢ φ) ∨ (T ⊢ ¬φ))
No, because an epistemological antinomie is not a "∀φ ∈ T", which >>>>>>> my verbal statement makes clear.
The elements of T are only those statements with a Truth Value in T. >>>>>>
Except that the definiton of ⊬ is "Does not PROVE", not is not true >>>>> (or you are quoting the wrong definition of incompleteness"
This seems to be your standard problem.
Alternatively, a system is incomplete if there exists a TRUE >>>>>>>>> statement which can not be proven in T
The set of (analytical) expressions of (formal or natural
language) have a complete semantic connection to their truth
maker axioms otherwise they are simply untrue. Copyright 2022 PL >>>>>>>> Olcott
Right, and that semantic connection can b infinite in length, and >>>>>>> thus not a proof.
It is not allowed to be infinite length within formal systems (and >>>>>> you know this) no proof in T means untrue in T.
Source of that claim?
For TRUTH (not proof)
Until you can document that, it isn't true, and you are just making
yourself to be a liar.
You said that infinite proofs are not allowed are you changing your
mind? Did you forget that you said this?
You seem to have a brain short between the concepts of Truth and Proof.
I said TRUTH allows an infinite connection, but PROOFS do not.
If an analytic expression of language is true or false there must be a
complete set of semantic connections making it true or false otherwise
it is not a truth bearer.
Right, but the connect set can be infinite.
Because formal systems are only allowed to have finite proofs formal
systems are not allowed to have infinite connections to their semantic
truth maker. Thus an expression is only true in a formal system iff it
is provable within this system. Otherwise this expression is untrue
which may or may not include false.
Right, FINITE PROOFS, says nothing about TRUTH.
On 1/12/2023 8:37 PM, Richard Damon wrote:
On 1/12/23 8:56 PM, olcott wrote:
Because formal systems are only allowed to have finite proofs formal
systems are not allowed to have infinite connections to their semantic
truth maker. Thus an expression is only true in a formal system iff it
is provable within this system. Otherwise this expression is untrue
which may or may not include false.
Right, FINITE PROOFS, says nothing about TRUTH.
That it ridiculously false. Expressions of language that are proven to
have a connection to their truth maker axioms are a subset of all truth
and comprise the entire body of analytical knowledge.
On 1/12/23 10:04 PM, olcott wrote:
On 1/12/2023 8:37 PM, Richard Damon wrote:
On 1/12/23 8:56 PM, olcott wrote:
Because formal systems are only allowed to have finite proofs formal
systems are not allowed to have infinite connections to their semantic >>>> truth maker. Thus an expression is only true in a formal system iff it >>>> is provable within this system. Otherwise this expression is untrue
which may or may not include false.
Right, FINITE PROOFS, says nothing about TRUTH.
That it ridiculously false. Expressions of language that are proven to
have a connection to their truth maker axioms are a subset of all truth
and comprise the entire body of analytical knowledge.
So, since proven statements are a SUBSET of all truth, what says that
all truths have to be proven.
On 1/12/2023 9:35 PM, Richard Damon wrote:
On 1/12/23 10:04 PM, olcott wrote:
On 1/12/2023 8:37 PM, Richard Damon wrote:
On 1/12/23 8:56 PM, olcott wrote:
Because formal systems are only allowed to have finite proofs formal >>>>> systems are not allowed to have infinite connections to their semantic >>>>> truth maker. Thus an expression is only true in a formal system iff it >>>>> is provable within this system. Otherwise this expression is untrue
which may or may not include false.
Right, FINITE PROOFS, says nothing about TRUTH.
That it ridiculously false. Expressions of language that are proven to
have a connection to their truth maker axioms are a subset of all truth
and comprise the entire body of analytical knowledge.
So, since proven statements are a SUBSET of all truth, what says that
all truths have to be proven.
Within a formal system where true requires finite proof to axioms the
lack of a finite proof to axioms means untrue.
On 1/12/23 10:52 PM, olcott wrote:
On 1/12/2023 9:35 PM, Richard Damon wrote:
On 1/12/23 10:04 PM, olcott wrote:
On 1/12/2023 8:37 PM, Richard Damon wrote:
On 1/12/23 8:56 PM, olcott wrote:
Because formal systems are only allowed to have finite proofs formal >>>>>> systems are not allowed to have infinite connections to their
semantic
truth maker. Thus an expression is only true in a formal system
iff it
is provable within this system. Otherwise this expression is untrue >>>>>> which may or may not include false.
Right, FINITE PROOFS, says nothing about TRUTH.
That it ridiculously false. Expressions of language that are proven to >>>> have a connection to their truth maker axioms are a subset of all truth >>>> and comprise the entire body of analytical knowledge.
So, since proven statements are a SUBSET of all truth, what says that
all truths have to be proven.
Within a formal system where true requires finite proof to axioms the
lack of a finite proof to axioms means untrue.
Source of claim?
On 1/12/2023 9:58 PM, Richard Damon wrote:
On 1/12/23 10:52 PM, olcott wrote:
On 1/12/2023 9:35 PM, Richard Damon wrote:
On 1/12/23 10:04 PM, olcott wrote:
On 1/12/2023 8:37 PM, Richard Damon wrote:
On 1/12/23 8:56 PM, olcott wrote:
Because formal systems are only allowed to have finite proofs formal >>>>>>> systems are not allowed to have infinite connections to their
semantic
truth maker. Thus an expression is only true in a formal system
iff it
is provable within this system. Otherwise this expression is untrue >>>>>>> which may or may not include false.
Right, FINITE PROOFS, says nothing about TRUTH.
That it ridiculously false. Expressions of language that are proven to >>>>> have a connection to their truth maker axioms are a subset of all
truth
and comprise the entire body of analytical knowledge.
So, since proven statements are a SUBSET of all truth, what says
that all truths have to be proven.
Within a formal system where true requires finite proof to axioms the
lack of a finite proof to axioms means untrue.
Source of claim?
How else can it possibly work?
I figure these things out on the basis of categorically exhaustive
reasoning.
On 1/13/23 10:42 AM, olcott wrote:
On 1/12/2023 9:58 PM, Richard Damon wrote:
On 1/12/23 10:52 PM, olcott wrote:
On 1/12/2023 9:35 PM, Richard Damon wrote:
On 1/12/23 10:04 PM, olcott wrote:
On 1/12/2023 8:37 PM, Richard Damon wrote:
On 1/12/23 8:56 PM, olcott wrote:
Because formal systems are only allowed to have finite proofs
formal
systems are not allowed to have infinite connections to their
semantic
truth maker. Thus an expression is only true in a formal system >>>>>>>> iff it
is provable within this system. Otherwise this expression is untrue >>>>>>>> which may or may not include false.
Right, FINITE PROOFS, says nothing about TRUTH.
That it ridiculously false. Expressions of language that are
proven to
have a connection to their truth maker axioms are a subset of all
truth
and comprise the entire body of analytical knowledge.
So, since proven statements are a SUBSET of all truth, what says
that all truths have to be proven.
Within a formal system where true requires finite proof to axioms the
lack of a finite proof to axioms means untrue.
Source of claim?
How else can it possibly work?
I figure these things out on the basis of categorically exhaustive
reasoning.
Good, so you admit that it isn't based on any REAL theoretical basis but
only due to your limited (and flawed) thinkng ability.
Note, you CAN'T do a categorically exhaustive reasoning on this problem,
as its is an INFINITE domain. You just don't seem to understand the
nature of infinity, so you don't see that.
The Truth value is just based on the existance of a set on connections
to the axioms. Formal system or not, and NOTHING limits that to being
finite.
On 1/13/2023 6:25 PM, Richard Damon wrote:
On 1/13/23 10:42 AM, olcott wrote:In other words you are saying that it is *TRUE IN THE FORMAL SYSTEM*
On 1/12/2023 9:58 PM, Richard Damon wrote:
On 1/12/23 10:52 PM, olcott wrote:
On 1/12/2023 9:35 PM, Richard Damon wrote:
On 1/12/23 10:04 PM, olcott wrote:
On 1/12/2023 8:37 PM, Richard Damon wrote:
On 1/12/23 8:56 PM, olcott wrote:
Because formal systems are only allowed to have finite proofs >>>>>>>>> formal
systems are not allowed to have infinite connections to their >>>>>>>>> semantic
truth maker. Thus an expression is only true in a formal system >>>>>>>>> iff it
is provable within this system. Otherwise this expression is >>>>>>>>> untrue
which may or may not include false.
Right, FINITE PROOFS, says nothing about TRUTH.
That it ridiculously false. Expressions of language that are
proven to
have a connection to their truth maker axioms are a subset of all >>>>>>> truth
and comprise the entire body of analytical knowledge.
So, since proven statements are a SUBSET of all truth, what says
that all truths have to be proven.
Within a formal system where true requires finite proof to axioms the >>>>> lack of a finite proof to axioms means untrue.
Source of claim?
How else can it possibly work?
I figure these things out on the basis of categorically exhaustive
reasoning.
Good, so you admit that it isn't based on any REAL theoretical basis
but only due to your limited (and flawed) thinkng ability.
Note, you CAN'T do a categorically exhaustive reasoning on this
problem, as its is an INFINITE domain. You just don't seem to
understand the nature of infinity, so you don't see that.
The Truth value is just based on the existance of a set on connections
to the axioms. Formal system or not, and NOTHING limits that to being
finite.
even if it is *NOT TRUE IN THE FORMAL SYSTEM*.
On 1/14/23 3:28 PM, olcott wrote:
On 1/13/2023 6:25 PM, Richard Damon wrote:
On 1/13/23 10:42 AM, olcott wrote:In other words you are saying that it is *TRUE IN THE FORMAL SYSTEM*
On 1/12/2023 9:58 PM, Richard Damon wrote:
On 1/12/23 10:52 PM, olcott wrote:
On 1/12/2023 9:35 PM, Richard Damon wrote:
On 1/12/23 10:04 PM, olcott wrote:
On 1/12/2023 8:37 PM, Richard Damon wrote:
On 1/12/23 8:56 PM, olcott wrote:
Because formal systems are only allowed to have finite proofs >>>>>>>>>> formal
systems are not allowed to have infinite connections to their >>>>>>>>>> semantic
truth maker. Thus an expression is only true in a formal
system iff it
is provable within this system. Otherwise this expression is >>>>>>>>>> untrue
which may or may not include false.
Right, FINITE PROOFS, says nothing about TRUTH.
That it ridiculously false. Expressions of language that are
proven to
have a connection to their truth maker axioms are a subset of
all truth
and comprise the entire body of analytical knowledge.
So, since proven statements are a SUBSET of all truth, what says >>>>>>> that all truths have to be proven.
Within a formal system where true requires finite proof to axioms the >>>>>> lack of a finite proof to axioms means untrue.
Source of claim?
How else can it possibly work?
I figure these things out on the basis of categorically exhaustive
reasoning.
Good, so you admit that it isn't based on any REAL theoretical basis
but only due to your limited (and flawed) thinkng ability.
Note, you CAN'T do a categorically exhaustive reasoning on this
problem, as its is an INFINITE domain. You just don't seem to
understand the nature of infinity, so you don't see that.
The Truth value is just based on the existance of a set on
connections to the axioms. Formal system or not, and NOTHING limits
that to being finite.
even if it is *NOT TRUE IN THE FORMAL SYSTEM*.
No, it is TRUE in the formal system, because it has a connection to its
Truth Makers, but is not PROVEN (or even PROVABLE) becaue that
connection in not finte, as required for a PROOF, but not for Truth.
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language of this
formal system is true unless this expression of language has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* ???
Becaue the formal system doesn't need to KNOW what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ???
On 1/14/23 4:48 PM, olcott wrote:
On 1/14/2023 3:16 PM, Richard Damon wrote:
On 1/14/23 3:28 PM, olcott wrote:
On 1/13/2023 6:25 PM, Richard Damon wrote:
On 1/13/23 10:42 AM, olcott wrote:In other words you are saying that it is *TRUE IN THE FORMAL SYSTEM*
On 1/12/2023 9:58 PM, Richard Damon wrote:
On 1/12/23 10:52 PM, olcott wrote:
On 1/12/2023 9:35 PM, Richard Damon wrote:
On 1/12/23 10:04 PM, olcott wrote:
On 1/12/2023 8:37 PM, Richard Damon wrote:
On 1/12/23 8:56 PM, olcott wrote:
Because formal systems are only allowed to have finite >>>>>>>>>>>> proofs formal
systems are not allowed to have infinite connections to >>>>>>>>>>>> their semantic
truth maker. Thus an expression is only true in a formal >>>>>>>>>>>> system iff it
is provable within this system. Otherwise this expression is >>>>>>>>>>>> untrue
which may or may not include false.
Right, FINITE PROOFS, says nothing about TRUTH.
That it ridiculously false. Expressions of language that are >>>>>>>>>> proven to
have a connection to their truth maker axioms are a subset of >>>>>>>>>> all truth
and comprise the entire body of analytical knowledge.
So, since proven statements are a SUBSET of all truth, what
says that all truths have to be proven.
Within a formal system where true requires finite proof to
axioms the
lack of a finite proof to axioms means untrue.
Source of claim?
How else can it possibly work?
I figure these things out on the basis of categorically exhaustive >>>>>> reasoning.
Good, so you admit that it isn't based on any REAL theoretical
basis but only due to your limited (and flawed) thinkng ability.
Note, you CAN'T do a categorically exhaustive reasoning on this
problem, as its is an INFINITE domain. You just don't seem to
understand the nature of infinity, so you don't see that.
The Truth value is just based on the existance of a set on
connections to the axioms. Formal system or not, and NOTHING limits
that to being finite.
even if it is *NOT TRUE IN THE FORMAL SYSTEM*.
No, it is TRUE in the formal system, because it has a connection to
its Truth Makers, but is not PROVEN (or even PROVABLE) becaue that
connection in not finte, as required for a PROOF, but not for Truth.
How does the formal system know that an expression of language of this
formal system is true unless this expression of language has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* ???
Becaue the formal system doesn't need to KNOW what is true.
On 1/14/2023 3:16 PM, Richard Damon wrote:
On 1/14/23 3:28 PM, olcott wrote:
On 1/13/2023 6:25 PM, Richard Damon wrote:
On 1/13/23 10:42 AM, olcott wrote:In other words you are saying that it is *TRUE IN THE FORMAL SYSTEM*
On 1/12/2023 9:58 PM, Richard Damon wrote:
On 1/12/23 10:52 PM, olcott wrote:
On 1/12/2023 9:35 PM, Richard Damon wrote:
On 1/12/23 10:04 PM, olcott wrote:
On 1/12/2023 8:37 PM, Richard Damon wrote:
On 1/12/23 8:56 PM, olcott wrote:
Because formal systems are only allowed to have finite proofs >>>>>>>>>>> formal
systems are not allowed to have infinite connections to their >>>>>>>>>>> semantic
truth maker. Thus an expression is only true in a formal >>>>>>>>>>> system iff it
is provable within this system. Otherwise this expression is >>>>>>>>>>> untrue
which may or may not include false.
Right, FINITE PROOFS, says nothing about TRUTH.
That it ridiculously false. Expressions of language that are >>>>>>>>> proven to
have a connection to their truth maker axioms are a subset of >>>>>>>>> all truth
and comprise the entire body of analytical knowledge.
So, since proven statements are a SUBSET of all truth, what says >>>>>>>> that all truths have to be proven.
Within a formal system where true requires finite proof to axioms >>>>>>> the
lack of a finite proof to axioms means untrue.
Source of claim?
How else can it possibly work?
I figure these things out on the basis of categorically exhaustive
reasoning.
Good, so you admit that it isn't based on any REAL theoretical basis
but only due to your limited (and flawed) thinkng ability.
Note, you CAN'T do a categorically exhaustive reasoning on this
problem, as its is an INFINITE domain. You just don't seem to
understand the nature of infinity, so you don't see that.
The Truth value is just based on the existance of a set on
connections to the axioms. Formal system or not, and NOTHING limits
that to being finite.
even if it is *NOT TRUE IN THE FORMAL SYSTEM*.
No, it is TRUE in the formal system, because it has a connection to
its Truth Makers, but is not PROVEN (or even PROVABLE) becaue that
connection in not finte, as required for a PROOF, but not for Truth.
How does the formal system know that an expression of language of this
formal system is true unless this expression of language has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* ???
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language of this >>>> formal system is true unless this expression of language has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* ???
Becaue the formal system doesn't need to KNOW what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true, doesn't
mean it can't be.
In fact, your statement just comes out of a simple application of the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true.
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language of this >>>>> formal system is true unless this expression of language has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* ???
Becaue the formal system doesn't need to KNOW what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true,
doesn't mean it can't be.
In fact, your statement just comes out of a simple application of the
addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true.
Unless a formal system has a syntactic connection from an expression of
its language to its truth maker axioms the expression is untrue in that formal system.
Try and show an expression of language that is true in a formal system
(not just true somewhere else) that does not have any connection to
truth maker axioms in this formal system. You must show why it is true
in this formal system not merely that it is true somewhere else.
On 1/14/23 6:19 PM, olcott wrote:
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language of
this
formal system is true unless this expression of language has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* ???
Becaue the formal system doesn't need to KNOW what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true,
doesn't mean it can't be.
In fact, your statement just comes out of a simple application of the
addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true.
Unless a formal system has a syntactic connection from an expression of
its language to its truth maker axioms the expression is untrue in that
formal system.
Right, but the connection can be infinite in length, and thus not provable.
Try and show an expression of language that is true in a formal system
(not just true somewhere else) that does not have any connection to
truth maker axioms in this formal system. You must show why it is true
in this formal system not merely that it is true somewhere else.
The connection might be infinite, and thus not SHOWABLE as a proof
strictly in the formal system.
If the connection exists as an infinite connection within the system,
then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can not
be proven within the formal system, it is still possible, that another system, related to that system, with more knowledge, might be able to
show that there does exist within the original formal system such an
infinte connection.
This is what happens to G in F and meta-F
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language of >>>>>>> this
formal system is true unless this expression of language has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* ???
Becaue the formal system doesn't need to KNOW what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true,
doesn't mean it can't be.
In fact, your statement just comes out of a simple application of
the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true.
Unless a formal system has a syntactic connection from an expression of
its language to its truth maker axioms the expression is untrue in that
formal system.
Right, but the connection can be infinite in length, and thus not
provable.
Thus not a connection within the formal system only connection outside
of the formal system.
Try and show an expression of language that is true in a formal system
(not just true somewhere else) that does not have any connection to
truth maker axioms in this formal system. You must show why it is true
in this formal system not merely that it is true somewhere else.
The connection might be infinite, and thus not SHOWABLE as a proof
strictly in the formal system.
If the connection exists as an infinite connection within the system,
then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can not
be proven within the formal system, it is still possible, that another
system, related to that system, with more knowledge, might be able to
show that there does exist within the original formal system such an
infinte connection.
This is what happens to G in F and meta-F
This is you fundamental misunderstanding about the way that truth works.
If the original system cannot possibly show that an expression of
language is true then it is not true in that formal system.
True means that there is a connection to truth maker axioms. True in a
formal system means a connection to truth maker axioms in this formal
system.
To do is the way that you are doing it would mean that homeless Bill is
not homeless because some entirely different person is not homeless.
On 1/14/23 6:58 PM, olcott wrote:
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language >>>>>>>> of this
formal system is true unless this expression of language has a >>>>>>>> connection to truth maker axioms *IN THIS FORMAL SYSTEM* ???
Becaue the formal system doesn't need to KNOW what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ??? >>>>>>
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true,
doesn't mean it can't be.
In fact, your statement just comes out of a simple application of
the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true.
Unless a formal system has a syntactic connection from an expression of >>>> its language to its truth maker axioms the expression is untrue in that >>>> formal system.
Right, but the connection can be infinite in length, and thus not
provable.
Thus not a connection within the formal system only connection outside
of the formal system.
Why? Where does it say "FINITE" connection needed. An infinite set of connections within the formal system IS a connection.
You just don't understand what you are talking about.
You are just speaking your natural language, the language of LIES.
Do you somehow mistakenly think that Formal Logic systems are limited to being finite? The difference between a Formal system and a non-Formal
System is that in a Formal Logic System you begin with the
Formalizatioin, the explicit stating of the rules and axioms that it is
built on.
Unless the formal system EXPLICITLY restricts itself to finite linkage,
by adding an axiom that the only things that are true are those that are provable, such a rule does not exist, and if you add such a rule to a
system, you limit its power or it becomes inconsistent.
Try and show an expression of language that is true in a formal system >>>> (not just true somewhere else) that does not have any connection to
truth maker axioms in this formal system. You must show why it is true >>>> in this formal system not merely that it is true somewhere else.
The connection might be infinite, and thus not SHOWABLE as a proof
strictly in the formal system.
If the connection exists as an infinite connection within the system,
then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can
not be proven within the formal system, it is still possible, that
another system, related to that system, with more knowledge, might be
able to show that there does exist within the original formal system
such an infinte connection.
This is what happens to G in F and meta-F
This is you fundamental misunderstanding about the way that truth works.
If the original system cannot possibly show that an expression of
language is true then it is not true in that formal system.
Nope. SHOWING (as in proving) is different than BEING.
On 1/14/2023 6:30 PM, Richard Damon wrote:
On 1/14/23 6:58 PM, olcott wrote:
This is you fundamental misunderstanding about the way that truth works. >>> If the original system cannot possibly show that an expression of
language is true then it is not true in that formal system.
Nope. SHOWING (as in proving) is different than BEING.
It is not true in PA that "Mary had a little lamb"
Do you understand this ?
Even if we make a Gödel number from the adjacent ASCII characters it is still not true in PA.
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if the truth
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language of >>>>>>> this
formal system is true unless this expression of language has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* ???
Becaue the formal system doesn't need to KNOW what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true,
doesn't mean it can't be.
In fact, your statement just comes out of a simple application of
the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true.
Unless a formal system has a syntactic connection from an expression of
its language to its truth maker axioms the expression is untrue in that
formal system.
Right, but the connection can be infinite in length, and thus not
provable.
Try and show an expression of language that is true in a formal system
(not just true somewhere else) that does not have any connection to
truth maker axioms in this formal system. You must show why it is true
in this formal system not merely that it is true somewhere else.
The connection might be infinite, and thus not SHOWABLE as a proof
strictly in the formal system.
If the connection exists as an infinite connection within the system,
then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can not
be proven within the formal system, it is still possible, that another
system, related to that system, with more knowledge, might be able to
show that there does exist within the original formal system such an
infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets a
specific requirement (expressed as a primative recursive relationship).
This statement turns out to be true, because it turns out that no
number g does meet that requirement, but it can't be proven in F that
this is true, because in F, to show this we need to test every natuarl
number, which requires an infinite number of steps (finite for each
number, but an infinite number of numbers to test).
In meta-F, we can do better, because due to additional knowledge in
meta-F, we can show that if a number g could be found, then that
number g could be converted into a proof, in F, of the statement G
(which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that no
proof of it can exist in F.
of G cannot even be expressed in F as long as the truth of G can be
expressed in meta-F.
On 1/14/23 6:19 PM, olcott wrote:
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language of
this
formal system is true unless this expression of language has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* ???
Becaue the formal system doesn't need to KNOW what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true,
doesn't mean it can't be.
In fact, your statement just comes out of a simple application of the
addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true.
Unless a formal system has a syntactic connection from an expression of
its language to its truth maker axioms the expression is untrue in that
formal system.
Right, but the connection can be infinite in length, and thus not provable.
Try and show an expression of language that is true in a formal system
(not just true somewhere else) that does not have any connection to
truth maker axioms in this formal system. You must show why it is true
in this formal system not merely that it is true somewhere else.
The connection might be infinite, and thus not SHOWABLE as a proof
strictly in the formal system.
If the connection exists as an infinite connection within the system,
then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can not
be proven within the formal system, it is still possible, that another system, related to that system, with more knowledge, might be able to
show that there does exist within the original formal system such an
infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets a
specific requirement (expressed as a primative recursive relationship).
This statement turns out to be true, because it turns out that no number
g does meet that requirement, but it can't be proven in F that this is
true, because in F, to show this we need to test every natuarl number,
which requires an infinite number of steps (finite for each number, but
an infinite number of numbers to test).
In meta-F, we can do better, because due to additional knowledge in
meta-F, we can show that if a number g could be found, then that number
g could be converted into a proof, in F, of the statement G (which says
that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that no
proof of it can exist in F.
On 1/15/23 11:15 AM, olcott wrote:That is false. It is not that G takes an infinite number of steps to
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if the truth
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language >>>>>>>> of this
formal system is true unless this expression of language has a >>>>>>>> connection to truth maker axioms *IN THIS FORMAL SYSTEM* ???
Becaue the formal system doesn't need to KNOW what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ??? >>>>>>
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true,
doesn't mean it can't be.
In fact, your statement just comes out of a simple application of
the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true.
Unless a formal system has a syntactic connection from an expression of >>>> its language to its truth maker axioms the expression is untrue in that >>>> formal system.
Right, but the connection can be infinite in length, and thus not
provable.
Try and show an expression of language that is true in a formal system >>>> (not just true somewhere else) that does not have any connection to
truth maker axioms in this formal system. You must show why it is true >>>> in this formal system not merely that it is true somewhere else.
The connection might be infinite, and thus not SHOWABLE as a proof
strictly in the formal system.
If the connection exists as an infinite connection within the system,
then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can
not be proven within the formal system, it is still possible, that
another system, related to that system, with more knowledge, might be
able to show that there does exist within the original formal system
such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets a
specific requirement (expressed as a primative recursive relationship).
This statement turns out to be true, because it turns out that no
number g does meet that requirement, but it can't be proven in F that
this is true, because in F, to show this we need to test every
natuarl number, which requires an infinite number of steps (finite
for each number, but an infinite number of numbers to test).
In meta-F, we can do better, because due to additional knowledge in
meta-F, we can show that if a number g could be found, then that
number g could be converted into a proof, in F, of the statement G
(which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that no
proof of it can exist in F.
of G cannot even be expressed in F as long as the truth of G can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite expression.
On 1/15/2023 11:41 AM, Richard Damon wrote:
On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:That is false. It is not that G takes an infinite number of steps to
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if the
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of
language of this
formal system is true unless this expression of language has a >>>>>>>>>>> connection to truth maker axioms *IN THIS FORMAL SYSTEM* ??? >>>>>>>>>>>
Becaue the formal system doesn't need to KNOW what is true. >>>>>>>>>>
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ??? >>>>>>>>>
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true, >>>>>>>> doesn't mean it can't be.
In fact, your statement just comes out of a simple application >>>>>>>> of the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true. >>>>>>>>
Unless a formal system has a syntactic connection from an
expression of
its language to its truth maker axioms the expression is untrue
in that
formal system.
Right, but the connection can be infinite in length, and thus not
provable.
Try and show an expression of language that is true in a formal
system
(not just true somewhere else) that does not have any connection to >>>>>>> truth maker axioms in this formal system. You must show why it is >>>>>>> true
in this formal system not merely that it is true somewhere else. >>>>>>>
The connection might be infinite, and thus not SHOWABLE as a proof >>>>>> strictly in the formal system.
If the connection exists as an infinite connection within the
system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can >>>>>> not be proven within the formal system, it is still possible, that >>>>>> another system, related to that system, with more knowledge, might >>>>>> be able to show that there does exist within the original formal
system such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets a >>>>>> specific requirement (expressed as a primative recursive
relationship).
This statement turns out to be true, because it turns out that no
number g does meet that requirement, but it can't be proven in F
that this is true, because in F, to show this we need to test
every natuarl number, which requires an infinite number of steps
(finite for each number, but an infinite number of numbers to test). >>>>>>
In meta-F, we can do better, because due to additional knowledge
in meta-F, we can show that if a number g could be found, then
that number g could be converted into a proof, in F, of the
statement G (which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that
no proof of it can exist in F.
truth
of G cannot even be expressed in F as long as the truth of G can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite
expression.
reach its truth maker axioms in F it is that even after an infinite
number of steps it never reaches is truth maker axioms in F because G is >>> simply untrue in F.
No, YOUR problem is you aren't actually talking about G in F.
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what?
Not true about being not true about being not true.
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be used for a similar
undecidability proof.
By using the Liar Paradox as a Gödel approved proxy for his proof we
refute his proof by this Gödel approved proxy.
No, all you have proved is that you are a LYING MORON.
Ad Hominem attacks are the tactic that people having no interest in any honest dialogue use when they realize that their reasoning has been
utterly defeated.
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:That is false. It is not that G takes an infinite number of steps to
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if the truth >>> of G cannot even be expressed in F as long as the truth of G can be
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language >>>>>>>>> of this
formal system is true unless this expression of language has a >>>>>>>>> connection to truth maker axioms *IN THIS FORMAL SYSTEM* ??? >>>>>>>>>
Becaue the formal system doesn't need to KNOW what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ??? >>>>>>>
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true,
doesn't mean it can't be.
In fact, your statement just comes out of a simple application of
the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true. >>>>>>
Unless a formal system has a syntactic connection from an
expression of
its language to its truth maker axioms the expression is untrue in
that
formal system.
Right, but the connection can be infinite in length, and thus not
provable.
Try and show an expression of language that is true in a formal system >>>>> (not just true somewhere else) that does not have any connection to
truth maker axioms in this formal system. You must show why it is true >>>>> in this formal system not merely that it is true somewhere else.
The connection might be infinite, and thus not SHOWABLE as a proof
strictly in the formal system.
If the connection exists as an infinite connection within the
system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can
not be proven within the formal system, it is still possible, that
another system, related to that system, with more knowledge, might
be able to show that there does exist within the original formal
system such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets a
specific requirement (expressed as a primative recursive relationship). >>>>
This statement turns out to be true, because it turns out that no
number g does meet that requirement, but it can't be proven in F
that this is true, because in F, to show this we need to test every
natuarl number, which requires an infinite number of steps (finite
for each number, but an infinite number of numbers to test).
In meta-F, we can do better, because due to additional knowledge in
meta-F, we can show that if a number g could be found, then that
number g could be converted into a proof, in F, of the statement G
(which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that no
proof of it can exist in F.
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite expression.
reach its truth maker axioms in F it is that even after an infinite
number of steps it never reaches is truth maker axioms in F because G is simply untrue in F.
"This sentence is not true"
Not true about what?
Not true about being not true.
Not true about being not true about what?
Not true about being not true about being not true.
Since Gödel said:
14 Every epistemological antinomy can likewise be used for a similar undecidability proof.
By using the Liar Paradox as a Gödel approved proxy for his proof we
refute his proof by this Gödel approved proxy.
On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:That is false. It is not that G takes an infinite number of steps to
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if the
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language >>>>>>>>>> of this
formal system is true unless this expression of language has a >>>>>>>>>> connection to truth maker axioms *IN THIS FORMAL SYSTEM* ??? >>>>>>>>>>
Becaue the formal system doesn't need to KNOW what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ??? >>>>>>>>
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true,
doesn't mean it can't be.
In fact, your statement just comes out of a simple application of >>>>>>> the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true. >>>>>>>
Unless a formal system has a syntactic connection from an
expression of
its language to its truth maker axioms the expression is untrue in >>>>>> that
formal system.
Right, but the connection can be infinite in length, and thus not
provable.
Try and show an expression of language that is true in a formal
system
(not just true somewhere else) that does not have any connection to >>>>>> truth maker axioms in this formal system. You must show why it is
true
in this formal system not merely that it is true somewhere else.
The connection might be infinite, and thus not SHOWABLE as a proof
strictly in the formal system.
If the connection exists as an infinite connection within the
system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can
not be proven within the formal system, it is still possible, that
another system, related to that system, with more knowledge, might
be able to show that there does exist within the original formal
system such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets a
specific requirement (expressed as a primative recursive
relationship).
This statement turns out to be true, because it turns out that no
number g does meet that requirement, but it can't be proven in F
that this is true, because in F, to show this we need to test every
natuarl number, which requires an infinite number of steps (finite
for each number, but an infinite number of numbers to test).
In meta-F, we can do better, because due to additional knowledge in
meta-F, we can show that if a number g could be found, then that
number g could be converted into a proof, in F, of the statement G
(which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that no
proof of it can exist in F.
truth
of G cannot even be expressed in F as long as the truth of G can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite expression.
reach its truth maker axioms in F it is that even after an infinite
number of steps it never reaches is truth maker axioms in F because G is
simply untrue in F.
No, YOUR problem is you aren't actually talking about G in F.
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what?
Not true about being not true about being not true.
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be used for a similar
undecidability proof.
By using the Liar Paradox as a Gödel approved proxy for his proof we
refute his proof by this Gödel approved proxy.
No, all you have proved is that you are a LYING MORON.
On 1/15/2023 11:58 AM, Richard Damon wrote:
On 1/15/23 12:47 PM, olcott wrote:All of your Ad Hominem attacks cannot possibly hide the fact that you
On 1/15/2023 11:41 AM, Richard Damon wrote:
On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:That is false. It is not that G takes an infinite number of steps to >>>>> reach its truth maker axioms in F it is that even after an infinite
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of >>>>>>>>>>>>> language of this
formal system is true unless this expression of language has a >>>>>>>>>>>>> connection to truth maker axioms *IN THIS FORMAL SYSTEM* ??? >>>>>>>>>>>>>
Becaue the formal system doesn't need to KNOW what is true. >>>>>>>>>>>>
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is >>>>>>>>>>> true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be
true, doesn't mean it can't be.
In fact, your statement just comes out of a simple application >>>>>>>>>> of the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be >>>>>>>>>> true.
Unless a formal system has a syntactic connection from an
expression of
its language to its truth maker axioms the expression is untrue >>>>>>>>> in that
formal system.
Right, but the connection can be infinite in length, and thus
not provable.
Try and show an expression of language that is true in a formal >>>>>>>>> system
(not just true somewhere else) that does not have any
connection to
truth maker axioms in this formal system. You must show why it >>>>>>>>> is true
in this formal system not merely that it is true somewhere else. >>>>>>>>>
The connection might be infinite, and thus not SHOWABLE as a
proof strictly in the formal system.
If the connection exists as an infinite connection within the
system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus >>>>>>>> can not be proven within the formal system, it is still
possible, that another system, related to that system, with more >>>>>>>> knowledge, might be able to show that there does exist within
the original formal system such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets >>>>>>>> a specific requirement (expressed as a primative recursive
relationship).
This statement turns out to be true, because it turns out that >>>>>>>> no number g does meet that requirement, but it can't be proven >>>>>>>> in F that this is true, because in F, to show this we need to
test every natuarl number, which requires an infinite number of >>>>>>>> steps (finite for each number, but an infinite number of numbers >>>>>>>> to test).
In meta-F, we can do better, because due to additional knowledge >>>>>>>> in meta-F, we can show that if a number g could be found, then >>>>>>>> that number g could be converted into a proof, in F, of the
statement G (which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that >>>>>>>> no proof of it can exist in F.
the truth
of G cannot even be expressed in F as long as the truth of G can be >>>>>>> expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite
expression.
number of steps it never reaches is truth maker axioms in F because
G is
simply untrue in F.
No, YOUR problem is you aren't actually talking about G in F.
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what?
Not true about being not true about being not true.
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be used for a
similar undecidability proof.
By using the Liar Paradox as a Gödel approved proxy for his proof we >>>>> refute his proof by this Gödel approved proxy.
No, all you have proved is that you are a LYING MORON.
Ad Hominem attacks are the tactic that people having no interest in any
honest dialogue use when they realize that their reasoning has been
utterly defeated.
And RED HERRING arguements don't work either.
You ARE a LYING MORON as you insist that Godel's G is a sentence that
is actually provably isn't.
You think it is because you are too stupid to actually read any of the
paper, so you take that comment that the statment is "based" on that
statement to mean it IS that statement.
are asserting this counter-factual statement:
when a valid proxy for an argument is defeated this does not defeat the original argument.
On 1/15/23 12:47 PM, olcott wrote:
On 1/15/2023 11:41 AM, Richard Damon wrote:
On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:That is false. It is not that G takes an infinite number of steps to
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if the >>>>>> truth
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of
language of this
formal system is true unless this expression of language has a >>>>>>>>>>>> connection to truth maker axioms *IN THIS FORMAL SYSTEM* ??? >>>>>>>>>>>>
Becaue the formal system doesn't need to KNOW what is true. >>>>>>>>>>>
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true >>>>>>>>>> ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true, >>>>>>>>> doesn't mean it can't be.
In fact, your statement just comes out of a simple application >>>>>>>>> of the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true. >>>>>>>>>
Unless a formal system has a syntactic connection from an
expression of
its language to its truth maker axioms the expression is untrue >>>>>>>> in that
formal system.
Right, but the connection can be infinite in length, and thus not >>>>>>> provable.
Try and show an expression of language that is true in a formal >>>>>>>> system
(not just true somewhere else) that does not have any connection to >>>>>>>> truth maker axioms in this formal system. You must show why it >>>>>>>> is true
in this formal system not merely that it is true somewhere else. >>>>>>>>
The connection might be infinite, and thus not SHOWABLE as a
proof strictly in the formal system.
If the connection exists as an infinite connection within the
system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus
can not be proven within the formal system, it is still possible, >>>>>>> that another system, related to that system, with more knowledge, >>>>>>> might be able to show that there does exist within the original
formal system such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets >>>>>>> a specific requirement (expressed as a primative recursive
relationship).
This statement turns out to be true, because it turns out that no >>>>>>> number g does meet that requirement, but it can't be proven in F >>>>>>> that this is true, because in F, to show this we need to test
every natuarl number, which requires an infinite number of steps >>>>>>> (finite for each number, but an infinite number of numbers to test). >>>>>>>
In meta-F, we can do better, because due to additional knowledge >>>>>>> in meta-F, we can show that if a number g could be found, then
that number g could be converted into a proof, in F, of the
statement G (which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that >>>>>>> no proof of it can exist in F.
of G cannot even be expressed in F as long as the truth of G can be >>>>>> expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite
expression.
reach its truth maker axioms in F it is that even after an infinite
number of steps it never reaches is truth maker axioms in F because
G is
simply untrue in F.
