Incompleteness is a property of a given Formal System, it says that
there exist a statement that is true in that system, but can not be
proven in that system.
Godel's proof build a meta-system that constructs a statement that
exists in the base system, and which is true in that base system, but
can not be proven in the base system, only the meta-system.
On 29/12/2025 13:37, Richard Damon wrote:
Incompleteness is a property of a given Formal System, it says that
there exist a statement that is true in that system, but can not be
proven in that system.
Godel's proof build a meta-system that constructs a statement that
exists in the base system, and which is true in that base system, but
can not be proven in the base system, only the meta-system.
What do you mean by "proven" here. Do you mean "derived" ? Normally we
say a proposition is true in a system when we mean the proposition is a theorem of the system; it has a derivation in the system: a finite
sequence of statements that each is derivable from the axioms and
earlier statements in the sequence by application of a deductive rule of
the system--where the deductive rules are transitive so that's also application of /some/ deductive rules--and which ends with the
proposition that's being derived.
If there is a derivation then it is provable in the base system, if
there /isn't/ a derivation then it is /not/ true in that base system
(which is different from saying its contrapositive is true in that base system).
On 29/12/2025 13:37, Richard Damon wrote:
Incompleteness is a property of a given Formal System, it says that
there exist a statement that is true in that system, but can not be
proven in that system.
What do you mean by "proven" here. Do you mean "derived" ?
In sci.logic Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
On 29/12/2025 13:37, Richard Damon wrote:
Incompleteness is a property of a given Formal System, it says that
there exist a statement that is true in that system, but can not be
proven in that system.
What do you mean by "proven" here. Do you mean "derived" ?
I think Richard misspoke slightly. The undecidable statement is
true *in the intended interpretation* of the formal system
(In Goedel's case, the natural numbers with addition and multiplication).
Truth "in the formal system" isn't really defined. You need an interpretation.
On 29/12/2025 13:37, Richard Damon wrote:
Incompleteness is a property of a given Formal System, it says that
there exist a statement that is true in that system, but can not be
proven in that system.
Godel's proof build a meta-system that constructs a statement that
exists in the base system, and which is true in that base system, but
can not be proven in the base system, only the meta-system.
What do you mean by "proven" here. Do you mean "derived" ? Normally we
say a proposition is true in a system when we mean the proposition is a theorem of the system; it has a derivation in the system: a finite
sequence of statements that each is derivable from the axioms and
earlier statements in the sequence by application of a deductive rule of
the system--where the deductive rules are transitive so that's also application of /some/ deductive rules--and which ends with the
proposition that's being derived.
If there is a derivation then it is provable in the base system, if
there /isn't/ a derivation then it is /not/ true in that base system
(which is different from saying its contrapositive is true in that base system).
In sci.logic Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
On 29/12/2025 13:37, Richard Damon wrote:
Incompleteness is a property of a given Formal System, it says that
there exist a statement that is true in that system, but can not be
proven in that system.
What do you mean by "proven" here. Do you mean "derived" ?
I think Richard misspoke slightly. The undecidable statement is
true *in the intended interpretation* of the formal system
(In Goedel's case, the natural numbers with addition and multiplication).
Truth "in the formal system" isn't really defined. You need an interpretation.
On 12/29/2025 1:21 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
On 29/12/2025 13:37, Richard Damon wrote:
Incompleteness is a property of a given Formal System, it says that
there exist a statement that is true in that system, but can not be
proven in that system.
What do you mean by "proven" here. Do you mean "derived" ?
I think Richard misspoke slightly. The undecidable statement is
true *in the intended interpretation* of the formal system
(In Goedel's case, the natural numbers with addition and multiplication).
Truth "in the formal system" isn't really defined. You need an
interpretation.
Unless (as I have been saying for at least a decade)
the formal language directly encodes all of its
semantics directly in its syntax. The Montague
Grammar of natural language semantics is the best
known example of this.
On 12/29/25 2:32 PM, olcott wrote:
On 12/29/2025 1:21 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
On 29/12/2025 13:37, Richard Damon wrote:
Incompleteness is a property of a given Formal System, it says that
there exist a statement that is true in that system, but can not be
proven in that system.
What do you mean by "proven" here. Do you mean "derived" ?
I think Richard misspoke slightly. The undecidable statement is
true *in the intended interpretation* of the formal system
(In Goedel's case, the natural numbers with addition and
multiplication).
Truth "in the formal system" isn't really defined. You need an
interpretation.
Unless (as I have been saying for at least a decade)
the formal language directly encodes all of its
semantics directly in its syntax. The Montague
Grammar of natural language semantics is the best
known example of this.
But it can't, as any system that defines symbols, can have something
outside it assign additional meaning to those symbols.
There may be SOME meaning within the system, but, with a sufficiently expressive system, additional meaning can be imposed.
On 12/29/25 2:21 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
On 29/12/2025 13:37, Richard Damon wrote:
Incompleteness is a property of a given Formal System, it says that
there exist a statement that is true in that system, but can not be
proven in that system.
What do you mean by "proven" here. Do you mean "derived" ?
I think Richard misspoke slightly. The undecidable statement is
true *in the intended interpretation* of the formal system
(In Goedel's case, the natural numbers with addition and multiplication).
Truth "in the formal system" isn't really defined. You need an
interpretation.
No, statements in a formal system are DEFINED to be true
if that> statement, referencing object defined in the system model,and related
by relationships defined in the system-a can be established starting with^^^^^^^^^^^^^^^^^^
the initial "facts" (axioms) of the system, and following the allowed
logical operations of the system.
On 12/29/25 2:32 PM, olcott wrote:
On 12/29/2025 1:21 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
On 29/12/2025 13:37, Richard Damon wrote:
Incompleteness is a property of a given Formal System, it says that
there exist a statement that is true in that system, but can not be
proven in that system.
What do you mean by "proven" here. Do you mean "derived" ?
I think Richard misspoke slightly. The undecidable statement is
true *in the intended interpretation* of the formal system
(In Goedel's case, the natural numbers with addition and
multiplication).
Truth "in the formal system" isn't really defined. You need an
interpretation.
Unless (as I have been saying for at least a decade)
the formal language directly encodes all of its
semantics directly in its syntax. The Montague
Grammar of natural language semantics is the best
known example of this.
But it can't, as any system that defines symbols, can have something
outside it assign additional meaning to those symbols.
There may be SOME meaning within the system, but, with a sufficiently expressive system, additional meaning can be imposed.
An Montague grammer is out of scope here, as we are talking FORMAL
langauges and system, not Natural Language,
Something which seems beyound your ability to understand, since you brainwashed youself to not understand the basics of this.--
On 12/29/2025 1:53 PM, Richard Damon wrote:
On 12/29/25 2:32 PM, olcott wrote:
On 12/29/2025 1:21 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
On 29/12/2025 13:37, Richard Damon wrote:
Incompleteness is a property of a given Formal System, it says that >>>>>> there exist a statement that is true in that system, but can not be >>>>>> proven in that system.
What do you mean by "proven" here. Do you mean "derived" ?
I think Richard misspoke slightly. The undecidable statement is
true *in the intended interpretation* of the formal system
(In Goedel's case, the natural numbers with addition and
multiplication).
Truth "in the formal system" isn't really defined. You need an
interpretation.
Unless (as I have been saying for at least a decade)
the formal language directly encodes all of its
semantics directly in its syntax. The Montague
Grammar of natural language semantics is the best
known example of this.
But it can't, as any system that defines symbols, can have something
outside it assign additional meaning to those symbols.
"true on the basis of meaning expressed in language"
can be expressed as relations between finite strings.
There may be SOME meaning within the system, but, with a sufficiently
expressive system, additional meaning can be imposed.
An Montague grammer is out of scope here, as we are talking FORMAL
langauges and system, not Natural Language,
"We are therefore confronted with a proposition which
asserts its own unprovability." (G||del 1931:39-41)
By using an enormously convoluted process with
G||del numbers hiding his actual claim:
There exists a sequence of inference steps from
the axioms of a formal system that prove that
they themselves do not exist.
readers are simply conned into believing that
G||del Incompleteness is coherent and true.
Something which seems beyound your ability to understand, since you
brainwashed youself to not understand the basics of this.
On 12/29/25 4:38 PM, olcott wrote:
On 12/29/2025 1:53 PM, Richard Damon wrote:
On 12/29/25 2:32 PM, olcott wrote:
On 12/29/2025 1:21 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
On 29/12/2025 13:37, Richard Damon wrote:
Incompleteness is a property of a given Formal System, it says that >>>>>>> there exist a statement that is true in that system, but can not be >>>>>>> proven in that system.
What do you mean by "proven" here. Do you mean "derived" ?
I think Richard misspoke slightly. The undecidable statement is
true *in the intended interpretation* of the formal system
(In Goedel's case, the natural numbers with addition and
multiplication).
Truth "in the formal system" isn't really defined. You need an
interpretation.
Unless (as I have been saying for at least a decade)
the formal language directly encodes all of its
semantics directly in its syntax. The Montague
Grammar of natural language semantics is the best
known example of this.
But it can't, as any system that defines symbols, can have something
outside it assign additional meaning to those symbols.
"true on the basis of meaning expressed in language"
can be expressed as relations between finite strings.
Try to do that.
There may be SOME meaning within the system, but, with a sufficiently
expressive system, additional meaning can be imposed.
An Montague grammer is out of scope here, as we are talking FORMAL
langauges and system, not Natural Language,
"We are therefore confronted with a proposition which
asserts its own unprovability." (G||del 1931:39-41)
Right, it is a statement in the meta-theory, commenting on it
unprovabiilty in the base theory.
Context seems to elude you, because it requires understand.
By using an enormously convoluted process with
G||del numbers hiding his actual claim:
There exists a sequence of inference steps from
the axioms of a formal system that prove that
they themselves do not exist.
Right, there is an INFININTE string of inference steps in the base
theory that shows that no FINITE string of inference steps to show it.
On 12/29/2025 5:06 PM, Richard Damon wrote:
On 12/29/25 4:38 PM, olcott wrote:
There exists a sequence of inference steps from
the axioms of a formal system that prove that
they themselves do not exist.
Right, there is an INFININTE string of inference steps in the base
theory that shows that no FINITE string of inference steps to show it.
Rene Descartes said: "I think therefore I never existed".
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
On 12/29/25 6:28 PM, olcott wrote:
On 12/29/2025 5:06 PM, Richard Damon wrote:
On 12/29/25 4:38 PM, olcott wrote:
There exists a sequence of inference steps from
the axioms of a formal system that prove that
they themselves do not exist.
Right, there is an INFININTE string of inference steps in the base
theory that shows that no FINITE string of inference steps to show it.
Rene Descartes said: "I think therefore I never existed".
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
In other words, you are just showing that you don't know what you are talking about and thus going into non-sense,
As I said, and you were too stupid to understand, there is a finite
sequence of steps in the META systen that show that there is an INFINITE sequence of steps in the system that show there is not a FINITE sequence
of steps in the system to prove it.
It seems to you, infinity is finite, and thus your mind is just ZERO.
Of course, you never let facts get in the way of your stupidity.
On 12/29/2025 9:51 PM, Richard Damon wrote:
On 12/29/25 6:28 PM, olcott wrote:
On 12/29/2025 5:06 PM, Richard Damon wrote:
On 12/29/25 4:38 PM, olcott wrote:
There exists a sequence of inference steps from
the axioms of a formal system that prove that
they themselves do not exist.
Right, there is an INFININTE string of inference steps in the base
theory that shows that no FINITE string of inference steps to show it. >>>>
Rene Descartes said: "I think therefore I never existed".
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
In other words, you are just showing that you don't know what you are
talking about and thus going into non-sense,
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41)
Correctly paraphrased as:
a sequence of inference steps from axioms
that assert that they themselves do not exist.
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
As I said, and you were too stupid to understand, there is a finite
sequence of steps in the META systen that show that there is an
INFINITE sequence of steps in the system that show there is not a
FINITE sequence of steps in the system to prove it.
It seems to you, infinity is finite, and thus your mind is just ZERO.
Of course, you never let facts get in the way of your stupidity.
On 12/29/25 11:35 PM, olcott wrote:
On 12/29/2025 9:51 PM, Richard Damon wrote:
On 12/29/25 6:28 PM, olcott wrote:
On 12/29/2025 5:06 PM, Richard Damon wrote:
On 12/29/25 4:38 PM, olcott wrote:
There exists a sequence of inference steps from
the axioms of a formal system that prove that
they themselves do not exist.
Right, there is an INFININTE string of inference steps in the base
theory that shows that no FINITE string of inference steps to show it. >>>>>
Rene Descartes said: "I think therefore I never existed".
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
In other words, you are just showing that you don't know what you are
talking about and thus going into non-sense,
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41)
Yes, you have said this before, and I have explained it, but apparently
you can't read.
Correctly paraphrased as:
a sequence of inference steps from axioms
that assert that they themselves do not exist.
Nope, as I have pointed out, you have missed the context, because you
are so stupid.
The statement, when looked at under the meaning that only exists in the meta-system, shows that in the meta-system there is a proof, a finite
series of steps, that shows that in the system, the statement in the
system does not have a proof, which is a finite series of steps IN THE SYSTEM (not the meta-system) but there is a infinite series of steps in
the system that make it true.
Thus, you show you can't tell the difference between an infinite series
of steps from a finitee series of step, thus you IQ must be 0 by that
scale.
And, you can't tell the difference between the Meta-system and the
system, which is like thinking your pet cat is a dog.
The fact you keep on repeating this, and never try to answer the error pointed out just means that you can't understand what an error is,
because to you truth, knowledge, fact, rules, don't mean anything
because you chose to make your self just stupid and ignorant.
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
As I said, and you were too stupid to understand, there is a finite
sequence of steps in the META systen that show that there is an
INFINITE sequence of steps in the system that show there is not a
FINITE sequence of steps in the system to prove it.
It seems to you, infinity is finite, and thus your mind is just ZERO.
Of course, you never let facts get in the way of your stupidity.
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41)
Correctly paraphrased as:
a sequence of inference steps from axioms
that assert that they themselves do not exist.
On 30/12/2025 04:35, olcott wrote:
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41)
Correctly paraphrased as:
a sequence of inference steps from axioms
that assert that they themselves do not exist.
No they don't. That's an interpretation outside the system. The axioms
merely force you to conclude that some symbol or other is not negation
and/or another one is not a reference to the system itself when fools
think they both /are/ those things.