No, YOUR problem is you aren't actually talking about G in F.
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what?
Not true about being not true about being not true.
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be used for a similar
undecidability proof.
By using the Liar Paradox as a Gödel approved proxy for his proof we
refute his proof by this Gödel approved proxy.
No, all you have proved is that you are a LYING MORON.
Ad Hominem attacks are the tactic that people having no interest in any
honest dialogue use when they realize that their reasoning has been
utterly defeated.
And RED HERRING arguements don't work either.
You ARE a LYING MORON as you insist that Godel's G is a sentence that is actually provably isn't.
You think it is because you are too stupid to actually read any of the
paper, so you take that comment that the statment is "based" on that statement to mean it IS that statement.
On 1/15/23 1:23 PM, olcott wrote:
On 1/15/2023 11:58 AM, Richard Damon wrote:
On 1/15/23 12:47 PM, olcott wrote:All of your Ad Hominem attacks cannot possibly hide the fact that you
On 1/15/2023 11:41 AM, Richard Damon wrote:
On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:That is false. It is not that G takes an infinite number of steps to >>>>>> reach its truth maker axioms in F it is that even after an infinite >>>>>> number of steps it never reaches is truth maker axioms in F
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if >>>>>>>> the truth
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of >>>>>>>>>>>>>> language of this
formal system is true unless this expression of language >>>>>>>>>>>>>> has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* ??? >>>>>>>>>>>>>>
Becaue the formal system doesn't need to KNOW what is true. >>>>>>>>>>>>>
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is >>>>>>>>>>>> true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be >>>>>>>>>>> true, doesn't mean it can't be.
In fact, your statement just comes out of a simple
application of the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be >>>>>>>>>>> true.
Unless a formal system has a syntactic connection from an
expression of
its language to its truth maker axioms the expression is
untrue in that
formal system.
Right, but the connection can be infinite in length, and thus >>>>>>>>> not provable.
Try and show an expression of language that is true in a
formal system
(not just true somewhere else) that does not have any
connection to
truth maker axioms in this formal system. You must show why it >>>>>>>>>> is true
in this formal system not merely that it is true somewhere else. >>>>>>>>>>
The connection might be infinite, and thus not SHOWABLE as a >>>>>>>>> proof strictly in the formal system.
If the connection exists as an infinite connection within the >>>>>>>>> system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus >>>>>>>>> can not be proven within the formal system, it is still
possible, that another system, related to that system, with
more knowledge, might be able to show that there does exist
within the original formal system such an infinte connection. >>>>>>>>>
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that
meets a specific requirement (expressed as a primative
recursive relationship).
This statement turns out to be true, because it turns out that >>>>>>>>> no number g does meet that requirement, but it can't be proven >>>>>>>>> in F that this is true, because in F, to show this we need to >>>>>>>>> test every natuarl number, which requires an infinite number of >>>>>>>>> steps (finite for each number, but an infinite number of
numbers to test).
In meta-F, we can do better, because due to additional
knowledge in meta-F, we can show that if a number g could be >>>>>>>>> found, then that number g could be converted into a proof, in >>>>>>>>> F, of the statement G (which says that such a number does not >>>>>>>>> exist).
Thus, in meta-F, we can prove that G is true, and also show
that no proof of it can exist in F.
of G cannot even be expressed in F as long as the truth of G can be >>>>>>>> expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite
expression.
because G is
simply untrue in F.
No, YOUR problem is you aren't actually talking about G in F.
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what?
Not true about being not true about being not true.
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be used for a
similar undecidability proof.
By using the Liar Paradox as a Gödel approved proxy for his proof we >>>>>> refute his proof by this Gödel approved proxy.
No, all you have proved is that you are a LYING MORON.
Ad Hominem attacks are the tactic that people having no interest in any >>>> honest dialogue use when they realize that their reasoning has been
utterly defeated.
And RED HERRING arguements don't work either.
You ARE a LYING MORON as you insist that Godel's G is a sentence that
is actually provably isn't.
You think it is because you are too stupid to actually read any of
the paper, so you take that comment that the statment is "based" on
that statement to mean it IS that statement.
are asserting this counter-factual statement:
when a valid proxy for an argument is defeated this does not defeat the
original argument.
Right, which is what YOU are doing, showing your arguement is INVALID.
On 1/15/2023 12:55 PM, Richard Damon wrote:
On 1/15/23 1:23 PM, olcott wrote:
On 1/15/2023 11:58 AM, Richard Damon wrote:
On 1/15/23 12:47 PM, olcott wrote:All of your Ad Hominem attacks cannot possibly hide the fact that you
On 1/15/2023 11:41 AM, Richard Damon wrote:
On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:That is false. It is not that G takes an infinite number of steps to >>>>>>> reach its truth maker axioms in F it is that even after an infinite >>>>>>> number of steps it never reaches is truth maker axioms in F
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if >>>>>>>>> the truth
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of >>>>>>>>>>>>>>> language of this
formal system is true unless this expression of language >>>>>>>>>>>>>>> has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* ??? >>>>>>>>>>>>>>>
Becaue the formal system doesn't need to KNOW what is true. >>>>>>>>>>>>>>
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is >>>>>>>>>>>>> true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be >>>>>>>>>>>> true, doesn't mean it can't be.
In fact, your statement just comes out of a simple
application of the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be >>>>>>>>>>>> true.
Unless a formal system has a syntactic connection from an >>>>>>>>>>> expression of
its language to its truth maker axioms the expression is >>>>>>>>>>> untrue in that
formal system.
Right, but the connection can be infinite in length, and thus >>>>>>>>>> not provable.
Try and show an expression of language that is true in a >>>>>>>>>>> formal system
(not just true somewhere else) that does not have any
connection to
truth maker axioms in this formal system. You must show why >>>>>>>>>>> it is true
in this formal system not merely that it is true somewhere else. >>>>>>>>>>>
The connection might be infinite, and thus not SHOWABLE as a >>>>>>>>>> proof strictly in the formal system.
If the connection exists as an infinite connection within the >>>>>>>>>> system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus >>>>>>>>>> can not be proven within the formal system, it is still
possible, that another system, related to that system, with >>>>>>>>>> more knowledge, might be able to show that there does exist >>>>>>>>>> within the original formal system such an infinte connection. >>>>>>>>>>
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that >>>>>>>>>> meets a specific requirement (expressed as a primative
recursive relationship).
This statement turns out to be true, because it turns out that >>>>>>>>>> no number g does meet that requirement, but it can't be proven >>>>>>>>>> in F that this is true, because in F, to show this we need to >>>>>>>>>> test every natuarl number, which requires an infinite number >>>>>>>>>> of steps (finite for each number, but an infinite number of >>>>>>>>>> numbers to test).
In meta-F, we can do better, because due to additional
knowledge in meta-F, we can show that if a number g could be >>>>>>>>>> found, then that number g could be converted into a proof, in >>>>>>>>>> F, of the statement G (which says that such a number does not >>>>>>>>>> exist).
Thus, in meta-F, we can prove that G is true, and also show >>>>>>>>>> that no proof of it can exist in F.
of G cannot even be expressed in F as long as the truth of G >>>>>>>>> can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite
expression.
because G is
simply untrue in F.
No, YOUR problem is you aren't actually talking about G in F.
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what?
Not true about being not true about being not true.
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be used for a
similar undecidability proof.
By using the Liar Paradox as a Gödel approved proxy for his proof we >>>>>>> refute his proof by this Gödel approved proxy.
No, all you have proved is that you are a LYING MORON.
Ad Hominem attacks are the tactic that people having no interest in
any
honest dialogue use when they realize that their reasoning has been
utterly defeated.
And RED HERRING arguements don't work either.
You ARE a LYING MORON as you insist that Godel's G is a sentence
that is actually provably isn't.
You think it is because you are too stupid to actually read any of
the paper, so you take that comment that the statment is "based" on
that statement to mean it IS that statement.
are asserting this counter-factual statement:
when a valid proxy for an argument is defeated this does not defeat the
original argument.
Right, which is what YOU are doing, showing your arguement is INVALID.
In other words you disagree that correctly refuting a valid proxy for an argument does correctly refute the original argument?
On 1/15/23 1:23 PM, olcott wrote:
On 1/15/2023 11:58 AM, Richard Damon wrote:
On 1/15/23 12:47 PM, olcott wrote:All of your Ad Hominem attacks cannot possibly hide the fact that you
On 1/15/2023 11:41 AM, Richard Damon wrote:
On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:That is false. It is not that G takes an infinite number of steps to >>>>>> reach its truth maker axioms in F it is that even after an infinite >>>>>> number of steps it never reaches is truth maker axioms in F
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if >>>>>>>> the truth
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of >>>>>>>>>>>>>> language of this
formal system is true unless this expression of language >>>>>>>>>>>>>> has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* ??? >>>>>>>>>>>>>>
Becaue the formal system doesn't need to KNOW what is true. >>>>>>>>>>>>>
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is >>>>>>>>>>>> true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be >>>>>>>>>>> true, doesn't mean it can't be.
In fact, your statement just comes out of a simple
application of the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be >>>>>>>>>>> true.
Unless a formal system has a syntactic connection from an
expression of
its language to its truth maker axioms the expression is
untrue in that
formal system.
Right, but the connection can be infinite in length, and thus >>>>>>>>> not provable.
Try and show an expression of language that is true in a
formal system
(not just true somewhere else) that does not have any
connection to
truth maker axioms in this formal system. You must show why it >>>>>>>>>> is true
in this formal system not merely that it is true somewhere else. >>>>>>>>>>
The connection might be infinite, and thus not SHOWABLE as a >>>>>>>>> proof strictly in the formal system.
If the connection exists as an infinite connection within the >>>>>>>>> system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus >>>>>>>>> can not be proven within the formal system, it is still
possible, that another system, related to that system, with
more knowledge, might be able to show that there does exist
within the original formal system such an infinte connection. >>>>>>>>>
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that
meets a specific requirement (expressed as a primative
recursive relationship).
This statement turns out to be true, because it turns out that >>>>>>>>> no number g does meet that requirement, but it can't be proven >>>>>>>>> in F that this is true, because in F, to show this we need to >>>>>>>>> test every natuarl number, which requires an infinite number of >>>>>>>>> steps (finite for each number, but an infinite number of
numbers to test).
In meta-F, we can do better, because due to additional
knowledge in meta-F, we can show that if a number g could be >>>>>>>>> found, then that number g could be converted into a proof, in >>>>>>>>> F, of the statement G (which says that such a number does not >>>>>>>>> exist).
Thus, in meta-F, we can prove that G is true, and also show
that no proof of it can exist in F.
of G cannot even be expressed in F as long as the truth of G can be >>>>>>>> expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite
expression.
because G is
simply untrue in F.
No, YOUR problem is you aren't actually talking about G in F.
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what?
Not true about being not true about being not true.
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be used for a
similar undecidability proof.
By using the Liar Paradox as a Gödel approved proxy for his proof we >>>>>> refute his proof by this Gödel approved proxy.
No, all you have proved is that you are a LYING MORON.
Ad Hominem attacks are the tactic that people having no interest in any >>>> honest dialogue use when they realize that their reasoning has been
utterly defeated.
And RED HERRING arguements don't work either.
You ARE a LYING MORON as you insist that Godel's G is a sentence that
is actually provably isn't.
You think it is because you are too stupid to actually read any of
the paper, so you take that comment that the statment is "based" on
that statement to mean it IS that statement.
are asserting this counter-factual statement:
when a valid proxy for an argument is defeated this does not defeat the
original argument.
Right, which is what YOU are doing, showing your arguement is INVALID.
On 1/15/23 2:00 PM, olcott wrote:So Gödel is wrong when he says:
On 1/15/2023 12:55 PM, Richard Damon wrote:
On 1/15/23 1:23 PM, olcott wrote:
On 1/15/2023 11:58 AM, Richard Damon wrote:
On 1/15/23 12:47 PM, olcott wrote:All of your Ad Hominem attacks cannot possibly hide the fact that you
On 1/15/2023 11:41 AM, Richard Damon wrote:
On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:That is false. It is not that G takes an infinite number of
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if >>>>>>>>>> the truth
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of >>>>>>>>>>>>>>>> language of this
formal system is true unless this expression of language >>>>>>>>>>>>>>>> has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* >>>>>>>>>>>>>>>> ???
Becaue the formal system doesn't need to KNOW what is true. >>>>>>>>>>>>>>>
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is >>>>>>>>>>>>>> true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be >>>>>>>>>>>>> true, doesn't mean it can't be.
In fact, your statement just comes out of a simple
application of the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to >>>>>>>>>>>>> be true.
Unless a formal system has a syntactic connection from an >>>>>>>>>>>> expression of
its language to its truth maker axioms the expression is >>>>>>>>>>>> untrue in that
formal system.
Right, but the connection can be infinite in length, and thus >>>>>>>>>>> not provable.
Try and show an expression of language that is true in a >>>>>>>>>>>> formal system
(not just true somewhere else) that does not have any
connection to
truth maker axioms in this formal system. You must show why >>>>>>>>>>>> it is true
in this formal system not merely that it is true somewhere >>>>>>>>>>>> else.
The connection might be infinite, and thus not SHOWABLE as a >>>>>>>>>>> proof strictly in the formal system.
If the connection exists as an infinite connection within the >>>>>>>>>>> system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which >>>>>>>>>>> thus can not be proven within the formal system, it is still >>>>>>>>>>> possible, that another system, related to that system, with >>>>>>>>>>> more knowledge, might be able to show that there does exist >>>>>>>>>>> within the original formal system such an infinte connection. >>>>>>>>>>>
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that >>>>>>>>>>> meets a specific requirement (expressed as a primative
recursive relationship).
This statement turns out to be true, because it turns out >>>>>>>>>>> that no number g does meet that requirement, but it can't be >>>>>>>>>>> proven in F that this is true, because in F, to show this we >>>>>>>>>>> need to test every natuarl number, which requires an infinite >>>>>>>>>>> number of steps (finite for each number, but an infinite >>>>>>>>>>> number of numbers to test).
In meta-F, we can do better, because due to additional
knowledge in meta-F, we can show that if a number g could be >>>>>>>>>>> found, then that number g could be converted into a proof, in >>>>>>>>>>> F, of the statement G (which says that such a number does not >>>>>>>>>>> exist).
Thus, in meta-F, we can prove that G is true, and also show >>>>>>>>>>> that no proof of it can exist in F.
of G cannot even be expressed in F as long as the truth of G >>>>>>>>>> can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite >>>>>>>>> expression.
steps to
reach its truth maker axioms in F it is that even after an infinite >>>>>>>> number of steps it never reaches is truth maker axioms in F
because G is
simply untrue in F.
No, YOUR problem is you aren't actually talking about G in F.
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what?
Not true about being not true about being not true.
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be used for a
similar undecidability proof.
By using the Liar Paradox as a Gödel approved proxy for his
proof we
refute his proof by this Gödel approved proxy.
No, all you have proved is that you are a LYING MORON.
Ad Hominem attacks are the tactic that people having no interest
in any
honest dialogue use when they realize that their reasoning has been >>>>>> utterly defeated.
And RED HERRING arguements don't work either.
You ARE a LYING MORON as you insist that Godel's G is a sentence
that is actually provably isn't.
You think it is because you are too stupid to actually read any of
the paper, so you take that comment that the statment is "based" on
that statement to mean it IS that statement.
are asserting this counter-factual statement:
when a valid proxy for an argument is defeated this does not defeat the >>>> original argument.
Right, which is what YOU are doing, showing your arguement is INVALID.
In other words you disagree that correctly refuting a valid proxy for an
argument does correctly refute the original argument?
VALID is the key word,
Yours isn't (I don't think you actually know the meaning of the words)
And you are an IDIOT to claim it is.
On 1/15/2023 12:55 PM, Richard Damon wrote:
On 1/15/23 1:23 PM, olcott wrote:In other words you disagree that correctly refuting a valid proxy for an argument does correctly refute the original argument?
On 1/15/2023 11:58 AM, Richard Damon wrote:
On 1/15/23 12:47 PM, olcott wrote:All of your Ad Hominem attacks cannot possibly hide the fact that you
On 1/15/2023 11:41 AM, Richard Damon wrote:
On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:That is false. It is not that G takes an infinite number of steps to >>>>>>> reach its truth maker axioms in F it is that even after an infinite >>>>>>> number of steps it never reaches is truth maker axioms in F
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if >>>>>>>>> the truth
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of >>>>>>>>>>>>>>> language of this
formal system is true unless this expression of language >>>>>>>>>>>>>>> has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* ??? >>>>>>>>>>>>>>>
Becaue the formal system doesn't need to KNOW what is true. >>>>>>>>>>>>>>
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is >>>>>>>>>>>>> true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be >>>>>>>>>>>> true, doesn't mean it can't be.
In fact, your statement just comes out of a simple
application of the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be >>>>>>>>>>>> true.
Unless a formal system has a syntactic connection from an >>>>>>>>>>> expression of
its language to its truth maker axioms the expression is >>>>>>>>>>> untrue in that
formal system.
Right, but the connection can be infinite in length, and thus >>>>>>>>>> not provable.
Try and show an expression of language that is true in a >>>>>>>>>>> formal system
(not just true somewhere else) that does not have any
connection to
truth maker axioms in this formal system. You must show why >>>>>>>>>>> it is true
in this formal system not merely that it is true somewhere else. >>>>>>>>>>>
The connection might be infinite, and thus not SHOWABLE as a >>>>>>>>>> proof strictly in the formal system.
If the connection exists as an infinite connection within the >>>>>>>>>> system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus >>>>>>>>>> can not be proven within the formal system, it is still
possible, that another system, related to that system, with >>>>>>>>>> more knowledge, might be able to show that there does exist >>>>>>>>>> within the original formal system such an infinte connection. >>>>>>>>>>
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that >>>>>>>>>> meets a specific requirement (expressed as a primative
recursive relationship).
This statement turns out to be true, because it turns out that >>>>>>>>>> no number g does meet that requirement, but it can't be proven >>>>>>>>>> in F that this is true, because in F, to show this we need to >>>>>>>>>> test every natuarl number, which requires an infinite number >>>>>>>>>> of steps (finite for each number, but an infinite number of >>>>>>>>>> numbers to test).
In meta-F, we can do better, because due to additional
knowledge in meta-F, we can show that if a number g could be >>>>>>>>>> found, then that number g could be converted into a proof, in >>>>>>>>>> F, of the statement G (which says that such a number does not >>>>>>>>>> exist).
Thus, in meta-F, we can prove that G is true, and also show >>>>>>>>>> that no proof of it can exist in F.
of G cannot even be expressed in F as long as the truth of G >>>>>>>>> can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite
expression.
because G is
simply untrue in F.
No, YOUR problem is you aren't actually talking about G in F.
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what?
Not true about being not true about being not true.
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be used for a
similar undecidability proof.
By using the Liar Paradox as a Gödel approved proxy for his proof we >>>>>>> refute his proof by this Gödel approved proxy.
No, all you have proved is that you are a LYING MORON.
Ad Hominem attacks are the tactic that people having no interest in
any
honest dialogue use when they realize that their reasoning has been
utterly defeated.
And RED HERRING arguements don't work either.
You ARE a LYING MORON as you insist that Godel's G is a sentence
that is actually provably isn't.
You think it is because you are too stupid to actually read any of
the paper, so you take that comment that the statment is "based" on
that statement to mean it IS that statement.
are asserting this counter-factual statement:
when a valid proxy for an argument is defeated this does not defeat the
original argument.
Right, which is what YOU are doing, showing your arguement is INVALID.
On 1/15/23 2:29 PM, olcott wrote:
On 1/15/2023 1:06 PM, Richard Damon wrote:
On 1/15/23 2:00 PM, olcott wrote:So Gödel is wrong when he says:
On 1/15/2023 12:55 PM, Richard Damon wrote:
On 1/15/23 1:23 PM, olcott wrote:
On 1/15/2023 11:58 AM, Richard Damon wrote:
On 1/15/23 12:47 PM, olcott wrote:All of your Ad Hominem attacks cannot possibly hide the fact that you >>>>>> are asserting this counter-factual statement:
On 1/15/2023 11:41 AM, Richard Damon wrote:
On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:That is false. It is not that G takes an infinite number of >>>>>>>>>> steps to
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even >>>>>>>>>>>> if the truth
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of >>>>>>>>>>>>>>>>>> language of this
formal system is true unless this expression of >>>>>>>>>>>>>>>>>> language has a
connection to truth maker axioms *IN THIS FORMAL >>>>>>>>>>>>>>>>>> SYSTEM* ???
Becaue the formal system doesn't need to KNOW what is >>>>>>>>>>>>>>>>> true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) >>>>>>>>>>>>>>>> is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be >>>>>>>>>>>>>>> true, doesn't mean it can't be.
In fact, your statement just comes out of a simple >>>>>>>>>>>>>>> application of the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to >>>>>>>>>>>>>>> be true.
Unless a formal system has a syntactic connection from an >>>>>>>>>>>>>> expression of
its language to its truth maker axioms the expression is >>>>>>>>>>>>>> untrue in that
formal system.
Right, but the connection can be infinite in length, and >>>>>>>>>>>>> thus not provable.
Try and show an expression of language that is true in a >>>>>>>>>>>>>> formal system
(not just true somewhere else) that does not have any >>>>>>>>>>>>>> connection to
truth maker axioms in this formal system. You must show >>>>>>>>>>>>>> why it is true
in this formal system not merely that it is true somewhere >>>>>>>>>>>>>> else.
The connection might be infinite, and thus not SHOWABLE as >>>>>>>>>>>>> a proof strictly in the formal system.
If the connection exists as an infinite connection within >>>>>>>>>>>>> the system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which >>>>>>>>>>>>> thus can not be proven within the formal system, it is >>>>>>>>>>>>> still possible, that another system, related to that >>>>>>>>>>>>> system, with more knowledge, might be able to show that >>>>>>>>>>>>> there does exist within the original formal system such an >>>>>>>>>>>>> infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that >>>>>>>>>>>>> meets a specific requirement (expressed as a primative >>>>>>>>>>>>> recursive relationship).
This statement turns out to be true, because it turns out >>>>>>>>>>>>> that no number g does meet that requirement, but it can't >>>>>>>>>>>>> be proven in F that this is true, because in F, to show >>>>>>>>>>>>> this we need to test every natuarl number, which requires >>>>>>>>>>>>> an infinite number of steps (finite for each number, but an >>>>>>>>>>>>> infinite number of numbers to test).
In meta-F, we can do better, because due to additional >>>>>>>>>>>>> knowledge in meta-F, we can show that if a number g could >>>>>>>>>>>>> be found, then that number g could be converted into a >>>>>>>>>>>>> proof, in F, of the statement G (which says that such a >>>>>>>>>>>>> number does not exist).
Thus, in meta-F, we can prove that G is true, and also show >>>>>>>>>>>>> that no proof of it can exist in F.
of G cannot even be expressed in F as long as the truth of G >>>>>>>>>>>> can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite >>>>>>>>>>> expression.
reach its truth maker axioms in F it is that even after an >>>>>>>>>> infinite
number of steps it never reaches is truth maker axioms in F >>>>>>>>>> because G is
simply untrue in F.
No, YOUR problem is you aren't actually talking about G in F. >>>>>>>>>
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what?
Not true about being not true about being not true.
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be used for a >>>>>>>>>> similar undecidability proof.
By using the Liar Paradox as a Gödel approved proxy for his >>>>>>>>>> proof we
refute his proof by this Gödel approved proxy.
No, all you have proved is that you are a LYING MORON.
Ad Hominem attacks are the tactic that people having no interest >>>>>>>> in any
honest dialogue use when they realize that their reasoning has been >>>>>>>> utterly defeated.
And RED HERRING arguements don't work either.
You ARE a LYING MORON as you insist that Godel's G is a sentence >>>>>>> that is actually provably isn't.
You think it is because you are too stupid to actually read any
of the paper, so you take that comment that the statment is
"based" on that statement to mean it IS that statement.
when a valid proxy for an argument is defeated this does not
defeat the
original argument.
Right, which is what YOU are doing, showing your arguement is INVALID. >>>>>
In other words you disagree that correctly refuting a valid proxy
for an
argument does correctly refute the original argument?
VALID is the key word,
Yours isn't (I don't think you actually know the meaning of the words)
And you are an IDIOT to claim it is.
14 Every epistemological antinomy can likewise be used for a
similar undecidability proof.
No, but you don't understand what he is saying.
On 1/15/2023 1:46 PM, Richard Damon wrote:
On 1/15/23 2:29 PM, olcott wrote:He is saying that every epistemological antinomy is a valid proxy for
On 1/15/2023 1:06 PM, Richard Damon wrote:
On 1/15/23 2:00 PM, olcott wrote:So Gödel is wrong when he says:
On 1/15/2023 12:55 PM, Richard Damon wrote:
On 1/15/23 1:23 PM, olcott wrote:
On 1/15/2023 11:58 AM, Richard Damon wrote:
On 1/15/23 12:47 PM, olcott wrote:All of your Ad Hominem attacks cannot possibly hide the fact that >>>>>>> you
On 1/15/2023 11:41 AM, Richard Damon wrote:
On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:That is false. It is not that G takes an infinite number of >>>>>>>>>>> steps to
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even >>>>>>>>>>>>> if the truth
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of >>>>>>>>>>>>>>>>>>> language of this
formal system is true unless this expression of >>>>>>>>>>>>>>>>>>> language has a
connection to truth maker axioms *IN THIS FORMAL >>>>>>>>>>>>>>>>>>> SYSTEM* ???
Becaue the formal system doesn't need to KNOW what is >>>>>>>>>>>>>>>>>> true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) >>>>>>>>>>>>>>>>> is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to >>>>>>>>>>>>>>>> be true, doesn't mean it can't be.
In fact, your statement just comes out of a simple >>>>>>>>>>>>>>>> application of the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN >>>>>>>>>>>>>>>> to be true.
Unless a formal system has a syntactic connection from an >>>>>>>>>>>>>>> expression of
its language to its truth maker axioms the expression is >>>>>>>>>>>>>>> untrue in that
formal system.
Right, but the connection can be infinite in length, and >>>>>>>>>>>>>> thus not provable.
Try and show an expression of language that is true in a >>>>>>>>>>>>>>> formal system
(not just true somewhere else) that does not have any >>>>>>>>>>>>>>> connection to
truth maker axioms in this formal system. You must show >>>>>>>>>>>>>>> why it is true
in this formal system not merely that it is true >>>>>>>>>>>>>>> somewhere else.
The connection might be infinite, and thus not SHOWABLE as >>>>>>>>>>>>>> a proof strictly in the formal system.
If the connection exists as an infinite connection within >>>>>>>>>>>>>> the system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which >>>>>>>>>>>>>> thus can not be proven within the formal system, it is >>>>>>>>>>>>>> still possible, that another system, related to that >>>>>>>>>>>>>> system, with more knowledge, might be able to show that >>>>>>>>>>>>>> there does exist within the original formal system such an >>>>>>>>>>>>>> infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that >>>>>>>>>>>>>> meets a specific requirement (expressed as a primative >>>>>>>>>>>>>> recursive relationship).
This statement turns out to be true, because it turns out >>>>>>>>>>>>>> that no number g does meet that requirement, but it can't >>>>>>>>>>>>>> be proven in F that this is true, because in F, to show >>>>>>>>>>>>>> this we need to test every natuarl number, which requires >>>>>>>>>>>>>> an infinite number of steps (finite for each number, but >>>>>>>>>>>>>> an infinite number of numbers to test).
In meta-F, we can do better, because due to additional >>>>>>>>>>>>>> knowledge in meta-F, we can show that if a number g could >>>>>>>>>>>>>> be found, then that number g could be converted into a >>>>>>>>>>>>>> proof, in F, of the statement G (which says that such a >>>>>>>>>>>>>> number does not exist).
Thus, in meta-F, we can prove that G is true, and also >>>>>>>>>>>>>> show that no proof of it can exist in F.
of G cannot even be expressed in F as long as the truth of >>>>>>>>>>>>> G can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite >>>>>>>>>>>> expression.
reach its truth maker axioms in F it is that even after an >>>>>>>>>>> infinite
number of steps it never reaches is truth maker axioms in F >>>>>>>>>>> because G is
simply untrue in F.
No, YOUR problem is you aren't actually talking about G in F. >>>>>>>>>>
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what?
Not true about being not true about being not true.
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be used for a >>>>>>>>>>> similar undecidability proof.
By using the Liar Paradox as a Gödel approved proxy for his >>>>>>>>>>> proof we
refute his proof by this Gödel approved proxy.
No, all you have proved is that you are a LYING MORON.
Ad Hominem attacks are the tactic that people having no
interest in any
honest dialogue use when they realize that their reasoning has >>>>>>>>> been
utterly defeated.
And RED HERRING arguements don't work either.
You ARE a LYING MORON as you insist that Godel's G is a sentence >>>>>>>> that is actually provably isn't.
You think it is because you are too stupid to actually read any >>>>>>>> of the paper, so you take that comment that the statment is
"based" on that statement to mean it IS that statement.
are asserting this counter-factual statement:
when a valid proxy for an argument is defeated this does not
defeat the
original argument.
Right, which is what YOU are doing, showing your arguement is
INVALID.
In other words you disagree that correctly refuting a valid proxy
for an
argument does correctly refute the original argument?
VALID is the key word,
Yours isn't (I don't think you actually know the meaning of the words) >>>>
And you are an IDIOT to claim it is.
14 Every epistemological antinomy can likewise be used for a
similar undecidability proof.
No, but you don't understand what he is saying.
his proof. He is not saying that his expression is not an
epistemological antinomy.
The analogy between this result and Richard’s antinomy leaps to the eye; there is also a close relationship with the “liar” antinomy,14 since the undecidable proposition [R(q); q] states precisely that q belongs to K,
i.e. according to (1), that [R(q); q] is not provable. We are therefore confronted with a proposition which asserts its own unprovability.
(Gödel 1931:43)
This <is> an isomorphism to a proposition that asserts its own untruth.
On 1/15/23 7:26 PM, olcott wrote:
On 1/15/2023 2:23 PM, Richard Damon wrote:
On 1/15/23 3:12 PM, olcott wrote:
On 1/15/2023 1:46 PM, Richard Damon wrote:
On 1/15/23 2:29 PM, olcott wrote:He is saying that every epistemological antinomy is a valid proxy for
On 1/15/2023 1:06 PM, Richard Damon wrote:
On 1/15/23 2:00 PM, olcott wrote:So Gödel is wrong when he says:
On 1/15/2023 12:55 PM, Richard Damon wrote:
On 1/15/23 1:23 PM, olcott wrote:
On 1/15/2023 11:58 AM, Richard Damon wrote:
On 1/15/23 12:47 PM, olcott wrote:All of your Ad Hominem attacks cannot possibly hide the fact >>>>>>>>>> that you
On 1/15/2023 11:41 AM, Richard Damon wrote:
On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:That is false. It is not that G takes an infinite number >>>>>>>>>>>>>> of steps to
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F >>>>>>>>>>>>>>>> even if the truth
On 1/14/2023 4:55 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>> On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>> On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression >>>>>>>>>>>>>>>>>>>>>> of language of this
formal system is true unless this expression of >>>>>>>>>>>>>>>>>>>>>> language has a
connection to truth maker axioms *IN THIS FORMAL >>>>>>>>>>>>>>>>>>>>>> SYSTEM* ???
Becaue the formal system doesn't need to KNOW what >>>>>>>>>>>>>>>>>>>>> is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + >>>>>>>>>>>>>>>>>>>> successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it >>>>>>>>>>>>>>>>>>> to be true, doesn't mean it can't be.
In fact, your statement just comes out of a simple >>>>>>>>>>>>>>>>>>> application of the addition AXIOMS of PA. >>>>>>>>>>>>>>>>>>>
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually >>>>>>>>>>>>>>>>>>> KNOWN to be true.
Unless a formal system has a syntactic connection from >>>>>>>>>>>>>>>>>> an expression of
its language to its truth maker axioms the expression >>>>>>>>>>>>>>>>>> is untrue in that
formal system.
Right, but the connection can be infinite in length, >>>>>>>>>>>>>>>>> and thus not provable.
Try and show an expression of language that is true in >>>>>>>>>>>>>>>>>> a formal system
(not just true somewhere else) that does not have any >>>>>>>>>>>>>>>>>> connection to
truth maker axioms in this formal system. You must >>>>>>>>>>>>>>>>>> show why it is true
in this formal system not merely that it is true >>>>>>>>>>>>>>>>>> somewhere else.
The connection might be infinite, and thus not SHOWABLE >>>>>>>>>>>>>>>>> as a proof strictly in the formal system.
If the connection exists as an infinite connection >>>>>>>>>>>>>>>>> within the system, then it is TRUE in the system. >>>>>>>>>>>>>>>>>
Note, that if there is such an infinite connection, >>>>>>>>>>>>>>>>> which thus can not be proven within the formal system, >>>>>>>>>>>>>>>>> it is still possible, that another system, related to >>>>>>>>>>>>>>>>> that system, with more knowledge, might be able to show >>>>>>>>>>>>>>>>> that there does exist within the original formal system >>>>>>>>>>>>>>>>> such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g >>>>>>>>>>>>>>>>> that meets a specific requirement (expressed as a >>>>>>>>>>>>>>>>> primative recursive relationship).
This statement turns out to be true, because it turns >>>>>>>>>>>>>>>>> out that no number g does meet that requirement, but it >>>>>>>>>>>>>>>>> can't be proven in F that this is true, because in F, >>>>>>>>>>>>>>>>> to show this we need to test every natuarl number, >>>>>>>>>>>>>>>>> which requires an infinite number of steps (finite for >>>>>>>>>>>>>>>>> each number, but an infinite number of numbers to test). >>>>>>>>>>>>>>>>>
In meta-F, we can do better, because due to additional >>>>>>>>>>>>>>>>> knowledge in meta-F, we can show that if a number g >>>>>>>>>>>>>>>>> could be found, then that number g could be converted >>>>>>>>>>>>>>>>> into a proof, in F, of the statement G (which says that >>>>>>>>>>>>>>>>> such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also >>>>>>>>>>>>>>>>> show that no proof of it can exist in F.
of G cannot even be expressed in F as long as the truth >>>>>>>>>>>>>>>> of G can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an >>>>>>>>>>>>>>> infinite expression.
reach its truth maker axioms in F it is that even after an >>>>>>>>>>>>>> infinite
number of steps it never reaches is truth maker axioms in >>>>>>>>>>>>>> F because G is
simply untrue in F.
No, YOUR problem is you aren't actually talking about G in F. >>>>>>>>>>>>>
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what?
Not true about being not true about being not true. >>>>>>>>>>>>>>
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be used for >>>>>>>>>>>>>> a similar undecidability proof.
By using the Liar Paradox as a Gödel approved proxy for >>>>>>>>>>>>>> his proof we
refute his proof by this Gödel approved proxy.
No, all you have proved is that you are a LYING MORON. >>>>>>>>>>>>>
Ad Hominem attacks are the tactic that people having no >>>>>>>>>>>> interest in any
honest dialogue use when they realize that their reasoning >>>>>>>>>>>> has been
utterly defeated.
And RED HERRING arguements don't work either.
You ARE a LYING MORON as you insist that Godel's G is a
sentence that is actually provably isn't.
You think it is because you are too stupid to actually read >>>>>>>>>>> any of the paper, so you take that comment that the statment >>>>>>>>>>> is "based" on that statement to mean it IS that statement. >>>>>>>>>>>
are asserting this counter-factual statement:
when a valid proxy for an argument is defeated this does not >>>>>>>>>> defeat the
original argument.
Right, which is what YOU are doing, showing your arguement is >>>>>>>>> INVALID.
In other words you disagree that correctly refuting a valid
proxy for an
argument does correctly refute the original argument?
VALID is the key word,
Yours isn't (I don't think you actually know the meaning of the
words)
And you are an IDIOT to claim it is.
14 Every epistemological antinomy can likewise be used for a
similar undecidability proof.
No, but you don't understand what he is saying.
his proof. He is not saying that his expression is not an
epistemological antinomy.
Nope, that isn't what he is saying. How could it be, the ACTUAL G is
a proven Truth Bearer, while the Liar's Paradox isn't
Your arguement just shows its inconsistency.
In part, because you don't actually understand what the sentence
actually is.
The analogy between this result and Richard’s antinomy leaps to the
eye;
there is also a close relationship with the “liar” antinomy,14 since >>>> the
undecidable proposition [R(q); q] states precisely that q belongs to K, >>>> i.e. according to (1), that [R(q); q] is not provable. We are therefore >>>> confronted with a proposition which asserts its own unprovability.
(Gödel 1931:43)
This <is> an isomorphism to a proposition that asserts its own untruth. >>>>
Nope, unless you erroneously think that statements about Truth ARE
statements about provability, that isn't an isomoprhism.
Note, you have even stated that *ALL* statements of the form
"statement x is provable" or "Statment x is not provable" are Truth
Bearers,
I have most definitely never said this or anything that could be
unintentionally misconstrued to mean this.
It is always the case that when-so-ever any expression of language only
refers to its own truth or provability that this expression is not a
truth bearer, thus not a member of any formal system of logic.
You admitted that it was TRUE that a statement could not be proven even
if the only way to show that it could not be proven was to check the
infinite set of all possible proofs to see that none of them were a proof.
On 1/15/23 3:12 PM, olcott wrote:
On 1/15/2023 1:46 PM, Richard Damon wrote:
On 1/15/23 2:29 PM, olcott wrote:He is saying that every epistemological antinomy is a valid proxy for
On 1/15/2023 1:06 PM, Richard Damon wrote:
On 1/15/23 2:00 PM, olcott wrote:So Gödel is wrong when he says:
On 1/15/2023 12:55 PM, Richard Damon wrote:
On 1/15/23 1:23 PM, olcott wrote:
On 1/15/2023 11:58 AM, Richard Damon wrote:
On 1/15/23 12:47 PM, olcott wrote:All of your Ad Hominem attacks cannot possibly hide the fact
On 1/15/2023 11:41 AM, Richard Damon wrote:
On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:That is false. It is not that G takes an infinite number of >>>>>>>>>>>> steps to
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F >>>>>>>>>>>>>> even if the truth
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>> On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression >>>>>>>>>>>>>>>>>>>> of language of this
formal system is true unless this expression of >>>>>>>>>>>>>>>>>>>> language has a
connection to truth maker axioms *IN THIS FORMAL >>>>>>>>>>>>>>>>>>>> SYSTEM* ???
Becaue the formal system doesn't need to KNOW what is >>>>>>>>>>>>>>>>>>> true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) >>>>>>>>>>>>>>>>>> is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to >>>>>>>>>>>>>>>>> be true, doesn't mean it can't be.
In fact, your statement just comes out of a simple >>>>>>>>>>>>>>>>> application of the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN >>>>>>>>>>>>>>>>> to be true.
Unless a formal system has a syntactic connection from >>>>>>>>>>>>>>>> an expression of
its language to its truth maker axioms the expression is >>>>>>>>>>>>>>>> untrue in that
formal system.
Right, but the connection can be infinite in length, and >>>>>>>>>>>>>>> thus not provable.
Try and show an expression of language that is true in a >>>>>>>>>>>>>>>> formal system
(not just true somewhere else) that does not have any >>>>>>>>>>>>>>>> connection to
truth maker axioms in this formal system. You must show >>>>>>>>>>>>>>>> why it is true
in this formal system not merely that it is true >>>>>>>>>>>>>>>> somewhere else.
The connection might be infinite, and thus not SHOWABLE >>>>>>>>>>>>>>> as a proof strictly in the formal system.
If the connection exists as an infinite connection within >>>>>>>>>>>>>>> the system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which >>>>>>>>>>>>>>> thus can not be proven within the formal system, it is >>>>>>>>>>>>>>> still possible, that another system, related to that >>>>>>>>>>>>>>> system, with more knowledge, might be able to show that >>>>>>>>>>>>>>> there does exist within the original formal system such >>>>>>>>>>>>>>> an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g >>>>>>>>>>>>>>> that meets a specific requirement (expressed as a >>>>>>>>>>>>>>> primative recursive relationship).
This statement turns out to be true, because it turns out >>>>>>>>>>>>>>> that no number g does meet that requirement, but it can't >>>>>>>>>>>>>>> be proven in F that this is true, because in F, to show >>>>>>>>>>>>>>> this we need to test every natuarl number, which requires >>>>>>>>>>>>>>> an infinite number of steps (finite for each number, but >>>>>>>>>>>>>>> an infinite number of numbers to test).
In meta-F, we can do better, because due to additional >>>>>>>>>>>>>>> knowledge in meta-F, we can show that if a number g could >>>>>>>>>>>>>>> be found, then that number g could be converted into a >>>>>>>>>>>>>>> proof, in F, of the statement G (which says that such a >>>>>>>>>>>>>>> number does not exist).
Thus, in meta-F, we can prove that G is true, and also >>>>>>>>>>>>>>> show that no proof of it can exist in F.
of G cannot even be expressed in F as long as the truth of >>>>>>>>>>>>>> G can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an
infinite expression.
reach its truth maker axioms in F it is that even after an >>>>>>>>>>>> infinite
number of steps it never reaches is truth maker axioms in F >>>>>>>>>>>> because G is
simply untrue in F.
No, YOUR problem is you aren't actually talking about G in F. >>>>>>>>>>>
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what?
Not true about being not true about being not true.
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be used for a >>>>>>>>>>>> similar undecidability proof.
By using the Liar Paradox as a Gödel approved proxy for his >>>>>>>>>>>> proof we
refute his proof by this Gödel approved proxy.
No, all you have proved is that you are a LYING MORON.
Ad Hominem attacks are the tactic that people having no
interest in any
honest dialogue use when they realize that their reasoning has >>>>>>>>>> been
utterly defeated.
And RED HERRING arguements don't work either.
You ARE a LYING MORON as you insist that Godel's G is a
sentence that is actually provably isn't.
You think it is because you are too stupid to actually read any >>>>>>>>> of the paper, so you take that comment that the statment is
"based" on that statement to mean it IS that statement.
that you
are asserting this counter-factual statement:
when a valid proxy for an argument is defeated this does not
defeat the
original argument.
Right, which is what YOU are doing, showing your arguement is
INVALID.
In other words you disagree that correctly refuting a valid proxy
for an
argument does correctly refute the original argument?
VALID is the key word,
Yours isn't (I don't think you actually know the meaning of the words) >>>>>
And you are an IDIOT to claim it is.
14 Every epistemological antinomy can likewise be used for a
similar undecidability proof.
No, but you don't understand what he is saying.
his proof. He is not saying that his expression is not an
epistemological antinomy.
Nope, that isn't what he is saying. How could it be, the ACTUAL G is a
proven Truth Bearer, while the Liar's Paradox isn't
Your arguement just shows its inconsistency.
In part, because you don't actually understand what the sentence
actually is.
The analogy between this result and Richard’s antinomy leaps to the eye; >> there is also a close relationship with the “liar” antinomy,14 since the >> undecidable proposition [R(q); q] states precisely that q belongs to K,
i.e. according to (1), that [R(q); q] is not provable. We are therefore
confronted with a proposition which asserts its own unprovability.
(Gödel 1931:43)
This <is> an isomorphism to a proposition that asserts its own untruth.
Nope, unless you erroneously think that statements about Truth ARE
statements about provability, that isn't an isomoprhism.
Note, you have even stated that *ALL* statements of the form "statement
x is provable" or "Statment x is not provable" are Truth Bearers,
On 1/15/2023 2:23 PM, Richard Damon wrote:
On 1/15/23 3:12 PM, olcott wrote:
On 1/15/2023 1:46 PM, Richard Damon wrote:
On 1/15/23 2:29 PM, olcott wrote:He is saying that every epistemological antinomy is a valid proxy for
On 1/15/2023 1:06 PM, Richard Damon wrote:
On 1/15/23 2:00 PM, olcott wrote:So Gödel is wrong when he says:
On 1/15/2023 12:55 PM, Richard Damon wrote:
On 1/15/23 1:23 PM, olcott wrote:
On 1/15/2023 11:58 AM, Richard Damon wrote:
On 1/15/23 12:47 PM, olcott wrote:All of your Ad Hominem attacks cannot possibly hide the fact >>>>>>>>> that you
On 1/15/2023 11:41 AM, Richard Damon wrote:
On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:That is false. It is not that G takes an infinite number of >>>>>>>>>>>>> steps to
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F >>>>>>>>>>>>>>> even if the truth
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>> On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression >>>>>>>>>>>>>>>>>>>>> of language of this
formal system is true unless this expression of >>>>>>>>>>>>>>>>>>>>> language has a
connection to truth maker axioms *IN THIS FORMAL >>>>>>>>>>>>>>>>>>>>> SYSTEM* ???
Becaue the formal system doesn't need to KNOW what >>>>>>>>>>>>>>>>>>>> is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + >>>>>>>>>>>>>>>>>>> successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to >>>>>>>>>>>>>>>>>> be true, doesn't mean it can't be.
In fact, your statement just comes out of a simple >>>>>>>>>>>>>>>>>> application of the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN >>>>>>>>>>>>>>>>>> to be true.
Unless a formal system has a syntactic connection from >>>>>>>>>>>>>>>>> an expression of
its language to its truth maker axioms the expression >>>>>>>>>>>>>>>>> is untrue in that
formal system.
Right, but the connection can be infinite in length, and >>>>>>>>>>>>>>>> thus not provable.
Try and show an expression of language that is true in >>>>>>>>>>>>>>>>> a formal system
(not just true somewhere else) that does not have any >>>>>>>>>>>>>>>>> connection to
truth maker axioms in this formal system. You must show >>>>>>>>>>>>>>>>> why it is true
in this formal system not merely that it is true >>>>>>>>>>>>>>>>> somewhere else.
The connection might be infinite, and thus not SHOWABLE >>>>>>>>>>>>>>>> as a proof strictly in the formal system.
If the connection exists as an infinite connection >>>>>>>>>>>>>>>> within the system, then it is TRUE in the system. >>>>>>>>>>>>>>>>
Note, that if there is such an infinite connection, >>>>>>>>>>>>>>>> which thus can not be proven within the formal system, >>>>>>>>>>>>>>>> it is still possible, that another system, related to >>>>>>>>>>>>>>>> that system, with more knowledge, might be able to show >>>>>>>>>>>>>>>> that there does exist within the original formal system >>>>>>>>>>>>>>>> such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g >>>>>>>>>>>>>>>> that meets a specific requirement (expressed as a >>>>>>>>>>>>>>>> primative recursive relationship).
This statement turns out to be true, because it turns >>>>>>>>>>>>>>>> out that no number g does meet that requirement, but it >>>>>>>>>>>>>>>> can't be proven in F that this is true, because in F, to >>>>>>>>>>>>>>>> show this we need to test every natuarl number, which >>>>>>>>>>>>>>>> requires an infinite number of steps (finite for each >>>>>>>>>>>>>>>> number, but an infinite number of numbers to test). >>>>>>>>>>>>>>>>
In meta-F, we can do better, because due to additional >>>>>>>>>>>>>>>> knowledge in meta-F, we can show that if a number g >>>>>>>>>>>>>>>> could be found, then that number g could be converted >>>>>>>>>>>>>>>> into a proof, in F, of the statement G (which says that >>>>>>>>>>>>>>>> such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also >>>>>>>>>>>>>>>> show that no proof of it can exist in F.
of G cannot even be expressed in F as long as the truth >>>>>>>>>>>>>>> of G can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an >>>>>>>>>>>>>> infinite expression.
reach its truth maker axioms in F it is that even after an >>>>>>>>>>>>> infinite
number of steps it never reaches is truth maker axioms in F >>>>>>>>>>>>> because G is
simply untrue in F.
No, YOUR problem is you aren't actually talking about G in F. >>>>>>>>>>>>
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what?
Not true about being not true about being not true.
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be used for >>>>>>>>>>>>> a similar undecidability proof.
By using the Liar Paradox as a Gödel approved proxy for his >>>>>>>>>>>>> proof we
refute his proof by this Gödel approved proxy.
No, all you have proved is that you are a LYING MORON. >>>>>>>>>>>>
Ad Hominem attacks are the tactic that people having no
interest in any
honest dialogue use when they realize that their reasoning >>>>>>>>>>> has been
utterly defeated.
And RED HERRING arguements don't work either.
You ARE a LYING MORON as you insist that Godel's G is a
sentence that is actually provably isn't.
You think it is because you are too stupid to actually read >>>>>>>>>> any of the paper, so you take that comment that the statment >>>>>>>>>> is "based" on that statement to mean it IS that statement. >>>>>>>>>>
are asserting this counter-factual statement:
when a valid proxy for an argument is defeated this does not >>>>>>>>> defeat the
original argument.
Right, which is what YOU are doing, showing your arguement is
INVALID.
In other words you disagree that correctly refuting a valid proxy >>>>>>> for an
argument does correctly refute the original argument?
VALID is the key word,
Yours isn't (I don't think you actually know the meaning of the
words)
And you are an IDIOT to claim it is.
14 Every epistemological antinomy can likewise be used for a
similar undecidability proof.
No, but you don't understand what he is saying.
his proof. He is not saying that his expression is not an
epistemological antinomy.
Nope, that isn't what he is saying. How could it be, the ACTUAL G is a
proven Truth Bearer, while the Liar's Paradox isn't
Your arguement just shows its inconsistency.
In part, because you don't actually understand what the sentence
actually is.
The analogy between this result and Richard’s antinomy leaps to the eye; >>> there is also a close relationship with the “liar” antinomy,14 since the
undecidable proposition [R(q); q] states precisely that q belongs to K,
i.e. according to (1), that [R(q); q] is not provable. We are therefore
confronted with a proposition which asserts its own unprovability.
(Gödel 1931:43)
This <is> an isomorphism to a proposition that asserts its own untruth.
Nope, unless you erroneously think that statements about Truth ARE
statements about provability, that isn't an isomoprhism.
Note, you have even stated that *ALL* statements of the form
"statement x is provable" or "Statment x is not provable" are Truth
Bearers,
I have most definitely never said this or anything that could be unintentionally misconstrued to mean this.
It is always the case that when-so-ever any expression of language only refers to its own truth or provability that this expression is not a
truth bearer, thus not a member of any formal system of logic.
On 1/15/2023 6:47 PM, Richard Damon wrote:
On 1/15/23 7:26 PM, olcott wrote:
On 1/15/2023 2:23 PM, Richard Damon wrote:
On 1/15/23 3:12 PM, olcott wrote:
On 1/15/2023 1:46 PM, Richard Damon wrote:
On 1/15/23 2:29 PM, olcott wrote:He is saying that every epistemological antinomy is a valid proxy for >>>>> his proof. He is not saying that his expression is not an
On 1/15/2023 1:06 PM, Richard Damon wrote:
On 1/15/23 2:00 PM, olcott wrote:So Gödel is wrong when he says:
On 1/15/2023 12:55 PM, Richard Damon wrote:
On 1/15/23 1:23 PM, olcott wrote:
On 1/15/2023 11:58 AM, Richard Damon wrote:
On 1/15/23 12:47 PM, olcott wrote:All of your Ad Hominem attacks cannot possibly hide the fact >>>>>>>>>>> that you
On 1/15/2023 11:41 AM, Richard Damon wrote:
On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:That is false. It is not that G takes an infinite number >>>>>>>>>>>>>>> of steps to
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F >>>>>>>>>>>>>>>>> even if the truth
On 1/14/2023 4:55 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>> On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>> On 1/14/23 4:48 PM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>> How does the formal system know that an >>>>>>>>>>>>>>>>>>>>>>> expression of language of this
formal system is true unless this expression of >>>>>>>>>>>>>>>>>>>>>>> language has a
connection to truth maker axioms *IN THIS FORMAL >>>>>>>>>>>>>>>>>>>>>>> SYSTEM* ???
Becaue the formal system doesn't need to KNOW what >>>>>>>>>>>>>>>>>>>>>> is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + >>>>>>>>>>>>>>>>>>>>> successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it >>>>>>>>>>>>>>>>>>>> to be true, doesn't mean it can't be.
In fact, your statement just comes out of a simple >>>>>>>>>>>>>>>>>>>> application of the addition AXIOMS of PA. >>>>>>>>>>>>>>>>>>>>
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually >>>>>>>>>>>>>>>>>>>> KNOWN to be true.
Unless a formal system has a syntactic connection >>>>>>>>>>>>>>>>>>> from an expression of
its language to its truth maker axioms the expression >>>>>>>>>>>>>>>>>>> is untrue in that
formal system.
Right, but the connection can be infinite in length, >>>>>>>>>>>>>>>>>> and thus not provable.
Try and show an expression of language that is true >>>>>>>>>>>>>>>>>>> in a formal system
(not just true somewhere else) that does not have any >>>>>>>>>>>>>>>>>>> connection to
truth maker axioms in this formal system. You must >>>>>>>>>>>>>>>>>>> show why it is true
in this formal system not merely that it is true >>>>>>>>>>>>>>>>>>> somewhere else.
The connection might be infinite, and thus not >>>>>>>>>>>>>>>>>> SHOWABLE as a proof strictly in the formal system. >>>>>>>>>>>>>>>>>>
If the connection exists as an infinite connection >>>>>>>>>>>>>>>>>> within the system, then it is TRUE in the system. >>>>>>>>>>>>>>>>>>
Note, that if there is such an infinite connection, >>>>>>>>>>>>>>>>>> which thus can not be proven within the formal system, >>>>>>>>>>>>>>>>>> it is still possible, that another system, related to >>>>>>>>>>>>>>>>>> that system, with more knowledge, might be able to >>>>>>>>>>>>>>>>>> show that there does exist within the original formal >>>>>>>>>>>>>>>>>> system such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g >>>>>>>>>>>>>>>>>> that meets a specific requirement (expressed as a >>>>>>>>>>>>>>>>>> primative recursive relationship).
This statement turns out to be true, because it turns >>>>>>>>>>>>>>>>>> out that no number g does meet that requirement, but >>>>>>>>>>>>>>>>>> it can't be proven in F that this is true, because in >>>>>>>>>>>>>>>>>> F, to show this we need to test every natuarl number, >>>>>>>>>>>>>>>>>> which requires an infinite number of steps (finite for >>>>>>>>>>>>>>>>>> each number, but an infinite number of numbers to test). >>>>>>>>>>>>>>>>>>
In meta-F, we can do better, because due to additional >>>>>>>>>>>>>>>>>> knowledge in meta-F, we can show that if a number g >>>>>>>>>>>>>>>>>> could be found, then that number g could be converted >>>>>>>>>>>>>>>>>> into a proof, in F, of the statement G (which says >>>>>>>>>>>>>>>>>> that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also >>>>>>>>>>>>>>>>>> show that no proof of it can exist in F.
of G cannot even be expressed in F as long as the truth >>>>>>>>>>>>>>>>> of G can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an >>>>>>>>>>>>>>>> infinite expression.
reach its truth maker axioms in F it is that even after >>>>>>>>>>>>>>> an infinite
number of steps it never reaches is truth maker axioms in >>>>>>>>>>>>>>> F because G is
simply untrue in F.
No, YOUR problem is you aren't actually talking about G in F. >>>>>>>>>>>>>>
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what?
Not true about being not true about being not true. >>>>>>>>>>>>>>>
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be used >>>>>>>>>>>>>>> for a similar undecidability proof.
By using the Liar Paradox as a Gödel approved proxy for >>>>>>>>>>>>>>> his proof we
refute his proof by this Gödel approved proxy.
No, all you have proved is that you are a LYING MORON. >>>>>>>>>>>>>>
Ad Hominem attacks are the tactic that people having no >>>>>>>>>>>>> interest in any
honest dialogue use when they realize that their reasoning >>>>>>>>>>>>> has been
utterly defeated.
And RED HERRING arguements don't work either.
You ARE a LYING MORON as you insist that Godel's G is a >>>>>>>>>>>> sentence that is actually provably isn't.
You think it is because you are too stupid to actually read >>>>>>>>>>>> any of the paper, so you take that comment that the statment >>>>>>>>>>>> is "based" on that statement to mean it IS that statement. >>>>>>>>>>>>
are asserting this counter-factual statement:
when a valid proxy for an argument is defeated this does not >>>>>>>>>>> defeat the
original argument.
Right, which is what YOU are doing, showing your arguement is >>>>>>>>>> INVALID.
In other words you disagree that correctly refuting a valid
proxy for an
argument does correctly refute the original argument?
VALID is the key word,
Yours isn't (I don't think you actually know the meaning of the >>>>>>>> words)
And you are an IDIOT to claim it is.
14 Every epistemological antinomy can likewise be used for a
similar undecidability proof.
No, but you don't understand what he is saying.
epistemological antinomy.
Nope, that isn't what he is saying. How could it be, the ACTUAL G is
a proven Truth Bearer, while the Liar's Paradox isn't
Your arguement just shows its inconsistency.
In part, because you don't actually understand what the sentence
actually is.
The analogy between this result and Richard’s antinomy leaps to the >>>>> eye;
there is also a close relationship with the “liar” antinomy,14
since the
undecidable proposition [R(q); q] states precisely that q belongs
to K,
i.e. according to (1), that [R(q); q] is not provable. We are
therefore
confronted with a proposition which asserts its own unprovability.
(Gödel 1931:43)
This <is> an isomorphism to a proposition that asserts its own
untruth.
Nope, unless you erroneously think that statements about Truth ARE
statements about provability, that isn't an isomoprhism.
Note, you have even stated that *ALL* statements of the form
"statement x is provable" or "Statment x is not provable" are Truth
Bearers,
I have most definitely never said this or anything that could be
unintentionally misconstrued to mean this.
It is always the case that when-so-ever any expression of language only
refers to its own truth or provability that this expression is not a
truth bearer, thus not a member of any formal system of logic.
You admitted that it was TRUE that a statement could not be proven
even if the only way to show that it could not be proven was to check
the infinite set of all possible proofs to see that none of them were
a proof.
This is not related to what I just said. Every expression of language is untrue unless it has a semantic connection to its truth maker axiom.
Expressions that only refer to their own truth or provability have a
vacuous truth object that are never truth bearers.
Epistemological antinomies are never truth bearers, thus Gödel admitted
the fallacious basis of his proof.
On 1/15/23 11:15 AM, olcott wrote:
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if the truth
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language >>>>>>>> of this
formal system is true unless this expression of language has a >>>>>>>> connection to truth maker axioms *IN THIS FORMAL SYSTEM* ???
Becaue the formal system doesn't need to KNOW what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ??? >>>>>>
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true,
doesn't mean it can't be.
In fact, your statement just comes out of a simple application of
the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true.
Unless a formal system has a syntactic connection from an expression of >>>> its language to its truth maker axioms the expression is untrue in that >>>> formal system.
Right, but the connection can be infinite in length, and thus not
provable.
Try and show an expression of language that is true in a formal system >>>> (not just true somewhere else) that does not have any connection to
truth maker axioms in this formal system. You must show why it is true >>>> in this formal system not merely that it is true somewhere else.
The connection might be infinite, and thus not SHOWABLE as a proof
strictly in the formal system.
If the connection exists as an infinite connection within the system,
then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can
not be proven within the formal system, it is still possible, that
another system, related to that system, with more knowledge, might be
able to show that there does exist within the original formal system
such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets a
specific requirement (expressed as a primative recursive relationship).
This statement turns out to be true, because it turns out that no
number g does meet that requirement, but it can't be proven in F that
this is true, because in F, to show this we need to test every
natuarl number, which requires an infinite number of steps (finite
for each number, but an infinite number of numbers to test).
In meta-F, we can do better, because due to additional knowledge in
meta-F, we can show that if a number g could be found, then that
number g could be converted into a proof, in F, of the statement G
(which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that no
proof of it can exist in F.
of G cannot even be expressed in F as long as the truth of G can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite expression.
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if the truth >> of G cannot even be expressed in F as long as the truth of G can be
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language >>>>>>>> of this
formal system is true unless this expression of language has a >>>>>>>> connection to truth maker axioms *IN THIS FORMAL SYSTEM* ??? >>>>>>>>
Becaue the formal system doesn't need to KNOW what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ??? >>>>>>
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true,
doesn't mean it can't be.
In fact, your statement just comes out of a simple application of >>>>> the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true. >>>>>
Unless a formal system has a syntactic connection from an expression of >>>> its language to its truth maker axioms the expression is untrue in that >>>> formal system.
Right, but the connection can be infinite in length, and thus not
provable.
Try and show an expression of language that is true in a formal system >>>> (not just true somewhere else) that does not have any connection to >>>> truth maker axioms in this formal system. You must show why it is true >>>> in this formal system not merely that it is true somewhere else.
The connection might be infinite, and thus not SHOWABLE as a proof
strictly in the formal system.
If the connection exists as an infinite connection within the system, >>> then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can
not be proven within the formal system, it is still possible, that
another system, related to that system, with more knowledge, might be >>> able to show that there does exist within the original formal system
such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets a
specific requirement (expressed as a primative recursive relationship). >>>
This statement turns out to be true, because it turns out that no
number g does meet that requirement, but it can't be proven in F that >>> this is true, because in F, to show this we need to test every
natuarl number, which requires an infinite number of steps (finite
for each number, but an infinite number of numbers to test).
In meta-F, we can do better, because due to additional knowledge in
meta-F, we can show that if a number g could be found, then that
number g could be converted into a proof, in F, of the statement G
(which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that no
proof of it can exist in F.
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite expression.
So you changed your mind about infinite proofs in formal systems?
We know the steps of the infinite proof for the Goldbach conjecture.
What are the infinite steps to show that a self-contradictory expression
is provable?
--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 1/15/23 7:56 PM, olcott wrote:Gödel was actually talking about the expression:
On 1/15/2023 6:47 PM, Richard Damon wrote:
On 1/15/23 7:26 PM, olcott wrote:
On 1/15/2023 2:23 PM, Richard Damon wrote:
On 1/15/23 3:12 PM, olcott wrote:
On 1/15/2023 1:46 PM, Richard Damon wrote:
On 1/15/23 2:29 PM, olcott wrote:He is saying that every epistemological antinomy is a valid proxy for >>>>>> his proof. He is not saying that his expression is not an
On 1/15/2023 1:06 PM, Richard Damon wrote:
On 1/15/23 2:00 PM, olcott wrote:So Gödel is wrong when he says:
On 1/15/2023 12:55 PM, Richard Damon wrote:
On 1/15/23 1:23 PM, olcott wrote:
On 1/15/2023 11:58 AM, Richard Damon wrote:
On 1/15/23 12:47 PM, olcott wrote:All of your Ad Hominem attacks cannot possibly hide the fact >>>>>>>>>>>> that you
On 1/15/2023 11:41 AM, Richard Damon wrote:
On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:That is false. It is not that G takes an infinite number >>>>>>>>>>>>>>>> of steps to
On 1/14/2023 5:42 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>> On 1/14/23 6:19 PM, olcott wrote:
So your basic line-of-reasoning is that G is true in F >>>>>>>>>>>>>>>>>> even if the truthOn 1/14/2023 4:55 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>> On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>>> On 1/14/23 4:48 PM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>> How does the formal system know that an >>>>>>>>>>>>>>>>>>>>>>>> expression of language of this >>>>>>>>>>>>>>>>>>>>>>>> formal system is true unless this expression of >>>>>>>>>>>>>>>>>>>>>>>> language has a
connection to truth maker axioms *IN THIS FORMAL >>>>>>>>>>>>>>>>>>>>>>>> SYSTEM* ???
Becaue the formal system doesn't need to KNOW >>>>>>>>>>>>>>>>>>>>>>> what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + >>>>>>>>>>>>>>>>>>>>>> successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it >>>>>>>>>>>>>>>>>>>>> to be true, doesn't mean it can't be. >>>>>>>>>>>>>>>>>>>>>
In fact, your statement just comes out of a simple >>>>>>>>>>>>>>>>>>>>> application of the addition AXIOMS of PA. >>>>>>>>>>>>>>>>>>>>>
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually >>>>>>>>>>>>>>>>>>>>> KNOWN to be true.
Unless a formal system has a syntactic connection >>>>>>>>>>>>>>>>>>>> from an expression of
its language to its truth maker axioms the >>>>>>>>>>>>>>>>>>>> expression is untrue in that
formal system.
Right, but the connection can be infinite in length, >>>>>>>>>>>>>>>>>>> and thus not provable.
Try and show an expression of language that is true >>>>>>>>>>>>>>>>>>>> in a formal system
(not just true somewhere else) that does not have >>>>>>>>>>>>>>>>>>>> any connection to
truth maker axioms in this formal system. You must >>>>>>>>>>>>>>>>>>>> show why it is true
in this formal system not merely that it is true >>>>>>>>>>>>>>>>>>>> somewhere else.
The connection might be infinite, and thus not >>>>>>>>>>>>>>>>>>> SHOWABLE as a proof strictly in the formal system. >>>>>>>>>>>>>>>>>>>
If the connection exists as an infinite connection >>>>>>>>>>>>>>>>>>> within the system, then it is TRUE in the system. >>>>>>>>>>>>>>>>>>>
Note, that if there is such an infinite connection, >>>>>>>>>>>>>>>>>>> which thus can not be proven within the formal >>>>>>>>>>>>>>>>>>> system, it is still possible, that another system, >>>>>>>>>>>>>>>>>>> related to that system, with more knowledge, might be >>>>>>>>>>>>>>>>>>> able to show that there does exist within the >>>>>>>>>>>>>>>>>>> original formal system such an infinte connection. >>>>>>>>>>>>>>>>>>>
This is what happens to G in F and meta-F >>>>>>>>>>>>>>>>>>>
G states that there does not exist a Natural Number g >>>>>>>>>>>>>>>>>>> that meets a specific requirement (expressed as a >>>>>>>>>>>>>>>>>>> primative recursive relationship).
This statement turns out to be true, because it turns >>>>>>>>>>>>>>>>>>> out that no number g does meet that requirement, but >>>>>>>>>>>>>>>>>>> it can't be proven in F that this is true, because in >>>>>>>>>>>>>>>>>>> F, to show this we need to test every natuarl number, >>>>>>>>>>>>>>>>>>> which requires an infinite number of steps (finite >>>>>>>>>>>>>>>>>>> for each number, but an infinite number of numbers to >>>>>>>>>>>>>>>>>>> test).
In meta-F, we can do better, because due to >>>>>>>>>>>>>>>>>>> additional knowledge in meta-F, we can show that if a >>>>>>>>>>>>>>>>>>> number g could be found, then that number g could be >>>>>>>>>>>>>>>>>>> converted into a proof, in F, of the statement G >>>>>>>>>>>>>>>>>>> (which says that such a number does not exist). >>>>>>>>>>>>>>>>>>>
Thus, in meta-F, we can prove that G is true, and >>>>>>>>>>>>>>>>>>> also show that no proof of it can exist in F. >>>>>>>>>>>>>>>>>>>
of G cannot even be expressed in F as long as the >>>>>>>>>>>>>>>>>> truth of G can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an >>>>>>>>>>>>>>>>> infinite expression.
reach its truth maker axioms in F it is that even after >>>>>>>>>>>>>>>> an infinite
number of steps it never reaches is truth maker axioms >>>>>>>>>>>>>>>> in F because G is
simply untrue in F.
No, YOUR problem is you aren't actually talking about G >>>>>>>>>>>>>>> in F.
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what?
Not true about being not true about being not true. >>>>>>>>>>>>>>>>
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be used >>>>>>>>>>>>>>>> for a similar undecidability proof.
By using the Liar Paradox as a Gödel approved proxy for >>>>>>>>>>>>>>>> his proof we
refute his proof by this Gödel approved proxy. >>>>>>>>>>>>>>>>
No, all you have proved is that you are a LYING MORON. >>>>>>>>>>>>>>>
Ad Hominem attacks are the tactic that people having no >>>>>>>>>>>>>> interest in any
honest dialogue use when they realize that their reasoning >>>>>>>>>>>>>> has been
utterly defeated.
And RED HERRING arguements don't work either.
You ARE a LYING MORON as you insist that Godel's G is a >>>>>>>>>>>>> sentence that is actually provably isn't.
You think it is because you are too stupid to actually read >>>>>>>>>>>>> any of the paper, so you take that comment that the
statment is "based" on that statement to mean it IS that >>>>>>>>>>>>> statement.
are asserting this counter-factual statement:
when a valid proxy for an argument is defeated this does not >>>>>>>>>>>> defeat the
original argument.
Right, which is what YOU are doing, showing your arguement is >>>>>>>>>>> INVALID.
In other words you disagree that correctly refuting a valid >>>>>>>>>> proxy for an
argument does correctly refute the original argument?
VALID is the key word,
Yours isn't (I don't think you actually know the meaning of the >>>>>>>>> words)
And you are an IDIOT to claim it is.
14 Every epistemological antinomy can likewise be used for a
similar undecidability proof.
No, but you don't understand what he is saying.
epistemological antinomy.
Nope, that isn't what he is saying. How could it be, the ACTUAL G
is a proven Truth Bearer, while the Liar's Paradox isn't
Your arguement just shows its inconsistency.
In part, because you don't actually understand what the sentence
actually is.
The analogy between this result and Richard’s antinomy leaps to
the eye;
there is also a close relationship with the “liar” antinomy,14 >>>>>> since the
undecidable proposition [R(q); q] states precisely that q belongs
to K,
i.e. according to (1), that [R(q); q] is not provable. We are
therefore
confronted with a proposition which asserts its own unprovability. >>>>>> (Gödel 1931:43)
This <is> an isomorphism to a proposition that asserts its own
untruth.
Nope, unless you erroneously think that statements about Truth ARE
statements about provability, that isn't an isomoprhism.
Note, you have even stated that *ALL* statements of the form
"statement x is provable" or "Statment x is not provable" are Truth
Bearers,
I have most definitely never said this or anything that could be
unintentionally misconstrued to mean this.
It is always the case that when-so-ever any expression of language only >>>> refers to its own truth or provability that this expression is not a
truth bearer, thus not a member of any formal system of logic.
You admitted that it was TRUE that a statement could not be proven
even if the only way to show that it could not be proven was to check
the infinite set of all possible proofs to see that none of them were
a proof.
This is not related to what I just said. Every expression of language is
untrue unless it has a semantic connection to its truth maker axiom.
I never disagreed with that, just that you keep on wavering between just saying there must be a connection, and then at times adding it must be a FINITE connection (which is actually only requried to be Proven)
G, the statment about the non-existance of a natural number that
satisfies the specified primative recursive relationship is TRUE,
because it IS connected to the truth maker axioms of math via an
infinite chain of steps.
Each natural number can be shown to not meet that requirement in a
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:So you changed your mind about infinite proofs in formal systems?
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if the truth >>> of G cannot even be expressed in F as long as the truth of G can be
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language >>>>>>>>> of this
formal system is true unless this expression of language has a >>>>>>>>> connection to truth maker axioms *IN THIS FORMAL SYSTEM* ??? >>>>>>>>>
Becaue the formal system doesn't need to KNOW what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ??? >>>>>>>
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true,
doesn't mean it can't be.
In fact, your statement just comes out of a simple application of
the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true. >>>>>>
Unless a formal system has a syntactic connection from an
expression of
its language to its truth maker axioms the expression is untrue in
that
formal system.
Right, but the connection can be infinite in length, and thus not
provable.
Try and show an expression of language that is true in a formal system >>>>> (not just true somewhere else) that does not have any connection to
truth maker axioms in this formal system. You must show why it is true >>>>> in this formal system not merely that it is true somewhere else.
The connection might be infinite, and thus not SHOWABLE as a proof
strictly in the formal system.
If the connection exists as an infinite connection within the
system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can
not be proven within the formal system, it is still possible, that
another system, related to that system, with more knowledge, might
be able to show that there does exist within the original formal
system such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets a
specific requirement (expressed as a primative recursive relationship). >>>>
This statement turns out to be true, because it turns out that no
number g does meet that requirement, but it can't be proven in F
that this is true, because in F, to show this we need to test every
natuarl number, which requires an infinite number of steps (finite
for each number, but an infinite number of numbers to test).
In meta-F, we can do better, because due to additional knowledge in
meta-F, we can show that if a number g could be found, then that
number g could be converted into a proof, in F, of the statement G
(which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that no
proof of it can exist in F.
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite expression.
We know the steps of the infinite proof for the Goldbach conjecture.
What are the infinite steps to show that a self-contradictory expression
is provable?
On 1/16/23 4:51 PM, olcott wrote:
On 1/15/2023 7:15 PM, Richard Damon wrote:
On 1/15/23 7:56 PM, olcott wrote:Gödel was actually talking about the expression:
On 1/15/2023 6:47 PM, Richard Damon wrote:
On 1/15/23 7:26 PM, olcott wrote:
On 1/15/2023 2:23 PM, Richard Damon wrote:
On 1/15/23 3:12 PM, olcott wrote:
On 1/15/2023 1:46 PM, Richard Damon wrote:
On 1/15/23 2:29 PM, olcott wrote:He is saying that every epistemological antinomy is a valid
On 1/15/2023 1:06 PM, Richard Damon wrote:
On 1/15/23 2:00 PM, olcott wrote:So Gödel is wrong when he says:
On 1/15/2023 12:55 PM, Richard Damon wrote:
On 1/15/23 1:23 PM, olcott wrote:
On 1/15/2023 11:58 AM, Richard Damon wrote:
On 1/15/23 12:47 PM, olcott wrote:All of your Ad Hominem attacks cannot possibly hide the >>>>>>>>>>>>>> fact that you
On 1/15/2023 11:41 AM, Richard Damon wrote:
On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>> On 1/15/23 11:15 AM, olcott wrote:
That is false. It is not that G takes an infinite >>>>>>>>>>>>>>>>>> number of steps toOn 1/14/2023 5:42 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>> On 1/14/23 6:19 PM, olcott wrote:
So your basic line-of-reasoning is that G is true in >>>>>>>>>>>>>>>>>>>> F even if the truthOn 1/14/2023 4:55 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>>> On 1/14/23 5:31 PM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 1/14/2023 4:26 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 1/14/23 4:48 PM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> How does the formal system know that an >>>>>>>>>>>>>>>>>>>>>>>>>> expression of language of this >>>>>>>>>>>>>>>>>>>>>>>>>> formal system is true unless this expression >>>>>>>>>>>>>>>>>>>>>>>>>> of language has a
connection to truth maker axioms *IN THIS >>>>>>>>>>>>>>>>>>>>>>>>>> FORMAL SYSTEM* ???
Becaue the formal system doesn't need to KNOW >>>>>>>>>>>>>>>>>>>>>>>>> what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + >>>>>>>>>>>>>>>>>>>>>>>> successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for >>>>>>>>>>>>>>>>>>>>>>> it to be true, doesn't mean it can't be. >>>>>>>>>>>>>>>>>>>>>>>
In fact, your statement just comes out of a >>>>>>>>>>>>>>>>>>>>>>> simple application of the addition AXIOMS of PA. >>>>>>>>>>>>>>>>>>>>>>>
a + 0 = a
a + Successor(b) = Successor(a + b) >>>>>>>>>>>>>>>>>>>>>>>
So it is a PROVABLE statement, and thus actually >>>>>>>>>>>>>>>>>>>>>>> KNOWN to be true.