On 12/29/2025 10:50 PM, Richard Damon wrote:
On 12/29/25 11:35 PM, olcott wrote:
On 12/29/2025 9:51 PM, Richard Damon wrote:
On 12/29/25 6:28 PM, olcott wrote:
On 12/29/2025 5:06 PM, Richard Damon wrote:
On 12/29/25 4:38 PM, olcott wrote:
There exists a sequence of inference steps from
the axioms of a formal system that prove that
they themselves do not exist.
Right, there is an INFININTE string of inference steps in the base >>>>>> theory that shows that no FINITE string of inference steps to show >>>>>> it.
Rene Descartes said: "I think therefore I never existed".
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
In other words, you are just showing that you don't know what you
are talking about and thus going into non-sense,
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41)
Yes, you have said this before, and I have explained it, but
apparently you can't read.
Correctly paraphrased as:
a sequence of inference steps from axioms
that assert that they themselves do not exist.
Nope, as I have pointed out, you have missed the context, because you
are so stupid.
a proposition which asserts its own unprovability.
The proof of such an propostion within the same
formal system would require a sequence of inference
steps that prove that they themselves do not exist.
The statement, when looked at under the meaning that only exists in
the meta-system, shows that in the meta-system there is a proof, a
finite series of steps, that shows that in the system, the statement
in the system does not have a proof, which is a finite series of steps
IN THE SYSTEM (not the meta-system) but there is a infinite series of
steps in the system that make it true.
Thus, you show you can't tell the difference between an infinite
series of steps from a finitee series of step, thus you IQ must be 0
by that scale.
And, you can't tell the difference between the Meta-system and the
system, which is like thinking your pet cat is a dog.
The fact you keep on repeating this, and never try to answer the error
pointed out just means that you can't understand what an error is,
because to you truth, knowledge, fact, rules, don't mean anything
because you chose to make your self just stupid and ignorant.
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
As I said, and you were too stupid to understand, there is a finite
sequence of steps in the META systen that show that there is an
INFINITE sequence of steps in the system that show there is not a
FINITE sequence of steps in the system to prove it.
It seems to you, infinity is finite, and thus your mind is just ZERO.
Of course, you never let facts get in the way of your stupidity.
On 12/29/2025 11:49 PM, Tristan Wibberley wrote:
On 30/12/2025 04:35, olcott wrote:
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41)
Correctly paraphrased as:
a sequence of inference steps from axioms
that assert that they themselves do not exist.
No they don't. That's an interpretation outside the system. The axioms
merely force you to conclude that some symbol or other is not negation
and/or another one is not a reference to the system itself when fools
think they both /are/ those things.
G := (F re4 G)
a sequence of inference steps in F from the axioms
of F that assert that they themselves do not exist in F.
On 12/30/25 12:33 AM, olcott wrote:
On 12/29/2025 10:50 PM, Richard Damon wrote:
On 12/29/25 11:35 PM, olcott wrote:
On 12/29/2025 9:51 PM, Richard Damon wrote:
On 12/29/25 6:28 PM, olcott wrote:
On 12/29/2025 5:06 PM, Richard Damon wrote:
On 12/29/25 4:38 PM, olcott wrote:
There exists a sequence of inference steps from
the axioms of a formal system that prove that
they themselves do not exist.
Right, there is an INFININTE string of inference steps in the
base theory that shows that no FINITE string of inference steps >>>>>>> to show it.
Rene Descartes said: "I think therefore I never existed".
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
In other words, you are just showing that you don't know what you
are talking about and thus going into non-sense,
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41)
Yes, you have said this before, and I have explained it, but
apparently you can't read.
Correctly paraphrased as:
a sequence of inference steps from axioms
that assert that they themselves do not exist.
Nope, as I have pointed out, you have missed the context, because you
are so stupid.
a proposition which asserts its own unprovability.
a proposition who has a meaning in the meta-system talking about its provability in the base system.
You just ignore context as that is just to complicated for you.
The proof of such an propostion within the same
formal system would require a sequence of inference
steps that prove that they themselves do not exist.
Which just shows you don't understand the concept of Formal Systems, and their meta-systems.
The proof was NOT in the same system, but in a meta-system built from
that system.
It shows, via a finite proof in the meta-system, that there does exist a sequence of infinite length in the system to show the statement is true,
but their can not be a finite length sequence in the system.
All you are doing is proving you are to stupid to understand this, as
you don't understand that two different systems ARE different systems,
but meta-system can know details of their base system, and that there is
a difference between infinite and finite. THis shows your intelegence to
be near zero.
The statement, when looked at under the meaning that only exists in
the meta-system, shows that in the meta-system there is a proof, a
finite series of steps, that shows that in the system, the statement
in the system does not have a proof, which is a finite series of
steps IN THE SYSTEM (not the meta-system) but there is a infinite
series of steps in the system that make it true.
Thus, you show you can't tell the difference between an infinite
series of steps from a finitee series of step, thus you IQ must be 0
by that scale.
And, you can't tell the difference between the Meta-system and the
system, which is like thinking your pet cat is a dog.
The fact you keep on repeating this, and never try to answer the
error pointed out just means that you can't understand what an error
is, because to you truth, knowledge, fact, rules, don't mean anything
because you chose to make your self just stupid and ignorant.
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
As I said, and you were too stupid to understand, there is a finite >>>>> sequence of steps in the META systen that show that there is an
INFINITE sequence of steps in the system that show there is not a
FINITE sequence of steps in the system to prove it.
It seems to you, infinity is finite, and thus your mind is just ZERO. >>>>>
Of course, you never let facts get in the way of your stupidity.
On 12/30/25 9:32 AM, olcott wrote:
On 12/29/2025 11:49 PM, Tristan Wibberley wrote:
On 30/12/2025 04:35, olcott wrote:
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41)
Correctly paraphrased as:
a sequence of inference steps from axioms
that assert that they themselves do not exist.
No they don't. That's an interpretation outside the system. The axioms
merely force you to conclude that some symbol or other is not negation
and/or another one is not a reference to the system itself when fools
think they both /are/ those things.
G := (F re4 G)
That isn't the statement of G, so you start with a lie.
a sequence of inference steps in F from the axioms
of F that assert that they themselves do not exist in F.
But that statement you are trying to start with isn't a statement in F,
but an interpretation of the statement in F as understood in MF.
All you are doing is showing you stupidity of not understanding context.
And thus you show you can't understand meaning, as meaning is based on context.
On 12/30/2025 8:32 AM, Richard Damon wrote:
On 12/30/25 12:33 AM, olcott wrote:
On 12/29/2025 10:50 PM, Richard Damon wrote:
On 12/29/25 11:35 PM, olcott wrote:
On 12/29/2025 9:51 PM, Richard Damon wrote:
On 12/29/25 6:28 PM, olcott wrote:
On 12/29/2025 5:06 PM, Richard Damon wrote:
On 12/29/25 4:38 PM, olcott wrote:
There exists a sequence of inference steps from
the axioms of a formal system that prove that
they themselves do not exist.
Right, there is an INFININTE string of inference steps in the >>>>>>>> base theory that shows that no FINITE string of inference steps >>>>>>>> to show it.
Rene Descartes said: "I think therefore I never existed".
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
In other words, you are just showing that you don't know what you >>>>>> are talking about and thus going into non-sense,
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41)
Yes, you have said this before, and I have explained it, but
apparently you can't read.
Correctly paraphrased as:
a sequence of inference steps from axioms
that assert that they themselves do not exist.
Nope, as I have pointed out, you have missed the context, because
you are so stupid.
a proposition which asserts its own unprovability.
a proposition who has a meaning in the meta-system talking about its
provability in the base system.
This sentence is not true: "This sentence is not true"
the outer sentence is true because the inner sentence
is semantically incoherent.
You just ignore context as that is just to complicated for you.
I focus on the details that everyone else has been
indoctrinated to ignore.
The proof of such an propostion within the same
formal system would require a sequence of inference
steps that prove that they themselves do not exist.
Which just shows you don't understand the concept of Formal Systems,
and their meta-systems.
This sentence is not true: "This sentence is not true"
the outer sentence is true because the inner sentence
is semantically incoherent.
Sentences that are semantically incoherent are not true.
This is ignored because a meta level version of the same
sentence can be made true on the basis of this incoherence.
G := (F re4 G)
a sequence of inference steps in F from the axioms
of F that assert that they themselves do not exist in F.
The proof was NOT in the same system, but in a meta-system built from
that system.
To hide the fact of the incoherence as was shown above.
It shows, via a finite proof in the meta-system, that there does exist
a sequence of infinite length in the system to show the statement is
true, but their can not be a finite length sequence in the system.
All you are doing is proving you are to stupid to understand this, as
The actual stupidity is how mathematicians believe that
the foundations of math are inherently infallible as if
they themselves are the actual mind of God.
you don't understand that two different systems ARE different systems,
but meta-system can know details of their base system, and that there
is a difference between infinite and finite. THis shows your
intelegence to be near zero.
The statement, when looked at under the meaning that only exists in
the meta-system, shows that in the meta-system there is a proof, a
finite series of steps, that shows that in the system, the statement
in the system does not have a proof, which is a finite series of
steps IN THE SYSTEM (not the meta-system) but there is a infinite
series of steps in the system that make it true.
Thus, you show you can't tell the difference between an infinite
series of steps from a finitee series of step, thus you IQ must be 0
by that scale.
And, you can't tell the difference between the Meta-system and the
system, which is like thinking your pet cat is a dog.
The fact you keep on repeating this, and never try to answer the
error pointed out just means that you can't understand what an error
is, because to you truth, knowledge, fact, rules, don't mean
anything because you chose to make your self just stupid and ignorant. >>>>
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
As I said, and you were too stupid to understand, there is a
finite sequence of steps in the META systen that show that there
is an INFINITE sequence of steps in the system that show there is >>>>>> not a FINITE sequence of steps in the system to prove it.
It seems to you, infinity is finite, and thus your mind is just ZERO. >>>>>>
Of course, you never let facts get in the way of your stupidity.
On 12/30/25 9:52 AM, olcott wrote:
On 12/30/2025 8:32 AM, Richard Damon wrote:
On 12/30/25 12:33 AM, olcott wrote:
On 12/29/2025 10:50 PM, Richard Damon wrote:
On 12/29/25 11:35 PM, olcott wrote:
On 12/29/2025 9:51 PM, Richard Damon wrote:
On 12/29/25 6:28 PM, olcott wrote:
On 12/29/2025 5:06 PM, Richard Damon wrote:
On 12/29/25 4:38 PM, olcott wrote:
There exists a sequence of inference steps from
the axioms of a formal system that prove that
they themselves do not exist.
Right, there is an INFININTE string of inference steps in the >>>>>>>>> base theory that shows that no FINITE string of inference steps >>>>>>>>> to show it.
Rene Descartes said: "I think therefore I never existed".
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
In other words, you are just showing that you don't know what you >>>>>>> are talking about and thus going into non-sense,
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41)
Yes, you have said this before, and I have explained it, but
apparently you can't read.
Correctly paraphrased as:
a sequence of inference steps from axioms
that assert that they themselves do not exist.
Nope, as I have pointed out, you have missed the context, because
you are so stupid.
a proposition which asserts its own unprovability.
a proposition who has a meaning in the meta-system talking about its
provability in the base system.
This sentence is not true: "This sentence is not true"
the outer sentence is true because the inner sentence
is semantically incoherent.
You just ignore context as that is just to complicated for you.
I focus on the details that everyone else has been
indoctrinated to ignore.
The proof of such an propostion within the same
formal system would require a sequence of inference
steps that prove that they themselves do not exist.
Which just shows you don't understand the concept of Formal Systems,
and their meta-systems.
This sentence is not true: "This sentence is not true"
the outer sentence is true because the inner sentence
is semantically incoherent.
In other words, you can't talk about the sentence you want to talk
about, so you do to soething irrelevent.
On 12/30/2025 9:14 AM, Richard Damon wrote:
On 12/30/25 9:52 AM, olcott wrote:
On 12/30/2025 8:32 AM, Richard Damon wrote:
On 12/30/25 12:33 AM, olcott wrote:
On 12/29/2025 10:50 PM, Richard Damon wrote:
On 12/29/25 11:35 PM, olcott wrote:
On 12/29/2025 9:51 PM, Richard Damon wrote:
On 12/29/25 6:28 PM, olcott wrote:
On 12/29/2025 5:06 PM, Richard Damon wrote:
On 12/29/25 4:38 PM, olcott wrote:
There exists a sequence of inference steps from
the axioms of a formal system that prove that
they themselves do not exist.
Right, there is an INFININTE string of inference steps in the >>>>>>>>>> base theory that shows that no FINITE string of inference >>>>>>>>>> steps to show it.
Rene Descartes said: "I think therefore I never existed".
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
In other words, you are just showing that you don't know what >>>>>>>> you are talking about and thus going into non-sense,
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41)
Yes, you have said this before, and I have explained it, but
apparently you can't read.
Correctly paraphrased as:
a sequence of inference steps from axioms
that assert that they themselves do not exist.
Nope, as I have pointed out, you have missed the context, because >>>>>> you are so stupid.
a proposition which asserts its own unprovability.
a proposition who has a meaning in the meta-system talking about its
provability in the base system.
This sentence is not true: "This sentence is not true"
the outer sentence is true because the inner sentence
is semantically incoherent.
You just ignore context as that is just to complicated for you.
I focus on the details that everyone else has been
indoctrinated to ignore.
The proof of such an propostion within the same
formal system would require a sequence of inference
steps that prove that they themselves do not exist.
Which just shows you don't understand the concept of Formal Systems,
and their meta-systems.
This sentence is not true: "This sentence is not true"
the outer sentence is true because the inner sentence
is semantically incoherent.
In other words, you can't talk about the sentence you want to talk
about, so you do to soething irrelevent.
Exactly the opposite Incompleteness and Undefinability
dishonestly dodge the fact the their actual sentences
are incoherent by using the meta-level.
This meta-level is correct to state that these sentences
are not provable and not true.
The meta-level never looks at why they are unprovable
and untrue. They are unprovable and untrue BECAUSE they
are semantically incoherent.
The proper treatment is to toss these sentences out as
incoherent. The proper treatment is not to create a
meta-level that simply ignores this incoherence.