Unless a formal system has a syntactic connection >>>>>>>>>>>>>>>>>>>>>> from an expression of
its language to its truth maker axioms the >>>>>>>>>>>>>>>>>>>>>> expression is untrue in that
formal system.
Right, but the connection can be infinite in >>>>>>>>>>>>>>>>>>>>> length, and thus not provable.
Try and show an expression of language that is >>>>>>>>>>>>>>>>>>>>>> true in a formal system
(not just true somewhere else) that does not have >>>>>>>>>>>>>>>>>>>>>> any connection to
truth maker axioms in this formal system. You must >>>>>>>>>>>>>>>>>>>>>> show why it is true
in this formal system not merely that it is true >>>>>>>>>>>>>>>>>>>>>> somewhere else.
The connection might be infinite, and thus not >>>>>>>>>>>>>>>>>>>>> SHOWABLE as a proof strictly in the formal system. >>>>>>>>>>>>>>>>>>>>>
If the connection exists as an infinite connection >>>>>>>>>>>>>>>>>>>>> within the system, then it is TRUE in the system. >>>>>>>>>>>>>>>>>>>>>
Note, that if there is such an infinite connection, >>>>>>>>>>>>>>>>>>>>> which thus can not be proven within the formal >>>>>>>>>>>>>>>>>>>>> system, it is still possible, that another system, >>>>>>>>>>>>>>>>>>>>> related to that system, with more knowledge, might >>>>>>>>>>>>>>>>>>>>> be able to show that there does exist within the >>>>>>>>>>>>>>>>>>>>> original formal system such an infinte connection. >>>>>>>>>>>>>>>>>>>>>
This is what happens to G in F and meta-F >>>>>>>>>>>>>>>>>>>>>
G states that there does not exist a Natural Number >>>>>>>>>>>>>>>>>>>>> g that meets a specific requirement (expressed as a >>>>>>>>>>>>>>>>>>>>> primative recursive relationship).
This statement turns out to be true, because it >>>>>>>>>>>>>>>>>>>>> turns out that no number g does meet that >>>>>>>>>>>>>>>>>>>>> requirement, but it can't be proven in F that this >>>>>>>>>>>>>>>>>>>>> is true, because in F, to show this we need to test >>>>>>>>>>>>>>>>>>>>> every natuarl number, which requires an infinite >>>>>>>>>>>>>>>>>>>>> number of steps (finite for each number, but an >>>>>>>>>>>>>>>>>>>>> infinite number of numbers to test). >>>>>>>>>>>>>>>>>>>>>
In meta-F, we can do better, because due to >>>>>>>>>>>>>>>>>>>>> additional knowledge in meta-F, we can show that if >>>>>>>>>>>>>>>>>>>>> a number g could be found, then that number g could >>>>>>>>>>>>>>>>>>>>> be converted into a proof, in F, of the statement G >>>>>>>>>>>>>>>>>>>>> (which says that such a number does not exist). >>>>>>>>>>>>>>>>>>>>>
Thus, in meta-F, we can prove that G is true, and >>>>>>>>>>>>>>>>>>>>> also show that no proof of it can exist in F. >>>>>>>>>>>>>>>>>>>>>
of G cannot even be expressed in F as long as the >>>>>>>>>>>>>>>>>>>> truth of G can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an >>>>>>>>>>>>>>>>>>> infinite expression.
reach its truth maker axioms in F it is that even >>>>>>>>>>>>>>>>>> after an infinite
number of steps it never reaches is truth maker axioms >>>>>>>>>>>>>>>>>> in F because G is
simply untrue in F.
No, YOUR problem is you aren't actually talking about G >>>>>>>>>>>>>>>>> in F.
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what?
Not true about being not true about being not true. >>>>>>>>>>>>>>>>>>
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be used >>>>>>>>>>>>>>>>>> for a similar undecidability proof.
By using the Liar Paradox as a Gödel approved proxy >>>>>>>>>>>>>>>>>> for his proof we
refute his proof by this Gödel approved proxy. >>>>>>>>>>>>>>>>>>
No, all you have proved is that you are a LYING MORON. >>>>>>>>>>>>>>>>>
Ad Hominem attacks are the tactic that people having no >>>>>>>>>>>>>>>> interest in any
honest dialogue use when they realize that their >>>>>>>>>>>>>>>> reasoning has been
utterly defeated.
And RED HERRING arguements don't work either.
You ARE a LYING MORON as you insist that Godel's G is a >>>>>>>>>>>>>>> sentence that is actually provably isn't.
You think it is because you are too stupid to actually >>>>>>>>>>>>>>> read any of the paper, so you take that comment that the >>>>>>>>>>>>>>> statment is "based" on that statement to mean it IS that >>>>>>>>>>>>>>> statement.
are asserting this counter-factual statement:
when a valid proxy for an argument is defeated this does >>>>>>>>>>>>>> not defeat the
original argument.
Right, which is what YOU are doing, showing your arguement >>>>>>>>>>>>> is INVALID.
In other words you disagree that correctly refuting a valid >>>>>>>>>>>> proxy for an
argument does correctly refute the original argument?
VALID is the key word,
Yours isn't (I don't think you actually know the meaning of >>>>>>>>>>> the words)
And you are an IDIOT to claim it is.
14 Every epistemological antinomy can likewise be used for a >>>>>>>>>> similar undecidability proof.
No, but you don't understand what he is saying.
proxy for
his proof. He is not saying that his expression is not an
epistemological antinomy.
Nope, that isn't what he is saying. How could it be, the ACTUAL G >>>>>>> is a proven Truth Bearer, while the Liar's Paradox isn't
Your arguement just shows its inconsistency.
In part, because you don't actually understand what the sentence >>>>>>> actually is.
The analogy between this result and Richard’s antinomy leaps to >>>>>>>> the eye;
there is also a close relationship with the “liar” antinomy,14 >>>>>>>> since the
undecidable proposition [R(q); q] states precisely that q
belongs to K,
i.e. according to (1), that [R(q); q] is not provable. We are
therefore
confronted with a proposition which asserts its own unprovability. >>>>>>>> (Gödel 1931:43)
This <is> an isomorphism to a proposition that asserts its own >>>>>>>> untruth.
Nope, unless you erroneously think that statements about Truth
ARE statements about provability, that isn't an isomoprhism.
Note, you have even stated that *ALL* statements of the form
"statement x is provable" or "Statment x is not provable" are
Truth Bearers,
I have most definitely never said this or anything that could be
unintentionally misconstrued to mean this.
It is always the case that when-so-ever any expression of language >>>>>> only
refers to its own truth or provability that this expression is not a >>>>>> truth bearer, thus not a member of any formal system of logic.
You admitted that it was TRUE that a statement could not be proven
even if the only way to show that it could not be proven was to
check the infinite set of all possible proofs to see that none of
them were a proof.
This is not related to what I just said. Every expression of
language is
untrue unless it has a semantic connection to its truth maker axiom.
I never disagreed with that, just that you keep on wavering between
just saying there must be a connection, and then at times adding it
must be a FINITE connection (which is actually only requried to be
Proven)
G, the statment about the non-existance of a natural number that
satisfies the specified primative recursive relationship is TRUE,
because it IS connected to the truth maker axioms of math via an
infinite chain of steps.
Each natural number can be shown to not meet that requirement in a
We are therefore confronted with a proposition which asserts its own
unprovability. (Gödel 1931:43)
No, that is a statement which is proven in Meta-F to have the identical
truth value of G. G doesn't SAY it is unprovable, but a natural
concesequence of G being True is that it is unprovable, and if it is provable, it can't be True. Since G must be True or False, if it is True
it IS unprovable, and if it is Provable, then it must be False, which is
a contradiction (since ALL provable statements are True), so that case
is impossible. Thus, G MUST be True but Unprovable.
If F includes an axiom that says all Truths are Provable, then F is
proved to be inconsistent.
He only used the whole natural numbers thing to be able to encode the
above expression in a language that did not have a provability
predicate.
No, F might well have a provability predicate, ies of the Natural Numbers.
(G) F ⊢ GF ↔ ¬ProvF(┌GF┐). // with Gödel number
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
Which if you read, agrees with Godel, that this sentence must be neither provable or disprovable, and agrees with the right conditions, can be
made True.
On 1/16/2023 7:51 PM, Richard Damon wrote:
On 1/16/23 4:51 PM, olcott wrote:
On 1/15/2023 7:15 PM, Richard Damon wrote:
On 1/15/23 7:56 PM, olcott wrote:Gödel was actually talking about the expression:
On 1/15/2023 6:47 PM, Richard Damon wrote:
On 1/15/23 7:26 PM, olcott wrote:
On 1/15/2023 2:23 PM, Richard Damon wrote:
On 1/15/23 3:12 PM, olcott wrote:
On 1/15/2023 1:46 PM, Richard Damon wrote:
On 1/15/23 2:29 PM, olcott wrote:He is saying that every epistemological antinomy is a valid
On 1/15/2023 1:06 PM, Richard Damon wrote:
On 1/15/23 2:00 PM, olcott wrote:So Gödel is wrong when he says:
On 1/15/2023 12:55 PM, Richard Damon wrote:
On 1/15/23 1:23 PM, olcott wrote:
On 1/15/2023 11:58 AM, Richard Damon wrote:
On 1/15/23 12:47 PM, olcott wrote:All of your Ad Hominem attacks cannot possibly hide the >>>>>>>>>>>>>>> fact that you
On 1/15/2023 11:41 AM, Richard Damon wrote: >>>>>>>>>>>>>>>>>> On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>> On 1/15/23 11:15 AM, olcott wrote:
That is false. It is not that G takes an infinite >>>>>>>>>>>>>>>>>>> number of steps toOn 1/14/2023 5:42 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>> On 1/14/23 6:19 PM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>> On 1/14/2023 4:55 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 1/14/23 5:31 PM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 1/14/2023 4:26 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 1/14/23 4:48 PM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> How does the formal system know that an >>>>>>>>>>>>>>>>>>>>>>>>>>> expression of language of this >>>>>>>>>>>>>>>>>>>>>>>>>>> formal system is true unless this expression >>>>>>>>>>>>>>>>>>>>>>>>>>> of language has a
So your basic line-of-reasoning is that G is true >>>>>>>>>>>>>>>>>>>>> in F even if the truthconnection to truth maker axioms *IN THIS >>>>>>>>>>>>>>>>>>>>>>>>>>> FORMAL SYSTEM* ???
Becaue the formal system doesn't need to KNOW >>>>>>>>>>>>>>>>>>>>>>>>>> what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + >>>>>>>>>>>>>>>>>>>>>>>>> successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for >>>>>>>>>>>>>>>>>>>>>>>> it to be true, doesn't mean it can't be. >>>>>>>>>>>>>>>>>>>>>>>>
In fact, your statement just comes out of a >>>>>>>>>>>>>>>>>>>>>>>> simple application of the addition AXIOMS of PA. >>>>>>>>>>>>>>>>>>>>>>>>
a + 0 = a
a + Successor(b) = Successor(a + b) >>>>>>>>>>>>>>>>>>>>>>>>
So it is a PROVABLE statement, and thus actually >>>>>>>>>>>>>>>>>>>>>>>> KNOWN to be true.
Unless a formal system has a syntactic connection >>>>>>>>>>>>>>>>>>>>>>> from an expression of
its language to its truth maker axioms the >>>>>>>>>>>>>>>>>>>>>>> expression is untrue in that
formal system.
Right, but the connection can be infinite in >>>>>>>>>>>>>>>>>>>>>> length, and thus not provable.
Try and show an expression of language that is >>>>>>>>>>>>>>>>>>>>>>> true in a formal system
(not just true somewhere else) that does not have >>>>>>>>>>>>>>>>>>>>>>> any connection to
truth maker axioms in this formal system. You >>>>>>>>>>>>>>>>>>>>>>> must show why it is true
in this formal system not merely that it is true >>>>>>>>>>>>>>>>>>>>>>> somewhere else.
The connection might be infinite, and thus not >>>>>>>>>>>>>>>>>>>>>> SHOWABLE as a proof strictly in the formal system. >>>>>>>>>>>>>>>>>>>>>>
If the connection exists as an infinite connection >>>>>>>>>>>>>>>>>>>>>> within the system, then it is TRUE in the system. >>>>>>>>>>>>>>>>>>>>>>
Note, that if there is such an infinite >>>>>>>>>>>>>>>>>>>>>> connection, which thus can not be proven within >>>>>>>>>>>>>>>>>>>>>> the formal system, it is still possible, that >>>>>>>>>>>>>>>>>>>>>> another system, related to that system, with more >>>>>>>>>>>>>>>>>>>>>> knowledge, might be able to show that there does >>>>>>>>>>>>>>>>>>>>>> exist within the original formal system such an >>>>>>>>>>>>>>>>>>>>>> infinte connection.
This is what happens to G in F and meta-F >>>>>>>>>>>>>>>>>>>>>>
G states that there does not exist a Natural >>>>>>>>>>>>>>>>>>>>>> Number g that meets a specific requirement >>>>>>>>>>>>>>>>>>>>>> (expressed as a primative recursive relationship). >>>>>>>>>>>>>>>>>>>>>>
This statement turns out to be true, because it >>>>>>>>>>>>>>>>>>>>>> turns out that no number g does meet that >>>>>>>>>>>>>>>>>>>>>> requirement, but it can't be proven in F that this >>>>>>>>>>>>>>>>>>>>>> is true, because in F, to show this we need to >>>>>>>>>>>>>>>>>>>>>> test every natuarl number, which requires an >>>>>>>>>>>>>>>>>>>>>> infinite number of steps (finite for each number, >>>>>>>>>>>>>>>>>>>>>> but an infinite number of numbers to test). >>>>>>>>>>>>>>>>>>>>>>
In meta-F, we can do better, because due to >>>>>>>>>>>>>>>>>>>>>> additional knowledge in meta-F, we can show that >>>>>>>>>>>>>>>>>>>>>> if a number g could be found, then that number g >>>>>>>>>>>>>>>>>>>>>> could be converted into a proof, in F, of the >>>>>>>>>>>>>>>>>>>>>> statement G (which says that such a number does >>>>>>>>>>>>>>>>>>>>>> not exist).
Thus, in meta-F, we can prove that G is true, and >>>>>>>>>>>>>>>>>>>>>> also show that no proof of it can exist in F. >>>>>>>>>>>>>>>>>>>>>>
of G cannot even be expressed in F as long as the >>>>>>>>>>>>>>>>>>>>> truth of G can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an >>>>>>>>>>>>>>>>>>>> infinite expression.
reach its truth maker axioms in F it is that even >>>>>>>>>>>>>>>>>>> after an infinite
number of steps it never reaches is truth maker >>>>>>>>>>>>>>>>>>> axioms in F because G is
simply untrue in F.
No, YOUR problem is you aren't actually talking about >>>>>>>>>>>>>>>>>> G in F.
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what? >>>>>>>>>>>>>>>>>>> Not true about being not true about being not true. >>>>>>>>>>>>>>>>>>>
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be >>>>>>>>>>>>>>>>>>> used for a similar undecidability proof. >>>>>>>>>>>>>>>>>>>
By using the Liar Paradox as a Gödel approved proxy >>>>>>>>>>>>>>>>>>> for his proof we
refute his proof by this Gödel approved proxy. >>>>>>>>>>>>>>>>>>>
No, all you have proved is that you are a LYING MORON. >>>>>>>>>>>>>>>>>>
Ad Hominem attacks are the tactic that people having no >>>>>>>>>>>>>>>>> interest in any
honest dialogue use when they realize that their >>>>>>>>>>>>>>>>> reasoning has been
utterly defeated.
And RED HERRING arguements don't work either.
You ARE a LYING MORON as you insist that Godel's G is a >>>>>>>>>>>>>>>> sentence that is actually provably isn't.
You think it is because you are too stupid to actually >>>>>>>>>>>>>>>> read any of the paper, so you take that comment that the >>>>>>>>>>>>>>>> statment is "based" on that statement to mean it IS that >>>>>>>>>>>>>>>> statement.
are asserting this counter-factual statement:
when a valid proxy for an argument is defeated this does >>>>>>>>>>>>>>> not defeat the
original argument.
Right, which is what YOU are doing, showing your arguement >>>>>>>>>>>>>> is INVALID.
In other words you disagree that correctly refuting a valid >>>>>>>>>>>>> proxy for an
argument does correctly refute the original argument? >>>>>>>>>>>>>
VALID is the key word,
Yours isn't (I don't think you actually know the meaning of >>>>>>>>>>>> the words)
And you are an IDIOT to claim it is.
14 Every epistemological antinomy can likewise be used for a >>>>>>>>>>> similar undecidability proof.
No, but you don't understand what he is saying.
proxy for
his proof. He is not saying that his expression is not an
epistemological antinomy.
Nope, that isn't what he is saying. How could it be, the ACTUAL >>>>>>>> G is a proven Truth Bearer, while the Liar's Paradox isn't
Your arguement just shows its inconsistency.
In part, because you don't actually understand what the sentence >>>>>>>> actually is.
The analogy between this result and Richard’s antinomy leaps to >>>>>>>>> the eye;
there is also a close relationship with the “liar” antinomy,14 >>>>>>>>> since the
undecidable proposition [R(q); q] states precisely that q
belongs to K,
i.e. according to (1), that [R(q); q] is not provable. We are >>>>>>>>> therefore
confronted with a proposition which asserts its own unprovability. >>>>>>>>> (Gödel 1931:43)
This <is> an isomorphism to a proposition that asserts its own >>>>>>>>> untruth.
Nope, unless you erroneously think that statements about Truth >>>>>>>> ARE statements about provability, that isn't an isomoprhism.
Note, you have even stated that *ALL* statements of the form
"statement x is provable" or "Statment x is not provable" are
Truth Bearers,
I have most definitely never said this or anything that could be >>>>>>> unintentionally misconstrued to mean this.
It is always the case that when-so-ever any expression of
language only
refers to its own truth or provability that this expression is not a >>>>>>> truth bearer, thus not a member of any formal system of logic.
You admitted that it was TRUE that a statement could not be proven >>>>>> even if the only way to show that it could not be proven was to
check the infinite set of all possible proofs to see that none of
them were a proof.
This is not related to what I just said. Every expression of
language is
untrue unless it has a semantic connection to its truth maker axiom.
I never disagreed with that, just that you keep on wavering between
just saying there must be a connection, and then at times adding it
must be a FINITE connection (which is actually only requried to be
Proven)
G, the statment about the non-existance of a natural number that
satisfies the specified primative recursive relationship is TRUE,
because it IS connected to the truth maker axioms of math via an
infinite chain of steps.
Each natural number can be shown to not meet that requirement in a
We are therefore confronted with a proposition which asserts its own
unprovability. (Gödel 1931:43)
No, that is a statement which is proven in Meta-F to have the
identical truth value of G. G doesn't SAY it is unprovable, but a
natural concesequence of G being True is that it is unprovable, and if
it is provable, it can't be True. Since G must be True or False, if it
is True it IS unprovable, and if it is Provable, then it must be
False, which is a contradiction (since ALL provable statements are
True), so that case is impossible. Thus, G MUST be True but Unprovable.
If F includes an axiom that says all Truths are Provable, then F is
proved to be inconsistent.
He only used the whole natural numbers thing to be able to encode the
above expression in a language that did not have a provability
predicate.
No, F might well have a provability predicate, ies of the Natural
Numbers.
Then it would not need any Gödel number.
(G) F ⊢ GF ↔ ¬ProvF(┌GF┐). // with Gödel number
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
*paraphrased as: GF ↔ (F ⊬ GF) // without Gödel number*
Which if you read, agrees with Godel, that this sentence must be
neither provable or disprovable, and agrees with the right conditions,
can be made True.
I am removing the Gödel number and showing what's left.
On 1/16/23 10:17 AM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:So you changed your mind about infinite proofs in formal systems?
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if the
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language >>>>>>>>>> of this
formal system is true unless this expression of language has a >>>>>>>>>> connection to truth maker axioms *IN THIS FORMAL SYSTEM* ??? >>>>>>>>>>
Becaue the formal system doesn't need to KNOW what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ??? >>>>>>>>
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true,
doesn't mean it can't be.
In fact, your statement just comes out of a simple application of >>>>>>> the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true. >>>>>>>
Unless a formal system has a syntactic connection from an
expression of
its language to its truth maker axioms the expression is untrue in >>>>>> that
formal system.
Right, but the connection can be infinite in length, and thus not
provable.
Try and show an expression of language that is true in a formal
system
(not just true somewhere else) that does not have any connection to >>>>>> truth maker axioms in this formal system. You must show why it is
true
in this formal system not merely that it is true somewhere else.
The connection might be infinite, and thus not SHOWABLE as a proof
strictly in the formal system.
If the connection exists as an infinite connection within the
system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can
not be proven within the formal system, it is still possible, that
another system, related to that system, with more knowledge, might
be able to show that there does exist within the original formal
system such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets a
specific requirement (expressed as a primative recursive
relationship).
This statement turns out to be true, because it turns out that no
number g does meet that requirement, but it can't be proven in F
that this is true, because in F, to show this we need to test every
natuarl number, which requires an infinite number of steps (finite
for each number, but an infinite number of numbers to test).
In meta-F, we can do better, because due to additional knowledge in
meta-F, we can show that if a number g could be found, then that
number g could be converted into a proof, in F, of the statement G
(which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that no
proof of it can exist in F.
truth
of G cannot even be expressed in F as long as the truth of G can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite expression. >>>
We know the steps of the infinite proof for the Goldbach conjecture.
No, because I am showing that G is TRUE, not PROVABLE. Truth can use
infinte sets oc connections, proofs can't. Only YOU have perposed that
we think about infinite proofs.
On 1/17/23 12:32 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
On 1/16/23 4:51 PM, olcott wrote:
On 1/15/2023 7:15 PM, Richard Damon wrote:
On 1/15/23 7:56 PM, olcott wrote:Gödel was actually talking about the expression:
On 1/15/2023 6:47 PM, Richard Damon wrote:I never disagreed with that, just that you keep on wavering between
On 1/15/23 7:26 PM, olcott wrote:
On 1/15/2023 2:23 PM, Richard Damon wrote:
On 1/15/23 3:12 PM, olcott wrote:
On 1/15/2023 1:46 PM, Richard Damon wrote:
On 1/15/23 2:29 PM, olcott wrote:He is saying that every epistemological antinomy is a valid >>>>>>>>>> proxy for
On 1/15/2023 1:06 PM, Richard Damon wrote:
On 1/15/23 2:00 PM, olcott wrote:So Gödel is wrong when he says:
On 1/15/2023 12:55 PM, Richard Damon wrote:
On 1/15/23 1:23 PM, olcott wrote:
On 1/15/2023 11:58 AM, Richard Damon wrote:
On 1/15/23 12:47 PM, olcott wrote:All of your Ad Hominem attacks cannot possibly hide the >>>>>>>>>>>>>>>> fact that you
On 1/15/2023 11:41 AM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>> On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>> On 1/15/23 11:15 AM, olcott wrote:
That is false. It is not that G takes an infinite >>>>>>>>>>>>>>>>>>>> number of steps toOn 1/14/2023 5:42 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>>> On 1/14/23 6:19 PM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 1/14/2023 4:55 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 1/14/23 5:31 PM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 1/14/2023 4:26 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 1/14/23 4:48 PM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> How does the formal system know that an >>>>>>>>>>>>>>>>>>>>>>>>>>>> expression of language of this >>>>>>>>>>>>>>>>>>>>>>>>>>>> formal system is true unless this expression >>>>>>>>>>>>>>>>>>>>>>>>>>>> of language has a
So your basic line-of-reasoning is that G is true >>>>>>>>>>>>>>>>>>>>>> in F even if the truthconnection to truth maker axioms *IN THIS >>>>>>>>>>>>>>>>>>>>>>>>>>>> FORMAL SYSTEM* ???
Becaue the formal system doesn't need to KNOW >>>>>>>>>>>>>>>>>>>>>>>>>>> what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + >>>>>>>>>>>>>>>>>>>>>>>>>> successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven >>>>>>>>>>>>>>>>>>>>>>>>> for it to be true, doesn't mean it can't be. >>>>>>>>>>>>>>>>>>>>>>>>>
In fact, your statement just comes out of a >>>>>>>>>>>>>>>>>>>>>>>>> simple application of the addition AXIOMS of PA. >>>>>>>>>>>>>>>>>>>>>>>>>
a + 0 = a
a + Successor(b) = Successor(a + b) >>>>>>>>>>>>>>>>>>>>>>>>>
So it is a PROVABLE statement, and thus >>>>>>>>>>>>>>>>>>>>>>>>> actually KNOWN to be true.
Unless a formal system has a syntactic >>>>>>>>>>>>>>>>>>>>>>>> connection from an expression of >>>>>>>>>>>>>>>>>>>>>>>> its language to its truth maker axioms the >>>>>>>>>>>>>>>>>>>>>>>> expression is untrue in that
formal system.
Right, but the connection can be infinite in >>>>>>>>>>>>>>>>>>>>>>> length, and thus not provable.
Try and show an expression of language that is >>>>>>>>>>>>>>>>>>>>>>>> true in a formal system
(not just true somewhere else) that does not >>>>>>>>>>>>>>>>>>>>>>>> have any connection to
truth maker axioms in this formal system. You >>>>>>>>>>>>>>>>>>>>>>>> must show why it is true
in this formal system not merely that it is true >>>>>>>>>>>>>>>>>>>>>>>> somewhere else.
The connection might be infinite, and thus not >>>>>>>>>>>>>>>>>>>>>>> SHOWABLE as a proof strictly in the formal system. >>>>>>>>>>>>>>>>>>>>>>>
If the connection exists as an infinite >>>>>>>>>>>>>>>>>>>>>>> connection within the system, then it is TRUE in >>>>>>>>>>>>>>>>>>>>>>> the system.
Note, that if there is such an infinite >>>>>>>>>>>>>>>>>>>>>>> connection, which thus can not be proven within >>>>>>>>>>>>>>>>>>>>>>> the formal system, it is still possible, that >>>>>>>>>>>>>>>>>>>>>>> another system, related to that system, with more >>>>>>>>>>>>>>>>>>>>>>> knowledge, might be able to show that there does >>>>>>>>>>>>>>>>>>>>>>> exist within the original formal system such an >>>>>>>>>>>>>>>>>>>>>>> infinte connection.
This is what happens to G in F and meta-F >>>>>>>>>>>>>>>>>>>>>>>
G states that there does not exist a Natural >>>>>>>>>>>>>>>>>>>>>>> Number g that meets a specific requirement >>>>>>>>>>>>>>>>>>>>>>> (expressed as a primative recursive relationship). >>>>>>>>>>>>>>>>>>>>>>>
This statement turns out to be true, because it >>>>>>>>>>>>>>>>>>>>>>> turns out that no number g does meet that >>>>>>>>>>>>>>>>>>>>>>> requirement, but it can't be proven in F that >>>>>>>>>>>>>>>>>>>>>>> this is true, because in F, to show this we need >>>>>>>>>>>>>>>>>>>>>>> to test every natuarl number, which requires an >>>>>>>>>>>>>>>>>>>>>>> infinite number of steps (finite for each number, >>>>>>>>>>>>>>>>>>>>>>> but an infinite number of numbers to test). >>>>>>>>>>>>>>>>>>>>>>>
In meta-F, we can do better, because due to >>>>>>>>>>>>>>>>>>>>>>> additional knowledge in meta-F, we can show that >>>>>>>>>>>>>>>>>>>>>>> if a number g could be found, then that number g >>>>>>>>>>>>>>>>>>>>>>> could be converted into a proof, in F, of the >>>>>>>>>>>>>>>>>>>>>>> statement G (which says that such a number does >>>>>>>>>>>>>>>>>>>>>>> not exist).
Thus, in meta-F, we can prove that G is true, and >>>>>>>>>>>>>>>>>>>>>>> also show that no proof of it can exist in F. >>>>>>>>>>>>>>>>>>>>>>>
of G cannot even be expressed in F as long as the >>>>>>>>>>>>>>>>>>>>>> truth of G can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an >>>>>>>>>>>>>>>>>>>>> infinite expression.
reach its truth maker axioms in F it is that even >>>>>>>>>>>>>>>>>>>> after an infinite
number of steps it never reaches is truth maker >>>>>>>>>>>>>>>>>>>> axioms in F because G is
simply untrue in F.
No, YOUR problem is you aren't actually talking about >>>>>>>>>>>>>>>>>>> G in F.
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what? >>>>>>>>>>>>>>>>>>>> Not true about being not true about being not true. >>>>>>>>>>>>>>>>>>>>
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be >>>>>>>>>>>>>>>>>>>> used for a similar undecidability proof. >>>>>>>>>>>>>>>>>>>>
By using the Liar Paradox as a Gödel approved proxy >>>>>>>>>>>>>>>>>>>> for his proof we
refute his proof by this Gödel approved proxy. >>>>>>>>>>>>>>>>>>>>
No, all you have proved is that you are a LYING MORON. >>>>>>>>>>>>>>>>>>>
Ad Hominem attacks are the tactic that people having >>>>>>>>>>>>>>>>>> no interest in any
honest dialogue use when they realize that their >>>>>>>>>>>>>>>>>> reasoning has been
utterly defeated.
And RED HERRING arguements don't work either. >>>>>>>>>>>>>>>>>
You ARE a LYING MORON as you insist that Godel's G is a >>>>>>>>>>>>>>>>> sentence that is actually provably isn't.
You think it is because you are too stupid to actually >>>>>>>>>>>>>>>>> read any of the paper, so you take that comment that >>>>>>>>>>>>>>>>> the statment is "based" on that statement to mean it IS >>>>>>>>>>>>>>>>> that statement.
are asserting this counter-factual statement:
when a valid proxy for an argument is defeated this does >>>>>>>>>>>>>>>> not defeat the
original argument.
Right, which is what YOU are doing, showing your >>>>>>>>>>>>>>> arguement is INVALID.
In other words you disagree that correctly refuting a >>>>>>>>>>>>>> valid proxy for an
argument does correctly refute the original argument? >>>>>>>>>>>>>>
VALID is the key word,
Yours isn't (I don't think you actually know the meaning of >>>>>>>>>>>>> the words)
And you are an IDIOT to claim it is.
14 Every epistemological antinomy can likewise be used for a >>>>>>>>>>>> similar undecidability proof.
No, but you don't understand what he is saying.
his proof. He is not saying that his expression is not an
epistemological antinomy.
Nope, that isn't what he is saying. How could it be, the ACTUAL >>>>>>>>> G is a proven Truth Bearer, while the Liar's Paradox isn't
Your arguement just shows its inconsistency.
In part, because you don't actually understand what the
sentence actually is.
The analogy between this result and Richard’s antinomy leaps >>>>>>>>>> to the eye;
there is also a close relationship with the “liar” antinomy,14 >>>>>>>>>> since the
undecidable proposition [R(q); q] states precisely that q
belongs to K,
i.e. according to (1), that [R(q); q] is not provable. We are >>>>>>>>>> therefore
confronted with a proposition which asserts its own
unprovability.
(Gödel 1931:43)
This <is> an isomorphism to a proposition that asserts its own >>>>>>>>>> untruth.
Nope, unless you erroneously think that statements about Truth >>>>>>>>> ARE statements about provability, that isn't an isomoprhism. >>>>>>>>>
Note, you have even stated that *ALL* statements of the form >>>>>>>>> "statement x is provable" or "Statment x is not provable" are >>>>>>>>> Truth Bearers,
I have most definitely never said this or anything that could be >>>>>>>> unintentionally misconstrued to mean this.
It is always the case that when-so-ever any expression of
language only
refers to its own truth or provability that this expression is >>>>>>>> not a
truth bearer, thus not a member of any formal system of logic. >>>>>>>>
You admitted that it was TRUE that a statement could not be
proven even if the only way to show that it could not be proven
was to check the infinite set of all possible proofs to see that >>>>>>> none of them were a proof.
This is not related to what I just said. Every expression of
language is
untrue unless it has a semantic connection to its truth maker axiom. >>>>>
just saying there must be a connection, and then at times adding it
must be a FINITE connection (which is actually only requried to be
Proven)
G, the statment about the non-existance of a natural number that
satisfies the specified primative recursive relationship is TRUE,
because it IS connected to the truth maker axioms of math via an
infinite chain of steps.
Each natural number can be shown to not meet that requirement in a
We are therefore confronted with a proposition which asserts its own
unprovability. (Gödel 1931:43)
No, that is a statement which is proven in Meta-F to have the
identical truth value of G. G doesn't SAY it is unprovable, but a
natural concesequence of G being True is that it is unprovable, and
if it is provable, it can't be True. Since G must be True or False,
if it is True it IS unprovable, and if it is Provable, then it must
be False, which is a contradiction (since ALL provable statements are
True), so that case is impossible. Thus, G MUST be True but Unprovable.
If F includes an axiom that says all Truths are Provable, then F is
proved to be inconsistent.
He only used the whole natural numbers thing to be able to encode the
above expression in a language that did not have a provability
predicate.
No, F might well have a provability predicate, ies of the Natural
Numbers.
Then it would not need any Gödel number.
Maybe, but the key is that it DOESN'T use the operator, so your
"special" rule based on using it doesn't apply.
Godel showed that Meta-F we can construct a calculation in Meta-F that
is the exact same caluclation in F that provides us proofs in Meta-F
based on what is simply a calculation in F.
Since it is just a calculation in F about the existance of a number
based on a computable function, in F the stateement ALWAYS has a Truth
Value, either such a number exsits or it doesn't.
Because of the DEFINED relationship between F and Meta-F, that truth
value transfers, and due to the extra axioms in Meta-F.
Arguing that in Meta-F we have an epistemological antinomy means that
your logic system makes the mathematics in F be able to create this same situation, which doesn't match the behavior of the Natural Numbers, so
your F doesn't meet the requirements for it.
YOU "PROOF" FAILS.
If you want to make the sort of claims you are doing, you need to show exactly which step in his proof does something wrong. You are not
allowed to rebut a proof by saying its answer must be wrong, or you
disagree with a footnote. You need to find an actaul erroneous step in
the proof itself.
Since you have shown you don't actually understand the proof at all,
this is probably impossible for you.
(G) F ⊢ GF ↔ ¬ProvF(┌GF┐). // with Gödel number
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
*paraphrased as: GF ↔ (F ⊬ GF) // without Gödel number*
Nope, it only "means" that in Meta F, not in F.
F doesn't have Truth Makers to establish that meaning, so that
"parapharse" is incorrect.
You are just showing you don't understand what logic does.
Which if you read, agrees with Godel, that this sentence must be
neither provable or disprovable, and agrees with the right
conditions, can be made True.
I am removing the Gödel number and showing what's left.
Remove the Godel Number, and NOTHING is left of the statement in F,
On Tuesday, January 17, 2023 at 8:40:02 AM UTC-8, _ Olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
On 1/16/23 10:17 AM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:So you changed your mind about infinite proofs in formal systems?
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if the >>>>>> truth
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language >>>>>>>>>>>> of this
formal system is true unless this expression of language has a >>>>>>>>>>>> connection to truth maker axioms *IN THIS FORMAL SYSTEM* ??? >>>>>>>>>>>>
Becaue the formal system doesn't need to KNOW what is true. >>>>>>>>>>>
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ??? >>>>>>>>>>
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true, >>>>>>>>> doesn't mean it can't be.
In fact, your statement just comes out of a simple application of >>>>>>>>> the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true. >>>>>>>>>
Unless a formal system has a syntactic connection from an
expression of
its language to its truth maker axioms the expression is untrue in >>>>>>>> that
formal system.
Right, but the connection can be infinite in length, and thus not >>>>>>> provable.
Try and show an expression of language that is true in a formal >>>>>>>> system
(not just true somewhere else) that does not have any connection to >>>>>>>> truth maker axioms in this formal system. You must show why it is >>>>>>>> true
in this formal system not merely that it is true somewhere else. >>>>>>>>
The connection might be infinite, and thus not SHOWABLE as a proof >>>>>>> strictly in the formal system.
If the connection exists as an infinite connection within the
system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can >>>>>>> not be proven within the formal system, it is still possible, that >>>>>>> another system, related to that system, with more knowledge, might >>>>>>> be able to show that there does exist within the original formal >>>>>>> system such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets a >>>>>>> specific requirement (expressed as a primative recursive
relationship).
This statement turns out to be true, because it turns out that no >>>>>>> number g does meet that requirement, but it can't be proven in F >>>>>>> that this is true, because in F, to show this we need to test every >>>>>>> natuarl number, which requires an infinite number of steps (finite >>>>>>> for each number, but an infinite number of numbers to test).
In meta-F, we can do better, because due to additional knowledge in >>>>>>> meta-F, we can show that if a number g could be found, then that >>>>>>> number g could be converted into a proof, in F, of the statement G >>>>>>> (which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that no >>>>>>> proof of it can exist in F.
of G cannot even be expressed in F as long as the truth of G can be >>>>>> expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite expression. >>>>>
We know the steps of the infinite proof for the Goldbach conjecture.
No, because I am showing that G is TRUE, not PROVABLE. Truth can use
infinte sets oc connections, proofs can't. Only YOU have perposed that
we think about infinite proofs.
Formal systems cannot ever use infinite connections from their
expressions of language to their truth maker axioms thus eliminating
these from consideration as any measure of true "in the system".
--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
"What you've written is almost like formal logic."