Tarski's metatheory-a-a-a-a-a-a-a Tarski's theory
This sentence is not true: "This sentence is not true" is true
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
In meta-F-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a In F
This sentence cannot be proven: "This sentence cannot be proven" is true
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
On 12/30/25 11:15 AM, olcott wrote:
On 12/30/2025 9:14 AM, Richard Damon wrote:
On 12/30/25 9:52 AM, olcott wrote:
On 12/30/2025 8:32 AM, Richard Damon wrote:
On 12/30/25 12:33 AM, olcott wrote:
On 12/29/2025 10:50 PM, Richard Damon wrote:
On 12/29/25 11:35 PM, olcott wrote:
On 12/29/2025 9:51 PM, Richard Damon wrote:Yes, you have said this before, and I have explained it, but
On 12/29/25 6:28 PM, olcott wrote:
On 12/29/2025 5:06 PM, Richard Damon wrote:
On 12/29/25 4:38 PM, olcott wrote:
There exists a sequence of inference steps from
the axioms of a formal system that prove that
they themselves do not exist.
Right, there is an INFININTE string of inference steps in the >>>>>>>>>>> base theory that shows that no FINITE string of inference >>>>>>>>>>> steps to show it.
Rene Descartes said: "I think therefore I never existed".
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
In other words, you are just showing that you don't know what >>>>>>>>> you are talking about and thus going into non-sense,
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41) >>>>>>>
apparently you can't read.
Correctly paraphrased as:
a sequence of inference steps from axioms
that assert that they themselves do not exist.
Nope, as I have pointed out, you have missed the context, because >>>>>>> you are so stupid.
a proposition which asserts its own unprovability.
a proposition who has a meaning in the meta-system talking about
its provability in the base system.
This sentence is not true: "This sentence is not true"
the outer sentence is true because the inner sentence
is semantically incoherent.
You just ignore context as that is just to complicated for you.
I focus on the details that everyone else has been
indoctrinated to ignore.
The proof of such an propostion within the same
formal system would require a sequence of inference
steps that prove that they themselves do not exist.
Which just shows you don't understand the concept of Formal
Systems, and their meta-systems.
This sentence is not true: "This sentence is not true"
the outer sentence is true because the inner sentence
is semantically incoherent.
In other words, you can't talk about the sentence you want to talk
about, so you do to soething irrelevent.
Exactly the opposite Incompleteness and Undefinability
dishonestly dodge the fact the their actual sentences
are incoherent by using the meta-level.
And what is incoherent about using a meta-level.
All a mete-level is, is to build a new Formal System, based on the base system that knows the basic properties of the base system.
For instance, the Rational Numbers can be considers a "meta" of the Integeres.
This meta-level is correct to state that these sentences
are not provable and not true.
The meta-level never looks at why they are unprovable
and untrue. They are unprovable and untrue BECAUSE they
are semantically incoherent.
No, the sentence of G was specifically constructed to have a coherent meaning in the base system, but you just are too stupid to understand that.
THe statment G, in the base system, as well as in the meta system is the claim that there exists no natural number g that satisifies a particular mathematical property expresses as a primative recursive relationship.
The mathematics of that is fully coherent in the base system, and WILL
have an answer of either yes or no, even if that system might not be
able to compute that answer.
In the meta-system, because of how the relationship was created, we see
that in adds meaning from the base system into numbers that inherently
only mean themselves. Just like we can form words with meaning from
letters that have no inherent meaning.
It seems you don't even understand how "meaning" works, so your core is based on a fundamental misunderstanding of what you talk about.
The proper treatment is to toss these sentences out as
incoherent. The proper treatment is not to create a
meta-level that simply ignores this incoherence.
But they aren't.
I guess to you, mathematics is just incoherent, and logic has to be kept primative.
In other words, you are just too stupid to be in the field.
Tarski's metatheory-a-a-a-a-a-a-a Tarski's theory
This sentence is not true: "This sentence is not true" is true
You just don't understand what Tarski is saying, as his proof build on
the concept that Godel uses, and Tarski shows that if we assume the existance of a predicate "True" that will return True if its input
sentance is actually True, but False otherwise (either the contradiction
of the sentence is true, so it is false, or the sentence doesn't have a truth value) then by the same "math" Godel uses, we can prove that the
liar must have a truth value.
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
Right, so by the proof, "True" as a predicate can't exist.
In meta-F-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a In F
This sentence cannot be proven: "This sentence cannot be proven" is true
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
And all you are doing is proving your ignornce of how logic works, since none of the system you are talking about can be modeled by Prolog.
Of course, YOU can't handle systems that can't be handled by Prolog as
you are just too stupid.
I will note again, the fact that you just refuse to even try to address--
any of the points, but just keep repeating your wrong opinion that it
can't be right shows that inside, you understand you have no grounds for your claims, and accept that you argument is baseless, but you still
just repeat it.
If you wanted to try to actually show an error in what I say, you would actually address my words and try to show an error, but that would
require you showing an understand of the field that you just don't have,
and would force you to reveal that you really have nothing to base your claims on.
I note that everything you say is based on your own (ignorant)
understanding of how logic works and you can't actually get to the meet
of any source to back you up.
At best, you look at minor offhand high level explainations that you mis-interprete.
On 12/30/2025 8:38 AM, Richard Damon wrote:
On 12/30/25 9:32 AM, olcott wrote:
On 12/29/2025 11:49 PM, Tristan Wibberley wrote:
On 30/12/2025 04:35, olcott wrote:
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41)
Correctly paraphrased as:
a sequence of inference steps from axioms
that assert that they themselves do not exist.
No they don't. That's an interpretation outside the system. The axioms >>>> merely force you to conclude that some symbol or other is not negation >>>> and/or another one is not a reference to the system itself when fools
think they both /are/ those things.
G := (F re4 G)
That isn't the statement of G, so you start with a lie.
a sequence of inference steps in F from the axioms
of F that assert that they themselves do not exist in F.
(F re4 G)
"re4" means that a sequence of inference steps from
F to G do not exist.
But that statement you are trying to start with isn't a statement in F,
Since is begins with F it is in F.
That people do not usually look at this degree
of detail do not mean that I am incorrect.
but an interpretation of the statement in F as understood in MF.
All you are doing is showing you stupidity of not understanding context.
All the I am doing is looking at these things at
the deeper level beyond indoctrination. I am directly
examining the foundations of logic and math.
Everyone else takes these as "given" as if from
the mind of God.
And thus you show you can't understand meaning, as meaning is based on
context.
I understand meaning better then anyone else.
"true on the basis of meaning expressed in language"
for this entire body is one giant semantic tautology.
On 12/30/25 10:10 AM, olcott wrote:
On 12/30/2025 8:38 AM, Richard Damon wrote:
On 12/30/25 9:32 AM, olcott wrote:
On 12/29/2025 11:49 PM, Tristan Wibberley wrote:
On 30/12/2025 04:35, olcott wrote:
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41)
Correctly paraphrased as:
a sequence of inference steps from axioms
that assert that they themselves do not exist.
No they don't. That's an interpretation outside the system. The axioms >>>>> merely force you to conclude that some symbol or other is not negation >>>>> and/or another one is not a reference to the system itself when fools >>>>> think they both /are/ those things.
G := (F re4 G)
That isn't the statement of G, so you start with a lie.
a sequence of inference steps in F from the axioms
of F that assert that they themselves do not exist in F.
(F re4 G)
"re4" means that a sequence of inference steps from
F to G do not exist.
Right, and there is, it is just an infinite sequence of steps.
On 12/30/2025 12:57 PM, Richard Damon wrote:
On 12/30/25 11:15 AM, olcott wrote:
On 12/30/2025 9:14 AM, Richard Damon wrote:
On 12/30/25 9:52 AM, olcott wrote:
On 12/30/2025 8:32 AM, Richard Damon wrote:
On 12/30/25 12:33 AM, olcott wrote:
On 12/29/2025 10:50 PM, Richard Damon wrote:
On 12/29/25 11:35 PM, olcott wrote:
On 12/29/2025 9:51 PM, Richard Damon wrote:Yes, you have said this before, and I have explained it, but
On 12/29/25 6:28 PM, olcott wrote:
On 12/29/2025 5:06 PM, Richard Damon wrote:
On 12/29/25 4:38 PM, olcott wrote:
There exists a sequence of inference steps from
the axioms of a formal system that prove that
they themselves do not exist.
Right, there is an INFININTE string of inference steps in >>>>>>>>>>>> the base theory that shows that no FINITE string of
inference steps to show it.
Rene Descartes said: "I think therefore I never existed". >>>>>>>>>>>
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
In other words, you are just showing that you don't know what >>>>>>>>>> you are talking about and thus going into non-sense,
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41) >>>>>>>>
apparently you can't read.
Correctly paraphrased as:
a sequence of inference steps from axioms
that assert that they themselves do not exist.
Nope, as I have pointed out, you have missed the context,
because you are so stupid.
a proposition which asserts its own unprovability.
a proposition who has a meaning in the meta-system talking about
its provability in the base system.
This sentence is not true: "This sentence is not true"
the outer sentence is true because the inner sentence
is semantically incoherent.
You just ignore context as that is just to complicated for you.
I focus on the details that everyone else has been
indoctrinated to ignore.
The proof of such an propostion within the same
formal system would require a sequence of inference
steps that prove that they themselves do not exist.
Which just shows you don't understand the concept of Formal
Systems, and their meta-systems.
This sentence is not true: "This sentence is not true"
the outer sentence is true because the inner sentence
is semantically incoherent.
In other words, you can't talk about the sentence you want to talk
about, so you do to soething irrelevent.
Exactly the opposite Incompleteness and Undefinability
dishonestly dodge the fact the their actual sentences
are incoherent by using the meta-level.
And what is incoherent about using a meta-level.
All a mete-level is, is to build a new Formal System, based on the
base system that knows the basic properties of the base system.
For instance, the Rational Numbers can be considers a "meta" of the
Integeres.
This meta-level is correct to state that these sentences
are not provable and not true.
The meta-level never looks at why they are unprovable
and untrue. They are unprovable and untrue BECAUSE they
are semantically incoherent.
No, the sentence of G was specifically constructed to have a coherent
meaning in the base system, but you just are too stupid to understand
that.
Why do you lie about this? Does lying give you cheap thrill?
...We are therefore confronted with a proposition which asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G||del, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And
Related Systems
On 12/30/25 2:01 PM, olcott wrote:
On 12/30/2025 12:57 PM, Richard Damon wrote:
On 12/30/25 11:15 AM, olcott wrote:
On 12/30/2025 9:14 AM, Richard Damon wrote:
On 12/30/25 9:52 AM, olcott wrote:
On 12/30/2025 8:32 AM, Richard Damon wrote:
On 12/30/25 12:33 AM, olcott wrote:
On 12/29/2025 10:50 PM, Richard Damon wrote:
On 12/29/25 11:35 PM, olcott wrote:
On 12/29/2025 9:51 PM, Richard Damon wrote:Yes, you have said this before, and I have explained it, but >>>>>>>>> apparently you can't read.
On 12/29/25 6:28 PM, olcott wrote:
On 12/29/2025 5:06 PM, Richard Damon wrote:
On 12/29/25 4:38 PM, olcott wrote:
There exists a sequence of inference steps from
the axioms of a formal system that prove that
they themselves do not exist.
Right, there is an INFININTE string of inference steps in >>>>>>>>>>>>> the base theory that shows that no FINITE string of >>>>>>>>>>>>> inference steps to show it.
Rene Descartes said: "I think therefore I never existed". >>>>>>>>>>>>
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
In other words, you are just showing that you don't know what >>>>>>>>>>> you are talking about and thus going into non-sense,
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41) >>>>>>>>>
Correctly paraphrased as:
a sequence of inference steps from axioms
that assert that they themselves do not exist.
Nope, as I have pointed out, you have missed the context,
because you are so stupid.
a proposition which asserts its own unprovability.
a proposition who has a meaning in the meta-system talking about >>>>>>> its provability in the base system.
This sentence is not true: "This sentence is not true"
the outer sentence is true because the inner sentence
is semantically incoherent.
You just ignore context as that is just to complicated for you.
I focus on the details that everyone else has been
indoctrinated to ignore.
The proof of such an propostion within the same
formal system would require a sequence of inference
steps that prove that they themselves do not exist.
Which just shows you don't understand the concept of Formal
Systems, and their meta-systems.
This sentence is not true: "This sentence is not true"
the outer sentence is true because the inner sentence
is semantically incoherent.
In other words, you can't talk about the sentence you want to talk
about, so you do to soething irrelevent.
Exactly the opposite Incompleteness and Undefinability
dishonestly dodge the fact the their actual sentences
are incoherent by using the meta-level.
And what is incoherent about using a meta-level.
All a mete-level is, is to build a new Formal System, based on the
base system that knows the basic properties of the base system.
For instance, the Rational Numbers can be considers a "meta" of the
Integeres.
This meta-level is correct to state that these sentences
are not provable and not true.
The meta-level never looks at why they are unprovable
and untrue. They are unprovable and untrue BECAUSE they
are semantically incoherent.
No, the sentence of G was specifically constructed to have a coherent
meaning in the base system, but you just are too stupid to understand
that.
Why do you lie about this? Does lying give you cheap thrill?
...We are therefore confronted with a proposition which asserts its
own unprovability. 15 rCa (G||del 1931:40-41)
And where does this say that is what the sentence is in the base system?
On 12/30/2025 1:04 PM, Richard Damon wrote:
On 12/30/25 10:10 AM, olcott wrote:
On 12/30/2025 8:38 AM, Richard Damon wrote:
On 12/30/25 9:32 AM, olcott wrote:
On 12/29/2025 11:49 PM, Tristan Wibberley wrote:
On 30/12/2025 04:35, olcott wrote:
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41)
Correctly paraphrased as:
a sequence of inference steps from axioms
that assert that they themselves do not exist.
No they don't. That's an interpretation outside the system. The
axioms
merely force you to conclude that some symbol or other is not
negation
and/or another one is not a reference to the system itself when fools >>>>>> think they both /are/ those things.
G := (F re4 G)
That isn't the statement of G, so you start with a lie.
a sequence of inference steps in F from the axioms
of F that assert that they themselves do not exist in F.
(F re4 G)
"re4" means that a sequence of inference steps from
F to G do not exist.
Right, and there is, it is just an infinite sequence of steps.
You are stupidly saying that something that does not exist
at all infinitely exists.