On 1/17/2023 5:02 AM, Richard Damon wrote:
On 1/17/23 12:32 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
On 1/16/23 4:51 PM, olcott wrote:
On 1/15/2023 7:15 PM, Richard Damon wrote:
On 1/15/23 7:56 PM, olcott wrote:Gödel was actually talking about the expression:
On 1/15/2023 6:47 PM, Richard Damon wrote:I never disagreed with that, just that you keep on wavering
On 1/15/23 7:26 PM, olcott wrote:
On 1/15/2023 2:23 PM, Richard Damon wrote:
On 1/15/23 3:12 PM, olcott wrote:
On 1/15/2023 1:46 PM, Richard Damon wrote:
On 1/15/23 2:29 PM, olcott wrote:He is saying that every epistemological antinomy is a valid >>>>>>>>>>> proxy for
On 1/15/2023 1:06 PM, Richard Damon wrote:
On 1/15/23 2:00 PM, olcott wrote:So Gödel is wrong when he says:
On 1/15/2023 12:55 PM, Richard Damon wrote:
On 1/15/23 1:23 PM, olcott wrote:
On 1/15/2023 11:58 AM, Richard Damon wrote: >>>>>>>>>>>>>>>>>> On 1/15/23 12:47 PM, olcott wrote:
All of your Ad Hominem attacks cannot possibly hide the >>>>>>>>>>>>>>>>> fact that youOn 1/15/2023 11:41 AM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>> On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>> On 1/15/23 11:15 AM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>> On 1/14/2023 5:42 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 1/14/23 6:19 PM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 1/14/2023 4:55 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 1/14/23 5:31 PM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 1/14/2023 4:26 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> On 1/14/23 4:48 PM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> How does the formal system know that an >>>>>>>>>>>>>>>>>>>>>>>>>>>>> expression of language of this >>>>>>>>>>>>>>>>>>>>>>>>>>>>> formal system is true unless this >>>>>>>>>>>>>>>>>>>>>>>>>>>>> expression of language has a >>>>>>>>>>>>>>>>>>>>>>>>>>>>> connection to truth maker axioms *IN THIS >>>>>>>>>>>>>>>>>>>>>>>>>>>>> FORMAL SYSTEM* ???
That is false. It is not that G takes an infinite >>>>>>>>>>>>>>>>>>>>> number of steps toSo your basic line-of-reasoning is that G is true >>>>>>>>>>>>>>>>>>>>>>> in F even if the truth
Becaue the formal system doesn't need to >>>>>>>>>>>>>>>>>>>>>>>>>>>> KNOW what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + >>>>>>>>>>>>>>>>>>>>>>>>>>> successor(0) is true ??? >>>>>>>>>>>>>>>>>>>>>>>>>>>
Why do you say that.
Just because truth doesn't NEED to be proven >>>>>>>>>>>>>>>>>>>>>>>>>> for it to be true, doesn't mean it can't be. >>>>>>>>>>>>>>>>>>>>>>>>>>
In fact, your statement just comes out of a >>>>>>>>>>>>>>>>>>>>>>>>>> simple application of the addition AXIOMS of PA. >>>>>>>>>>>>>>>>>>>>>>>>>>
a + 0 = a
a + Successor(b) = Successor(a + b) >>>>>>>>>>>>>>>>>>>>>>>>>>
So it is a PROVABLE statement, and thus >>>>>>>>>>>>>>>>>>>>>>>>>> actually KNOWN to be true. >>>>>>>>>>>>>>>>>>>>>>>>>>
Unless a formal system has a syntactic >>>>>>>>>>>>>>>>>>>>>>>>> connection from an expression of >>>>>>>>>>>>>>>>>>>>>>>>> its language to its truth maker axioms the >>>>>>>>>>>>>>>>>>>>>>>>> expression is untrue in that >>>>>>>>>>>>>>>>>>>>>>>>> formal system.
Right, but the connection can be infinite in >>>>>>>>>>>>>>>>>>>>>>>> length, and thus not provable. >>>>>>>>>>>>>>>>>>>>>>>>
Try and show an expression of language that is >>>>>>>>>>>>>>>>>>>>>>>>> true in a formal system
(not just true somewhere else) that does not >>>>>>>>>>>>>>>>>>>>>>>>> have any connection to
truth maker axioms in this formal system. You >>>>>>>>>>>>>>>>>>>>>>>>> must show why it is true
in this formal system not merely that it is >>>>>>>>>>>>>>>>>>>>>>>>> true somewhere else.
The connection might be infinite, and thus not >>>>>>>>>>>>>>>>>>>>>>>> SHOWABLE as a proof strictly in the formal system. >>>>>>>>>>>>>>>>>>>>>>>>
If the connection exists as an infinite >>>>>>>>>>>>>>>>>>>>>>>> connection within the system, then it is TRUE in >>>>>>>>>>>>>>>>>>>>>>>> the system.
Note, that if there is such an infinite >>>>>>>>>>>>>>>>>>>>>>>> connection, which thus can not be proven within >>>>>>>>>>>>>>>>>>>>>>>> the formal system, it is still possible, that >>>>>>>>>>>>>>>>>>>>>>>> another system, related to that system, with >>>>>>>>>>>>>>>>>>>>>>>> more knowledge, might be able to show that there >>>>>>>>>>>>>>>>>>>>>>>> does exist within the original formal system >>>>>>>>>>>>>>>>>>>>>>>> such an infinte connection.
This is what happens to G in F and meta-F >>>>>>>>>>>>>>>>>>>>>>>>
G states that there does not exist a Natural >>>>>>>>>>>>>>>>>>>>>>>> Number g that meets a specific requirement >>>>>>>>>>>>>>>>>>>>>>>> (expressed as a primative recursive relationship). >>>>>>>>>>>>>>>>>>>>>>>>
This statement turns out to be true, because it >>>>>>>>>>>>>>>>>>>>>>>> turns out that no number g does meet that >>>>>>>>>>>>>>>>>>>>>>>> requirement, but it can't be proven in F that >>>>>>>>>>>>>>>>>>>>>>>> this is true, because in F, to show this we need >>>>>>>>>>>>>>>>>>>>>>>> to test every natuarl number, which requires an >>>>>>>>>>>>>>>>>>>>>>>> infinite number of steps (finite for each >>>>>>>>>>>>>>>>>>>>>>>> number, but an infinite number of numbers to test). >>>>>>>>>>>>>>>>>>>>>>>>
In meta-F, we can do better, because due to >>>>>>>>>>>>>>>>>>>>>>>> additional knowledge in meta-F, we can show that >>>>>>>>>>>>>>>>>>>>>>>> if a number g could be found, then that number g >>>>>>>>>>>>>>>>>>>>>>>> could be converted into a proof, in F, of the >>>>>>>>>>>>>>>>>>>>>>>> statement G (which says that such a number does >>>>>>>>>>>>>>>>>>>>>>>> not exist).
Thus, in meta-F, we can prove that G is true, >>>>>>>>>>>>>>>>>>>>>>>> and also show that no proof of it can exist in F. >>>>>>>>>>>>>>>>>>>>>>>>
of G cannot even be expressed in F as long as the >>>>>>>>>>>>>>>>>>>>>>> truth of G can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is >>>>>>>>>>>>>>>>>>>>>> an infinite expression.
reach its truth maker axioms in F it is that even >>>>>>>>>>>>>>>>>>>>> after an infinite
number of steps it never reaches is truth maker >>>>>>>>>>>>>>>>>>>>> axioms in F because G is
simply untrue in F.
No, YOUR problem is you aren't actually talking >>>>>>>>>>>>>>>>>>>> about G in F.
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what? >>>>>>>>>>>>>>>>>>>>> Not true about being not true about being not true. >>>>>>>>>>>>>>>>>>>>>
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be >>>>>>>>>>>>>>>>>>>>> used for a similar undecidability proof. >>>>>>>>>>>>>>>>>>>>>
By using the Liar Paradox as a Gödel approved proxy >>>>>>>>>>>>>>>>>>>>> for his proof we
refute his proof by this Gödel approved proxy. >>>>>>>>>>>>>>>>>>>>>
No, all you have proved is that you are a LYING MORON. >>>>>>>>>>>>>>>>>>>>
Ad Hominem attacks are the tactic that people having >>>>>>>>>>>>>>>>>>> no interest in any
honest dialogue use when they realize that their >>>>>>>>>>>>>>>>>>> reasoning has been
utterly defeated.
And RED HERRING arguements don't work either. >>>>>>>>>>>>>>>>>>
You ARE a LYING MORON as you insist that Godel's G is >>>>>>>>>>>>>>>>>> a sentence that is actually provably isn't. >>>>>>>>>>>>>>>>>>
You think it is because you are too stupid to actually >>>>>>>>>>>>>>>>>> read any of the paper, so you take that comment that >>>>>>>>>>>>>>>>>> the statment is "based" on that statement to mean it >>>>>>>>>>>>>>>>>> IS that statement.
are asserting this counter-factual statement: >>>>>>>>>>>>>>>>>
when a valid proxy for an argument is defeated this >>>>>>>>>>>>>>>>> does not defeat the
original argument.
Right, which is what YOU are doing, showing your >>>>>>>>>>>>>>>> arguement is INVALID.
In other words you disagree that correctly refuting a >>>>>>>>>>>>>>> valid proxy for an
argument does correctly refute the original argument? >>>>>>>>>>>>>>>
VALID is the key word,
Yours isn't (I don't think you actually know the meaning >>>>>>>>>>>>>> of the words)
And you are an IDIOT to claim it is.
14 Every epistemological antinomy can likewise be used for a >>>>>>>>>>>>> similar undecidability proof.
No, but you don't understand what he is saying.
his proof. He is not saying that his expression is not an >>>>>>>>>>> epistemological antinomy.
Nope, that isn't what he is saying. How could it be, the
ACTUAL G is a proven Truth Bearer, while the Liar's Paradox isn't >>>>>>>>>>
Your arguement just shows its inconsistency.
In part, because you don't actually understand what the
sentence actually is.
The analogy between this result and Richard’s antinomy leaps >>>>>>>>>>> to the eye;
there is also a close relationship with the “liar”
antinomy,14 since the
undecidable proposition [R(q); q] states precisely that q >>>>>>>>>>> belongs to K,
i.e. according to (1), that [R(q); q] is not provable. We are >>>>>>>>>>> therefore
confronted with a proposition which asserts its own
unprovability.
(Gödel 1931:43)
This <is> an isomorphism to a proposition that asserts its >>>>>>>>>>> own untruth.
Nope, unless you erroneously think that statements about Truth >>>>>>>>>> ARE statements about provability, that isn't an isomoprhism. >>>>>>>>>>
Note, you have even stated that *ALL* statements of the form >>>>>>>>>> "statement x is provable" or "Statment x is not provable" are >>>>>>>>>> Truth Bearers,
I have most definitely never said this or anything that could be >>>>>>>>> unintentionally misconstrued to mean this.
It is always the case that when-so-ever any expression of
language only
refers to its own truth or provability that this expression is >>>>>>>>> not a
truth bearer, thus not a member of any formal system of logic. >>>>>>>>>
You admitted that it was TRUE that a statement could not be
proven even if the only way to show that it could not be proven >>>>>>>> was to check the infinite set of all possible proofs to see that >>>>>>>> none of them were a proof.
This is not related to what I just said. Every expression of
language is
untrue unless it has a semantic connection to its truth maker axiom. >>>>>>
between just saying there must be a connection, and then at times
adding it must be a FINITE connection (which is actually only
requried to be Proven)
G, the statment about the non-existance of a natural number that
satisfies the specified primative recursive relationship is TRUE,
because it IS connected to the truth maker axioms of math via an
infinite chain of steps.
Each natural number can be shown to not meet that requirement in a
We are therefore confronted with a proposition which asserts its own >>>>> unprovability. (Gödel 1931:43)
No, that is a statement which is proven in Meta-F to have the
identical truth value of G. G doesn't SAY it is unprovable, but a
natural concesequence of G being True is that it is unprovable, and
if it is provable, it can't be True. Since G must be True or False,
if it is True it IS unprovable, and if it is Provable, then it must
be False, which is a contradiction (since ALL provable statements
are True), so that case is impossible. Thus, G MUST be True but
Unprovable.
If F includes an axiom that says all Truths are Provable, then F is
proved to be inconsistent.
He only used the whole natural numbers thing to be able to encode the >>>>> above expression in a language that did not have a provability
predicate.
No, F might well have a provability predicate, ies of the Natural
Numbers.
Then it would not need any Gödel number.
Maybe, but the key is that it DOESN'T use the operator, so your
"special" rule based on using it doesn't apply.
Godel showed that Meta-F we can construct a calculation in Meta-F that
is the exact same caluclation in F that provides us proofs in Meta-F
based on what is simply a calculation in F.
Since it is just a calculation in F about the existance of a number
based on a computable function, in F the stateement ALWAYS has a Truth
Value, either such a number exsits or it doesn't.
Because of the DEFINED relationship between F and Meta-F, that truth
value transfers, and due to the extra axioms in Meta-F.
Arguing that in Meta-F we have an epistemological antinomy means that
your logic system makes the mathematics in F be able to create this
same situation, which doesn't match the behavior of the Natural
Numbers, so your F doesn't meet the requirements for it.
YOU "PROOF" FAILS.
If you want to make the sort of claims you are doing, you need to show
exactly which step in his proof does something wrong. You are not
allowed to rebut a proof by saying its answer must be wrong, or you
disagree with a footnote. You need to find an actaul erroneous step in
the proof itself.
Since you have shown you don't actually understand the proof at all,
this is probably impossible for you.
*paraphrased as: GF ↔ (F ⊬ GF) // without Gödel number*
(G) F ⊢ GF ↔ ¬ProvF(┌GF┐). // with Gödel number
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom >>>
Nope, it only "means" that in Meta F, not in F.
F doesn't have Truth Makers to establish that meaning, so that
"parapharse" is incorrect.
You are just showing you don't understand what logic does.
Which if you read, agrees with Godel, that this sentence must be
neither provable or disprovable, and agrees with the right
conditions, can be made True.
I am removing the Gödel number and showing what's left.
Remove the Godel Number, and NOTHING is left of the statement in F,
Here is what remains: GF ↔ (F ⊬ GF)
If any mathematician had noticed this then they would have noticed that Gödel did not prove that formal systems are incomplete. He only proved
that some of the expressions of language of a formal system are simply
untrue which is a mere triviality that everyone already knew.
On 1/17/2023 10:44 AM, Jeffrey Rubard wrote:
On Tuesday, January 17, 2023 at 8:40:02 AM UTC-8, _ Olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
On 1/16/23 10:17 AM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:No, because I am showing that G is TRUE, not PROVABLE. Truth can use
On 1/15/23 11:15 AM, olcott wrote:So you changed your mind about infinite proofs in formal systems?
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if the >>>>>> truth
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language >>>>>>>>>>>> of this
formal system is true unless this expression of language has a >>>>>>>>>>>> connection to truth maker axioms *IN THIS FORMAL SYSTEM* ??? >>>>>>>>>>>>
Becaue the formal system doesn't need to KNOW what is true. >>>>>>>>>>>
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true, >>>>>>>>> doesn't mean it can't be.
In fact, your statement just comes out of a simple application of >>>>>>>>> the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true. >>>>>>>>>
Unless a formal system has a syntactic connection from an
expression of
its language to its truth maker axioms the expression is untrue in >>>>>>>> that
formal system.
Right, but the connection can be infinite in length, and thus not >>>>>>> provable.
Try and show an expression of language that is true in a formal >>>>>>>> system
(not just true somewhere else) that does not have any connection to >>>>>>>> truth maker axioms in this formal system. You must show why it is >>>>>>>> true
in this formal system not merely that it is true somewhere else. >>>>>>>>
The connection might be infinite, and thus not SHOWABLE as a proof >>>>>>> strictly in the formal system.
If the connection exists as an infinite connection within the >>>>>>> system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can >>>>>>> not be proven within the formal system, it is still possible, that >>>>>>> another system, related to that system, with more knowledge, might >>>>>>> be able to show that there does exist within the original formal >>>>>>> system such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets a >>>>>>> specific requirement (expressed as a primative recursive
relationship).
This statement turns out to be true, because it turns out that no >>>>>>> number g does meet that requirement, but it can't be proven in F >>>>>>> that this is true, because in F, to show this we need to test every >>>>>>> natuarl number, which requires an infinite number of steps (finite >>>>>>> for each number, but an infinite number of numbers to test).
In meta-F, we can do better, because due to additional knowledge in >>>>>>> meta-F, we can show that if a number g could be found, then that >>>>>>> number g could be converted into a proof, in F, of the statement G >>>>>>> (which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that no >>>>>>> proof of it can exist in F.
of G cannot even be expressed in F as long as the truth of G can be >>>>>> expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite expression.
We know the steps of the infinite proof for the Goldbach conjecture. >>>
infinte sets oc connections, proofs can't. Only YOU have perposed that >>> we think about infinite proofs.
Formal systems cannot ever use infinite connections from their
expressions of language to their truth maker axioms thus eliminating
these from consideration as any measure of true "in the system".
--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius >> hits a target no one else can see." Arthur Schopenhauer
"What you've written is almost like formal logic."
I am filling in a key detail about the way that "true" in formal logic actually works. Wittgenstein first pointed this out and no mathematician
has ever noticed. https://www.liarparadox.org/Wittgenstein.pdf
If any mathematician had noticed this then they would have noticed that Gödel did not prove that formal systems are incomplete. He only proved
that some of the expressions of language of a formal system are simply untrue which is a mere triviality that everyone already knew.
--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On Tuesday, January 17, 2023 at 9:25:11 AM UTC-8, olcott wrote:
On 1/17/2023 10:44 AM, Jeffrey Rubard wrote:
On Tuesday, January 17, 2023 at 8:40:02 AM UTC-8, _ Olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
On 1/16/23 10:17 AM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:No, because I am showing that G is TRUE, not PROVABLE. Truth can use >>> infinte sets oc connections, proofs can't. Only YOU have perposed that >>> we think about infinite proofs.
On 1/15/23 11:15 AM, olcott wrote:So you changed your mind about infinite proofs in formal systems? >>>> We know the steps of the infinite proof for the Goldbach conjecture. >>>
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if the >>>>>> truth
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language
of this
formal system is true unless this expression of language has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* ??? >>>>>>>>>>>>
Becaue the formal system doesn't need to KNOW what is true. >>>>>>>>>>>
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true, >>>>>>>>> doesn't mean it can't be.
In fact, your statement just comes out of a simple application of
the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true.
Unless a formal system has a syntactic connection from an >>>>>>>> expression of
its language to its truth maker axioms the expression is untrue in
that
formal system.
Right, but the connection can be infinite in length, and thus not >>>>>>> provable.
Try and show an expression of language that is true in a formal >>>>>>>> system
(not just true somewhere else) that does not have any connection to
truth maker axioms in this formal system. You must show why it is >>>>>>>> true
in this formal system not merely that it is true somewhere else. >>>>>>>>
The connection might be infinite, and thus not SHOWABLE as a proof >>>>>>> strictly in the formal system.
If the connection exists as an infinite connection within the >>>>>>> system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can >>>>>>> not be proven within the formal system, it is still possible, that >>>>>>> another system, related to that system, with more knowledge, might >>>>>>> be able to show that there does exist within the original formal >>>>>>> system such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets a >>>>>>> specific requirement (expressed as a primative recursive
relationship).
This statement turns out to be true, because it turns out that no >>>>>>> number g does meet that requirement, but it can't be proven in F >>>>>>> that this is true, because in F, to show this we need to test every
natuarl number, which requires an infinite number of steps (finite >>>>>>> for each number, but an infinite number of numbers to test). >>>>>>>
In meta-F, we can do better, because due to additional knowledge in
meta-F, we can show that if a number g could be found, then that >>>>>>> number g could be converted into a proof, in F, of the statement G >>>>>>> (which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that no
proof of it can exist in F.
of G cannot even be expressed in F as long as the truth of G can be >>>>>> expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite expression.
Formal systems cannot ever use infinite connections from their
expressions of language to their truth maker axioms thus eliminating
these from consideration as any measure of true "in the system".
--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius >> hits a target no one else can see." Arthur Schopenhauer
"What you've written is almost like formal logic."
I am filling in a key detail about the way that "true" in formal logic actually works. Wittgenstein first pointed this out and no mathematician has ever noticed. https://www.liarparadox.org/Wittgenstein.pdf
If any mathematician had noticed this then they would have noticed that Gödel did not prove that formal systems are incomplete. He only proved that some of the expressions of language of a formal system are simply untrue which is a mere triviality that everyone already knew.This is just really, really poor.
--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius hits a target no one else can see." Arthur Schopenhauer
On Tuesday, January 17, 2023 at 9:25:11 AM UTC-8, olcott wrote:
On 1/17/2023 10:44 AM, Jeffrey Rubard wrote:
On Tuesday, January 17, 2023 at 8:40:02 AM UTC-8, _ Olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
On 1/16/23 10:17 AM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:No, because I am showing that G is TRUE, not PROVABLE. Truth can use >>> infinte sets oc connections, proofs can't. Only YOU have perposed that >>> we think about infinite proofs.
On 1/15/23 11:15 AM, olcott wrote:So you changed your mind about infinite proofs in formal systems? >>>> We know the steps of the infinite proof for the Goldbach conjecture. >>>
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if the >>>>>> truth
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language
of this
formal system is true unless this expression of language has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* ??? >>>>>>>>>>>>
Becaue the formal system doesn't need to KNOW what is true. >>>>>>>>>>>
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true, >>>>>>>>> doesn't mean it can't be.
In fact, your statement just comes out of a simple application of
the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true.
Unless a formal system has a syntactic connection from an >>>>>>>> expression of
its language to its truth maker axioms the expression is untrue in
that
formal system.
Right, but the connection can be infinite in length, and thus not >>>>>>> provable.
Try and show an expression of language that is true in a formal >>>>>>>> system
(not just true somewhere else) that does not have any connection to
truth maker axioms in this formal system. You must show why it is >>>>>>>> true
in this formal system not merely that it is true somewhere else. >>>>>>>>
The connection might be infinite, and thus not SHOWABLE as a proof >>>>>>> strictly in the formal system.
If the connection exists as an infinite connection within the >>>>>>> system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can >>>>>>> not be proven within the formal system, it is still possible, that >>>>>>> another system, related to that system, with more knowledge, might >>>>>>> be able to show that there does exist within the original formal >>>>>>> system such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets a >>>>>>> specific requirement (expressed as a primative recursive
relationship).
This statement turns out to be true, because it turns out that no >>>>>>> number g does meet that requirement, but it can't be proven in F >>>>>>> that this is true, because in F, to show this we need to test every
natuarl number, which requires an infinite number of steps (finite >>>>>>> for each number, but an infinite number of numbers to test). >>>>>>>
In meta-F, we can do better, because due to additional knowledge in
meta-F, we can show that if a number g could be found, then that >>>>>>> number g could be converted into a proof, in F, of the statement G >>>>>>> (which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that no
proof of it can exist in F.
of G cannot even be expressed in F as long as the truth of G can be >>>>>> expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite expression.
Formal systems cannot ever use infinite connections from their
expressions of language to their truth maker axioms thus eliminating
these from consideration as any measure of true "in the system".
--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius >> hits a target no one else can see." Arthur Schopenhauer
"What you've written is almost like formal logic."
I am filling in a key detail about the way that "true" in formal logic actually works. Wittgenstein first pointed this out and no mathematician has ever noticed. https://www.liarparadox.org/Wittgenstein.pdf
If any mathematician had noticed this then they would have noticed that Gödel did not prove that formal systems are incomplete. He only proved that some of the expressions of language of a formal system are simply untrue which is a mere triviality that everyone already knew.This is just really, really poor.
--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius hits a target no one else can see." Arthur Schopenhauer
On Tuesday, January 17, 2023 at 6:46:00 PM UTC-6, Jeffrey Rubard wrote:
On Tuesday, January 17, 2023 at 9:25:11 AM UTC-8, olcott wrote:
On 1/17/2023 10:44 AM, Jeffrey Rubard wrote:
On Tuesday, January 17, 2023 at 8:40:02 AM UTC-8, _ Olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
On 1/16/23 10:17 AM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:So you changed your mind about infinite proofs in formal systems? >>>> We know the steps of the infinite proof for the Goldbach conjecture.
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if the
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language
of this
formal system is true unless this expression of language has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* ???
Becaue the formal system doesn't need to KNOW what is true. >>>>>>>>>>>
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true,
doesn't mean it can't be.
In fact, your statement just comes out of a simple application of
the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true.
Unless a formal system has a syntactic connection from an >>>>>>>> expression of
its language to its truth maker axioms the expression is untrue in
that
formal system.
Right, but the connection can be infinite in length, and thus not
provable.
Try and show an expression of language that is true in a formal >>>>>>>> system
(not just true somewhere else) that does not have any connection to
truth maker axioms in this formal system. You must show why it is
true
in this formal system not merely that it is true somewhere else.
The connection might be infinite, and thus not SHOWABLE as a proof
strictly in the formal system.
If the connection exists as an infinite connection within the >>>>>>> system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can
not be proven within the formal system, it is still possible, that
another system, related to that system, with more knowledge, might
be able to show that there does exist within the original formal >>>>>>> system such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets a
specific requirement (expressed as a primative recursive
relationship).
This statement turns out to be true, because it turns out that no
number g does meet that requirement, but it can't be proven in F >>>>>>> that this is true, because in F, to show this we need to test every
natuarl number, which requires an infinite number of steps (finite
for each number, but an infinite number of numbers to test). >>>>>>>
In meta-F, we can do better, because due to additional knowledge in
meta-F, we can show that if a number g could be found, then that >>>>>>> number g could be converted into a proof, in F, of the statement G
(which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that no
proof of it can exist in F.
truth
of G cannot even be expressed in F as long as the truth of G can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite expression.
No, because I am showing that G is TRUE, not PROVABLE. Truth can use >>> infinte sets oc connections, proofs can't. Only YOU have perposed that
we think about infinite proofs.
Formal systems cannot ever use infinite connections from their
expressions of language to their truth maker axioms thus eliminating >> these from consideration as any measure of true "in the system".
--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
"What you've written is almost like formal logic."
I am filling in a key detail about the way that "true" in formal logic actually works. Wittgenstein first pointed this out and no mathematician has ever noticed. https://www.liarparadox.org/Wittgenstein.pdf
Jeffrey, I'll second that emotion.If any mathematician had noticed this then they would have noticed that Gödel did not prove that formal systems are incomplete. He only proved that some of the expressions of language of a formal system are simply untrue which is a mere triviality that everyone already knew.This is just really, really poor.
--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius hits a target no one else can see." Arthur Schopenhauer
On Wednesday, January 18, 2023 at 11:11:45 PM UTC-6, Don Stockbauer wrote:
On Tuesday, January 17, 2023 at 6:46:00 PM UTC-6, Jeffrey Rubard wrote:
On Tuesday, January 17, 2023 at 9:25:11 AM UTC-8, olcott wrote:
On 1/17/2023 10:44 AM, Jeffrey Rubard wrote:
On Tuesday, January 17, 2023 at 8:40:02 AM UTC-8, _ Olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
On 1/16/23 10:17 AM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:So you changed your mind about infinite proofs in formal systems? >>>> We know the steps of the infinite proof for the Goldbach conjecture.
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if the
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language
of this
formal system is true unless this expression of language has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* ???
Becaue the formal system doesn't need to KNOW what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true,
doesn't mean it can't be.
In fact, your statement just comes out of a simple application of
the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true.
Unless a formal system has a syntactic connection from an >>>>>>>> expression of
its language to its truth maker axioms the expression is untrue in
that
formal system.
Right, but the connection can be infinite in length, and thus not
provable.
Try and show an expression of language that is true in a formal
system
(not just true somewhere else) that does not have any connection to
truth maker axioms in this formal system. You must show why it is
true
in this formal system not merely that it is true somewhere else.
The connection might be infinite, and thus not SHOWABLE as a proof
strictly in the formal system.
If the connection exists as an infinite connection within the >>>>>>> system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can
not be proven within the formal system, it is still possible, that
another system, related to that system, with more knowledge, might
be able to show that there does exist within the original formal
system such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets a
specific requirement (expressed as a primative recursive >>>>>>> relationship).
This statement turns out to be true, because it turns out that no
number g does meet that requirement, but it can't be proven in F
that this is true, because in F, to show this we need to test every
natuarl number, which requires an infinite number of steps (finite
for each number, but an infinite number of numbers to test). >>>>>>>
In meta-F, we can do better, because due to additional knowledge in
meta-F, we can show that if a number g could be found, then that
number g could be converted into a proof, in F, of the statement G
(which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that no
proof of it can exist in F.
truth
of G cannot even be expressed in F as long as the truth of G can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite expression.
No, because I am showing that G is TRUE, not PROVABLE. Truth can use
infinte sets oc connections, proofs can't. Only YOU have perposed that
we think about infinite proofs.
Formal systems cannot ever use infinite connections from their
expressions of language to their truth maker axioms thus eliminating
these from consideration as any measure of true "in the system".
--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
"What you've written is almost like formal logic."
I am filling in a key detail about the way that "true" in formal logic actually works. Wittgenstein first pointed this out and no mathematician
has ever noticed. https://www.liarparadox.org/Wittgenstein.pdf
these people could be out raising corn, doing something useful.Jeffrey, I'll second that emotion.If any mathematician had noticed this then they would have noticed thatThis is just really, really poor.
Gödel did not prove that formal systems are incomplete. He only proved
that some of the expressions of language of a formal system are simply untrue which is a mere triviality that everyone already knew.
--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 1/17/23 12:07 PM, olcott wrote:
On 1/17/2023 5:02 AM, Richard Damon wrote:
On 1/17/23 12:32 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
On 1/16/23 4:51 PM, olcott wrote:
On 1/15/2023 7:15 PM, Richard Damon wrote:
On 1/15/23 7:56 PM, olcott wrote:
On 1/15/2023 6:47 PM, Richard Damon wrote:
On 1/15/23 7:26 PM, olcott wrote:
On 1/15/2023 2:23 PM, Richard Damon wrote:
On 1/15/23 3:12 PM, olcott wrote:
On 1/15/2023 1:46 PM, Richard Damon wrote:
On 1/15/23 2:29 PM, olcott wrote:He is saying that every epistemological antinomy is a valid >>>>>>>>>>>> proxy for
On 1/15/2023 1:06 PM, Richard Damon wrote:
On 1/15/23 2:00 PM, olcott wrote:So Gödel is wrong when he says:
On 1/15/2023 12:55 PM, Richard Damon wrote:
On 1/15/23 1:23 PM, olcott wrote:
On 1/15/2023 11:58 AM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>> On 1/15/23 12:47 PM, olcott wrote:
All of your Ad Hominem attacks cannot possibly hide >>>>>>>>>>>>>>>>>> the fact that youOn 1/15/2023 11:41 AM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>> On 1/15/23 12:31 PM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>>> On 1/15/23 11:15 AM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>> On 1/14/2023 5:42 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 1/14/23 6:19 PM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>> On 1/14/2023 4:55 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>> On 1/14/23 5:31 PM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>> On 1/14/2023 4:26 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 1/14/23 4:48 PM, olcott wrote: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> How does the formal system know that an >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> expression of language of this >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> formal system is true unless this >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> expression of language has a >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> connection to truth maker axioms *IN THIS >>>>>>>>>>>>>>>>>>>>>>>>>>>>>> FORMAL SYSTEM* ??? >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
That is false. It is not that G takes an infinite >>>>>>>>>>>>>>>>>>>>>> number of steps toSo your basic line-of-reasoning is that G is >>>>>>>>>>>>>>>>>>>>>>>> true in F even if the truth
Becaue the formal system doesn't need to >>>>>>>>>>>>>>>>>>>>>>>>>>>>> KNOW what is true.
So PA has no idea that: >>>>>>>>>>>>>>>>>>>>>>>>>>>> successor(successor(0)) == successor(0) + >>>>>>>>>>>>>>>>>>>>>>>>>>>> successor(0) is true ??? >>>>>>>>>>>>>>>>>>>>>>>>>>>>
Why do you say that.
Just because truth doesn't NEED to be proven >>>>>>>>>>>>>>>>>>>>>>>>>>> for it to be true, doesn't mean it can't be. >>>>>>>>>>>>>>>>>>>>>>>>>>>
In fact, your statement just comes out of a >>>>>>>>>>>>>>>>>>>>>>>>>>> simple application of the addition AXIOMS of PA. >>>>>>>>>>>>>>>>>>>>>>>>>>>
a + 0 = a
a + Successor(b) = Successor(a + b) >>>>>>>>>>>>>>>>>>>>>>>>>>>
So it is a PROVABLE statement, and thus >>>>>>>>>>>>>>>>>>>>>>>>>>> actually KNOWN to be true. >>>>>>>>>>>>>>>>>>>>>>>>>>>
Unless a formal system has a syntactic >>>>>>>>>>>>>>>>>>>>>>>>>> connection from an expression of >>>>>>>>>>>>>>>>>>>>>>>>>> its language to its truth maker axioms the >>>>>>>>>>>>>>>>>>>>>>>>>> expression is untrue in that >>>>>>>>>>>>>>>>>>>>>>>>>> formal system.
Right, but the connection can be infinite in >>>>>>>>>>>>>>>>>>>>>>>>> length, and thus not provable. >>>>>>>>>>>>>>>>>>>>>>>>>
Try and show an expression of language that is >>>>>>>>>>>>>>>>>>>>>>>>>> true in a formal system
(not just true somewhere else) that does not >>>>>>>>>>>>>>>>>>>>>>>>>> have any connection to
truth maker axioms in this formal system. You >>>>>>>>>>>>>>>>>>>>>>>>>> must show why it is true
in this formal system not merely that it is >>>>>>>>>>>>>>>>>>>>>>>>>> true somewhere else.
The connection might be infinite, and thus not >>>>>>>>>>>>>>>>>>>>>>>>> SHOWABLE as a proof strictly in the formal system. >>>>>>>>>>>>>>>>>>>>>>>>>
If the connection exists as an infinite >>>>>>>>>>>>>>>>>>>>>>>>> connection within the system, then it is TRUE >>>>>>>>>>>>>>>>>>>>>>>>> in the system.
Note, that if there is such an infinite >>>>>>>>>>>>>>>>>>>>>>>>> connection, which thus can not be proven within >>>>>>>>>>>>>>>>>>>>>>>>> the formal system, it is still possible, that >>>>>>>>>>>>>>>>>>>>>>>>> another system, related to that system, with >>>>>>>>>>>>>>>>>>>>>>>>> more knowledge, might be able to show that >>>>>>>>>>>>>>>>>>>>>>>>> there does exist within the original formal >>>>>>>>>>>>>>>>>>>>>>>>> system such an infinte connection. >>>>>>>>>>>>>>>>>>>>>>>>>
This is what happens to G in F and meta-F >>>>>>>>>>>>>>>>>>>>>>>>>
G states that there does not exist a Natural >>>>>>>>>>>>>>>>>>>>>>>>> Number g that meets a specific requirement >>>>>>>>>>>>>>>>>>>>>>>>> (expressed as a primative recursive relationship). >>>>>>>>>>>>>>>>>>>>>>>>>
This statement turns out to be true, because it >>>>>>>>>>>>>>>>>>>>>>>>> turns out that no number g does meet that >>>>>>>>>>>>>>>>>>>>>>>>> requirement, but it can't be proven in F that >>>>>>>>>>>>>>>>>>>>>>>>> this is true, because in F, to show this we >>>>>>>>>>>>>>>>>>>>>>>>> need to test every natuarl number, which >>>>>>>>>>>>>>>>>>>>>>>>> requires an infinite number of steps (finite >>>>>>>>>>>>>>>>>>>>>>>>> for each number, but an infinite number of >>>>>>>>>>>>>>>>>>>>>>>>> numbers to test).
In meta-F, we can do better, because due to >>>>>>>>>>>>>>>>>>>>>>>>> additional knowledge in meta-F, we can show >>>>>>>>>>>>>>>>>>>>>>>>> that if a number g could be found, then that >>>>>>>>>>>>>>>>>>>>>>>>> number g could be converted into a proof, in F, >>>>>>>>>>>>>>>>>>>>>>>>> of the statement G (which says that such a >>>>>>>>>>>>>>>>>>>>>>>>> number does not exist).
Thus, in meta-F, we can prove that G is true, >>>>>>>>>>>>>>>>>>>>>>>>> and also show that no proof of it can exist in F. >>>>>>>>>>>>>>>>>>>>>>>>>
of G cannot even be expressed in F as long as >>>>>>>>>>>>>>>>>>>>>>>> the truth of G can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is >>>>>>>>>>>>>>>>>>>>>>> an infinite expression.
reach its truth maker axioms in F it is that even >>>>>>>>>>>>>>>>>>>>>> after an infinite
number of steps it never reaches is truth maker >>>>>>>>>>>>>>>>>>>>>> axioms in F because G is
simply untrue in F.
No, YOUR problem is you aren't actually talking >>>>>>>>>>>>>>>>>>>>> about G in F.
"This sentence is not true"
Which isn't G.
Not true about what?
Not true about being not true.
Not true about being not true about what? >>>>>>>>>>>>>>>>>>>>>> Not true about being not true about being not true. >>>>>>>>>>>>>>>>>>>>>>
So you are just a MORON.
Since Gödel said:
14 Every epistemological antinomy can likewise be >>>>>>>>>>>>>>>>>>>>>> used for a similar undecidability proof. >>>>>>>>>>>>>>>>>>>>>>
By using the Liar Paradox as a Gödel approved >>>>>>>>>>>>>>>>>>>>>> proxy for his proof we
refute his proof by this Gödel approved proxy. >>>>>>>>>>>>>>>>>>>>>>
No, all you have proved is that you are a LYING MORON. >>>>>>>>>>>>>>>>>>>>>
Ad Hominem attacks are the tactic that people having >>>>>>>>>>>>>>>>>>>> no interest in any
honest dialogue use when they realize that their >>>>>>>>>>>>>>>>>>>> reasoning has been
utterly defeated.