On 12/30/2025 1:10 PM, Richard Damon wrote:
On 12/30/25 2:01 PM, olcott wrote:
On 12/30/2025 12:57 PM, Richard Damon wrote:
On 12/30/25 11:15 AM, olcott wrote:
On 12/30/2025 9:14 AM, Richard Damon wrote:
On 12/30/25 9:52 AM, olcott wrote:
On 12/30/2025 8:32 AM, Richard Damon wrote:
On 12/30/25 12:33 AM, olcott wrote:
On 12/29/2025 10:50 PM, Richard Damon wrote:
On 12/29/25 11:35 PM, olcott wrote:
On 12/29/2025 9:51 PM, Richard Damon wrote:Yes, you have said this before, and I have explained it, but >>>>>>>>>> apparently you can't read.
On 12/29/25 6:28 PM, olcott wrote:
On 12/29/2025 5:06 PM, Richard Damon wrote:
On 12/29/25 4:38 PM, olcott wrote:
There exists a sequence of inference steps from
the axioms of a formal system that prove that
they themselves do not exist.
Right, there is an INFININTE string of inference steps in >>>>>>>>>>>>>> the base theory that shows that no FINITE string of >>>>>>>>>>>>>> inference steps to show it.
Rene Descartes said: "I think therefore I never existed". >>>>>>>>>>>>>
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
There is no sequence of inference steps that
prove they themselves do not exist.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
That is all that G||del ever proved.
In other words, you are just showing that you don't know >>>>>>>>>>>> what you are talking about and thus going into non-sense, >>>>>>>>>>>>
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41) >>>>>>>>>>
Correctly paraphrased as:
a sequence of inference steps from axioms
that assert that they themselves do not exist.
Nope, as I have pointed out, you have missed the context, >>>>>>>>>> because you are so stupid.
a proposition which asserts its own unprovability.
a proposition who has a meaning in the meta-system talking about >>>>>>>> its provability in the base system.
This sentence is not true: "This sentence is not true"
the outer sentence is true because the inner sentence
is semantically incoherent.
You just ignore context as that is just to complicated for you. >>>>>>>>
I focus on the details that everyone else has been
indoctrinated to ignore.
The proof of such an propostion within the same
formal system would require a sequence of inference
steps that prove that they themselves do not exist.
Which just shows you don't understand the concept of Formal
Systems, and their meta-systems.
This sentence is not true: "This sentence is not true"
the outer sentence is true because the inner sentence
is semantically incoherent.
In other words, you can't talk about the sentence you want to talk >>>>>> about, so you do to soething irrelevent.
Exactly the opposite Incompleteness and Undefinability
dishonestly dodge the fact the their actual sentences
are incoherent by using the meta-level.
And what is incoherent about using a meta-level.
All a mete-level is, is to build a new Formal System, based on the
base system that knows the basic properties of the base system.
For instance, the Rational Numbers can be considers a "meta" of the
Integeres.
This meta-level is correct to state that these sentences
are not provable and not true.
The meta-level never looks at why they are unprovable
and untrue. They are unprovable and untrue BECAUSE they
are semantically incoherent.
No, the sentence of G was specifically constructed to have a
coherent meaning in the base system, but you just are too stupid to
understand that.
Why do you lie about this? Does lying give you cheap thrill?
...We are therefore confronted with a proposition which asserts its
own unprovability. 15 rCa (G||del 1931:40-41)
And where does this say that is what the sentence is in the base system?
That <is> the summation of his whole proof dip shit.
G := (F re4 G)
a sequence of inference steps in F from the axioms
of F that assert that they themselves do not exist in F.
On 30/12/2025 14:32, olcott wrote:
G := (F re4 G)
a sequence of inference steps in F from the axioms
of F that assert that they themselves do not exist in F.
You suppose that's what the symbols mean. Yet you know that supposition
is inadmissible per-Se. Cognitive dissonance in action.
You rely on the delusion that the internal sensation of defining a
symbol actually has that effect on your mindspace and also on the
continued hallucination that the symbol is then stably so defined when
you later introspect your mind-space.
On 12/30/2025 2:22 PM, Tristan Wibberley wrote:
On 30/12/2025 14:32, olcott wrote:
G := (F re4 G)
a sequence of inference steps in F from the axioms
of F that assert that they themselves do not exist in F.
You suppose that's what the symbols mean. Yet you know that supposition
is inadmissible per-Se. Cognitive dissonance in action.
The symbols *mean* a self-contradictory expression of language
the same sort of thing as: "this sentence is not true".
You rely on the delusion that the internal sensation of defining a
symbol actually has that effect on your mindspace and also on the
continued hallucination that the symbol is then stably so defined when
you later introspect your mind-space.
The symbols *mean* a self-contradictory expression of language
the same sort of thing as: "this sentence is not true".
On 12/30/25 3:35 PM, olcott wrote:
On 12/30/2025 2:22 PM, Tristan Wibberley wrote:
On 30/12/2025 14:32, olcott wrote:
G := (F re4 G)
a sequence of inference steps in F from the axioms
of F that assert that they themselves do not exist in F.
You suppose that's what the symbols mean. Yet you know that supposition
is inadmissible per-Se. Cognitive dissonance in action.
The symbols *mean* a self-contradictory expression of language
the same sort of thing as: "this sentence is not true".
But it doesn't, as it is satisfiable by a statement that is true but unprovable, which just mean the statement is established true by an
infinite chain of infernce
On 30/12/2025 20:35, olcott wrote:
The symbols *mean* a self-contradictory expression of language
the same sort of thing as: "this sentence is not true".
Not per-Se. Formally, it depends on the full nature of the system
they're in.
On 30/12/2025 20:59, Richard Damon wrote:
On 12/30/25 3:35 PM, olcott wrote:
On 12/30/2025 2:22 PM, Tristan Wibberley wrote:
On 30/12/2025 14:32, olcott wrote:
G := (F re4 G)
a sequence of inference steps in F from the axioms
of F that assert that they themselves do not exist in F.
You suppose that's what the symbols mean. Yet you know that supposition >>>> is inadmissible per-Se. Cognitive dissonance in action.
The symbols *mean* a self-contradictory expression of language
the same sort of thing as: "this sentence is not true".
But it doesn't, as it is satisfiable by a statement that is true but
unprovable, which just mean the statement is established true by an
infinite chain of infernce
Are you using a finite derivation in the meta-system of the limit of a converging sequence of finite derivations of increasing length (whose terminals may or may not be the statement being proved but the limit of
whose terminals /is/)?
And thus you say the statement is true thereby exemplifying a point from which we may inductively infer a meaning for "true"? Is that "true of
the system in the meta-system" ?
On 12/30/2025 3:34 PM, Tristan Wibberley wrote:
On 30/12/2025 20:59, Richard Damon wrote:
On 12/30/25 3:35 PM, olcott wrote:
On 12/30/2025 2:22 PM, Tristan Wibberley wrote:
On 30/12/2025 14:32, olcott wrote:
G := (F re4 G)
a sequence of inference steps in F from the axioms
of F that assert that they themselves do not exist in F.
You suppose that's what the symbols mean. Yet you know that
supposition
is inadmissible per-Se. Cognitive dissonance in action.
The symbols *mean* a self-contradictory expression of language
the same sort of thing as: "this sentence is not true".
But it doesn't, as it is satisfiable by a statement that is true but
unprovable, which just mean the statement is established true by an
infinite chain of infernce
Are you using a finite derivation in the meta-system of the limit of a
converging sequence of finite derivations of increasing length (whose
terminals may or may not be the statement being proved but the limit of
whose terminals /is/)?
And thus you say the statement is true thereby exemplifying a point from
which we may inductively infer a meaning for "true"? Is that "true of
the system in the meta-system" ?
True in the system can only really mean provable
from the axioms of this same system any other
meaning is nonsense.
On 30/12/2025 20:59, Richard Damon wrote:
On 12/30/25 3:35 PM, olcott wrote:
On 12/30/2025 2:22 PM, Tristan Wibberley wrote:
On 30/12/2025 14:32, olcott wrote:
G := (F re4 G)
a sequence of inference steps in F from the axioms
of F that assert that they themselves do not exist in F.
You suppose that's what the symbols mean. Yet you know that supposition >>>> is inadmissible per-Se. Cognitive dissonance in action.
The symbols *mean* a self-contradictory expression of language
the same sort of thing as: "this sentence is not true".
But it doesn't, as it is satisfiable by a statement that is true but
unprovable, which just mean the statement is established true by an
infinite chain of infernce
Are you using a finite derivation in the meta-system of the limit of a converging sequence of finite derivations of increasing length (whose terminals may or may not be the statement being proved but the limit of
whose terminals /is/)?
And thus you say the statement is true thereby exemplifying a point from which we may inductively infer a meaning for "true"? Is that "true of
the system in the meta-system" ?
On 12/30/2025 3:26 PM, Tristan Wibberley wrote:
On 30/12/2025 20:35, olcott wrote:
The symbols *mean* a self-contradictory expression of language
the same sort of thing as: "this sentence is not true".
Not per-Se. Formally, it depends on the full nature of the system
they're in.
Sure and we could define a "black cat" as a
{fifteen story office building eating a sandwich}
Within the pure semantics of the actual underlying
meanings any expression of language that means:
{a sequence of inference steps in F from the axioms
-aof F that assert that they themselves do not exist in F}
is semantically incoherent.
On 12/29/25 2:21 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
On 29/12/2025 13:37, Richard Damon wrote:
Incompleteness is a property of a given Formal System, it says that
there exist a statement that is true in that system, but can not be
proven in that system.
What do you mean by "proven" here. Do you mean "derived" ?
I think Richard misspoke slightly. The undecidable statement is
true *in the intended interpretation* of the formal system
(In Goedel's case, the natural numbers with addition and multiplication).
Truth "in the formal system" isn't really defined. You need an interpretation.
No, statements in a formal system are DEFINED to be true, if that
statement, referencing object defined in the system model, and related
by relationships defined in the system can be established starting with
the initial "facts" (axioms) of the system, and following the allowed logical operations of the system.
THus in the formal system of addition of Natural Numbers, the statement
2 + 3 = 5 is a true statement, as it can be derived from the operations
in the system.
Some Formal System include a "model" that define
interpreations, though another layer above can be added (as Godel did).
In sci.logic Richard Damon <Richard@damon-family.org> wrote:
On 12/29/25 2:21 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
On 29/12/2025 13:37, Richard Damon wrote:
Incompleteness is a property of a given Formal System, it says that
there exist a statement that is true in that system, but can not be
proven in that system.
What do you mean by "proven" here. Do you mean "derived" ?
I think Richard misspoke slightly. The undecidable statement is
true *in the intended interpretation* of the formal system
(In Goedel's case, the natural numbers with addition and multiplication). >>>
Truth "in the formal system" isn't really defined. You need an
interpretation.
By the way when I wrote "Richard misspoke slightly" I should have
added "but that doesn't invalidate his argument". Sorry about that.
No, statements in a formal system are DEFINED to be true, if that
statement, referencing object defined in the system model, and related
by relationships defined in the system can be established starting with
the initial "facts" (axioms) of the system, and following the allowed
logical operations of the system.
That's provability, not truth.
THus in the formal system of addition of Natural Numbers, the statement
2 + 3 = 5 is a true statement, as it can be derived from the operations
in the system.
A statement provable in the system, and a true statement about natural numbers.
Some Formal System include a "model" that define
interpreations, though another layer above can be added (as Godel did).
I dunno, I always saw the models as separate from the formal system.
(That said, an intended model often provides the motivation for the
formal system.)
[ ... ]
Then he defines a new system "P" which he uses to get even more muddled, leaves out the crucial elements of his proof because it's too easy to
get wrong,
and Stephen Meyer says he does get it wrong; he seems to be
the only person in the world that ever checked.
No, provability requires a FINITE sequence to be showable.
Truth can be established by an INFINITE sequence.
On 29/12/2025 19:53, Richard Damon wrote:
On 12/29/25 2:32 PM, olcott wrote:
On 12/29/2025 1:21 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
On 29/12/2025 13:37, Richard Damon wrote:
Incompleteness is a property of a given Formal System, it says that >>>>>> there exist a statement that is true in that system, but can not be >>>>>> proven in that system.
What do you mean by "proven" here. Do you mean "derived" ?
I think Richard misspoke slightly. The undecidable statement is
true *in the intended interpretation* of the formal system
(In Goedel's case, the natural numbers with addition and
multiplication).
Truth "in the formal system" isn't really defined. You need an
interpretation.
Unless (as I have been saying for at least a decade)
the formal language directly encodes all of its
semantics directly in its syntax. The Montague
Grammar of natural language semantics is the best
known example of this.
But it can't, as any system that defines symbols, can have something
outside it assign additional meaning to those symbols.
Ontology suggests ways to *apply* a system. The system itself works
without additional meaning just as it does with. That's the point of
formal systems.
There may be SOME meaning within the system, but, with a sufficiently
expressive system, additional meaning can be imposed.
additional meaning is given to an embedding or extension (which is pretty-much a special-case of embedding) of a system, not to the system itself.
In the case of G||del's preamble, he defines an extension of PM (I should suppose he was using 2nd ed. in 1931 from his untruths about PM if
applied to 1st. ed.) That extension is inconsistent (or, better, I
think, indiscriminate). his referent there for PM slides between PM and
the derived system as he writes and he gets muddled taking a half-formed conclusion about one, assuming and completing it for the other.
Then he defines a new system "P" which he uses to get even more muddled, leaves out the crucial elements of his proof because it's too easy to
get wrong, and Stephen Meyer says he does get it wrong; he seems to be
the only person in the world that ever checked.
In sci.logic Richard Damon <Richard@damon-family.org> wrote:
No, provability requires a FINITE sequence to be showable.
Truth can be established by an INFINITE sequence.
You're going off the beaten paths. Does that work
if you quantify over real numbers? Just curious.
In sci.logic Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
[ ... ]
Then he defines a new system "P" which he uses to get even more muddled,
leaves out the crucial elements of his proof because it's too easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch that ever
lived!
and Stephen Meyer says he does get it wrong; he seems to be
the only person in the world that ever checked.
People have misunderstood G||del and proved it by their comments.
I don't know who Stephen Meyer is; my money is on G||del.
On 12/31/2025 3:16 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
[ ... ]
Then he defines a new system "P" which he uses to get even more muddled, >>> leaves out the crucial elements of his proof because it's too easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch that ever
lived!
and Stephen Meyer says he does get it wrong; he seems to be
the only person in the world that ever checked.
People have misunderstood G||del and proved it by their comments.
I don't know who Stephen Meyer is; my money is on G||del.
G||del proved that there cannot possibly exist any
sequence of inference steps in F prove that they
themselves do not exist.