And RED HERRING arguements don't work either. >>>>>>>>>>>>>>>>>>>
You ARE a LYING MORON as you insist that Godel's G is >>>>>>>>>>>>>>>>>>> a sentence that is actually provably isn't. >>>>>>>>>>>>>>>>>>>
You think it is because you are too stupid to >>>>>>>>>>>>>>>>>>> actually read any of the paper, so you take that >>>>>>>>>>>>>>>>>>> comment that the statment is "based" on that >>>>>>>>>>>>>>>>>>> statement to mean it IS that statement.
are asserting this counter-factual statement: >>>>>>>>>>>>>>>>>>
when a valid proxy for an argument is defeated this >>>>>>>>>>>>>>>>>> does not defeat the
original argument.
Right, which is what YOU are doing, showing your >>>>>>>>>>>>>>>>> arguement is INVALID.
In other words you disagree that correctly refuting a >>>>>>>>>>>>>>>> valid proxy for an
argument does correctly refute the original argument? >>>>>>>>>>>>>>>>
VALID is the key word,
Yours isn't (I don't think you actually know the meaning >>>>>>>>>>>>>>> of the words)
And you are an IDIOT to claim it is.
14 Every epistemological antinomy can likewise be used for a >>>>>>>>>>>>>> similar undecidability proof.
No, but you don't understand what he is saying.
his proof. He is not saying that his expression is not an >>>>>>>>>>>> epistemological antinomy.
Nope, that isn't what he is saying. How could it be, the >>>>>>>>>>> ACTUAL G is a proven Truth Bearer, while the Liar's Paradox >>>>>>>>>>> isn't
Your arguement just shows its inconsistency.
In part, because you don't actually understand what the
sentence actually is.
The analogy between this result and Richard’s antinomy leaps >>>>>>>>>>>> to the eye;
there is also a close relationship with the “liar” >>>>>>>>>>>> antinomy,14 since the
undecidable proposition [R(q); q] states precisely that q >>>>>>>>>>>> belongs to K,
i.e. according to (1), that [R(q); q] is not provable. We >>>>>>>>>>>> are therefore
confronted with a proposition which asserts its own
unprovability.
(Gödel 1931:43)
This <is> an isomorphism to a proposition that asserts its >>>>>>>>>>>> own untruth.
Nope, unless you erroneously think that statements about >>>>>>>>>>> Truth ARE statements about provability, that isn't an
isomoprhism.
Note, you have even stated that *ALL* statements of the form >>>>>>>>>>> "statement x is provable" or "Statment x is not provable" are >>>>>>>>>>> Truth Bearers,
I have most definitely never said this or anything that could be >>>>>>>>>> unintentionally misconstrued to mean this.
It is always the case that when-so-ever any expression of
language only
refers to its own truth or provability that this expression is >>>>>>>>>> not a
truth bearer, thus not a member of any formal system of logic. >>>>>>>>>>
You admitted that it was TRUE that a statement could not be
proven even if the only way to show that it could not be proven >>>>>>>>> was to check the infinite set of all possible proofs to see
that none of them were a proof.
This is not related to what I just said. Every expression of
language is
untrue unless it has a semantic connection to its truth maker
axiom.
I never disagreed with that, just that you keep on wavering
between just saying there must be a connection, and then at times >>>>>>> adding it must be a FINITE connection (which is actually only
requried to be Proven)
G, the statment about the non-existance of a natural number that >>>>>>> satisfies the specified primative recursive relationship is TRUE, >>>>>>> because it IS connected to the truth maker axioms of math via an >>>>>>> infinite chain of steps.
Each natural number can be shown to not meet that requirement in a >>>>>> Gödel was actually talking about the expression:
We are therefore confronted with a proposition which asserts its own >>>>>> unprovability. (Gödel 1931:43)
No, that is a statement which is proven in Meta-F to have the
identical truth value of G. G doesn't SAY it is unprovable, but a
natural concesequence of G being True is that it is unprovable, and
if it is provable, it can't be True. Since G must be True or False,
if it is True it IS unprovable, and if it is Provable, then it must
be False, which is a contradiction (since ALL provable statements
are True), so that case is impossible. Thus, G MUST be True but
Unprovable.
If F includes an axiom that says all Truths are Provable, then F is
proved to be inconsistent.
He only used the whole natural numbers thing to be able to encode the >>>>>> above expression in a language that did not have a provability
predicate.
No, F might well have a provability predicate, ies of the Natural
Numbers.
Then it would not need any Gödel number.
Maybe, but the key is that it DOESN'T use the operator, so your
"special" rule based on using it doesn't apply.
Godel showed that Meta-F we can construct a calculation in Meta-F
that is the exact same caluclation in F that provides us proofs in
Meta-F based on what is simply a calculation in F.
Since it is just a calculation in F about the existance of a number
based on a computable function, in F the stateement ALWAYS has a
Truth Value, either such a number exsits or it doesn't.
Because of the DEFINED relationship between F and Meta-F, that truth
value transfers, and due to the extra axioms in Meta-F.
Arguing that in Meta-F we have an epistemological antinomy means that
your logic system makes the mathematics in F be able to create this
same situation, which doesn't match the behavior of the Natural
Numbers, so your F doesn't meet the requirements for it.
YOU "PROOF" FAILS.
If you want to make the sort of claims you are doing, you need to
show exactly which step in his proof does something wrong. You are
not allowed to rebut a proof by saying its answer must be wrong, or
you disagree with a footnote. You need to find an actaul erroneous
step in the proof itself.
Since you have shown you don't actually understand the proof at all,
this is probably impossible for you.
*paraphrased as: GF ↔ (F ⊬ GF) // without Gödel number*
(G) F ⊢ GF ↔ ¬ProvF(┌GF┐). // with Gödel number
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom >>>>
Nope, it only "means" that in Meta F, not in F.
F doesn't have Truth Makers to establish that meaning, so that
"parapharse" is incorrect.
You are just showing you don't understand what logic does.
Which if you read, agrees with Godel, that this sentence must be
neither provable or disprovable, and agrees with the right
conditions, can be made True.
I am removing the Gödel number and showing what's left.
Remove the Godel Number, and NOTHING is left of the statement in F,
Here is what remains: GF ↔ (F ⊬ GF)
Nope IT CAN'T be that in F, as in F it doesn't talk about proving at all.
On Thursday, January 19, 2023 at 5:35:08 AM UTC-8, Don Stockbauer wrote:
On Wednesday, January 18, 2023 at 11:11:45 PM UTC-6, Don Stockbauer wrote:
On Tuesday, January 17, 2023 at 6:46:00 PM UTC-6, Jeffrey Rubard wrote:
On Tuesday, January 17, 2023 at 9:25:11 AM UTC-8, olcott wrote:
On 1/17/2023 10:44 AM, Jeffrey Rubard wrote:
On Tuesday, January 17, 2023 at 8:40:02 AM UTC-8, _ Olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
On 1/16/23 10:17 AM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:So you changed your mind about infinite proofs in formal systems?
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if the
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language
of this
formal system is true unless this expression of language has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* ???
Becaue the formal system doesn't need to KNOW what is true.
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true,
doesn't mean it can't be.
In fact, your statement just comes out of a simple application of
the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true.
Unless a formal system has a syntactic connection from an >>>>>>>> expression of
its language to its truth maker axioms the expression is untrue in
that
formal system.
Right, but the connection can be infinite in length, and thus not
provable.
Try and show an expression of language that is true in a formal
system
(not just true somewhere else) that does not have any connection to
truth maker axioms in this formal system. You must show why it is
true
in this formal system not merely that it is true somewhere else.
The connection might be infinite, and thus not SHOWABLE as a proof
strictly in the formal system.
If the connection exists as an infinite connection within the
system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can
not be proven within the formal system, it is still possible, that
another system, related to that system, with more knowledge, might
be able to show that there does exist within the original formal
system such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets a
specific requirement (expressed as a primative recursive >>>>>>> relationship).
This statement turns out to be true, because it turns out that no
number g does meet that requirement, but it can't be proven in F
that this is true, because in F, to show this we need to test every
natuarl number, which requires an infinite number of steps (finite
for each number, but an infinite number of numbers to test). >>>>>>>
In meta-F, we can do better, because due to additional knowledge in
meta-F, we can show that if a number g could be found, then that
number g could be converted into a proof, in F, of the statement G
(which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that no
proof of it can exist in F.
truth
of G cannot even be expressed in F as long as the truth of G can be
expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite expression.
We know the steps of the infinite proof for the Goldbach conjecture.
No, because I am showing that G is TRUE, not PROVABLE. Truth can use
infinte sets oc connections, proofs can't. Only YOU have perposed that
we think about infinite proofs.
Formal systems cannot ever use infinite connections from their
expressions of language to their truth maker axioms thus eliminating
these from consideration as any measure of true "in the system".
--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
"What you've written is almost like formal logic."
I am filling in a key detail about the way that "true" in formal logic
actually works. Wittgenstein first pointed this out and no mathematician
has ever noticed. https://www.liarparadox.org/Wittgenstein.pdf
these people could be out raising corn, doing something useful.Jeffrey, I'll second that emotion.If any mathematician had noticed this then they would have noticed thatThis is just really, really poor.
Gödel did not prove that formal systems are incomplete. He only proved
that some of the expressions of language of a formal system are simply
untrue which is a mere triviality that everyone already knew.
--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
You're full of shit, but the writing on logic *is* bad.
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
On 1/16/23 10:17 AM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:
On 1/15/23 11:15 AM, olcott wrote:So you changed your mind about infinite proofs in formal systems?
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if the >>>>>> truth
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of
language of this
formal system is true unless this expression of language has a >>>>>>>>>>>> connection to truth maker axioms *IN THIS FORMAL SYSTEM* ??? >>>>>>>>>>>>
Becaue the formal system doesn't need to KNOW what is true. >>>>>>>>>>>
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true >>>>>>>>>> ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true, >>>>>>>>> doesn't mean it can't be.
In fact, your statement just comes out of a simple application >>>>>>>>> of the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true. >>>>>>>>>
Unless a formal system has a syntactic connection from an
expression of
its language to its truth maker axioms the expression is untrue >>>>>>>> in that
formal system.
Right, but the connection can be infinite in length, and thus not >>>>>>> provable.
Try and show an expression of language that is true in a formal >>>>>>>> system
(not just true somewhere else) that does not have any connection to >>>>>>>> truth maker axioms in this formal system. You must show why it >>>>>>>> is true
in this formal system not merely that it is true somewhere else. >>>>>>>>
The connection might be infinite, and thus not SHOWABLE as a
proof strictly in the formal system.
If the connection exists as an infinite connection within the
system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus
can not be proven within the formal system, it is still possible, >>>>>>> that another system, related to that system, with more knowledge, >>>>>>> might be able to show that there does exist within the original
formal system such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets >>>>>>> a specific requirement (expressed as a primative recursive
relationship).
This statement turns out to be true, because it turns out that no >>>>>>> number g does meet that requirement, but it can't be proven in F >>>>>>> that this is true, because in F, to show this we need to test
every natuarl number, which requires an infinite number of steps >>>>>>> (finite for each number, but an infinite number of numbers to test). >>>>>>>
In meta-F, we can do better, because due to additional knowledge >>>>>>> in meta-F, we can show that if a number g could be found, then
that number g could be converted into a proof, in F, of the
statement G (which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that >>>>>>> no proof of it can exist in F.
of G cannot even be expressed in F as long as the truth of G can be >>>>>> expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite
expression.
We know the steps of the infinite proof for the Goldbach conjecture.
No, because I am showing that G is TRUE, not PROVABLE. Truth can use
infinte sets oc connections, proofs can't. Only YOU have perposed
that we think about infinite proofs.
Formal systems cannot ever use infinite connections from their
expressions of language to their truth maker axioms thus eliminating
these from consideration as any measure of true "in the system".
Source? or is this just another of your made up "Facts"
WHERE in the definition of a "Formal System" does it say that the
connecti0on must be finite.
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 12:07 PM, olcott wrote:
On 1/17/2023 5:02 AM, Richard Damon wrote:
On 1/17/23 12:32 AM, olcott wrote:
Remove the Godel Number, and NOTHING is left of the statement in F,
Here is what remains: GF ↔ (F ⊬ GF)
Nope IT CAN'T be that in F, as in F it doesn't talk about proving at all.
In other words you did not bother to pay attention to this: ¬ProvF
2.5 The First Incompleteness Theorem—Proof Completed
To complete the proof, the Diagonalization Lemma is applied to the
negated provability predicate ¬ProvF(x): this gives a sentence G F such
that
(G) F ⊢ GF ↔ ¬ProvF(┌GF┐)
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
On Thursday, January 19, 2023 at 5:35:08 AM UTC-8, Don Stockbauer wrote:
On Wednesday, January 18, 2023 at 11:11:45 PM UTC-6, Don Stockbauer wrote: >>> On Tuesday, January 17, 2023 at 6:46:00 PM UTC-6, Jeffrey Rubard wrote: >>>> On Tuesday, January 17, 2023 at 9:25:11 AM UTC-8, olcott wrote:
these people could be out raising corn, doing something useful.Jeffrey, I'll second that emotion.On 1/17/2023 10:44 AM, Jeffrey Rubard wrote:This is just really, really poor.
On Tuesday, January 17, 2023 at 8:40:02 AM UTC-8, _ Olcott wrote: >>>>>>> On 1/16/2023 7:51 PM, Richard Damon wrote:
On 1/16/23 10:17 AM, olcott wrote:
On 1/15/2023 11:05 AM, Richard Damon wrote:No, because I am showing that G is TRUE, not PROVABLE. Truth can use >>>>>>>> infinte sets oc connections, proofs can't. Only YOU have perposed that >>>>>>>> we think about infinite proofs.
On 1/15/23 11:15 AM, olcott wrote:So you changed your mind about infinite proofs in formal systems? >>>>>>>>> We know the steps of the infinite proof for the Goldbach conjecture. >>>>>>>>
On 1/14/2023 5:42 PM, Richard Damon wrote:
On 1/14/23 6:19 PM, olcott wrote:So your basic line-of-reasoning is that G is true in F even if the >>>>>>>>>>> truth
On 1/14/2023 4:55 PM, Richard Damon wrote:
On 1/14/23 5:31 PM, olcott wrote:
On 1/14/2023 4:26 PM, Richard Damon wrote:
On 1/14/23 4:48 PM, olcott wrote:
How does the formal system know that an expression of language
of this
formal system is true unless this expression of language has a
connection to truth maker axioms *IN THIS FORMAL SYSTEM* ??? >>>>>>>>>>>>>>>>>
Becaue the formal system doesn't need to KNOW what is true. >>>>>>>>>>>>>>>>
So PA has no idea that:
successor(successor(0)) == successor(0) + successor(0) is true ???
Why do you say that.
Just because truth doesn't NEED to be proven for it to be true, >>>>>>>>>>>>>> doesn't mean it can't be.
In fact, your statement just comes out of a simple application of
the addition AXIOMS of PA.
a + 0 = a
a + Successor(b) = Successor(a + b)
So it is a PROVABLE statement, and thus actually KNOWN to be true.
Unless a formal system has a syntactic connection from an >>>>>>>>>>>>> expression of
its language to its truth maker axioms the expression is untrue in
that
formal system.
Right, but the connection can be infinite in length, and thus not >>>>>>>>>>>> provable.
Try and show an expression of language that is true in a formal >>>>>>>>>>>>> system
(not just true somewhere else) that does not have any connection to
truth maker axioms in this formal system. You must show why it is >>>>>>>>>>>>> true
in this formal system not merely that it is true somewhere else. >>>>>>>>>>>>>
The connection might be infinite, and thus not SHOWABLE as a proof >>>>>>>>>>>> strictly in the formal system.
If the connection exists as an infinite connection within the >>>>>>>>>>>> system, then it is TRUE in the system.
Note, that if there is such an infinite connection, which thus can >>>>>>>>>>>> not be proven within the formal system, it is still possible, that >>>>>>>>>>>> another system, related to that system, with more knowledge, might >>>>>>>>>>>> be able to show that there does exist within the original formal >>>>>>>>>>>> system such an infinte connection.
This is what happens to G in F and meta-F
G states that there does not exist a Natural Number g that meets a >>>>>>>>>>>> specific requirement (expressed as a primative recursive >>>>>>>>>>>> relationship).
This statement turns out to be true, because it turns out that no >>>>>>>>>>>> number g does meet that requirement, but it can't be proven in F >>>>>>>>>>>> that this is true, because in F, to show this we need to test every
natuarl number, which requires an infinite number of steps (finite >>>>>>>>>>>> for each number, but an infinite number of numbers to test). >>>>>>>>>>>>
In meta-F, we can do better, because due to additional knowledge in
meta-F, we can show that if a number g could be found, then that >>>>>>>>>>>> number g could be converted into a proof, in F, of the statement G >>>>>>>>>>>> (which says that such a number does not exist).
Thus, in meta-F, we can prove that G is true, and also show that no
proof of it can exist in F.
of G cannot even be expressed in F as long as the truth of G can be >>>>>>>>>>> expressed in meta-F.
Expect that it CAN be expressed in F, it just is an infinite expression.
Formal systems cannot ever use infinite connections from their
expressions of language to their truth maker axioms thus eliminating >>>>>>> these from consideration as any measure of true "in the system".
--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius >>>>>>> hits a target no one else can see." Arthur Schopenhauer
"What you've written is almost like formal logic."
I am filling in a key detail about the way that "true" in formal logic >>>>> actually works. Wittgenstein first pointed this out and no mathematician >>>>> has ever noticed. https://www.liarparadox.org/Wittgenstein.pdf
If any mathematician had noticed this then they would have noticed that >>>>> Gödel did not prove that formal systems are incomplete. He only proved >>>>> that some of the expressions of language of a formal system are simply >>>>> untrue which is a mere triviality that everyone already knew.
--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius >>>>> hits a target no one else can see." Arthur Schopenhauer
You're full of shit, but the writing on logic *is* bad.
On 1/19/23 2:09 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 12:07 PM, olcott wrote:
On 1/17/2023 5:02 AM, Richard Damon wrote:
On 1/17/23 12:32 AM, olcott wrote:
Remove the Godel Number, and NOTHING is left of the statement in F,
Here is what remains: GF ↔ (F ⊬ GF)
Nope IT CAN'T be that in F, as in F it doesn't talk about proving at all. >>
In other words you did not bother to pay attention to this: ¬ProvF
2.5 The First Incompleteness Theorem—Proof Completed
To complete the proof, the Diagonalization Lemma is applied to the
negated provability predicate ¬ProvF(x): this gives a sentence G F such that
(G) F ⊢ GF ↔ ¬ProvF(┌GF┐)
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
So, you are going by SUMMARIES of the proof, rather than the proof
itself. A guess you just don't know how to read an actual proof itself.
And, it seems you are missing that this logic is bing done in META-F,
not F, so it doesn't say what is happening in F. (Do you even understand
the difference?)_)
I see that you are not using your definiton of Truth anymore, at least
when it isn't convienient for you.
This statment is NOT based on a connection to the truth makers IN F, as
you are claiming that something can be true in F (What G means in F)
even though there is absolutely NO connection to that in F (only from
this particular Meata-F)
On 1/19/23 2:12 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
No, because I am showing that G is TRUE, not PROVABLE. Truth can
use infinte sets oc connections, proofs can't. Only YOU have
perposed that we think about infinite proofs.
Formal systems cannot ever use infinite connections from their
expressions of language to their truth maker axioms thus eliminating
these from consideration as any measure of true "in the system".
Source? or is this just another of your made up "Facts"
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite connections to Truth.
WHERE in the definition of a "Formal System" does it say that the
connecti0on must be finite.
You said that formal system cannot have infinite proofs.
Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections, proofs can not.
On 1/19/2023 8:34 PM, Richard Damon wrote:
On 1/19/23 2:12 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
No, because I am showing that G is TRUE, not PROVABLE. Truth can
use infinte sets oc connections, proofs can't. Only YOU have
perposed that we think about infinite proofs.
Formal systems cannot ever use infinite connections from their
expressions of language to their truth maker axioms thus eliminating >>>>> these from consideration as any measure of true "in the system".
Source? or is this just another of your made up "Facts"
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite connections
to Truth.
WHERE in the definition of a "Formal System" does it say that the
connecti0on must be finite.
You said that formal system cannot have infinite proofs.
Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections, proofs can not.
Truth *in a formal system* cannot be based on infinite connections
because formal systems are not allowed to have infinite connections.
Haskell Curry establishes that truth in a theory (AKA formal system) is anchored in the elementary theorems (AKA axioms) of this formal system.
A theory (over (f) is defined as a conceptual class of these elementary statements. Let::t be such a theory. Then the elementary statements
which belong to ::t we shall call the elementary theorems of::t; we also
say that these elementary statements are true for::t. Thus, given ::t,
an elementary theorem is an elementary statement which is true. A theory
is thus a way of picking out from the statements of (f a certain
subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Perhaps you believe that you are enormously much brighter than Haskell
Curry ?
The statements of (f are called elementary statements to distinguish them from other stateents which we may form from them
On 1/20/2023 2:46 PM, Richard Damon wrote:
On 1/20/23 2:31 PM, olcott wrote:
On 1/19/2023 8:34 PM, Richard Damon wrote:
On 1/19/23 2:12 PM, olcott wrote:Truth *in a formal system* cannot be based on infinite connections
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
No, because I am showing that G is TRUE, not PROVABLE. Truth can >>>>>>>> use infinte sets oc connections, proofs can't. Only YOU have
perposed that we think about infinite proofs.
Formal systems cannot ever use infinite connections from their
expressions of language to their truth maker axioms thus eliminating >>>>>>> these from consideration as any measure of true "in the system". >>>>>>>
Source? or is this just another of your made up "Facts"
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite connections
to Truth.
WHERE in the definition of a "Formal System" does it say that the
connecti0on must be finite.
You said that formal system cannot have infinite proofs.
Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections, proofs can not. >>>
because formal systems are not allowed to have infinite connections.
Says Who ***FOR TRUTH***
You reference does not provide that data, so I guess you are just
making it up, and thus showing you to be a LIAR.
Haskell Curry establishes that truth in a theory (AKA formal system)
is anchored in the elementary theorems (AKA axioms) of this formal
system.
Right, ANCHORED TO, not limited to. Statments other than the
elementary theorems are True, and they are true if they have a
connection (not limited to finite) to these Truths.
Where does he say True statements must have a FINITE connection to the
elementary theorems.
A theory (over (f) is defined as a conceptual class of these elementary
statements. Let::t be such a theory. Then the elementary statements
which belong to ::t we shall call the elementary theorems of::t; we also >>> say that these elementary statements are true for::t. Thus, given ::t,
an elementary theorem is an elementary statement which is true. A theory >>> is thus a way of picking out from the statements of (f a certain
subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Perhaps you believe that you are enormously much brighter than
Haskell Curry ?
You don't understand what he is saying,
He is saying these statements are True in F, as a given.
Wrongo !!!
The terminology which has just been used implies that the
elementary statements are not such that their truth and
falsity are known to us without reference to::t.
On 1/20/23 2:31 PM, olcott wrote:
On 1/19/2023 8:34 PM, Richard Damon wrote:
On 1/19/23 2:12 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
No, because I am showing that G is TRUE, not PROVABLE. Truth can >>>>>>> use infinte sets oc connections, proofs can't. Only YOU have
perposed that we think about infinite proofs.
Formal systems cannot ever use infinite connections from their
expressions of language to their truth maker axioms thus eliminating >>>>>> these from consideration as any measure of true "in the system".
Source? or is this just another of your made up "Facts"
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite connections
to Truth.
WHERE in the definition of a "Formal System" does it say that the
connecti0on must be finite.
You said that formal system cannot have infinite proofs.
Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections, proofs can not.
Truth *in a formal system* cannot be based on infinite connections
because formal systems are not allowed to have infinite connections.
Says Who ***FOR TRUTH***
You reference does not provide that data, so I guess you are just making
it up, and thus showing you to be a LIAR.
Haskell Curry establishes that truth in a theory (AKA formal system)
is anchored in the elementary theorems (AKA axioms) of this formal
system.
Right, ANCHORED TO, not limited to. Statments other than the elementary theorems are True, and they are true if they have a connection (not
limited to finite) to these Truths.
Where does he say True statements must have a FINITE connection to the elementary theorems.
A theory (over (f) is defined as a conceptual class of these elementary
statements. Let::t be such a theory. Then the elementary statements
which belong to ::t we shall call the elementary theorems of::t; we also
say that these elementary statements are true for::t. Thus, given ::t,
an elementary theorem is an elementary statement which is true. A theory
is thus a way of picking out from the statements of (f a certain
subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Perhaps you believe that you are enormously much brighter than Haskell
Curry ?
You don't understand what he is saying,
He is saying these statements are True in F, as a given.
On 1/20/23 5:02 PM, olcott wrote:
On 1/20/2023 2:46 PM, Richard Damon wrote:
On 1/20/23 2:31 PM, olcott wrote:
On 1/19/2023 8:34 PM, Richard Damon wrote:
On 1/19/23 2:12 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
No, because I am showing that G is TRUE, not PROVABLE. Truth >>>>>>>>> can use infinte sets oc connections, proofs can't. Only YOU
have perposed that we think about infinite proofs.
Formal systems cannot ever use infinite connections from their >>>>>>>> expressions of language to their truth maker axioms thus
eliminating
these from consideration as any measure of true "in the system". >>>>>>>>
Source? or is this just another of your made up "Facts"
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite
connections to Truth.
WHERE in the definition of a "Formal System" does it say that the >>>>>>> connecti0on must be finite.
You said that formal system cannot have infinite proofs.
Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections, proofs can
not.
Truth *in a formal system* cannot be based on infinite connections
because formal systems are not allowed to have infinite connections.
Says Who ***FOR TRUTH***
You reference does not provide that data, so I guess you are just
making it up, and thus showing you to be a LIAR.
Haskell Curry establishes that truth in a theory (AKA formal system)
is anchored in the elementary theorems (AKA axioms) of this formal
system.
Right, ANCHORED TO, not limited to. Statments other than the
elementary theorems are True, and they are true if they have a
connection (not limited to finite) to these Truths.
Where does he say True statements must have a FINITE connection to
the elementary theorems.
A theory (over (f) is defined as a conceptual class of these elementary >>>> statements. Let::t be such a theory. Then the elementary statements
which belong to ::t we shall call the elementary theorems of::t; we
also
say that these elementary statements are true for::t. Thus, given ::t, >>>> an elementary theorem is an elementary statement which is true. A
theory
is thus a way of picking out from the statements of (f a certain
subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Perhaps you believe that you are enormously much brighter than
Haskell Curry ?
You don't understand what he is saying,
He is saying these statements are True in F, as a given.
Wrongo !!!
The terminology which has just been used implies that the
elementary statements are not such that their truth and
falsity are known to us without reference to::t.
Right, they aren't just true in the Statement class, but are only
considerdd true because we are in the Theory F.
On 1/20/2023 4:09 PM, Richard Damon wrote:
On 1/20/23 5:02 PM, olcott wrote:
On 1/20/2023 2:46 PM, Richard Damon wrote:
On 1/20/23 2:31 PM, olcott wrote:
On 1/19/2023 8:34 PM, Richard Damon wrote:
On 1/19/23 2:12 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
No, because I am showing that G is TRUE, not PROVABLE. Truth >>>>>>>>>> can use infinte sets oc connections, proofs can't. Only YOU >>>>>>>>>> have perposed that we think about infinite proofs.
Formal systems cannot ever use infinite connections from their >>>>>>>>> expressions of language to their truth maker axioms thus
eliminating
these from consideration as any measure of true "in the system". >>>>>>>>>
Source? or is this just another of your made up "Facts"
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite
connections to Truth.
WHERE in the definition of a "Formal System" does it say that
the connecti0on must be finite.
You said that formal system cannot have infinite proofs.
Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections, proofs can >>>>>> not.
Truth *in a formal system* cannot be based on infinite connections
because formal systems are not allowed to have infinite connections.
Says Who ***FOR TRUTH***
You reference does not provide that data, so I guess you are just
making it up, and thus showing you to be a LIAR.
Haskell Curry establishes that truth in a theory (AKA formal
system) is anchored in the elementary theorems (AKA axioms) of this
formal system.
Right, ANCHORED TO, not limited to. Statments other than the
elementary theorems are True, and they are true if they have a
connection (not limited to finite) to these Truths.
Where does he say True statements must have a FINITE connection to
the elementary theorems.
A theory (over (f) is defined as a conceptual class of these
elementary
statements. Let::t be such a theory. Then the elementary statements
which belong to ::t we shall call the elementary theorems of::t; we
also
say that these elementary statements are true for::t. Thus, given ::t, >>>>> an elementary theorem is an elementary statement which is true. A
theory
is thus a way of picking out from the statements of (f a certain
subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Perhaps you believe that you are enormously much brighter than
Haskell Curry ?
You don't understand what he is saying,
He is saying these statements are True in F, as a given.
Wrongo !!!
The terminology which has just been used implies that the
elementary statements are not such that their truth and
falsity are known to us without reference to::t.
Right, they aren't just true in the Statement class, but are only
considerdd true because we are in the Theory F.
F is not the theory T is the theory.
On 1/20/23 5:16 PM, olcott wrote:
On 1/20/2023 4:09 PM, Richard Damon wrote:
On 1/20/23 5:02 PM, olcott wrote:
On 1/20/2023 2:46 PM, Richard Damon wrote:
On 1/20/23 2:31 PM, olcott wrote:
On 1/19/2023 8:34 PM, Richard Damon wrote:Says Who ***FOR TRUTH***
On 1/19/23 2:12 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
No, because I am showing that G is TRUE, not PROVABLE. Truth >>>>>>>>>>> can use infinte sets oc connections, proofs can't. Only YOU >>>>>>>>>>> have perposed that we think about infinite proofs.
Formal systems cannot ever use infinite connections from their >>>>>>>>>> expressions of language to their truth maker axioms thus
eliminating
these from consideration as any measure of true "in the system". >>>>>>>>>>
Source? or is this just another of your made up "Facts"
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite
connections to Truth.
WHERE in the definition of a "Formal System" does it say that >>>>>>>>> the connecti0on must be finite.
You said that formal system cannot have infinite proofs.
Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections, proofs
can not.
Truth *in a formal system* cannot be based on infinite connections >>>>>> because formal systems are not allowed to have infinite connections. >>>>>
You reference does not provide that data, so I guess you are just
making it up, and thus showing you to be a LIAR.
Haskell Curry establishes that truth in a theory (AKA formal
system) is anchored in the elementary theorems (AKA axioms) of
this formal system.
Right, ANCHORED TO, not limited to. Statments other than the
elementary theorems are True, and they are true if they have a
connection (not limited to finite) to these Truths.
Where does he say True statements must have a FINITE connection to
the elementary theorems.
A theory (over (f) is defined as a conceptual class of these
elementary
statements. Let::t be such a theory. Then the elementary statements >>>>>> which belong to ::t we shall call the elementary theorems of::t;
we also
say that these elementary statements are true for::t. Thus, given
::t,
an elementary theorem is an elementary statement which is true. A
theory
is thus a way of picking out from the statements of (f a certain
subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Perhaps you believe that you are enormously much brighter than
Haskell Curry ?
You don't understand what he is saying,
He is saying these statements are True in F, as a given.
Wrongo !!!
The terminology which has just been used implies that the
elementary statements are not such that their truth and
falsity are known to us without reference to::t.
Right, they aren't just true in the Statement class, but are only
considerdd true because we are in the Theory F.
F is not the theory T is the theory.
Red Herring.
F is the Theory in Godels descussion.
On 1/20/2023 4:54 PM, Richard Damon wrote:
On 1/20/23 5:16 PM, olcott wrote:
On 1/20/2023 4:09 PM, Richard Damon wrote:
On 1/20/23 5:02 PM, olcott wrote:
On 1/20/2023 2:46 PM, Richard Damon wrote:
On 1/20/23 2:31 PM, olcott wrote:
On 1/19/2023 8:34 PM, Richard Damon wrote:
On 1/19/23 2:12 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
No, because I am showing that G is TRUE, not PROVABLE. Truth >>>>>>>>>>>> can use infinte sets oc connections, proofs can't. Only YOU >>>>>>>>>>>> have perposed that we think about infinite proofs.
Formal systems cannot ever use infinite connections from their >>>>>>>>>>> expressions of language to their truth maker axioms thus >>>>>>>>>>> eliminating
these from consideration as any measure of true "in the system". >>>>>>>>>>>
Source? or is this just another of your made up "Facts"
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite
connections to Truth.
WHERE in the definition of a "Formal System" does it say that >>>>>>>>>> the connecti0on must be finite.
You said that formal system cannot have infinite proofs.
Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections, proofs >>>>>>>> can not.
Truth *in a formal system* cannot be based on infinite
connections because formal systems are not allowed to have
infinite connections.
Says Who ***FOR TRUTH***
You reference does not provide that data, so I guess you are just
making it up, and thus showing you to be a LIAR.
Haskell Curry establishes that truth in a theory (AKA formal
system) is anchored in the elementary theorems (AKA axioms) of
this formal system.
Right, ANCHORED TO, not limited to. Statments other than the
elementary theorems are True, and they are true if they have a
connection (not limited to finite) to these Truths.
Where does he say True statements must have a FINITE connection to >>>>>> the elementary theorems.
A theory (over (f) is defined as a conceptual class of these
elementary
statements. Let::t be such a theory. Then the elementary statements >>>>>>> which belong to ::t we shall call the elementary theorems of::t; >>>>>>> we also
say that these elementary statements are true for::t. Thus, given >>>>>>> ::t,
an elementary theorem is an elementary statement which is true. A >>>>>>> theory
is thus a way of picking out from the statements of (f a certain >>>>>>> subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Perhaps you believe that you are enormously much brighter than
Haskell Curry ?
You don't understand what he is saying,
He is saying these statements are True in F, as a given.
Wrongo !!!
The terminology which has just been used implies that the
elementary statements are not such that their truth and
falsity are known to us without reference to::t.
Right, they aren't just true in the Statement class, but are only
considerdd true because we are in the Theory F.
F is not the theory T is the theory.
Red Herring.
F is the Theory in Godels descussion.
It is not a red herring at all. Curry proves that the mathematical
notion of incompleteness itself is incoherent in that Curry sustains Wittgenstein's notion of true in a formal system.
That G is unprovable in F merely means that G is untrue in F a triviality.
On 1/20/23 7:29 PM, olcott wrote:
On 1/20/2023 4:54 PM, Richard Damon wrote:
On 1/20/23 5:16 PM, olcott wrote:
On 1/20/2023 4:09 PM, Richard Damon wrote:
On 1/20/23 5:02 PM, olcott wrote:
On 1/20/2023 2:46 PM, Richard Damon wrote:
On 1/20/23 2:31 PM, olcott wrote:
On 1/19/2023 8:34 PM, Richard Damon wrote:
On 1/19/23 2:12 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
No, because I am showing that G is TRUE, not PROVABLE. Truth >>>>>>>>>>>> can use infinte sets oc connections, proofs can't. Only YOU >>>>>>>>>>>> have perposed that we think about infinite proofs.
Formal systems cannot ever use infinite connections from their >>>>>>>>>>> expressions of language to their truth maker axioms thus >>>>>>>>>>> eliminating
these from consideration as any measure of true "in the system". >>>>>>>>>>>
Source? or is this just another of your made up "Facts"
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite
connections to Truth.
WHERE in the definition of a "Formal System" does it say that >>>>>>>>>> the connecti0on must be finite.
You said that formal system cannot have infinite proofs.
Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections, proofs >>>>>>>> can not.
Truth *in a formal system* cannot be based on infinite
connections because formal systems are not allowed to have
infinite connections.
Says Who ***FOR TRUTH***
You reference does not provide that data, so I guess you are just >>>>>> making it up, and thus showing you to be a LIAR.
Haskell Curry establishes that truth in a theory (AKA formal
system) is anchored in the elementary theorems (AKA axioms) of >>>>>>> this formal system.
Right, ANCHORED TO, not limited to. Statments other than the
elementary theorems are True, and they are true if they have a
connection (not limited to finite) to these Truths.
Where does he say True statements must have a FINITE connection to >>>>>> the elementary theorems.
A theory (over (f) is defined as a conceptual class of these
elementary
statements. Let::t be such a theory. Then the elementary statements >>>>>>> which belong to ::t we shall call the elementary theorems of::t; >>>>>>> we also
say that these elementary statements are true for::t. Thus, given >>>>>>> ::t,
an elementary theorem is an elementary statement which is true. A >>>>>>> theory
is thus a way of picking out from the statements of (f a certain >>>>>>> subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Perhaps you believe that you are enormously much brighter than >>>>>>> Haskell Curry ?
You don't understand what he is saying,
He is saying these statements are True in F, as a given.
Wrongo !!!
The terminology which has just been used implies that the
elementary statements are not such that their truth and
falsity are known to us without reference to::t.
Right, they aren't just true in the Statement class, but are only
considerdd true because we are in the Theory F.
F is not the theory T is the theory.
Red Herring.
F is the Theory in Godels descussion.
It is not a red herring at all. Curry proves that the mathematical
notion of incompleteness itself is incoherent in that Curry sustains Wittgenstein's notion of true in a formal system.
That G is unprovable in F merely means that G is untrue in F a triviality.
Please point out WHERE in the page you have cited that he does this.
Remeber, the CLASS of statemeents he talks about as "Elementary
Statments" that he talks about is NOT a "exhaustive" list of statements
that can be formed, but a base set to start from.
This is clear from the line you have highlighted pointing out that these statements of are called elementary statements to distinguish them from
other statements which we may form from them.