He admitted this himself:
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
On 29/12/2025 19:53, Richard Damon wrote:
On 12/29/25 2:32 PM, olcott wrote:
On 12/29/2025 1:21 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
On 29/12/2025 13:37, Richard Damon wrote:
Incompleteness is a property of a given Formal System, it says that >>>>>> there exist a statement that is true in that system, but can not be >>>>>> proven in that system.
What do you mean by "proven" here. Do you mean "derived" ?
I think Richard misspoke slightly. The undecidable statement is
true *in the intended interpretation* of the formal system
(In Goedel's case, the natural numbers with addition and
multiplication).
Truth "in the formal system" isn't really defined. You need an
interpretation.
Unless (as I have been saying for at least a decade)
the formal language directly encodes all of its
semantics directly in its syntax. The Montague
Grammar of natural language semantics is the best
known example of this.
But it can't, as any system that defines symbols, can have something
outside it assign additional meaning to those symbols.
Ontology suggests ways to *apply* a system. The system itself works
without additional meaning just as it does with. That's the point of
formal systems.
There may be SOME meaning within the system, but, with a sufficiently
expressive system, additional meaning can be imposed.
additional meaning is given to an embedding or extension (which is pretty-much a special-case of embedding) of a system, not to the system itself.
In the case of G||del's preamble, he defines an extension of PM (I should suppose he was using 2nd ed. in 1931 from his untruths about PM if
applied to 1st. ed.) That extension is inconsistent (or, better, I
think, indiscriminate). his referent there for PM slides between PM and
the derived system as he writes and he gets muddled taking a half-formed conclusion about one, assuming and completing it for the other.
Then he defines a new system "P" which he uses to get even more muddled, leaves out the crucial elements of his proof because it's too easy to
get wrong, and Stephen Meyer says he does get it wrong; he seems to be
the only person in the world that ever checked.
On 12/31/25 4:52 PM, olcott wrote:
On 12/31/2025 3:16 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
[ ... ]
Then he defines a new system "P" which he uses to get even more
muddled,
leaves out the crucial elements of his proof because it's too easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch that ever
lived!
and Stephen Meyer says he does get it wrong; he seems to be
the only person in the world that ever checked.
People have misunderstood G||del and proved it by their comments.
I don't know who Stephen Meyer is; my money is on G||del.
G||del proved that there cannot possibly exist any
sequence of inference steps in F prove that they
themselves do not exist.
No *FINITE* sequence of inference steps.
He also proves there *IS* an infinite sequence of steps
He admitted this himself:
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41)
And proofs are finite.
And that statement is made in the Meta System, and is talking about the
base system.
All you are doing is proving that you are an idiot, and maybe in your
case there isn't a difference between You and a deterministic machine,
as you are stuck in your bad programming.
It seems you hae a broken CPU.
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
On 12/29/2025 2:20 PM, Tristan Wibberley wrote:
On 29/12/2025 19:53, Richard Damon wrote:
On 12/29/25 2:32 PM, olcott wrote:
On 12/29/2025 1:21 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
On 29/12/2025 13:37, Richard Damon wrote:
Incompleteness is a property of a given Formal System, it says that >>>>>>> there exist a statement that is true in that system, but can not be >>>>>>> proven in that system.
What do you mean by "proven" here. Do you mean "derived" ?
I think Richard misspoke slightly. The undecidable statement is
true *in the intended interpretation* of the formal system
(In Goedel's case, the natural numbers with addition and
multiplication).
Truth "in the formal system" isn't really defined. You need an
interpretation.
Unless (as I have been saying for at least a decade)
the formal language directly encodes all of its
semantics directly in its syntax. The Montague
Grammar of natural language semantics is the best
known example of this.
But it can't, as any system that defines symbols, can have something
outside it assign additional meaning to those symbols.
Ontology suggests ways to *apply* a system. The system itself works
without additional meaning just as it does with. That's the point of
formal systems.
There may be SOME meaning within the system, but, with a sufficiently
expressive system, additional meaning can be imposed.
additional meaning is given to an embedding or extension (which is
pretty-much a special-case of embedding) of a system, not to the system
itself.
In the case of G||del's preamble, he defines an extension of PM (I should
suppose he was using 2nd ed. in 1931 from his untruths about PM if
applied to 1st. ed.) That extension is inconsistent (or, better, I
think, indiscriminate). his referent there for PM slides between PM and
the derived system as he writes and he gets muddled taking a half-formed
conclusion about one, assuming and completing it for the other.
Then he defines a new system "P" which he uses to get even more muddled,
leaves out the crucial elements of his proof because it's too easy to
get wrong, and Stephen Meyer says he does get it wrong; he seems to be
the only person in the world that ever checked.
G||del proved that there cannot possibly exist any
sequence of inference steps in F prove that they
themselves do not exist.
He admitted this himself:
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
On 12/31/2025 3:56 PM, Richard Damon wrote:
On 12/31/25 4:52 PM, olcott wrote:
On 12/31/2025 3:16 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
[ ... ]
Then he defines a new system "P" which he uses to get even more
muddled,
leaves out the crucial elements of his proof because it's too easy to >>>>> get wrong,
G||del, muddled? He was the most meticulous sonovabitch that ever
lived!
and Stephen Meyer says he does get it wrong; he seems to be
the only person in the world that ever checked.
People have misunderstood G||del and proved it by their comments.
I don't know who Stephen Meyer is; my money is on G||del.
G||del proved that there cannot possibly exist any
sequence of inference steps in F prove that they
themselves do not exist.
No *FINITE* sequence of inference steps.
Nothing can prove that itself does not
exist because that forms proof that it
does exist, dumbo.
He also proves there *IS* an infinite sequence of steps
He admitted this himself:
...We are therefore confronted with a proposition
which asserts its own unprovability. 15 rCa (G||del 1931:40-41)
And proofs are finite.
And that statement is made in the Meta System, and is talking about
the base system.
All you are doing is proving that you are an idiot, and maybe in your
case there isn't a difference between You and a deterministic machine,
as you are stuck in your bad programming.
It seems you hae a broken CPU.
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
On 12/31/25 4:59 PM, olcott wrote:
On 12/31/2025 3:56 PM, Richard Damon wrote:
On 12/31/25 4:52 PM, olcott wrote:
On 12/31/2025 3:16 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
[ ... ]
Then he defines a new system "P" which he uses to get even more
muddled,
leaves out the crucial elements of his proof because it's too easy to >>>>>> get wrong,
G||del, muddled? He was the most meticulous sonovabitch that ever
lived!
and Stephen Meyer says he does get it wrong; he seems to be
the only person in the world that ever checked.
People have misunderstood G||del and proved it by their comments.
I don't know who Stephen Meyer is; my money is on G||del.
G||del proved that there cannot possibly exist any
sequence of inference steps in F prove that they
themselves do not exist.
No *FINITE* sequence of inference steps.
Nothing can prove that itself does not
exist because that forms proof that it
does exist, dumbo.
So you are just ignoring context because you are stupid.
The statement, with the added information of the meta-system proves (by
a proof in the meta system) that the statment is true.
On 12/31/2025 4:09 PM, Richard Damon wrote:
On 12/31/25 4:59 PM, olcott wrote:
On 12/31/2025 3:56 PM, Richard Damon wrote:
On 12/31/25 4:52 PM, olcott wrote:
On 12/31/2025 3:16 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
[ ... ]
Then he defines a new system "P" which he uses to get even more >>>>>>> muddled,
leaves out the crucial elements of his proof because it's too
easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch that ever
lived!
and Stephen Meyer says he does get it wrong; he seems to be
the only person in the world that ever checked.
People have misunderstood G||del and proved it by their comments.
I don't know who Stephen Meyer is; my money is on G||del.
G||del proved that there cannot possibly exist any
sequence of inference steps in F prove that they
themselves do not exist.
No *FINITE* sequence of inference steps.
Nothing can prove that itself does not
exist because that forms proof that it
does exist, dumbo.
So you are just ignoring context because you are stupid.
The statement, with the added information of the meta-system proves
(by a proof in the meta system) that the statment is true.
Something else can prove that X cannot prove that
X does not exist, AKA your meta-system.
Nothing can directly prove that itself does not
exist because this forms proof that it does exist.
On 12/31/25 5:42 PM, olcott wrote:
On 12/31/2025 4:09 PM, Richard Damon wrote:
On 12/31/25 4:59 PM, olcott wrote:
On 12/31/2025 3:56 PM, Richard Damon wrote:
On 12/31/25 4:52 PM, olcott wrote:
On 12/31/2025 3:16 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
[ ... ]
Then he defines a new system "P" which he uses to get even more >>>>>>>> muddled,
leaves out the crucial elements of his proof because it's too >>>>>>>> easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch that ever >>>>>>> lived!
and Stephen Meyer says he does get it wrong; he seems to be
the only person in the world that ever checked.
People have misunderstood G||del and proved it by their comments. >>>>>>> I don't know who Stephen Meyer is; my money is on G||del.
G||del proved that there cannot possibly exist any
sequence of inference steps in F prove that they
themselves do not exist.
No *FINITE* sequence of inference steps.
Nothing can prove that itself does not
exist because that forms proof that it
does exist, dumbo.
So you are just ignoring context because you are stupid.
The statement, with the added information of the meta-system proves
(by a proof in the meta system) that the statment is true.
Something else can prove that X cannot prove that
X does not exist, AKA your meta-system.
Nothing can directly prove that itself does not
exist because this forms proof that it does exist.
Nope, got a source for that?
Why does my explanation not work?
Can you even put my explaination imto your own words to show that you understand it.--
The statement G, under the interpreation provided by M certainly can
prove that the system without M can't prove it.
It seems you think that X is as big as X+1
Sorry, you are just showing that you brain has self-distructed itself
and left you with no ability to reason.
On 12/31/2025 4:48 PM, Richard Damon wrote:
On 12/31/25 5:42 PM, olcott wrote:
On 12/31/2025 4:09 PM, Richard Damon wrote:
On 12/31/25 4:59 PM, olcott wrote:
On 12/31/2025 3:56 PM, Richard Damon wrote:
On 12/31/25 4:52 PM, olcott wrote:
On 12/31/2025 3:16 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
[ ... ]
Then he defines a new system "P" which he uses to get even more >>>>>>>>> muddled,
leaves out the crucial elements of his proof because it's too >>>>>>>>> easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch that ever >>>>>>>> lived!
and Stephen Meyer says he does get it wrong; he seems to be
the only person in the world that ever checked.
People have misunderstood G||del and proved it by their comments. >>>>>>>> I don't know who Stephen Meyer is; my money is on G||del.
G||del proved that there cannot possibly exist any
sequence of inference steps in F prove that they
themselves do not exist.
No *FINITE* sequence of inference steps.
Nothing can prove that itself does not
exist because that forms proof that it
does exist, dumbo.
So you are just ignoring context because you are stupid.
The statement, with the added information of the meta-system proves
(by a proof in the meta system) that the statment is true.
Something else can prove that X cannot prove that
X does not exist, AKA your meta-system.
Nothing can directly prove that itself does not
exist because this forms proof that it does exist.
Nope, got a source for that?
Why does my explanation not work?
It is not that your explanation doesn't work.
It is that it ignores the root cause of why
G is unprovable in F.
If you disagree then provide a correct
proof that you yourself never existed.
If you can't see how this is impossible
you must by very dumb.
Since you have proved that you are quite
smart then any disagreement would most
likely be a lie, a mere head game.
Can you even put my explaination imto your own words to show that you
understand it.
The statement G, under the interpreation provided by M certainly can
prove that the system without M can't prove it.
It seems you think that X is as big as X+1
Sorry, you are just showing that you brain has self-distructed itself
and left you with no ability to reason.
On 12/31/25 6:08 PM, olcott wrote:
On 12/31/2025 4:48 PM, Richard Damon wrote:
On 12/31/25 5:42 PM, olcott wrote:
On 12/31/2025 4:09 PM, Richard Damon wrote:
On 12/31/25 4:59 PM, olcott wrote:
On 12/31/2025 3:56 PM, Richard Damon wrote:
On 12/31/25 4:52 PM, olcott wrote:
On 12/31/2025 3:16 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
[ ... ]
Then he defines a new system "P" which he uses to get even >>>>>>>>>> more muddled,
leaves out the crucial elements of his proof because it's too >>>>>>>>>> easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch that ever >>>>>>>>> lived!
and Stephen Meyer says he does get it wrong; he seems to be >>>>>>>>>> the only person in the world that ever checked.
People have misunderstood G||del and proved it by their comments. >>>>>>>>> I don't know who Stephen Meyer is; my money is on G||del.
G||del proved that there cannot possibly exist any
sequence of inference steps in F prove that they
themselves do not exist.
No *FINITE* sequence of inference steps.
Nothing can prove that itself does not
exist because that forms proof that it
does exist, dumbo.
So you are just ignoring context because you are stupid.
The statement, with the added information of the meta-system proves >>>>> (by a proof in the meta system) that the statment is true.
Something else can prove that X cannot prove that
X does not exist, AKA your meta-system.
Nothing can directly prove that itself does not
exist because this forms proof that it does exist.
Nope, got a source for that?
Why does my explanation not work?
It is not that your explanation doesn't work.
It is that it ignores the root cause of why
G is unprovable in F.
So, how do you think you can prove it in F?
On 12/31/2025 5:27 PM, Richard Damon wrote:
On 12/31/25 6:08 PM, olcott wrote:
On 12/31/2025 4:48 PM, Richard Damon wrote:
On 12/31/25 5:42 PM, olcott wrote:
On 12/31/2025 4:09 PM, Richard Damon wrote:
On 12/31/25 4:59 PM, olcott wrote:
On 12/31/2025 3:56 PM, Richard Damon wrote:
On 12/31/25 4:52 PM, olcott wrote:
On 12/31/2025 3:16 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote: >>>>>>>>>>
[ ... ]
Then he defines a new system "P" which he uses to get even >>>>>>>>>>> more muddled,
leaves out the crucial elements of his proof because it's too >>>>>>>>>>> easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch that ever >>>>>>>>>> lived!
and Stephen Meyer says he does get it wrong; he seems to be >>>>>>>>>>> the only person in the world that ever checked.
People have misunderstood G||del and proved it by their comments. >>>>>>>>>> I don't know who Stephen Meyer is; my money is on G||del.
G||del proved that there cannot possibly exist any
sequence of inference steps in F prove that they
themselves do not exist.
No *FINITE* sequence of inference steps.
Nothing can prove that itself does not
exist because that forms proof that it
does exist, dumbo.
So you are just ignoring context because you are stupid.
The statement, with the added information of the meta-system
proves (by a proof in the meta system) that the statment is true.
Something else can prove that X cannot prove that
X does not exist, AKA your meta-system.
Nothing can directly prove that itself does not
exist because this forms proof that it does exist.
Nope, got a source for that?
Why does my explanation not work?
It is not that your explanation doesn't work.
It is that it ignores the root cause of why
G is unprovable in F.
So, how do you think you can prove it in F?
Nothing can prove that itself does not exist.
Any such proof would be self-refuting.
On 12/31/25 7:23 PM, olcott wrote:
On 12/31/2025 5:27 PM, Richard Damon wrote:
On 12/31/25 6:08 PM, olcott wrote:
On 12/31/2025 4:48 PM, Richard Damon wrote:
On 12/31/25 5:42 PM, olcott wrote:
On 12/31/2025 4:09 PM, Richard Damon wrote:
On 12/31/25 4:59 PM, olcott wrote:
On 12/31/2025 3:56 PM, Richard Damon wrote:
On 12/31/25 4:52 PM, olcott wrote:
On 12/31/2025 3:16 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote: >>>>>>>>>>>
[ ... ]
Then he defines a new system "P" which he uses to get even >>>>>>>>>>>> more muddled,
leaves out the crucial elements of his proof because it's >>>>>>>>>>>> too easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch that ever >>>>>>>>>>> lived!
and Stephen Meyer says he does get it wrong; he seems to be >>>>>>>>>>>> the only person in the world that ever checked.
People have misunderstood G||del and proved it by their comments. >>>>>>>>>>> I don't know who Stephen Meyer is; my money is on G||del. >>>>>>>>>>>
G||del proved that there cannot possibly exist any
sequence of inference steps in F prove that they
themselves do not exist.
No *FINITE* sequence of inference steps.
Nothing can prove that itself does not
exist because that forms proof that it
does exist, dumbo.
So you are just ignoring context because you are stupid.
The statement, with the added information of the meta-system
proves (by a proof in the meta system) that the statment is true. >>>>>>>
Something else can prove that X cannot prove that
X does not exist, AKA your meta-system.
Nothing can directly prove that itself does not
exist because this forms proof that it does exist.
Nope, got a source for that?
Why does my explanation not work?
It is not that your explanation doesn't work.
It is that it ignores the root cause of why
G is unprovable in F.
So, how do you think you can prove it in F?
Nothing can prove that itself does not exist.
Any such proof would be self-refuting.
But it isn't the PROOF that does the proving, it is the statement.
THe statement G exist, and it is True.
Because it is true, and can be proven with the additional knowledge and tools-a of the meta-system, it shows that without the addtional knowledge and tools you can't make the proof.--
It seems you don't understand that the base system and the meta system
are different.
Boy, are you stupid.
On 12/31/2025 6:35 PM, Richard Damon wrote:
On 12/31/25 7:23 PM, olcott wrote:
On 12/31/2025 5:27 PM, Richard Damon wrote:
On 12/31/25 6:08 PM, olcott wrote:
On 12/31/2025 4:48 PM, Richard Damon wrote:
On 12/31/25 5:42 PM, olcott wrote:
On 12/31/2025 4:09 PM, Richard Damon wrote:
On 12/31/25 4:59 PM, olcott wrote:
On 12/31/2025 3:56 PM, Richard Damon wrote:
On 12/31/25 4:52 PM, olcott wrote:
On 12/31/2025 3:16 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote: >>>>>>>>>>>>
[ ... ]
Then he defines a new system "P" which he uses to get even >>>>>>>>>>>>> more muddled,
leaves out the crucial elements of his proof because it's >>>>>>>>>>>>> too easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch that >>>>>>>>>>>> ever
lived!
and Stephen Meyer says he does get it wrong; he seems to be >>>>>>>>>>>>> the only person in the world that ever checked.
People have misunderstood G||del and proved it by their >>>>>>>>>>>> comments.
I don't know who Stephen Meyer is; my money is on G||del. >>>>>>>>>>>>
G||del proved that there cannot possibly exist any
sequence of inference steps in F prove that they
themselves do not exist.
No *FINITE* sequence of inference steps.
Nothing can prove that itself does not
exist because that forms proof that it
does exist, dumbo.
So you are just ignoring context because you are stupid.
The statement, with the added information of the meta-system
proves (by a proof in the meta system) that the statment is true. >>>>>>>>
Something else can prove that X cannot prove that
X does not exist, AKA your meta-system.
Nothing can directly prove that itself does not
exist because this forms proof that it does exist.
Nope, got a source for that?
Why does my explanation not work?
It is not that your explanation doesn't work.
It is that it ignores the root cause of why
G is unprovable in F.
So, how do you think you can prove it in F?
Nothing can prove that itself does not exist.
Any such proof would be self-refuting.
But it isn't the PROOF that does the proving, it is the statement.
THe statement G exist, and it is True.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
When we name this proposition G then a proof of G
would be a sequence of inference steps that prove
that they themselves do not exist.
Anything that asserts its own non-existence
is necessarily incorrect.
Because it is true, and can be proven with the additional knowledge
and tools-a of the meta-system, it shows that without the addtional
knowledge and tools you can't make the proof.
It seems you don't understand that the base system and the meta system
are different.
Boy, are you stupid.
On 12/31/25 8:04 PM, olcott wrote:
On 12/31/2025 6:35 PM, Richard Damon wrote:
On 12/31/25 7:23 PM, olcott wrote:
On 12/31/2025 5:27 PM, Richard Damon wrote:
On 12/31/25 6:08 PM, olcott wrote:
On 12/31/2025 4:48 PM, Richard Damon wrote:
On 12/31/25 5:42 PM, olcott wrote:
On 12/31/2025 4:09 PM, Richard Damon wrote:
On 12/31/25 4:59 PM, olcott wrote:
On 12/31/2025 3:56 PM, Richard Damon wrote:
On 12/31/25 4:52 PM, olcott wrote:
On 12/31/2025 3:16 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote: >>>>>>>>>>>>>
[ ... ]
Then he defines a new system "P" which he uses to get even >>>>>>>>>>>>>> more muddled,
leaves out the crucial elements of his proof because it's >>>>>>>>>>>>>> too easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch that >>>>>>>>>>>>> ever
lived!
and Stephen Meyer says he does get it wrong; he seems to be >>>>>>>>>>>>>> the only person in the world that ever checked.
People have misunderstood G||del and proved it by their >>>>>>>>>>>>> comments.
I don't know who Stephen Meyer is; my money is on G||del. >>>>>>>>>>>>>
G||del proved that there cannot possibly exist any
sequence of inference steps in F prove that they
themselves do not exist.
No *FINITE* sequence of inference steps.
Nothing can prove that itself does not
exist because that forms proof that it
does exist, dumbo.
So you are just ignoring context because you are stupid.
The statement, with the added information of the meta-system >>>>>>>>> proves (by a proof in the meta system) that the statment is true. >>>>>>>>>
Something else can prove that X cannot prove that
X does not exist, AKA your meta-system.
Nothing can directly prove that itself does not
exist because this forms proof that it does exist.
Nope, got a source for that?
Why does my explanation not work?
It is not that your explanation doesn't work.
It is that it ignores the root cause of why
G is unprovable in F.
So, how do you think you can prove it in F?
Nothing can prove that itself does not exist.
Any such proof would be self-refuting.
But it isn't the PROOF that does the proving, it is the statement.
THe statement G exist, and it is True.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
You keep on repeating that, but show you don't know what it means,
proving your stupidity.
On 12/31/2025 7:29 PM, Richard Damon wrote:
On 12/31/25 8:04 PM, olcott wrote:
On 12/31/2025 6:35 PM, Richard Damon wrote:
On 12/31/25 7:23 PM, olcott wrote:
On 12/31/2025 5:27 PM, Richard Damon wrote:
On 12/31/25 6:08 PM, olcott wrote:
On 12/31/2025 4:48 PM, Richard Damon wrote:
On 12/31/25 5:42 PM, olcott wrote:
On 12/31/2025 4:09 PM, Richard Damon wrote:
On 12/31/25 4:59 PM, olcott wrote:
On 12/31/2025 3:56 PM, Richard Damon wrote:
On 12/31/25 4:52 PM, olcott wrote:
On 12/31/2025 3:16 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote: >>>>>>>>>>>>>>
[ ... ]
Then he defines a new system "P" which he uses to get >>>>>>>>>>>>>>> even more muddled,
leaves out the crucial elements of his proof because it's >>>>>>>>>>>>>>> too easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch >>>>>>>>>>>>>> that ever
lived!
and Stephen Meyer says he does get it wrong; he seems to be >>>>>>>>>>>>>>> the only person in the world that ever checked.
People have misunderstood G||del and proved it by their >>>>>>>>>>>>>> comments.
I don't know who Stephen Meyer is; my money is on G||del. >>>>>>>>>>>>>>
G||del proved that there cannot possibly exist any
sequence of inference steps in F prove that they
themselves do not exist.
No *FINITE* sequence of inference steps.
Nothing can prove that itself does not
exist because that forms proof that it
does exist, dumbo.
So you are just ignoring context because you are stupid.
The statement, with the added information of the meta-system >>>>>>>>>> proves (by a proof in the meta system) that the statment is true. >>>>>>>>>>
Something else can prove that X cannot prove that
X does not exist, AKA your meta-system.
Nothing can directly prove that itself does not
exist because this forms proof that it does exist.
Nope, got a source for that?
Why does my explanation not work?
It is not that your explanation doesn't work.
It is that it ignores the root cause of why
G is unprovable in F.
So, how do you think you can prove it in F?
Nothing can prove that itself does not exist.
Any such proof would be self-refuting.
But it isn't the PROOF that does the proving, it is the statement.
THe statement G exist, and it is True.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
You keep on repeating that, but show you don't know what it means,
proving your stupidity.
It can only mean one thing when taken 100% literally.
a proposition which asserts its own unprovability.
G says that itself is unprovable
G says that itself has no sequence of inference
steps that prove that they themselves do not exist.
It say nothing at all about any meta-system.
On 12/31/25 9:15 PM, olcott wrote:
On 12/31/2025 7:29 PM, Richard Damon wrote:
On 12/31/25 8:04 PM, olcott wrote:
On 12/31/2025 6:35 PM, Richard Damon wrote:
On 12/31/25 7:23 PM, olcott wrote:
On 12/31/2025 5:27 PM, Richard Damon wrote:
On 12/31/25 6:08 PM, olcott wrote:
On 12/31/2025 4:48 PM, Richard Damon wrote:
On 12/31/25 5:42 PM, olcott wrote:
On 12/31/2025 4:09 PM, Richard Damon wrote:
On 12/31/25 4:59 PM, olcott wrote:
On 12/31/2025 3:56 PM, Richard Damon wrote:
On 12/31/25 4:52 PM, olcott wrote:
On 12/31/2025 3:16 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote: >>>>>>>>>>>>>>>
[ ... ]
Then he defines a new system "P" which he uses to get >>>>>>>>>>>>>>>> even more muddled,
leaves out the crucial elements of his proof because >>>>>>>>>>>>>>>> it's too easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch >>>>>>>>>>>>>>> that ever
lived!
and Stephen Meyer says he does get it wrong; he seems to be >>>>>>>>>>>>>>>> the only person in the world that ever checked. >>>>>>>>>>>>>>>People have misunderstood G||del and proved it by their >>>>>>>>>>>>>>> comments.
I don't know who Stephen Meyer is; my money is on G||del. >>>>>>>>>>>>>>>
G||del proved that there cannot possibly exist any >>>>>>>>>>>>>> sequence of inference steps in F prove that they
themselves do not exist.
No *FINITE* sequence of inference steps.
Nothing can prove that itself does not
exist because that forms proof that it
does exist, dumbo.
So you are just ignoring context because you are stupid. >>>>>>>>>>>
The statement, with the added information of the meta-system >>>>>>>>>>> proves (by a proof in the meta system) that the statment is >>>>>>>>>>> true.
Something else can prove that X cannot prove that
X does not exist, AKA your meta-system.
Nothing can directly prove that itself does not
exist because this forms proof that it does exist.
Nope, got a source for that?
Why does my explanation not work?
It is not that your explanation doesn't work.
It is that it ignores the root cause of why
G is unprovable in F.
So, how do you think you can prove it in F?
Nothing can prove that itself does not exist.
Any such proof would be self-refuting.
But it isn't the PROOF that does the proving, it is the statement.
THe statement G exist, and it is True.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
You keep on repeating that, but show you don't know what it means,
proving your stupidity.
It can only mean one thing when taken 100% literally.
The problem is language is not to be taken "100% literally", and thus
you just show you don't understand how words have meaning.
--Sure it does, as it is in the section talking about an analysis in the meta-syste
a proposition which asserts its own unprovability.
G says that itself is unprovable
G says that itself has no sequence of inference
steps that prove that they themselves do not exist.
It say nothing at all about any meta-system.
I guess you are just proving you are a total idiot with no understanding
of the structure of language, which is why your goal of trying to base
your logic on words and their meaning is so hilarious, since you neve runderstod the nature of language in the first place.
In sci.logic Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
[ ... ]
Then he defines a new system "P" which he uses to get even more muddled,
leaves out the crucial elements of his proof because it's too easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch that ever
lived!
and Stephen Meyer says he does get it wrong; he seems to be
the only person in the world that ever checked.
People have misunderstood G||del and proved it by their comments.
I don't know who Stephen Meyer is; my money is on G||del.