Then the "Elementary Theorems" are a SUBSET of these, that are defined
to be True in the Theory. Thus, these also are not a complete listing of
all true statements in the Theory, but only the set a base truths that
we are working from (in addition to the contensive statements that are
true indepentent of the Theory).
NOTHING on that page limits "True" statements to those things that are provable or only having a FINITE connection to those Elemetary Theories.
All this shows is that you don't understand what you are reading, or are
just lying.
On Friday, January 20, 2023 at 5:24:00 PM UTC-8, Richard Damon wrote:
On 1/20/23 7:29 PM, olcott wrote:
On 1/20/2023 4:54 PM, Richard Damon wrote:
On 1/20/23 5:16 PM, olcott wrote:
On 1/20/2023 4:09 PM, Richard Damon wrote:
On 1/20/23 5:02 PM, olcott wrote:
On 1/20/2023 2:46 PM, Richard Damon wrote:
On 1/20/23 2:31 PM, olcott wrote:
On 1/19/2023 8:34 PM, Richard Damon wrote:
On 1/19/23 2:12 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
No, because I am showing that G is TRUE, not PROVABLE. Truth >>>>>>>>>>>> can use infinte sets oc connections, proofs can't. Only YOU >>>>>>>>>>>> have perposed that we think about infinite proofs. >>>>>>>>>>>>
Formal systems cannot ever use infinite connections from their >>>>>>>>>>> expressions of language to their truth maker axioms thus >>>>>>>>>>> eliminating
these from consideration as any measure of true "in the system".
Source? or is this just another of your made up "Facts" >>>>>>>>>>
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite
connections to Truth.
WHERE in the definition of a "Formal System" does it say that >>>>>>>>>> the connecti0on must be finite.
You said that formal system cannot have infinite proofs. >>>>>>>>> Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections, proofs >>>>>>>> can not.
Truth *in a formal system* cannot be based on infinite
connections because formal systems are not allowed to have
infinite connections.
Says Who ***FOR TRUTH***
You reference does not provide that data, so I guess you are just >>>>>> making it up, and thus showing you to be a LIAR.
Haskell Curry establishes that truth in a theory (AKA formal >>>>>>> system) is anchored in the elementary theorems (AKA axioms) of >>>>>>> this formal system.
Right, ANCHORED TO, not limited to. Statments other than the
elementary theorems are True, and they are true if they have a >>>>>> connection (not limited to finite) to these Truths.
Where does he say True statements must have a FINITE connection to >>>>>> the elementary theorems.
A theory (over (f) is defined as a conceptual class of these >>>>>>> elementary
statements. Let::t be such a theory. Then the elementary statements
which belong to ::t we shall call the elementary theorems of::t; >>>>>>> we also
say that these elementary statements are true for::t. Thus, given >>>>>>> ::t,
an elementary theorem is an elementary statement which is true. A >>>>>>> theory
is thus a way of picking out from the statements of (f a certain >>>>>>> subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Perhaps you believe that you are enormously much brighter than >>>>>>> Haskell Curry ?
You don't understand what he is saying,
He is saying these statements are True in F, as a given.
Wrongo !!!
The terminology which has just been used implies that the
elementary statements are not such that their truth and
falsity are known to us without reference to::t.
Right, they aren't just true in the Statement class, but are only >>>> considerdd true because we are in the Theory F.
F is not the theory T is the theory.
Red Herring.
F is the Theory in Godels descussion.
It is not a red herring at all. Curry proves that the mathematical notion of incompleteness itself is incoherent in that Curry sustains Wittgenstein's notion of true in a formal system.
That G is unprovable in F merely means that G is untrue in F a triviality.
Please point out WHERE in the page you have cited that he does this.
Remeber, the CLASS of statemeents he talks about as "Elementary
Statments" that he talks about is NOT a "exhaustive" list of statements that can be formed, but a base set to start from.
This is clear from the line you have highlighted pointing out that these statements of are called elementary statements to distinguish them from other statements which we may form from them.
Then the "Elementary Theorems" are a SUBSET of these, that are defined
to be True in the Theory. Thus, these also are not a complete listing of all true statements in the Theory, but only the set a base truths that
we are working from (in addition to the contensive statements that are true indepentent of the Theory).
NOTHING on that page limits "True" statements to those things that are provable or only having a FINITE connection to those Elemetary Theories.
All this shows is that you don't understand what you are reading, or are just lying."They're just lying?"
On 1/20/23 5:16 PM, olcott wrote:
On 1/20/2023 4:09 PM, Richard Damon wrote:
On 1/20/23 5:02 PM, olcott wrote:
On 1/20/2023 2:46 PM, Richard Damon wrote:
On 1/20/23 2:31 PM, olcott wrote:
On 1/19/2023 8:34 PM, Richard Damon wrote:Says Who ***FOR TRUTH***
On 1/19/23 2:12 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
No, because I am showing that G is TRUE, not PROVABLE. Truth >>>>>>>>>>> can use infinte sets oc connections, proofs can't. Only YOU >>>>>>>>>>> have perposed that we think about infinite proofs.
Formal systems cannot ever use infinite connections from their >>>>>>>>>> expressions of language to their truth maker axioms thus
eliminating
these from consideration as any measure of true "in the system". >>>>>>>>>>
Source? or is this just another of your made up "Facts"
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite
connections to Truth.
WHERE in the definition of a "Formal System" does it say that >>>>>>>>> the connecti0on must be finite.
You said that formal system cannot have infinite proofs.
Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections, proofs
can not.
Truth *in a formal system* cannot be based on infinite connections >>>>>> because formal systems are not allowed to have infinite connections. >>>>>
You reference does not provide that data, so I guess you are just
making it up, and thus showing you to be a LIAR.
Haskell Curry establishes that truth in a theory (AKA formal
system) is anchored in the elementary theorems (AKA axioms) of
this formal system.
Right, ANCHORED TO, not limited to. Statments other than the
elementary theorems are True, and they are true if they have a
connection (not limited to finite) to these Truths.
Where does he say True statements must have a FINITE connection to
the elementary theorems.
A theory (over (f) is defined as a conceptual class of these
elementary
statements. Let::t be such a theory. Then the elementary statements >>>>>> which belong to ::t we shall call the elementary theorems of::t;
we also
say that these elementary statements are true for::t. Thus, given
::t,
an elementary theorem is an elementary statement which is true. A
theory
is thus a way of picking out from the statements of (f a certain
subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Perhaps you believe that you are enormously much brighter than
Haskell Curry ?
You don't understand what he is saying,
He is saying these statements are True in F, as a given.
Wrongo !!!
The terminology which has just been used implies that the
elementary statements are not such that their truth and
falsity are known to us without reference to::t.
Right, they aren't just true in the Statement class, but are only
considerdd true because we are in the Theory F.
F is not the theory T is the theory.
Red Herring.
F is the Theory in Godels descussion.
On 1/20/2023 4:54 PM, Richard Damon wrote:
On 1/20/23 5:16 PM, olcott wrote:
On 1/20/2023 4:09 PM, Richard Damon wrote:
On 1/20/23 5:02 PM, olcott wrote:
On 1/20/2023 2:46 PM, Richard Damon wrote:
On 1/20/23 2:31 PM, olcott wrote:
On 1/19/2023 8:34 PM, Richard Damon wrote:
On 1/19/23 2:12 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
No, because I am showing that G is TRUE, not PROVABLE. Truth >>>>>>>>>>>> can use infinte sets oc connections, proofs can't. Only YOU >>>>>>>>>>>> have perposed that we think about infinite proofs.
Formal systems cannot ever use infinite connections from their >>>>>>>>>>> expressions of language to their truth maker axioms thus >>>>>>>>>>> eliminating
these from consideration as any measure of true "in the system". >>>>>>>>>>>
Source? or is this just another of your made up "Facts"
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite
connections to Truth.
WHERE in the definition of a "Formal System" does it say that >>>>>>>>>> the connecti0on must be finite.
You said that formal system cannot have infinite proofs.
Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections, proofs >>>>>>>> can not.
Truth *in a formal system* cannot be based on infinite
connections because formal systems are not allowed to have
infinite connections.
Says Who ***FOR TRUTH***
You reference does not provide that data, so I guess you are just
making it up, and thus showing you to be a LIAR.
Haskell Curry establishes that truth in a theory (AKA formal
system) is anchored in the elementary theorems (AKA axioms) of
this formal system.
Right, ANCHORED TO, not limited to. Statments other than the
elementary theorems are True, and they are true if they have a
connection (not limited to finite) to these Truths.
Where does he say True statements must have a FINITE connection to >>>>>> the elementary theorems.
A theory (over (f) is defined as a conceptual class of these
elementary
statements. Let::t be such a theory. Then the elementary statements >>>>>>> which belong to ::t we shall call the elementary theorems of::t; >>>>>>> we also
say that these elementary statements are true for::t. Thus, given >>>>>>> ::t,
an elementary theorem is an elementary statement which is true. A >>>>>>> theory
is thus a way of picking out from the statements of (f a certain >>>>>>> subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Perhaps you believe that you are enormously much brighter than
Haskell Curry ?
You don't understand what he is saying,
He is saying these statements are True in F, as a given.
Wrongo !!!
The terminology which has just been used implies that the
elementary statements are not such that their truth and
falsity are known to us without reference to::t.
Right, they aren't just true in the Statement class, but are only
considerdd true because we are in the Theory F.
F is not the theory T is the theory.
Red Herring.
F is the Theory in Godels descussion.
https://en.wikipedia.org/wiki/Metamathematics
LP := ~True(LP) is untrue yet that does not make it true.
When we examine this at the meta level we escape the self-contradiction
and can say that it is true that LP is untrue.
https://plato.stanford.edu/entries/tarski-truth/#195DefOff
It looks like model theory is required to determine the truth of
some mathematical expressions, this had it origins in Tarski's
definition of truth.
∃n ∈ ℕ (N > 3) // does not seem to need model theory
∃G ∈ F (G ↔ (F ⊬ G)) // does not seem to need model theory
G is true in F iff it cannot be shown that G is true in F
On 1/20/2023 4:54 PM, Richard Damon wrote:
On 1/20/23 5:16 PM, olcott wrote:
On 1/20/2023 4:09 PM, Richard Damon wrote:
On 1/20/23 5:02 PM, olcott wrote:
On 1/20/2023 2:46 PM, Richard Damon wrote:
On 1/20/23 2:31 PM, olcott wrote:
On 1/19/2023 8:34 PM, Richard Damon wrote:
On 1/19/23 2:12 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
No, because I am showing that G is TRUE, not PROVABLE. Truth >>>>>>>>>>>> can use infinte sets oc connections, proofs can't. Only YOU >>>>>>>>>>>> have perposed that we think about infinite proofs.
Formal systems cannot ever use infinite connections from their >>>>>>>>>>> expressions of language to their truth maker axioms thus >>>>>>>>>>> eliminating
these from consideration as any measure of true "in the system". >>>>>>>>>>>
Source? or is this just another of your made up "Facts"
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite
connections to Truth.
WHERE in the definition of a "Formal System" does it say that >>>>>>>>>> the connecti0on must be finite.
You said that formal system cannot have infinite proofs.
Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections, proofs >>>>>>>> can not.
Truth *in a formal system* cannot be based on infinite
connections because formal systems are not allowed to have
infinite connections.
Says Who ***FOR TRUTH***
You reference does not provide that data, so I guess you are just
making it up, and thus showing you to be a LIAR.
Haskell Curry establishes that truth in a theory (AKA formal
system) is anchored in the elementary theorems (AKA axioms) of
this formal system.
Right, ANCHORED TO, not limited to. Statments other than the
elementary theorems are True, and they are true if they have a
connection (not limited to finite) to these Truths.
Where does he say True statements must have a FINITE connection to >>>>>> the elementary theorems.
A theory (over (f) is defined as a conceptual class of these
elementary
statements. Let::t be such a theory. Then the elementary statements >>>>>>> which belong to ::t we shall call the elementary theorems of::t; >>>>>>> we also
say that these elementary statements are true for::t. Thus, given >>>>>>> ::t,
an elementary theorem is an elementary statement which is true. A >>>>>>> theory
is thus a way of picking out from the statements of (f a certain >>>>>>> subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Perhaps you believe that you are enormously much brighter than
Haskell Curry ?
You don't understand what he is saying,
He is saying these statements are True in F, as a given.
Wrongo !!!
The terminology which has just been used implies that the
elementary statements are not such that their truth and
falsity are known to us without reference to::t.
Right, they aren't just true in the Statement class, but are only
considerdd true because we are in the Theory F.
F is not the theory T is the theory.
Red Herring.
F is the Theory in Godels descussion.
https://en.wikipedia.org/wiki/Metamathematics
LP := ~True(LP) is untrue yet that does not make it true.
When we examine this at the meta level we escape the self-contradiction
and can say that it is true that LP is untrue.
https://plato.stanford.edu/entries/tarski-truth/#195DefOff
It looks like model theory is required to determine the truth of
some mathematical expressions, this had it origins in Tarski's
definition of truth.
∃n ∈ ℕ (N > 3) // does not seem to need model theory
∃G ∈ F (G ↔ (F ⊬ G)) // does not seem to need model theory
G is true in F iff it cannot be shown that G is true in F
On 1/23/23 10:18 AM, olcott wrote:
On 1/20/2023 4:54 PM, Richard Damon wrote:
On 1/20/23 5:16 PM, olcott wrote:
On 1/20/2023 4:09 PM, Richard Damon wrote:
On 1/20/23 5:02 PM, olcott wrote:
On 1/20/2023 2:46 PM, Richard Damon wrote:
On 1/20/23 2:31 PM, olcott wrote:
On 1/19/2023 8:34 PM, Richard Damon wrote:
On 1/19/23 2:12 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
No, because I am showing that G is TRUE, not PROVABLE. >>>>>>>>>>>>> Truth can use infinte sets oc connections, proofs can't. >>>>>>>>>>>>> Only YOU have perposed that we think about infinite proofs. >>>>>>>>>>>>>
Formal systems cannot ever use infinite connections from their >>>>>>>>>>>> expressions of language to their truth maker axioms thus >>>>>>>>>>>> eliminating
these from consideration as any measure of true "in the >>>>>>>>>>>> system".
Source? or is this just another of your made up "Facts"
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite
connections to Truth.
WHERE in the definition of a "Formal System" does it say that >>>>>>>>>>> the connecti0on must be finite.
You said that formal system cannot have infinite proofs.
Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections, proofs >>>>>>>>> can not.
Truth *in a formal system* cannot be based on infinite
connections because formal systems are not allowed to have
infinite connections.
Says Who ***FOR TRUTH***
You reference does not provide that data, so I guess you are just >>>>>>> making it up, and thus showing you to be a LIAR.
Haskell Curry establishes that truth in a theory (AKA formal
system) is anchored in the elementary theorems (AKA axioms) of >>>>>>>> this formal system.
Right, ANCHORED TO, not limited to. Statments other than the
elementary theorems are True, and they are true if they have a
connection (not limited to finite) to these Truths.
Where does he say True statements must have a FINITE connection
to the elementary theorems.
A theory (over (f) is defined as a conceptual class of these
elementary
statements. Let::t be such a theory. Then the elementary statements >>>>>>>> which belong to ::t we shall call the elementary theorems of::t; >>>>>>>> we also
say that these elementary statements are true for::t. Thus,
given ::t,
an elementary theorem is an elementary statement which is true. >>>>>>>> A theory
is thus a way of picking out from the statements of (f a certain >>>>>>>> subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Perhaps you believe that you are enormously much brighter than >>>>>>>> Haskell Curry ?
You don't understand what he is saying,
He is saying these statements are True in F, as a given.
Wrongo !!!
The terminology which has just been used implies that the
elementary statements are not such that their truth and
falsity are known to us without reference to::t.
Right, they aren't just true in the Statement class, but are only
considerdd true because we are in the Theory F.
F is not the theory T is the theory.
Red Herring.
F is the Theory in Godels descussion.
https://en.wikipedia.org/wiki/Metamathematics
LP := ~True(LP) is untrue yet that does not make it true.
When we examine this at the meta level we escape the self-contradiction
and can say that it is true that LP is untrue.
Excpet that untrue is not ~True() in classical logic, which makes
statements either True or False, or makes them Not a Truth Bearer, which makes them not in the domain of the True predicate.
You need to move to tri-value logic to do this, at which point you loose
the relationship that ~True(x) -> False(x)
Note, most of mathematics is based on the two-value logic system.
https://plato.stanford.edu/entries/tarski-truth/#195DefOff
It looks like model theory is required to determine the truth of
some mathematical expressions, this had it origins in Tarski's
definition of truth.
∃n ∈ ℕ (N > 3) // does not seem to need model theory
∃G ∈ F (G ↔ (F ⊬ G)) // does not seem to need model theory
∃ is a symbol out of model theory, so hard to not need model theory.
Quoting from your reference:
Model theory by contrast works with three levels of symbol. There are
the logical constants ( = , ¬ , & for example), the variables (as
before), and between these a middle group of symbols which have no fixed meaning but get a meaning through being applied to a particular
structure. The symbols of this middle group include the nonlogical
constants of the language, such as relation symbols, function symbols
and constant individual symbols. They also include the quantifier
symbols ∀ and ∃, since we need to refer to the structure to see what set they range over.
G is true in F iff it cannot be shown that G is true in F
Nope, you don't understand what G is. The Definition of G in F does NOT
refer in any way determinable in F to the statement G.
On 1/23/2023 10:51 AM, Richard Damon wrote:
On 1/23/23 10:18 AM, olcott wrote:
On 1/20/2023 4:54 PM, Richard Damon wrote:
On 1/20/23 5:16 PM, olcott wrote:
On 1/20/2023 4:09 PM, Richard Damon wrote:
On 1/20/23 5:02 PM, olcott wrote:
On 1/20/2023 2:46 PM, Richard Damon wrote:
On 1/20/23 2:31 PM, olcott wrote:
On 1/19/2023 8:34 PM, Richard Damon wrote:
On 1/19/23 2:12 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
No, because I am showing that G is TRUE, not PROVABLE. >>>>>>>>>>>>>> Truth can use infinte sets oc connections, proofs can't. >>>>>>>>>>>>>> Only YOU have perposed that we think about infinite proofs. >>>>>>>>>>>>>>
Formal systems cannot ever use infinite connections from their >>>>>>>>>>>>> expressions of language to their truth maker axioms thus >>>>>>>>>>>>> eliminating
these from consideration as any measure of true "in the >>>>>>>>>>>>> system".
Source? or is this just another of your made up "Facts" >>>>>>>>>>>>
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite
connections to Truth.
WHERE in the definition of a "Formal System" does it say >>>>>>>>>>>> that the connecti0on must be finite.
You said that formal system cannot have infinite proofs. >>>>>>>>>>> Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections, proofs >>>>>>>>>> can not.
Truth *in a formal system* cannot be based on infinite
connections because formal systems are not allowed to have
infinite connections.
Says Who ***FOR TRUTH***
You reference does not provide that data, so I guess you are
just making it up, and thus showing you to be a LIAR.
Haskell Curry establishes that truth in a theory (AKA formal >>>>>>>>> system) is anchored in the elementary theorems (AKA axioms) of >>>>>>>>> this formal system.
Right, ANCHORED TO, not limited to. Statments other than the
elementary theorems are True, and they are true if they have a >>>>>>>> connection (not limited to finite) to these Truths.
Where does he say True statements must have a FINITE connection >>>>>>>> to the elementary theorems.
A theory (over (f) is defined as a conceptual class of these >>>>>>>>> elementary
statements. Let::t be such a theory. Then the elementary
statements
which belong to ::t we shall call the elementary theorems
of::t; we also
say that these elementary statements are true for::t. Thus,
given ::t,
an elementary theorem is an elementary statement which is true. >>>>>>>>> A theory
is thus a way of picking out from the statements of (f a certain >>>>>>>>> subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Perhaps you believe that you are enormously much brighter than >>>>>>>>> Haskell Curry ?
You don't understand what he is saying,
He is saying these statements are True in F, as a given.
Wrongo !!!
The terminology which has just been used implies that the >>>>>>> elementary statements are not such that their truth and
falsity are known to us without reference to::t.
Right, they aren't just true in the Statement class, but are only
considerdd true because we are in the Theory F.
F is not the theory T is the theory.
Red Herring.
F is the Theory in Godels descussion.
https://en.wikipedia.org/wiki/Metamathematics
LP := ~True(LP) is untrue yet that does not make it true.
When we examine this at the meta level we escape the self-contradiction
and can say that it is true that LP is untrue.
Excpet that untrue is not ~True() in classical logic, which makes
statements either True or False, or makes them Not a Truth Bearer,
which makes them not in the domain of the True predicate.
You need to move to tri-value logic to do this, at which point you
loose the relationship that ~True(x) -> False(x)
True / false and not a truth bearer.
Note, most of mathematics is based on the two-value logic system.
Thus forcing it to classify "not a truth bearer" incorrectly.
If all you have is a hammer the unscrewing a screw becomes quite
destructive.
https://plato.stanford.edu/entries/tarski-truth/#195DefOff
It looks like model theory is required to determine the truth of
some mathematical expressions, this had it origins in Tarski's
definition of truth.
∃n ∈ ℕ (N > 3) // does not seem to need model theory
∃G ∈ F (G ↔ (F ⊬ G)) // does not seem to need model theory
∃ is a symbol out of model theory, so hard to not need model theory.
Quoting from your reference:
Model theory by contrast works with three levels of symbol. There are
the logical constants ( = , ¬ , & for example), the variables (as
before), and between these a middle group of symbols which have no
fixed meaning but get a meaning through being applied to a particular
structure. The symbols of this middle group include the nonlogical
constants of the language, such as relation symbols, function symbols
and constant individual symbols. They also include the quantifier
symbols ∀ and ∃, since we need to refer to the structure to see what >> set they range over.
I just showed how to explicitly specify what they range over: ∃n ∈ ℕ
G is true in F iff it cannot be shown that G is true in F
Nope, you don't understand what G is. The Definition of G in F does
NOT refer in any way determinable in F to the statement G.
∃n ∈ ℕ (n > 3) // Is this true or false?
How do you know?
Generically how does ascertain that that any logic expression is true or false?
Most generically an analytical expression of formal or natural language
is only true if it has a semantic connection to its truth maker axioms.
The "truth maker axioms" of natural language are the definition of the meaning of its words.
The truth maker axioms for the above expression is the definition of the ordered set of natural numbers:
https://www.britannica.com/science/Peano-axioms
On 1/23/23 5:23 PM, olcott wrote:
On 1/23/2023 10:51 AM, Richard Damon wrote:
On 1/23/23 10:18 AM, olcott wrote:
On 1/20/2023 4:54 PM, Richard Damon wrote:
On 1/20/23 5:16 PM, olcott wrote:
On 1/20/2023 4:09 PM, Richard Damon wrote:
On 1/20/23 5:02 PM, olcott wrote:
On 1/20/2023 2:46 PM, Richard Damon wrote:
On 1/20/23 2:31 PM, olcott wrote:
On 1/19/2023 8:34 PM, Richard Damon wrote:
On 1/19/23 2:12 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
No, because I am showing that G is TRUE, not PROVABLE. >>>>>>>>>>>>>>> Truth can use infinte sets oc connections, proofs can't. >>>>>>>>>>>>>>> Only YOU have perposed that we think about infinite proofs. >>>>>>>>>>>>>>>
Formal systems cannot ever use infinite connections from >>>>>>>>>>>>>> their
expressions of language to their truth maker axioms thus >>>>>>>>>>>>>> eliminating
these from consideration as any measure of true "in the >>>>>>>>>>>>>> system".
Source? or is this just another of your made up "Facts" >>>>>>>>>>>>>
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite >>>>>>>>>>> connections to Truth.
WHERE in the definition of a "Formal System" does it say >>>>>>>>>>>>> that the connecti0on must be finite.
You said that formal system cannot have infinite proofs. >>>>>>>>>>>> Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections,
proofs can not.
Truth *in a formal system* cannot be based on infinite
connections because formal systems are not allowed to have >>>>>>>>>> infinite connections.
Says Who ***FOR TRUTH***
You reference does not provide that data, so I guess you are >>>>>>>>> just making it up, and thus showing you to be a LIAR.
Haskell Curry establishes that truth in a theory (AKA formal >>>>>>>>>> system) is anchored in the elementary theorems (AKA axioms) of >>>>>>>>>> this formal system.
Right, ANCHORED TO, not limited to. Statments other than the >>>>>>>>> elementary theorems are True, and they are true if they have a >>>>>>>>> connection (not limited to finite) to these Truths.
Where does he say True statements must have a FINITE connection >>>>>>>>> to the elementary theorems.
A theory (over (f) is defined as a conceptual class of these >>>>>>>>>> elementary
statements. Let::t be such a theory. Then the elementary
statements
which belong to ::t we shall call the elementary theorems
of::t; we also
say that these elementary statements are true for::t. Thus, >>>>>>>>>> given ::t,
an elementary theorem is an elementary statement which is
true. A theory
is thus a way of picking out from the statements of (f a certain >>>>>>>>>> subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Perhaps you believe that you are enormously much brighter than >>>>>>>>>> Haskell Curry ?
You don't understand what he is saying,
He is saying these statements are True in F, as a given.
Wrongo !!!
The terminology which has just been used implies that the >>>>>>>> elementary statements are not such that their truth and >>>>>>>> falsity are known to us without reference to::t.
Right, they aren't just true in the Statement class, but are only >>>>>>> considerdd true because we are in the Theory F.
F is not the theory T is the theory.
Red Herring.
F is the Theory in Godels descussion.
https://en.wikipedia.org/wiki/Metamathematics
LP := ~True(LP) is untrue yet that does not make it true.
When we examine this at the meta level we escape the self-contradiction >>>> and can say that it is true that LP is untrue.
Excpet that untrue is not ~True() in classical logic, which makes
statements either True or False, or makes them Not a Truth Bearer,
which makes them not in the domain of the True predicate.
You need to move to tri-value logic to do this, at which point you
loose the relationship that ~True(x) -> False(x)
True / false and not a truth bearer.
That is your TRI-value logic.
Note, most of mathematics is based on the two-value logic system.
Thus forcing it to classify "not a truth bearer" incorrectly.
If all you have is a hammer the unscrewing a screw becomes quite
destructive.
Nope, a "statement" can be well formed, and thus MUST be a "Truth
Bearer" or it isn't and is NOT a "Truth Bearer"
By ignoring that mathematically defined statement ARE "Truth Bearers",
you logic system is just broken.
https://plato.stanford.edu/entries/tarski-truth/#195DefOff
It looks like model theory is required to determine the truth of
some mathematical expressions, this had it origins in Tarski's
definition of truth.
∃n ∈ ℕ (N > 3) // does not seem to need model theory >>>> ∃G ∈ F (G ↔ (F ⊬ G)) // does not seem to need model theory
∃ is a symbol out of model theory, so hard to not need model theory.
Quoting from your reference:
Model theory by contrast works with three levels of symbol. There are
the logical constants ( = , ¬ , & for example), the variables (as
before), and between these a middle group of symbols which have no
fixed meaning but get a meaning through being applied to a particular
structure. The symbols of this middle group include the nonlogical
constants of the language, such as relation symbols, function symbols
and constant individual symbols. They also include the quantifier
symbols ∀ and ∃, since we need to refer to the structure to see what >>> set they range over.
I just showed how to explicitly specify what they range over: ∃n ∈ ℕ
Which means you are using "Model Theory"
Maybe you don't understand those words.
G is true in F iff it cannot be shown that G is true in F
Nope, you don't understand what G is. The Definition of G in F does
NOT refer in any way determinable in F to the statement G.
∃n ∈ ℕ (n > 3) // Is this true or false?
How do you know?
Simple, 4 exists (S(S(S(S(0)))), 4 > 3, 4 ∈ ℕ, thus the statement is True. Like many (but not all) True statements, it can be proven.
Generically how does ascertain that that any logic expression is true or
false?
Note, "Ascertain" means you are talking about KNOWLEDGE, not Truth.
Truth doesn't need to be ascertained to be true, it just is.
It needs to be ascertained to be KNOWN.
It is a TRUE statement that either all even numbers greater than 2 are
the sum of 2 primes or there exists at least one that is not. We don't
know which one of them is true right now, but we do know that one of
them is.
This seems to be one of your core problems, confusing what can be known
to be true with what IS true.
Most generically an analytical expression of formal or natural
language is only true if it has a semantic connection to its truth
maker axioms.
Right, but that connection might not be known, or might even be infinite.
It is only KNOWLEDGE or PROOF that requires a finite connection.
The "truth maker axioms" of natural language are the definition of the
meaning of its words.
No, the accepted Truth Maker Axioms of the Theory (not what their words
mean in Natural Language) determine what is true.
Your reliance on "Natural Language" is what has actually been proven to
lead to problems.
The truth maker axioms for the above expression is the definition of
the ordered set of natural numbers:
https://www.britannica.com/science/Peano-axioms
You understand that Godel showed that under the Peano-axioms, he proved
that their exists truths that can not be proven.
It becomes a
consequence of the induction axiom that allows him to be able to define
the primative recursive relationship that shows that you can not prove
within the theory that no nmber exists that matches that theory, and
also create an extention to that theory (that is used to create that relationship) that allows us to actually prove that statment must be
true, and also that no proof of this can exist in the base theory.
The induction property that proves it only comes in the extension (the meta-theory) and is not in the base theory,
so the base theory can't
make the proof, but can evaluate for every term, thus making the
INFINITE chain that makes it true in the Theory.
Peano ARITHMATIC changed that induction axiom to a first order logic definition, weaking what the theory can do, but allows it to appear to
be complete, but NOT express ALL the properties of the Natural Numbers.
I beleive that it becomes the (or at least one of the) largest logic
system that retains "Completeness" while sitll being "Consistent". (But
it can't prove itself to be consistent)
This is of course, over you head, so you wil either deny it or just
ignore the refuation and go off on some other tack.
On 1/23/2023 4:54 PM, Richard Damon wrote:
On 1/23/23 5:23 PM, olcott wrote:
On 1/23/2023 10:51 AM, Richard Damon wrote:
On 1/23/23 10:18 AM, olcott wrote:
On 1/20/2023 4:54 PM, Richard Damon wrote:
On 1/20/23 5:16 PM, olcott wrote:
On 1/20/2023 4:09 PM, Richard Damon wrote:
On 1/20/23 5:02 PM, olcott wrote:
On 1/20/2023 2:46 PM, Richard Damon wrote:
On 1/20/23 2:31 PM, olcott wrote:
On 1/19/2023 8:34 PM, Richard Damon wrote:
On 1/19/23 2:12 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
No, because I am showing that G is TRUE, not PROVABLE. >>>>>>>>>>>>>>>> Truth can use infinte sets oc connections, proofs can't. >>>>>>>>>>>>>>>> Only YOU have perposed that we think about infinite proofs. >>>>>>>>>>>>>>>>
Formal systems cannot ever use infinite connections from >>>>>>>>>>>>>>> their
expressions of language to their truth maker axioms thus >>>>>>>>>>>>>>> eliminating
these from consideration as any measure of true "in the >>>>>>>>>>>>>>> system".
Source? or is this just another of your made up "Facts" >>>>>>>>>>>>>>
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite >>>>>>>>>>>> connections to Truth.
WHERE in the definition of a "Formal System" does it say >>>>>>>>>>>>>> that the connecti0on must be finite.
You said that formal system cannot have infinite proofs. >>>>>>>>>>>>> Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections, >>>>>>>>>>>> proofs can not.
Truth *in a formal system* cannot be based on infinite
connections because formal systems are not allowed to have >>>>>>>>>>> infinite connections.
Says Who ***FOR TRUTH***
You reference does not provide that data, so I guess you are >>>>>>>>>> just making it up, and thus showing you to be a LIAR.
Haskell Curry establishes that truth in a theory (AKA formal >>>>>>>>>>> system) is anchored in the elementary theorems (AKA axioms) >>>>>>>>>>> of this formal system.
Right, ANCHORED TO, not limited to. Statments other than the >>>>>>>>>> elementary theorems are True, and they are true if they have a >>>>>>>>>> connection (not limited to finite) to these Truths.
Where does he say True statements must have a FINITE
connection to the elementary theorems.
A theory (over (f) is defined as a conceptual class of these >>>>>>>>>>> elementary
statements. Let::t be such a theory. Then the elementary >>>>>>>>>>> statements
which belong to ::t we shall call the elementary theorems >>>>>>>>>>> of::t; we also
say that these elementary statements are true for::t. Thus, >>>>>>>>>>> given ::t,
an elementary theorem is an elementary statement which is >>>>>>>>>>> true. A theory
is thus a way of picking out from the statements of (f a certain >>>>>>>>>>> subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Perhaps you believe that you are enormously much brighter >>>>>>>>>>> than Haskell Curry ?
You don't understand what he is saying,
He is saying these statements are True in F, as a given.
Wrongo !!!
The terminology which has just been used implies that the >>>>>>>>> elementary statements are not such that their truth and >>>>>>>>> falsity are known to us without reference to::t.
Right, they aren't just true in the Statement class, but are
only considerdd true because we are in the Theory F.
F is not the theory T is the theory.
Red Herring.
F is the Theory in Godels descussion.
https://en.wikipedia.org/wiki/Metamathematics
LP := ~True(LP) is untrue yet that does not make it true.
When we examine this at the meta level we escape the
self-contradiction
and can say that it is true that LP is untrue.
Excpet that untrue is not ~True() in classical logic, which makes
statements either True or False, or makes them Not a Truth Bearer,
which makes them not in the domain of the True predicate.
You need to move to tri-value logic to do this, at which point you
loose the relationship that ~True(x) -> False(x)
True / false and not a truth bearer.
That is your TRI-value logic.
It is true by logical necessity.
Every expression of language must necessarily be
true, false, neither true nor false.
Note, most of mathematics is based on the two-value logic system.
Thus forcing it to classify "not a truth bearer" incorrectly.
If all you have is a hammer the unscrewing a screw becomes quite
destructive.
Nope, a "statement" can be well formed, and thus MUST be a "Truth
Bearer" or it isn't and is NOT a "Truth Bearer"
By ignoring that mathematically defined statement ARE "Truth Bearers",
you logic system is just broken.
Which means you are using "Model Theory"
https://plato.stanford.edu/entries/tarski-truth/#195DefOff
It looks like model theory is required to determine the truth of
some mathematical expressions, this had it origins in Tarski's
definition of truth.
∃n ∈ ℕ (N > 3) // does not seem to need model theory >>>>> ∃G ∈ F (G ↔ (F ⊬ G)) // does not seem to need model theory
∃ is a symbol out of model theory, so hard to not need model theory. >>>>
Quoting from your reference:
Model theory by contrast works with three levels of symbol. There
are the logical constants ( = , ¬ , & for example), the variables
(as before), and between these a middle group of symbols which have
no fixed meaning but get a meaning through being applied to a
particular structure. The symbols of this middle group include the
nonlogical constants of the language, such as relation symbols,
function symbols and constant individual symbols. They also include
the quantifier symbols ∀ and ∃, since we need to refer to the
structure to see what set they range over.
I just showed how to explicitly specify what they range over: ∃n ∈ ℕ >>
Maybe you don't understand those words.
Model theory is used to define things that are not otherwise defined.
When they are otherwise defined there is no need for model theory.
G is true in F iff it cannot be shown that G is true in F
Nope, you don't understand what G is. The Definition of G in F does
NOT refer in any way determinable in F to the statement G.
∃n ∈ ℕ (n > 3) // Is this true or false?
How do you know?
Simple, 4 exists (S(S(S(S(0)))), 4 > 3, 4 ∈ ℕ, thus the statement is
True. Like many (but not all) True statements, it can be proven.
Generically how does ascertain that that any logic expression is true or >>> false?
Note, "Ascertain" means you are talking about KNOWLEDGE, not Truth.
Truth doesn't need to be ascertained to be true, it just is.
It needs to be ascertained to be KNOWN.
It is a TRUE statement that either all even numbers greater than 2 are
the sum of 2 primes or there exists at least one that is not. We don't
know which one of them is true right now, but we do know that one of
them is.
This seems to be one of your core problems, confusing what can be
known to be true with what IS true.
Most generically an analytical expression of formal or natural
language is only true if it has a semantic connection to its truth
maker axioms.
Right, but that connection might not be known, or might even be infinite.
It is only KNOWLEDGE or PROOF that requires a finite connection.
The "truth maker axioms" of natural language are the definition of the
meaning of its words.
No, the accepted Truth Maker Axioms of the Theory (not what their
words mean in Natural Language) determine what is true.
of natural language such as English
of natural language such as English
of natural language such as English
of natural language such as English
Your reliance on "Natural Language" is what has actually been proven
to lead to problems.
The entire body of all analytical knowledge can only be expressed using language. Hardly any of this is currently expressed using formal
language. All knowledge is necessarily true by definition.
The truth maker axioms for the above expression is the definition of
the ordered set of natural numbers:
https://www.britannica.com/science/Peano-axioms
You understand that Godel showed that under the Peano-axioms, he
proved that their exists truths that can not be proven.
We can make the Gödel number of "I just ate some chicken" using the
adjacent ASCII values. This too cannot be proven in the Peano-axioms.
It becomes a consequence of the induction axiom that allows him to be
able to define the primative recursive relationship that shows that
you can not prove within the theory that no nmber exists that matches
that theory, and also create an extention to that theory (that is used
to create that relationship) that allows us to actually prove that
statment must be true, and also that no proof of this can exist in the
base theory.
Its a mere gimmick.
He acknowledged that the Liar Paradox forms an equivalent proof.