On 12/31/2025 8:48 PM, Richard Damon wrote:
On 12/31/25 9:15 PM, olcott wrote:
On 12/31/2025 7:29 PM, Richard Damon wrote:
On 12/31/25 8:04 PM, olcott wrote:
On 12/31/2025 6:35 PM, Richard Damon wrote:
On 12/31/25 7:23 PM, olcott wrote:
On 12/31/2025 5:27 PM, Richard Damon wrote:
On 12/31/25 6:08 PM, olcott wrote:
On 12/31/2025 4:48 PM, Richard Damon wrote:
On 12/31/25 5:42 PM, olcott wrote:
On 12/31/2025 4:09 PM, Richard Damon wrote:
On 12/31/25 4:59 PM, olcott wrote:
On 12/31/2025 3:56 PM, Richard Damon wrote:
On 12/31/25 4:52 PM, olcott wrote:
On 12/31/2025 3:16 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote: >>>>>>>>>>>>>>>>
[ ... ]
Then he defines a new system "P" which he uses to get >>>>>>>>>>>>>>>>> even more muddled,
leaves out the crucial elements of his proof because >>>>>>>>>>>>>>>>> it's too easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch >>>>>>>>>>>>>>>> that ever
lived!
and Stephen Meyer says he does get it wrong; he seems >>>>>>>>>>>>>>>>> to bePeople have misunderstood G||del and proved it by their >>>>>>>>>>>>>>>> comments.
the only person in the world that ever checked. >>>>>>>>>>>>>>>>
I don't know who Stephen Meyer is; my money is on G||del. >>>>>>>>>>>>>>>>
G||del proved that there cannot possibly exist any >>>>>>>>>>>>>>> sequence of inference steps in F prove that they >>>>>>>>>>>>>>> themselves do not exist.
No *FINITE* sequence of inference steps.
Nothing can prove that itself does not
exist because that forms proof that it
does exist, dumbo.
So you are just ignoring context because you are stupid. >>>>>>>>>>>>
The statement, with the added information of the meta-system >>>>>>>>>>>> proves (by a proof in the meta system) that the statment is >>>>>>>>>>>> true.
Something else can prove that X cannot prove that
X does not exist, AKA your meta-system.
Nothing can directly prove that itself does not
exist because this forms proof that it does exist.
Nope, got a source for that?
Why does my explanation not work?
It is not that your explanation doesn't work.
It is that it ignores the root cause of why
G is unprovable in F.
So, how do you think you can prove it in F?
Nothing can prove that itself does not exist.
Any such proof would be self-refuting.
But it isn't the PROOF that does the proving, it is the statement. >>>>>>
THe statement G exist, and it is True.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
You keep on repeating that, but show you don't know what it means,
proving your stupidity.
It can only mean one thing when taken 100% literally.
The problem is language is not to be taken "100% literally", and thus
you just show you don't understand how words have meaning.
Formal mathematical specifications are taken literally or incorrectly.
a proposition which asserts its own unprovability.
G says that itself is unprovable
G says that itself has no sequence of inference
steps that prove that they themselves do not exist.
"a proposition which asserts its own unprovability."
says nothing at all about any meta-system.
Sure it does, as it is in the section talking about an analysis in the
a proposition which asserts its own unprovability.
G says that itself is unprovable
G says that itself has no sequence of inference
steps that prove that they themselves do not exist.
It say nothing at all about any meta-system.
meta-syste
I guess you are just proving you are a total idiot with no
understanding of the structure of language, which is why your goal of
trying to base your logic on words and their meaning is so hilarious,
since you neve runderstod the nature of language in the first place.
On 31/12/2025 21:16, Pierre Asselin wrote:
In sci.logic Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
[ ... ]
Then he defines a new system "P" which he uses to get even more muddled, >>> leaves out the crucial elements of his proof because it's too easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch that ever
lived!
Have you heard about his musings on God?
and Stephen Meyer says he does get it wrong; he seems to be
the only person in the world that ever checked.
People have misunderstood G||del and proved it by their comments.
I don't know who Stephen Meyer is; my money is on G||del.
I misremembered, it was James Meyer. He has a website on it http://www.jamesrmeyer.com . He's very angry about people telling him
he's wrong but who never checked like he did because they keep telling
him reasons it's right that he's certain are not reflected in the actual work.
On 1/1/26 5:41 AM, Tristan Wibberley wrote:
On 31/12/2025 21:16, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
[ ... ]
Then he defines a new system "P" which he uses to get even more
muddled,
leaves out the crucial elements of his proof because it's too easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch that ever
lived!
Have you heard about his musings on God?
and Stephen Meyer says he does get it wrong; he seems to be
the only person in the world that ever checked.
People have misunderstood G||del and proved it by their comments.
I don't know who Stephen Meyer is; my money is on G||del.
I misremembered, it was James Meyer. He has a website on it
http://www.jamesrmeyer.com . He's very angry about people telling him
he's wrong but who never checked like he did because they keep telling
him reasons it's right that he's certain are not reflected in the actual
work.
In other words, since he doesn't understand it, it must be wrong.
Since his page begins with a rejection of the axiom of Choice, and the example he gives, it shows a limitation in his ability to understand the nature of infinite systems.--
To expect that infinite systems behave just like we see finite systems
work is a funamental error.
Yes, it seems to create paradoxes, but those paradoxes are only apparent
due to the lack of understanding about the actual nature of infinite sets.
So, how do you think you can prove it in F?
THe statement G exist
On 12/31/25 8:04 PM, olcott wrote:
On 12/31/2025 6:35 PM, Richard Damon wrote:
On 12/31/25 7:23 PM, olcott wrote:
On 12/31/2025 5:27 PM, Richard Damon wrote:
On 12/31/25 6:08 PM, olcott wrote:
On 12/31/2025 4:48 PM, Richard Damon wrote:
On 12/31/25 5:42 PM, olcott wrote:
On 12/31/2025 4:09 PM, Richard Damon wrote:
On 12/31/25 4:59 PM, olcott wrote:
On 12/31/2025 3:56 PM, Richard Damon wrote:
On 12/31/25 4:52 PM, olcott wrote:
On 12/31/2025 3:16 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote: >>>>>>>>>>>>>
[ ... ]
Then he defines a new system "P" which he uses to get even >>>>>>>>>>>>>> more muddled,
leaves out the crucial elements of his proof because it's >>>>>>>>>>>>>> too easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch that >>>>>>>>>>>>> ever
lived!
and Stephen Meyer says he does get it wrong; he seems to be >>>>>>>>>>>>>> the only person in the world that ever checked.
People have misunderstood G||del and proved it by their >>>>>>>>>>>>> comments.
I don't know who Stephen Meyer is; my money is on G||del. >>>>>>>>>>>>>
G||del proved that there cannot possibly exist any
sequence of inference steps in F prove that they
themselves do not exist.
No *FINITE* sequence of inference steps.
Nothing can prove that itself does not
exist because that forms proof that it
does exist, dumbo.
So you are just ignoring context because you are stupid.
The statement, with the added information of the meta-system >>>>>>>>> proves (by a proof in the meta system) that the statment is true. >>>>>>>>>
Something else can prove that X cannot prove that
X does not exist, AKA your meta-system.
Nothing can directly prove that itself does not
exist because this forms proof that it does exist.
Nope, got a source for that?
Why does my explanation not work?
It is not that your explanation doesn't work.
It is that it ignores the root cause of why
G is unprovable in F.
So, how do you think you can prove it in F?
Nothing can prove that itself does not exist.
Any such proof would be self-refuting.
But it isn't the PROOF that does the proving, it is the statement.
THe statement G exist, and it is True.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
You keep on repeating that, but show you don't know what it means,
proving your stupidity.
When we name this proposition G then a proof of G
would be a sequence of inference steps that prove
that they themselves do not exist.
Right, we name the proposition G.
Then we form a set of steps in M, the Meta-system that proves that G is
True.
On 01/01/2026 01:29, Richard Damon wrote:
On 12/31/25 8:04 PM, olcott wrote:
On 12/31/2025 6:35 PM, Richard Damon wrote:
On 12/31/25 7:23 PM, olcott wrote:
On 12/31/2025 5:27 PM, Richard Damon wrote:
On 12/31/25 6:08 PM, olcott wrote:
On 12/31/2025 4:48 PM, Richard Damon wrote:
On 12/31/25 5:42 PM, olcott wrote:
On 12/31/2025 4:09 PM, Richard Damon wrote:
On 12/31/25 4:59 PM, olcott wrote:
On 12/31/2025 3:56 PM, Richard Damon wrote:
On 12/31/25 4:52 PM, olcott wrote:
On 12/31/2025 3:16 PM, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote: >>>>>>>>>>>>>>
[ ... ]
Then he defines a new system "P" which he uses to get even >>>>>>>>>>>>>>> more muddled,
leaves out the crucial elements of his proof because it's >>>>>>>>>>>>>>> too easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch that >>>>>>>>>>>>>> ever
lived!
and Stephen Meyer says he does get it wrong; he seems to be >>>>>>>>>>>>>>> the only person in the world that ever checked.
People have misunderstood G||del and proved it by their >>>>>>>>>>>>>> comments.
I don't know who Stephen Meyer is; my money is on G||del. >>>>>>>>>>>>>>
G||del proved that there cannot possibly exist any
sequence of inference steps in F prove that they
themselves do not exist.
No *FINITE* sequence of inference steps.
Nothing can prove that itself does not
exist because that forms proof that it
does exist, dumbo.
So you are just ignoring context because you are stupid.
The statement, with the added information of the meta-system >>>>>>>>>> proves (by a proof in the meta system) that the statment is true. >>>>>>>>>>
Something else can prove that X cannot prove that
X does not exist, AKA your meta-system.
Nothing can directly prove that itself does not
exist because this forms proof that it does exist.
Nope, got a source for that?
Why does my explanation not work?
It is not that your explanation doesn't work.
It is that it ignores the root cause of why
G is unprovable in F.
So, how do you think you can prove it in F?
Nothing can prove that itself does not exist.
Any such proof would be self-refuting.
But it isn't the PROOF that does the proving, it is the statement.
THe statement G exist, and it is True.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
You keep on repeating that, but show you don't know what it means,
proving your stupidity.
When we name this proposition G then a proof of G
would be a sequence of inference steps that prove
that they themselves do not exist.
Right, we name the proposition G.
Then we form a set of steps in M, the Meta-system that proves that G is
True.
But then all you've done is show that there exists a meta-system in
which a non-theorem of the base-system equal to "reo G" in the meta-system
is a theorem. Well, yes, there exist lots of them - one with "reo G" as
its sole axiom and the deduction rule that from "reo G" deduce the non-theorem of the base-system, for example.
Is this just a case of "can't see the woods through the trees" ?
On 01/01/2026 00:35, Richard Damon wrote:
THe statement G exist
Ah, I'm not so easily convinced
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
But it IS a theorem of the base system, as it uses ONLY the mathematical operations definable in the base system. What makes you think it isn't a Theorem in the base system.
On 1/1/26 5:13 PM, Tristan Wibberley wrote:
On 01/01/2026 00:35, Richard Damon wrote:
THe statement G exist
Ah, I'm not so easily convinced
What did he do that might allow it not to exist?
He constructs it by the rules of F, and shows that for it to not be
true, F must be inconsistant.
You can't just complain that you don't think something exists, when it
was constructed by the system.
On 1/1/26 5:12 PM, Tristan Wibberley wrote:
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
The base system. Depending on which discussion of Godel's paper you are reading it is described with different names.
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the mathematical
operations definable in the base system. What makes you think it isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't think
the base system were incomplete.
On 01/01/2026 22:42, Richard Damon wrote:
On 1/1/26 5:13 PM, Tristan Wibberley wrote:
On 01/01/2026 00:35, Richard Damon wrote:
THe statement G exist
Ah, I'm not so easily convinced
What did he do that might allow it not to exist?
He constructs it by the rules of F, and shows that for it to not be
true, F must be inconsistant.
You can't just complain that you don't think something exists, when it
was constructed by the system.
There's no symbol "G" in the system.
On 01/01/2026 22:43, Richard Damon wrote:
On 1/1/26 5:12 PM, Tristan Wibberley wrote:
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
The base system. Depending on which discussion of Godel's paper you are
reading it is described with different names.
His own 1931 paper calls it P. What's the probleme with that name?
Unless the discussion you're reading goes to the lengths that G||del went
to then it's inevitably wrong, and if it /does/ go to those lengths then
why wouldn't it require you to read G||del's paper as part of it?
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the mathematical >>> operations definable in the base system. What makes you think it isn't a >>> Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't think
the base system were incomplete.
It has no PROOF in the base system.
The statement is surely a statement in the base system.
It is shown to be true there, by a proof in the meta-system.
I think you are confused about what you talk about.
On 1/1/26 6:17 PM, Tristan Wibberley wrote:
On 01/01/2026 22:42, Richard Damon wrote:
On 1/1/26 5:13 PM, Tristan Wibberley wrote:
On 01/01/2026 00:35, Richard Damon wrote:
THe statement G exist
Ah, I'm not so easily convinced
What did he do that might allow it not to exist?
He constructs it by the rules of F, and shows that for it to not be
true, F must be inconsistant.
You can't just complain that you don't think something exists, when it
was constructed by the system.
There's no symbol "G" in the system.
Sure there is, as system allow the creation of names for objects in them.
Name a system that meets the basic requirements that doesn't allow the creation of a "name" for a statement in the system.
On 1/1/26 6:19 PM, Tristan Wibberley wrote:
On 01/01/2026 22:43, Richard Damon wrote:
On 1/1/26 5:12 PM, Tristan Wibberley wrote:
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
The base system. Depending on which discussion of Godel's paper you are
reading it is described with different names.
His own 1931 paper calls it P. What's the probleme with that name?
Unless the discussion you're reading goes to the lengths that G||del went
to then it's inevitably wrong, and if it /does/ go to those lengths then
why wouldn't it require you to read G||del's paper as part of it?
I use F here, as that is what Olcott calls it.
It isn't worth fighting him over that name.
On 01/01/2026 23:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the mathematical >>>> operations definable in the base system. What makes you think it isn't a >>>> Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't think
the base system were incomplete.
It has no PROOF in the base system.
The statement is surely a statement in the base system.
It is shown to be true there, by a proof in the meta-system.
I think you are confused about what you talk about.
No you're confused. You think its a theorem when its merely a statement.
On 01/01/2026 23:50, Richard Damon wrote:
On 1/1/26 6:17 PM, Tristan Wibberley wrote:
On 01/01/2026 22:42, Richard Damon wrote:
On 1/1/26 5:13 PM, Tristan Wibberley wrote:
On 01/01/2026 00:35, Richard Damon wrote:
THe statement G exist
Ah, I'm not so easily convinced
What did he do that might allow it not to exist?
He constructs it by the rules of F, and shows that for it to not be
true, F must be inconsistant.
You can't just complain that you don't think something exists, when it >>>> was constructed by the system.
There's no symbol "G" in the system.
Sure there is, as system allow the creation of names for objects in them.
Name a system that meets the basic requirements that doesn't allow the
creation of a "name" for a statement in the system.