The induction property that proves it only comes in the extension (the
meta-theory) and is not in the base theory,
"This sentence is not true" is self-evidently untrue yet that does not
make the sentence true within the scope of self-contradiction.
so the base theory can't make the proof, but can evaluate for every
term, thus making the INFINITE chain that makes it true in the Theory.
It is not an infinite chain, it is simply that the sentence is true
outside of the scope of self-contradiction and impossible to evaluate
within the scope of self-contradiction.
Peano ARITHMATIC changed that induction axiom to a first order logic
definition, weaking what the theory can do, but allows it to appear to
be complete, but NOT express ALL the properties of the Natural Numbers.
Natural numbers themselves never had the property of provability.
*The five Peano axioms are*
(1) Zero is a natural number.
(2) Every natural number has a successor in the natural numbers.
(3) Zero is not the successor of any natural number.
(4) If the successor of two natural numbers is the same, then the two
original numbers are the same.
(5) If a set contains zero and the successor of every number is in the
set, then the set contains the natural numbers.
https://www.britannica.com/science/Peano-axioms
I beleive that it becomes the (or at least one of the) largest logic
system that retains "Completeness" while sitll being "Consistent".
(But it can't prove itself to be consistent)
This is of course, over you head, so you wil either deny it or just
ignore the refuation and go off on some other tack.
On 1/23/23 6:39 PM, olcott wrote:
On 1/23/2023 4:54 PM, Richard Damon wrote:
On 1/23/23 5:23 PM, olcott wrote:
On 1/23/2023 10:51 AM, Richard Damon wrote:
On 1/23/23 10:18 AM, olcott wrote:
On 1/20/2023 4:54 PM, Richard Damon wrote:
On 1/20/23 5:16 PM, olcott wrote:
On 1/20/2023 4:09 PM, Richard Damon wrote:
On 1/20/23 5:02 PM, olcott wrote:
On 1/20/2023 2:46 PM, Richard Damon wrote:
On 1/20/23 2:31 PM, olcott wrote:
On 1/19/2023 8:34 PM, Richard Damon wrote:
On 1/19/23 2:12 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
No, because I am showing that G is TRUE, not PROVABLE. >>>>>>>>>>>>>>>>> Truth can use infinte sets oc connections, proofs >>>>>>>>>>>>>>>>> can't. Only YOU have perposed that we think about >>>>>>>>>>>>>>>>> infinite proofs.
Formal systems cannot ever use infinite connections from >>>>>>>>>>>>>>>> their
expressions of language to their truth maker axioms thus >>>>>>>>>>>>>>>> eliminating
these from consideration as any measure of true "in the >>>>>>>>>>>>>>>> system".
Source? or is this just another of your made up "Facts" >>>>>>>>>>>>>>>
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite >>>>>>>>>>>>> connections to Truth.
WHERE in the definition of a "Formal System" does it say >>>>>>>>>>>>>>> that the connecti0on must be finite.
You said that formal system cannot have infinite proofs. >>>>>>>>>>>>>> Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections, >>>>>>>>>>>>> proofs can not.
Truth *in a formal system* cannot be based on infinite >>>>>>>>>>>> connections because formal systems are not allowed to have >>>>>>>>>>>> infinite connections.
Says Who ***FOR TRUTH***
You reference does not provide that data, so I guess you are >>>>>>>>>>> just making it up, and thus showing you to be a LIAR.
Haskell Curry establishes that truth in a theory (AKA formal >>>>>>>>>>>> system) is anchored in the elementary theorems (AKA axioms) >>>>>>>>>>>> of this formal system.
Right, ANCHORED TO, not limited to. Statments other than the >>>>>>>>>>> elementary theorems are True, and they are true if they have >>>>>>>>>>> a connection (not limited to finite) to these Truths.
Where does he say True statements must have a FINITE
connection to the elementary theorems.
A theory (over (f) is defined as a conceptual class of these >>>>>>>>>>>> elementary
statements. Let::t be such a theory. Then the elementary >>>>>>>>>>>> statements
which belong to ::t we shall call the elementary theorems >>>>>>>>>>>> of::t; we also
say that these elementary statements are true for::t. Thus, >>>>>>>>>>>> given ::t,
an elementary theorem is an elementary statement which is >>>>>>>>>>>> true. A theory
is thus a way of picking out from the statements of (f a >>>>>>>>>>>> certain
subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Perhaps you believe that you are enormously much brighter >>>>>>>>>>>> than Haskell Curry ?
You don't understand what he is saying,
He is saying these statements are True in F, as a given.
Wrongo !!!
The terminology which has just been used implies that the >>>>>>>>>> elementary statements are not such that their truth and >>>>>>>>>> falsity are known to us without reference to::t.
Right, they aren't just true in the Statement class, but are >>>>>>>>> only considerdd true because we are in the Theory F.
F is not the theory T is the theory.
Red Herring.
F is the Theory in Godels descussion.
https://en.wikipedia.org/wiki/Metamathematics
LP := ~True(LP) is untrue yet that does not make it true.
When we examine this at the meta level we escape the
self-contradiction
and can say that it is true that LP is untrue.
Excpet that untrue is not ~True() in classical logic, which makes
statements either True or False, or makes them Not a Truth Bearer,
which makes them not in the domain of the True predicate.
You need to move to tri-value logic to do this, at which point you
loose the relationship that ~True(x) -> False(x)
True / false and not a truth bearer.
That is your TRI-value logic.
It is true by logical necessity.
Every expression of language must necessarily be
true, false, neither true nor false.
Nope. You can also use a two level division like you actually talk about.
Statments are either Truth Bearers or they are Not
Truth Bearers are either True or they are False.
Note, most of mathematics is based on the two-value logic system.
Thus forcing it to classify "not a truth bearer" incorrectly.
If all you have is a hammer the unscrewing a screw becomes quite
destructive.
Nope, a "statement" can be well formed, and thus MUST be a "Truth
Bearer" or it isn't and is NOT a "Truth Bearer"
By ignoring that mathematically defined statement ARE "Truth
Bearers", you logic system is just broken.
Which means you are using "Model Theory"
https://plato.stanford.edu/entries/tarski-truth/#195DefOff
It looks like model theory is required to determine the truth of
some mathematical expressions, this had it origins in Tarski's
definition of truth.
∃n ∈ ℕ (N > 3) // does not seem to need model theory >>>>>> ∃G ∈ F (G ↔ (F ⊬ G)) // does not seem to need model theory >>>>>>
∃ is a symbol out of model theory, so hard to not need model theory. >>>>>
Quoting from your reference:
Model theory by contrast works with three levels of symbol. There
are the logical constants ( = , ¬ , & for example), the variables
(as before), and between these a middle group of symbols which have
no fixed meaning but get a meaning through being applied to a
particular structure. The symbols of this middle group include the
nonlogical constants of the language, such as relation symbols,
function symbols and constant individual symbols. They also include
the quantifier symbols ∀ and ∃, since we need to refer to the
structure to see what set they range over.
I just showed how to explicitly specify what they range over: ∃n ∈ ℕ >>>
Maybe you don't understand those words.
Model theory is used to define things that are not otherwise defined.
When they are otherwise defined there is no need for model theory.
So, you don't understand what Model Theory is (or mixing different definitions)
G is true in F iff it cannot be shown that G is true in F
Nope, you don't understand what G is. The Definition of G in F does
NOT refer in any way determinable in F to the statement G.
∃n ∈ ℕ (n > 3) // Is this true or false?
How do you know?
Simple, 4 exists (S(S(S(S(0)))), 4 > 3, 4 ∈ ℕ, thus the statement is >>> True. Like many (but not all) True statements, it can be proven.
Generically how does ascertain that that any logic expression is
true or
false?
Note, "Ascertain" means you are talking about KNOWLEDGE, not Truth.
Truth doesn't need to be ascertained to be true, it just is.
It needs to be ascertained to be KNOWN.
It is a TRUE statement that either all even numbers greater than 2
are the sum of 2 primes or there exists at least one that is not. We
don't know which one of them is true right now, but we do know that
one of them is.
This seems to be one of your core problems, confusing what can be
known to be true with what IS true.
Most generically an analytical expression of formal or natural
language is only true if it has a semantic connection to its truth
maker axioms.
Right, but that connection might not be known, or might even be
infinite.
It is only KNOWLEDGE or PROOF that requires a finite connection.
The "truth maker axioms" of natural language are the definition of the >>>> meaning of its words.
No, the accepted Truth Maker Axioms of the Theory (not what their
words mean in Natural Language) determine what is true.
of natural language such as English
of natural language such as English
of natural language such as English
of natural language such as English
Which has been proven unsuitable for logic.
Given the statement:
If this sentence is true, Peter Olcott is a moron.
This is a valid logical statement of natural language form.
By the meaning of the words, it is TRUE, because if the sentence IS
true, then by the DEFINITION of True, it must be actually True.
Thus, since it HAS been proven true, its implication must be correct,
and thus YOU ARE A MORON.
The "flaw" in the statement is that Natural Language isn't suitable to
fully express logic.
Your reliance on "Natural Language" is what has actually been proven
to lead to problems.
The entire body of all analytical knowledge can only be expressed using
language. Hardly any of this is currently expressed using formal
language. All knowledge is necessarily true by definition.
Right, TECHNICAL/FORMAL language, not NATURAL language.
You are incorrect that hardly any of this is expressed using formal
language, and that is a major part of your problem. Words that are words
in "Natuaral Language" are frequently refined to a formal definition for particular usage. If you don't understand that formal definition, or
even more important WHICH formal definition is needed for a given
statement, you won't understand it.
And All Knowledge being necessarily true is NOT a universal definition,
in fact, one of the problems of the study of knowledge is how to avoid
the introduction into "Knowledge" of things that we THINK are True but
are actually incorrect. We WANT everything that we (think we) know to be actually true, but factually, since there IS a human element in the aquisition of knowledge, there is a possibility of error and of thinking
we know something that isn't true.
The truth maker axioms for the above expression is the definition of
the ordered set of natural numbers:
https://www.britannica.com/science/Peano-axioms
You understand that Godel showed that under the Peano-axioms, he
proved that their exists truths that can not be proven.
We can make the Gödel number of "I just ate some chicken" using the
adjacent ASCII values. This too cannot be proven in the Peano-axioms.
Nope. You don't understand what Godel does. Not understanding something
does not make it not true, you are just serving your Herring with Red
Sauce.
Yes, you can set up a system where you use Godel's math to create a
number that represents the statement "I just ate some chicken", but that statement has nothing to do with Godel's proof.
The fact you can throw out Red Herring that means nothing doesn't
discount the proof, it just proves you don't understand what you are
talking about.
It becomes a consequence of the induction axiom that allows him to be
able to define the primative recursive relationship that shows that
you can not prove within the theory that no nmber exists that matches
that theory, and also create an extention to that theory (that is
used to create that relationship) that allows us to actually prove
that statment must be true, and also that no proof of this can exist
in the base theory.
Its a mere gimmick.
He acknowledged that the Liar Paradox forms an equivalent proof.
No, it isn't.
If you think it is, then SHOW that it is.
But to do that,
you need to understand what he did and where he "just used a gimmick"
All your statments show is that you just don't understand what he is
saying and are such a pathological liar that you will make up excuses to cover that.
On 1/23/2023 6:38 PM, Richard Damon wrote:
On 1/23/23 6:39 PM, olcott wrote:
On 1/23/2023 4:54 PM, Richard Damon wrote:
On 1/23/23 5:23 PM, olcott wrote:
On 1/23/2023 10:51 AM, Richard Damon wrote:
On 1/23/23 10:18 AM, olcott wrote:
On 1/20/2023 4:54 PM, Richard Damon wrote:
On 1/20/23 5:16 PM, olcott wrote:
On 1/20/2023 4:09 PM, Richard Damon wrote:
On 1/20/23 5:02 PM, olcott wrote:
On 1/20/2023 2:46 PM, Richard Damon wrote:
On 1/20/23 2:31 PM, olcott wrote:Wrongo !!!
On 1/19/2023 8:34 PM, Richard Damon wrote:
On 1/19/23 2:12 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote:
No, because I am showing that G is TRUE, not PROVABLE. >>>>>>>>>>>>>>>>>> Truth can use infinte sets oc connections, proofs >>>>>>>>>>>>>>>>>> can't. Only YOU have perposed that we think about >>>>>>>>>>>>>>>>>> infinite proofs.
Formal systems cannot ever use infinite connections >>>>>>>>>>>>>>>>> from their
expressions of language to their truth maker axioms >>>>>>>>>>>>>>>>> thus eliminating
these from consideration as any measure of true "in the >>>>>>>>>>>>>>>>> system".
Source? or is this just another of your made up "Facts" >>>>>>>>>>>>>>>>
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite >>>>>>>>>>>>>> connections to Truth.
WHERE in the definition of a "Formal System" does it say >>>>>>>>>>>>>>>> that the connecti0on must be finite.
You said that formal system cannot have infinite proofs. >>>>>>>>>>>>>>> Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections, >>>>>>>>>>>>>> proofs can not.
Truth *in a formal system* cannot be based on infinite >>>>>>>>>>>>> connections because formal systems are not allowed to have >>>>>>>>>>>>> infinite connections.
Says Who ***FOR TRUTH***
You reference does not provide that data, so I guess you are >>>>>>>>>>>> just making it up, and thus showing you to be a LIAR.
Haskell Curry establishes that truth in a theory (AKA >>>>>>>>>>>>> formal system) is anchored in the elementary theorems (AKA >>>>>>>>>>>>> axioms) of this formal system.
Right, ANCHORED TO, not limited to. Statments other than the >>>>>>>>>>>> elementary theorems are True, and they are true if they have >>>>>>>>>>>> a connection (not limited to finite) to these Truths.
Where does he say True statements must have a FINITE
connection to the elementary theorems.
A theory (over (f) is defined as a conceptual class of >>>>>>>>>>>>> these elementary
statements. Let::t be such a theory. Then the elementary >>>>>>>>>>>>> statements
which belong to ::t we shall call the elementary theorems >>>>>>>>>>>>> of::t; we also
say that these elementary statements are true for::t. Thus, >>>>>>>>>>>>> given ::t,
an elementary theorem is an elementary statement which is >>>>>>>>>>>>> true. A theory
is thus a way of picking out from the statements of (f a >>>>>>>>>>>>> certain
subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Perhaps you believe that you are enormously much brighter >>>>>>>>>>>>> than Haskell Curry ?
You don't understand what he is saying,
He is saying these statements are True in F, as a given. >>>>>>>>>>>
The terminology which has just been used implies that the >>>>>>>>>>> elementary statements are not such that their truth and >>>>>>>>>>> falsity are known to us without reference to::t.
Right, they aren't just true in the Statement class, but are >>>>>>>>>> only considerdd true because we are in the Theory F.
F is not the theory T is the theory.
Red Herring.
F is the Theory in Godels descussion.
https://en.wikipedia.org/wiki/Metamathematics
LP := ~True(LP) is untrue yet that does not make it true.
When we examine this at the meta level we escape the
self-contradiction
and can say that it is true that LP is untrue.
Excpet that untrue is not ~True() in classical logic, which makes
statements either True or False, or makes them Not a Truth Bearer, >>>>>> which makes them not in the domain of the True predicate.
You need to move to tri-value logic to do this, at which point you >>>>>> loose the relationship that ~True(x) -> False(x)
True / false and not a truth bearer.
That is your TRI-value logic.
It is true by logical necessity.
Every expression of language must necessarily be
true, false, neither true nor false.
Nope. You can also use a two level division like you actually talk about.
Statments are either Truth Bearers or they are Not
Truth Bearers are either True or they are False.
Is this a trick? Did you just agree with me?
Note, most of mathematics is based on the two-value logic system.
Thus forcing it to classify "not a truth bearer" incorrectly.
If all you have is a hammer the unscrewing a screw becomes quite
destructive.
Nope, a "statement" can be well formed, and thus MUST be a "Truth
Bearer" or it isn't and is NOT a "Truth Bearer"
By ignoring that mathematically defined statement ARE "Truth
Bearers", you logic system is just broken.
https://plato.stanford.edu/entries/tarski-truth/#195DefOff
It looks like model theory is required to determine the truth of >>>>>>> some mathematical expressions, this had it origins in Tarski's
definition of truth.
∃n ∈ ℕ (N > 3) // does not seem to need model theory >>>>>>> ∃G ∈ F (G ↔ (F ⊬ G)) // does not seem to need model theory >>>>>>>
∃ is a symbol out of model theory, so hard to not need model theory. >>>>>>
Quoting from your reference:
Model theory by contrast works with three levels of symbol. There
are the logical constants ( = , ¬ , & for example), the variables >>>>>> (as before), and between these a middle group of symbols which
have no fixed meaning but get a meaning through being applied to a >>>>>> particular structure. The symbols of this middle group include the >>>>>> nonlogical constants of the language, such as relation symbols,
function symbols and constant individual symbols. They also
include the quantifier symbols ∀ and ∃, since we need to refer to >>>>>> the structure to see what set they range over.
I just showed how to explicitly specify what they range over: ∃n ∈ ℕ
Which means you are using "Model Theory"
Maybe you don't understand those words.
Model theory is used to define things that are not otherwise defined.
When they are otherwise defined there is no need for model theory.
So, you don't understand what Model Theory is (or mixing different
definitions)
The most succinct definition seems to be saying that model theory merely defines the semantics of elements expressions of language.
Sometimes we write or speak a sentence S that expresses nothing either
true or false, because some crucial information is missing about what
the words mean.
If we go on to add this information, so that S comes to express a true
or false statement, we are said to interpret S, and the added
information is called an interpretation of S.
If the interpretation I happens to make S state something true, we say
that I is a model of S, or that I satisfies S, in symbols ‘I ⊨ S’.
Another way of saying that I is a model of S is to say that S is true in
I, and so we have the notion of model-theoretic truth, which is truth in
a particular interpretation.
https://plato.stanford.edu/entries/model-theory/#Basic
G is true in F iff it cannot be shown that G is true in F
Nope, you don't understand what G is. The Definition of G in F
does NOT refer in any way determinable in F to the statement G.
∃n ∈ ℕ (n > 3) // Is this true or false?
How do you know?
Simple, 4 exists (S(S(S(S(0)))), 4 > 3, 4 ∈ ℕ, thus the statement is >>>> True. Like many (but not all) True statements, it can be proven.
Generically how does ascertain that that any logic expression is
true or
false?
Note, "Ascertain" means you are talking about KNOWLEDGE, not Truth.
Truth doesn't need to be ascertained to be true, it just is.
It needs to be ascertained to be KNOWN.
It is a TRUE statement that either all even numbers greater than 2
are the sum of 2 primes or there exists at least one that is not. We
don't know which one of them is true right now, but we do know that
one of them is.
This seems to be one of your core problems, confusing what can be
known to be true with what IS true.
Most generically an analytical expression of formal or natural
language is only true if it has a semantic connection to its truth
maker axioms.
Right, but that connection might not be known, or might even be
infinite.
It is only KNOWLEDGE or PROOF that requires a finite connection.
The "truth maker axioms" of natural language are the definition of the >>>>> meaning of its words.
No, the accepted Truth Maker Axioms of the Theory (not what their
words mean in Natural Language) determine what is true.
of natural language such as English
of natural language such as English
of natural language such as English
of natural language such as English
Which has been proven unsuitable for logic.
Given the statement:
If this sentence is true, Peter Olcott is a moron.
This is a valid logical statement of natural language form.
By the meaning of the words, it is TRUE, because if the sentence IS
true, then by the DEFINITION of True, it must be actually True.
Thus, since it HAS been proven true, its implication must be correct,
and thus YOU ARE A MORON.
The "flaw" in the statement is that Natural Language isn't suitable to
fully express logic.
Montague Grammar made great strides in formalizing natural language semantics.
Your reliance on "Natural Language" is what has actually been proven
to lead to problems.
The entire body of all analytical knowledge can only be expressed using
language. Hardly any of this is currently expressed using formal
language. All knowledge is necessarily true by definition.
Right, TECHNICAL/FORMAL language, not NATURAL language.
You are incorrect that hardly any of this is expressed using formal
language, and that is a major part of your problem. Words that are
words in "Natuaral Language" are frequently refined to a formal
definition for particular usage. If you don't understand that formal
definition, or even more important WHICH formal definition is needed
for a given statement, you won't understand it.
Of the sum total of all of analytical human knowledge far less than 1%
has been formalized.
And All Knowledge being necessarily true is NOT a universal
definition, in fact, one of the problems of the study of knowledge is
how to avoid the introduction into "Knowledge" of things that we THINK
are True but are actually incorrect. We WANT everything that we (think
we) know to be actually true, but factually, since there IS a human
element in the aquisition of knowledge, there is a possibility of
error and of thinking we know something that isn't true.
Once truth has been properly formalized the discerning truth from
falsehood or presumption is a mere computation.
The truth maker axioms for the above expression is the definition
of the ordered set of natural numbers:
https://www.britannica.com/science/Peano-axioms
You understand that Godel showed that under the Peano-axioms, he
proved that their exists truths that can not be proven.
We can make the Gödel number of "I just ate some chicken" using the
adjacent ASCII values. This too cannot be proven in the Peano-axioms.
Nope. You don't understand what Godel does. Not understanding
something does not make it not true, you are just serving your Herring
with Red Sauce.
Yes, you can set up a system where you use Godel's math to create a
number that represents the statement "I just ate some chicken", but
that statement has nothing to do with Godel's proof.
The fact you can throw out Red Herring that means nothing doesn't
discount the proof, it just proves you don't understand what you are
talking about.
It becomes a consequence of the induction axiom that allows him to
be able to define the primative recursive relationship that shows
that you can not prove within the theory that no nmber exists that
matches that theory, and also create an extention to that theory
(that is used to create that relationship) that allows us to
actually prove that statment must be true, and also that no proof of
this can exist in the base theory.
Its a mere gimmick.
He acknowledged that the Liar Paradox forms an equivalent proof.
No, it isn't.
He said that it is. That you reject this because you want to stay in
rebuttal mode is no actual rebuttal.
If you think it is, then SHOW that it is.
He said that it is.
He said that it is.
He said that it is.
He said that it is.
He said that it is.
But to do that, you need to understand what he did and where he "just
used a gimmick"
All your statments show is that you just don't understand what he is
saying and are such a pathological liar that you will make up excuses
to cover that.
When you lack a proper rebuttal you resort to ad Hominem.
This may be quite convincing to gullible fools.
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
Rejects all formal systems as incomplete on the basis that they cannot
prove that the Liar Paradox is true.
This is where he says that he uses the Liar Paradox as his basis: https://www.liarparadox.org/247_248.pdf
This is his whole proof:
https://www.liarparadox.org/Tarski_275_276.pdf
Tarski concluded that truth cannot be formally defined and he did this
on the basis that he could not prove that the Liar Paradox is true
within the formal system where it remains self-contradictory.
He was able to prove that it is true outside of the formal system where
it is self-contradictory.
On 1/23/23 10:46 PM, olcott wrote:
On 1/23/2023 6:38 PM, Richard Damon wrote:
On 1/23/23 6:39 PM, olcott wrote:
On 1/23/2023 4:54 PM, Richard Damon wrote:
On 1/23/23 5:23 PM, olcott wrote:
On 1/23/2023 10:51 AM, Richard Damon wrote:
On 1/23/23 10:18 AM, olcott wrote:
On 1/20/2023 4:54 PM, Richard Damon wrote:
On 1/20/23 5:16 PM, olcott wrote:
On 1/20/2023 4:09 PM, Richard Damon wrote:
On 1/20/23 5:02 PM, olcott wrote:
On 1/20/2023 2:46 PM, Richard Damon wrote:
On 1/20/23 2:31 PM, olcott wrote:Wrongo !!!
On 1/19/2023 8:34 PM, Richard Damon wrote:
On 1/19/23 2:12 PM, olcott wrote:
On 1/17/2023 5:44 PM, Richard Damon wrote:
On 1/17/23 11:39 AM, olcott wrote:
On 1/16/2023 7:51 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>> No, because I am showing that G is TRUE, not PROVABLE. >>>>>>>>>>>>>>>>>> Truth can use infinte sets oc connections, proofs >>>>>>>>>>>>>>>>>> can't. Only YOU have perposed that we think about >>>>>>>>>>>>>>>>>> infinite proofs.
Formal systems cannot ever use infinite connections >>>>>>>>>>>>>>>>> from their
expressions of language to their truth maker axioms >>>>>>>>>>>>>>>>> thus eliminating
these from consideration as any measure of true "in the >>>>>>>>>>>>>>>>> system".
Source? or is this just another of your made up "Facts" >>>>>>>>>>>>>>>>
You can't even remember that you said this?
No, I said they can't have infinite PROOFS, not infinite >>>>>>>>>>>>>> connections to Truth.
WHERE in the definition of a "Formal System" does it say >>>>>>>>>>>>>>>> that the connecti0on must be finite.
You said that formal system cannot have infinite proofs. >>>>>>>>>>>>>>> Did you change your mind?
Right ***PROOF*** not ***TRUTH***
Truth can be based on an infinite chain of connections, >>>>>>>>>>>>>> proofs can not.
Truth *in a formal system* cannot be based on infinite >>>>>>>>>>>>> connections because formal systems are not allowed to have >>>>>>>>>>>>> infinite connections.
Says Who ***FOR TRUTH***
You reference does not provide that data, so I guess you are >>>>>>>>>>>> just making it up, and thus showing you to be a LIAR. >>>>>>>>>>>>
Haskell Curry establishes that truth in a theory (AKA >>>>>>>>>>>>> formal system) is anchored in the elementary theorems (AKA >>>>>>>>>>>>> axioms) of this formal system.
Right, ANCHORED TO, not limited to. Statments other than the >>>>>>>>>>>> elementary theorems are True, and they are true if they have >>>>>>>>>>>> a connection (not limited to finite) to these Truths. >>>>>>>>>>>>
Where does he say True statements must have a FINITE >>>>>>>>>>>> connection to the elementary theorems.
A theory (over (f) is defined as a conceptual class of >>>>>>>>>>>>> these elementary
statements. Let::t be such a theory. Then the elementary >>>>>>>>>>>>> statements
which belong to ::t we shall call the elementary theorems >>>>>>>>>>>>> of::t; we also
say that these elementary statements are true for::t. Thus, >>>>>>>>>>>>> given ::t,
an elementary theorem is an elementary statement which is >>>>>>>>>>>>> true. A theory
is thus a way of picking out from the statements of (f a >>>>>>>>>>>>> certain
subclass of true statements.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Perhaps you believe that you are enormously much brighter >>>>>>>>>>>>> than Haskell Curry ?
You don't understand what he is saying,
He is saying these statements are True in F, as a given. >>>>>>>>>>>
The terminology which has just been used implies that the >>>>>>>>>>> elementary statements are not such that their truth and >>>>>>>>>>> falsity are known to us without reference to::t.
Right, they aren't just true in the Statement class, but are >>>>>>>>>> only considerdd true because we are in the Theory F.
F is not the theory T is the theory.
Red Herring.
F is the Theory in Godels descussion.
https://en.wikipedia.org/wiki/Metamathematics
LP := ~True(LP) is untrue yet that does not make it true.
When we examine this at the meta level we escape the
self-contradiction
and can say that it is true that LP is untrue.
Excpet that untrue is not ~True() in classical logic, which makes >>>>>> statements either True or False, or makes them Not a Truth Bearer, >>>>>> which makes them not in the domain of the True predicate.
You need to move to tri-value logic to do this, at which point you >>>>>> loose the relationship that ~True(x) -> False(x)
True / false and not a truth bearer.
That is your TRI-value logic.
It is true by logical necessity.
Every expression of language must necessarily be
true, false, neither true nor false.
Nope. You can also use a two level division like you actually talk about. >>
Statments are either Truth Bearers or they are Not
Truth Bearers are either True or they are False.
Is this a trick? Did you just agree with me?No. there are two DISTINT but connected binary values.
Is it a Truth Bearer, and if so, what is the Truth Value.
Once it has been made a Truth Bearer, the "untrue/unfalse" state is not available.
Your "Tri-value" system seems to look the fact that some statements are,
by definition, Truth Bearers.
This just shows your lack of understanding.
Note, most of mathematics is based on the two-value logic system. >>>>>>
Thus forcing it to classify "not a truth bearer" incorrectly.
If all you have is a hammer the unscrewing a screw becomes quite
destructive.
Nope, a "statement" can be well formed, and thus MUST be a "Truth
Bearer" or it isn't and is NOT a "Truth Bearer"
By ignoring that mathematically defined statement ARE "Truth
Bearers", you logic system is just broken.
https://plato.stanford.edu/entries/tarski-truth/#195DefOff
It looks like model theory is required to determine the truth of >>>>>>> some mathematical expressions, this had it origins in Tarski's >>>>>>> definition of truth.
∃n ∈ ℕ (N > 3) // does not seem to need model theory >>>>>>> ∃G ∈ F (G ↔ (F ⊬ G)) // does not seem to need model theory >>>>>>>
∃ is a symbol out of model theory, so hard to not need model theory.
Quoting from your reference:
Model theory by contrast works with three levels of symbol. There >>>>>> are the logical constants ( = , ¬ , & for example), the variables >>>>>> (as before), and between these a middle group of symbols which
have no fixed meaning but get a meaning through being applied to a >>>>>> particular structure. The symbols of this middle group include the >>>>>> nonlogical constants of the language, such as relation symbols, >>>>>> function symbols and constant individual symbols. They also
include the quantifier symbols ∀ and ∃, since we need to refer to
the structure to see what set they range over.
I just showed how to explicitly specify what they range over: ∃n ∈ ℕ
Which means you are using "Model Theory"
Maybe you don't understand those words.
Model theory is used to define things that are not otherwise defined. >>> When they are otherwise defined there is no need for model theory.
So, you don't understand what Model Theory is (or mixing different
definitions)
The most succinct definition seems to be saying that model theory merely defines the semantics of elements expressions of language.No, perhaps the issue is that logic doesn't actual deal with "semantics"
the way you want to talk about it. Meaning is NOT based on the "words"
used to describe it, but the descrete properties attached to it.
Items are defined as parts of classes.
Cats are not mammels by the "meaning" of the words, but because the
label "cats" is attached to a class of things, and "Mammels" is attached
to another class of things, and the class of things called cats is
defined to be a subset of the class of things called mammels.
Sometimes we write or speak a sentence S that expresses nothing either true or false, because some crucial information is missing about what
the words mean.
Which makes it not a logical statemen.
If we go on to add this information, so that S comes to express a true
or false statement, we are said to interpret S, and the added
information is called an interpretation of S.
If the interpretation I happens to make S state something true, we say that I is a model of S, or that I satisfies S, in symbols ‘I ⊨ S’.
Another way of saying that I is a model of S is to say that S is true in I, and so we have the notion of model-theoretic truth, which is truth in
a particular interpretation.
https://plato.stanford.edu/entries/model-theory/#BasicMaybe that is the Philosphical view of model theory, but doesn't sound
like what Mathematics calls Model Theory.
I would have to study it more to see how compatible it is with the model theory used by mathematitians.
As I said, one of your problems is you think that you can use Natural Language meaning of words, or don't understand that Technical Meanings
can differ between Fields, and you need to use the rght definition.
G is true in F iff it cannot be shown that G is true in F
Nope, you don't understand what G is. The Definition of G in F
does NOT refer in any way determinable in F to the statement G. >>>>>>
∃n ∈ ℕ (n > 3) // Is this true or false?
How do you know?
Simple, 4 exists (S(S(S(S(0)))), 4 > 3, 4 ∈ ℕ, thus the statement is
True. Like many (but not all) True statements, it can be proven.
Generically how does ascertain that that any logic expression is
true or
false?
Note, "Ascertain" means you are talking about KNOWLEDGE, not Truth. >>>>
Truth doesn't need to be ascertained to be true, it just is.
It needs to be ascertained to be KNOWN.
It is a TRUE statement that either all even numbers greater than 2
are the sum of 2 primes or there exists at least one that is not. We >>>> don't know which one of them is true right now, but we do know that >>>> one of them is.
This seems to be one of your core problems, confusing what can be
known to be true with what IS true.
Most generically an analytical expression of formal or natural
language is only true if it has a semantic connection to its truth >>>>> maker axioms.
Right, but that connection might not be known, or might even be
infinite.
It is only KNOWLEDGE or PROOF that requires a finite connection.
The "truth maker axioms" of natural language are the definition of the >>>>> meaning of its words.
No, the accepted Truth Maker Axioms of the Theory (not what their
words mean in Natural Language) determine what is true.
of natural language such as English
of natural language such as English
of natural language such as English
of natural language such as English
Which has been proven unsuitable for logic.
Given the statement:
If this sentence is true, Peter Olcott is a moron.
This is a valid logical statement of natural language form.
By the meaning of the words, it is TRUE, because if the sentence IS
true, then by the DEFINITION of True, it must be actually True.
Thus, since it HAS been proven true, its implication must be correct,
and thus YOU ARE A MORON.
The "flaw" in the statement is that Natural Language isn't suitable to
fully express logic.
Montague Grammar made great strides in formalizing natural language semantics.Then it must not be "Natural Language" any more, you CAN'T "formalise" a language and keep it Natural, as the definition of Natural Language is
the language evolved naturally without concious planning or premeditation.
Your reliance on "Natural Language" is what has actually been proven >>>> to lead to problems.
The entire body of all analytical knowledge can only be expressed using >>> language. Hardly any of this is currently expressed using formal
language. All knowledge is necessarily true by definition.
Right, TECHNICAL/FORMAL language, not NATURAL language.
You are incorrect that hardly any of this is expressed using formal
language, and that is a major part of your problem. Words that are
words in "Natuaral Language" are frequently refined to a formal
definition for particular usage. If you don't understand that formal
definition, or even more important WHICH formal definition is needed
for a given statement, you won't understand it.
Of the sum total of all of analytical human knowledge far less than 1%
has been formalized.
And All Knowledge being necessarily true is NOT a universal
definition, in fact, one of the problems of the study of knowledge is
how to avoid the introduction into "Knowledge" of things that we THINK
are True but are actually incorrect. We WANT everything that we (think
we) know to be actually true, but factually, since there IS a human
element in the aquisition of knowledge, there is a possibility of
error and of thinking we know something that isn't true.
Once truth has been properly formalized the discerning truth from falsehood or presumption is a mere computation.But it has been proven that you can't formaize truth in a manner that it
can be universally tested.
The truth maker axioms for the above expression is the definition >>>>> of the ordered set of natural numbers:
https://www.britannica.com/science/Peano-axioms
You understand that Godel showed that under the Peano-axioms, he
proved that their exists truths that can not be proven.
We can make the Gödel number of "I just ate some chicken" using the
adjacent ASCII values. This too cannot be proven in the Peano-axioms.
Nope. You don't understand what Godel does. Not understanding
something does not make it not true, you are just serving your Herring
with Red Sauce.
Yes, you can set up a system where you use Godel's math to create a
number that represents the statement "I just ate some chicken", but
that statement has nothing to do with Godel's proof.
The fact you can throw out Red Herring that means nothing doesn't
discount the proof, it just proves you don't understand what you are
talking about.
It becomes a consequence of the induction axiom that allows him to
be able to define the primative recursive relationship that shows
that you can not prove within the theory that no nmber exists that
matches that theory, and also create an extention to that theory
(that is used to create that relationship) that allows us to
actually prove that statment must be true, and also that no proof of >>>> this can exist in the base theory.
Its a mere gimmick.
He acknowledged that the Liar Paradox forms an equivalent proof.
No, it isn't.
He said that it is. That you reject this because you want to stay in rebuttal mode is no actual rebuttal.No, he said he used it to build the proof, and a similar antinomy could
be used as well.
Just like a recipe for a cake might say that a similar cake could be
made with other flowers, that doesn't make the cake the equivalent of flower.
If you think it is, then SHOW that it is.
He said that it is.No, he did NOT say it was equivalent, it was USED it.
He said that it is.
He said that it is.
He said that it is.
He said that it is.
Read what he said again.
But to do that, you need to understand what he did and where he "just
used a gimmick"
All your statments show is that you just don't understand what he is
saying and are such a pathological liar that you will make up excuses
to cover that.
When you lack a proper rebuttal you resort to ad Hominem.
This may be quite convincing to gullible fools.
What proper rebutal?
You claim Godel has said something that means what it doesn't
The conventional definition of incompleteness:No, becaue for that definition φ must be a statment that is a Truth Bearer.
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
This is the problem of combining the two step boolian value into a
single tri-value, you lost the definitoin of the domain of the statement.
THe other version of the Definition is that there exists a statment that
is True which can not be proven (or by negation, a false statement that
can not be refuted)
Rejects all formal systems as incomplete on the basis that they cannot prove that the Liar Paradox is true.Nope, The Liar Paradox is not in the required domain of the definition
This is where he says that he uses the Liar Paradox as his basis: https://www.liarparadox.org/247_248.pdfFirstt, you just switch topics mid-stream, we WERE talking about Godel
and his statment G, and now you are talking about Tarski and his proof
of no definition of Truth.
If you think these are the same thing, you are even dumber that I thought.
Yes, the proofs are similar, but there ARE subtle differences to get to their different conclusions.
Note, the page you are pointing to never says he uses the Liar's Paradox
to actually BUILD the proof, but that it is shown that the existance of
a "Definition of Truth" (which I am not sure you understand what he is talking about) would create the ability to prove that the Liar's Paradox
was a True STatement, which implies a contradiction, and thus the
premise, the existance of the Defintion of Truth, can not be in a system that is stipulated to be consistent.
This is his whole proof:
https://www.liarparadox.org/Tarski_275_276.pdf
Tarski concluded that truth cannot be formally defined and he did this
on the basis that he could not prove that the Liar Paradox is true
within the formal system where it remains self-contradictory.
He was able to prove that it is true outside of the formal system where
it is self-contradictory.
Nope, you don't understand his proof.
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