Nope. The name is not a statement of the system, it's a statement of a related system such as a meta-system or extension.
On 01/01/2026 23:52, Richard Damon wrote:
On 1/1/26 6:19 PM, Tristan Wibberley wrote:
On 01/01/2026 22:43, Richard Damon wrote:
On 1/1/26 5:12 PM, Tristan Wibberley wrote:
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
The base system. Depending on which discussion of Godel's paper you are >>>> reading it is described with different names.
His own 1931 paper calls it P. What's the probleme with that name?
Unless the discussion you're reading goes to the lengths that G||del went >>> to then it's inevitably wrong, and if it /does/ go to those lengths then >>> why wouldn't it require you to read G||del's paper as part of it?
I use F here, as that is what Olcott calls it.
It isn't worth fighting him over that name.
It sure is, his F is not G||del's P. It's Russel and Whitehead's PM.
On 01/01/2026 23:50, Richard Damon wrote:
On 1/1/26 6:17 PM, Tristan Wibberley wrote:
On 01/01/2026 22:42, Richard Damon wrote:
On 1/1/26 5:13 PM, Tristan Wibberley wrote:
On 01/01/2026 00:35, Richard Damon wrote:
THe statement G exist
Ah, I'm not so easily convinced
What did he do that might allow it not to exist?
He constructs it by the rules of F, and shows that for it to not be
true, F must be inconsistant.
You can't just complain that you don't think something exists, when it >>>> was constructed by the system.
There's no symbol "G" in the system.
Sure there is, as system allow the creation of names for objects in them.
Name a system that meets the basic requirements that doesn't allow the
creation of a "name" for a statement in the system.
Nope. The name is not a statement of the system, it's a statement of a related system such as a meta-system or extension.
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
On 1/1/2026 4:12 PM, Tristan Wibberley wrote:
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
F reo G_F rao -4Prov_F(riLG_FriY)
F proves that: G_F is equivalent to G_F is not provable in F https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
reaG ree WFF(F) (G rao (F re4 G))
There exists a G in F that is logically
equivalent to its own unprovability in F
reaG ree WFF(F) (G := (F re4 G))
There exists a G in F that asserts its own unprovability in F
The proof of G in F would seem to require a sequence
of inference steps in F that prove that they themselves
do not exist.
On 1/1/26 9:07 PM, olcott wrote:
On 1/1/2026 4:12 PM, Tristan Wibberley wrote:
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
F reo G_F rao -4Prov_F(riLG_FriY)
F proves that: G_F is equivalent to G_F is not provable in F
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
reaG ree WFF(F) (G rao (F re4 G))
There exists a G in F that is logically
equivalent to its own unprovability in F
reaG ree WFF(F) (G := (F re4 G))
There exists a G in F that asserts its own unprovability in F
The proof of G in F would seem to require a sequence
of inference steps in F that prove that they themselves
do not exist.
But that isn't what G is in the proof, so you are just using a bad reference.
I guess you are just showing that you think lying is correct logic.--
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the mathematical >>> operations definable in the base system. What makes you think it isn't a >>> Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't think
the base system were incomplete.
It has no PROOF in the base system.
The statement is surely a statement in the base system.
It is shown to be true there, by a proof in the meta-system.
I think you are confused about what you talk about.
On 1/1/2026 8:25 PM, Richard Damon wrote:
On 1/1/26 9:07 PM, olcott wrote:
On 1/1/2026 4:12 PM, Tristan Wibberley wrote:
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
F reo G_F rao -4Prov_F(riLG_FriY)
F proves that: G_F is equivalent to G_F is not provable in F
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
reaG ree WFF(F) (G rao (F re4 G))
There exists a G in F that is logically
equivalent to its own unprovability in F
reaG ree WFF(F) (G := (F re4 G))
There exists a G in F that asserts its own unprovability in F
The proof of G in F would seem to require a sequence
of inference steps in F that prove that they themselves
do not exist.
But that isn't what G is in the proof, so you are just using a bad
reference.
That you do not know exactly how semantics works in
linguistics (making sure to ignore all context) is
not my mistake. The reason that Ludwig Wittgenstein
was never understood is that none of his detractors
understood how language itself really works. Not
knowing how language really works results in
undetected muddled thinking.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G asserts its own unprovability.
Is what the above means semantically.
The proof of G does semantically entail a sequence
of inference steps that prove that they themselves
do not exist.
I guess you are just showing that you think lying is correct logic.
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the
mathematical
operations definable in the base system. What makes you think it
isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't think
the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a
statement which can be proven in a particular system.
The statement is surely a statement in the base system.
Sure, but being a statement and being a theorem are two different things.
It is shown to be true there, by a proof in the meta-system.
It is shown to be true in the base system, but only within the
metasystem. Within the base system it cannot be so shown, which
precludes it from being a theorem. That's the entire point of G||del:
truth and theoremhood cannot be made to coincide except in very trivial systems.
I think you are confused about what you talk about.
I think you are both somewhat confused here.
Andr|-
On 1/1/2026 8:25 PM, Richard Damon wrote:
On 1/1/26 9:07 PM, olcott wrote:
On 1/1/2026 4:12 PM, Tristan Wibberley wrote:
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
F reo G_F rao -4Prov_F(riLG_FriY)
F proves that: G_F is equivalent to G_F is not provable in F
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
reaG ree WFF(F) (G rao (F re4 G))
There exists a G in F that is logically
equivalent to its own unprovability in F
reaG ree WFF(F) (G := (F re4 G))
There exists a G in F that asserts its own unprovability in F
The proof of G in F would seem to require a sequence
of inference steps in F that prove that they themselves
do not exist.
But that isn't what G is in the proof, so you are just using a bad
reference.
That you do not know exactly how semantics works in
linguistics (making sure to ignore all context) is
not my mistake. The reason that Ludwig Wittgenstein
was never understood is that none of his detractors
understood how language itself really works. Not
knowing how language really works results in
undetected muddled thinking.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G asserts its own unprovability.
Is what the above means semantically.
The proof of G does semantically entail a sequence
of inference steps that prove that they themselves
do not exist.
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the
mathematical
operations definable in the base system. What makes you think it
isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't think
the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a
statement which can be proven in a particular system.
The statement is surely a statement in the base system.
Sure, but being a statement and being a theorem are two different things.
It is shown to be true there, by a proof in the meta-system.
It is shown to be true in the base system, but only within the
metasystem. Within the base system it cannot be so shown, which
precludes it from being a theorem. That's the entire point of G||del:
truth and theoremhood cannot be made to coincide except in very trivial systems.
I think you are confused about what you talk about.
I think you are both somewhat confused here.
Andr|-
On 1/1/2026 8:38 PM, olcott wrote:
On 1/1/2026 8:25 PM, Richard Damon wrote:
On 1/1/26 9:07 PM, olcott wrote:
On 1/1/2026 4:12 PM, Tristan Wibberley wrote:
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
F reo G_F rao -4Prov_F(riLG_FriY)
F proves that: G_F is equivalent to G_F is not provable in F
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom >>>>
reaG ree WFF(F) (G rao (F re4 G))
There exists a G in F that is logically
equivalent to its own unprovability in F
reaG ree WFF(F) (G := (F re4 G))
There exists a G in F that asserts its own unprovability in F
The proof of G in F would seem to require a sequence
of inference steps in F that prove that they themselves
do not exist.
But that isn't what G is in the proof, so you are just using a bad
reference.
That you do not know exactly how semantics works in
linguistics (making sure to ignore all context) is
not my mistake. The reason that Ludwig Wittgenstein
was never understood is that none of his detractors
understood how language itself really works. Not
knowing how language really works results in
undetected muddled thinking.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G asserts its own unprovability.
Is what the above means semantically.
The proof of G does semantically entail a sequence
of inference steps that prove that they themselves
do not exist.
Ludwig Wittgenstein
8. I imagine someone asking my advice; he says:
"I have constructed a proposition (1 will use
'P' to designate it) in Russell's symbolism,
and by means of certain definitions and
transformations it can be so interpreted that
it says: 'P is not provable in Russell's system'.
Must I not say that this proposition on the one
hand is true, and on the other hand is unprovable?
For suppose it were false; then it is true that
it is provable. And that surely cannot be And
if it is proved, then it is proved that it is
not provable. Thus it can only be true, but
unprovable. " Just as we ask: " 'provable'
in what system?", so we must also ask:"
'true' in what system?" '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
On 1/1/2026 8:45 PM, Andr|- G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the
mathematical
operations definable in the base system. What makes you think it
isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't think
the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a
statement which can be proven in a particular system.
This is the kind of clarity that we need.
True in the base system essentially means
a theorem of the base system.
The statement is surely a statement in the base system.
Sure, but being a statement and being a theorem are two different things.
It is shown to be true there, by a proof in the meta-system.
It is shown to be true in the base system, but only within the
metasystem. Within the base system it cannot be so shown, which
precludes it from being a theorem. That's the entire point of G||del:
truth and theoremhood cannot be made to coincide except in very
trivial systems.
I think you are confused about what you talk about.
I think you are both somewhat confused here.
Andr|-
On 1/1/26 9:38 PM, olcott wrote:
On 1/1/2026 8:25 PM, Richard Damon wrote:
On 1/1/26 9:07 PM, olcott wrote:
On 1/1/2026 4:12 PM, Tristan Wibberley wrote:
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
F reo G_F rao -4Prov_F(riLG_FriY)
F proves that: G_F is equivalent to G_F is not provable in F
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom >>>>
reaG ree WFF(F) (G rao (F re4 G))
There exists a G in F that is logically
equivalent to its own unprovability in F
reaG ree WFF(F) (G := (F re4 G))
There exists a G in F that asserts its own unprovability in F
The proof of G in F would seem to require a sequence
of inference steps in F that prove that they themselves
do not exist.
But that isn't what G is in the proof, so you are just using a bad
reference.
That you do not know exactly how semantics works in
linguistics (making sure to ignore all context) is
not my mistake. The reason that Ludwig Wittgenstein
was never understood is that none of his detractors
understood how language itself really works. Not
knowing how language really works results in
undetected muddled thinking.
No, YOU don't know how semantics work, or linqustics.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
Which is a statement in NATURAL LANGUAGE and you need to use Natural Language "rules" to interpret it.
And thus each word need to include its context.
The proposition exists in both the base system and the meta system.
The assertion is just in the meta system, which understand the "hidden" meaning of the relationship that the statement is based on.
The unprovabiliyt is just in the base system, which doesn't know this meaning.
If you don't understand that you can't read a coded message without the
code book, you are just stupid.
G asserts its own unprovability.
Is what the above means semantically.
The proof of G does semantically entail a sequence
of inference steps that prove that they themselves
do not exist.
I two different systems.
I guuess to you cats are dog, Calulus is just 1st grade arithmatic.
Of course, it seems you can't understand either due to your stupidity.
I guess you are just showing that you think lying is correct logic.
On 1/1/26 10:33 PM, olcott wrote:
On 1/1/2026 8:45 PM, Andr|- G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the
mathematical
operations definable in the base system. What makes you think it
isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't think >>>>> the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a
statement which can be proven in a particular system.
This is the kind of clarity that we need.
True in the base system essentially means
a theorem of the base system.
Which s I explained, it is by at least the very normal definition.
It is a statement of fact in the base system.
And, that fact in the base system has been proven by a proof in some
system that knows of the base system.
If you want to limit a "Theorem" to only be a something provable in the
base system then it is merely a True Statement in the base system, which
the system can not be proven.
On 1/1/2026 8:52 PM, Richard Damon wrote:
On 1/1/26 9:38 PM, olcott wrote:
On 1/1/2026 8:25 PM, Richard Damon wrote:
On 1/1/26 9:07 PM, olcott wrote:
On 1/1/2026 4:12 PM, Tristan Wibberley wrote:
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
F reo G_F rao -4Prov_F(riLG_FriY)
F proves that: G_F is equivalent to G_F is not provable in F
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom >>>>>
reaG ree WFF(F) (G rao (F re4 G))
There exists a G in F that is logically
equivalent to its own unprovability in F
reaG ree WFF(F) (G := (F re4 G))
There exists a G in F that asserts its own unprovability in F
The proof of G in F would seem to require a sequence
of inference steps in F that prove that they themselves
do not exist.
But that isn't what G is in the proof, so you are just using a bad
reference.
That you do not know exactly how semantics works in
linguistics (making sure to ignore all context) is
not my mistake. The reason that Ludwig Wittgenstein
was never understood is that none of his detractors
understood how language itself really works. Not
knowing how language really works results in
undetected muddled thinking.
No, YOU don't know how semantics work, or linqustics.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
Which is a statement in NATURAL LANGUAGE and you need to use Natural
Language "rules" to interpret it.
I have taken "interpretation" as a twisted lie since
I was 14. Semantics of linguistics agrees.
It has always been the exact meanings that are specified.
it has never been the way that people twist this in
their mind.
a proposition which asserts its own unprovability
Does not mean a box of chocolates crushed on the floor.
It only means exactly one thing.
And thus each word need to include its context.
Linguistic Semantics is required to exclude context
Context is only included in linguistic pragmatics.
Your lack of knowledge never has been my mistake.
The proposition exists in both the base system and the meta system.
The assertion is just in the meta system, which understand the
"hidden" meaning of the relationship that the statement is based on.
The unprovabiliyt is just in the base system, which doesn't know this
meaning.
If you don't understand that you can't read a coded message without
the code book, you are just stupid.
G asserts its own unprovability.
Is what the above means semantically.
The proof of G does semantically entail a sequence
of inference steps that prove that they themselves
do not exist.
I two different systems.
I guuess to you cats are dog, Calulus is just 1st grade arithmatic.
Of course, it seems you can't understand either due to your stupidity.
I guess you are just showing that you think lying is correct logic.
On 1/1/2026 9:45 PM, Richard Damon wrote:
On 1/1/26 10:33 PM, olcott wrote:
On 1/1/2026 8:45 PM, Andr|- G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the
mathematical
operations definable in the base system. What makes you think it >>>>>>> isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't think >>>>>> the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a
statement which can be proven in a particular system.
This is the kind of clarity that we need.
True in the base system essentially means
a theorem of the base system.
Which s I explained, it is by at least the very normal definition.
It is a statement of fact in the base system.
And, that fact in the base system has been proven by a proof in some
system that knows of the base system.
Has always been irrelevant.
Truth in the base system has always
actually been theorems of the base system.
That is the way that
"true on the basis of meaning expressed in language"
has always worked. When math diverged math erred.
If you want to limit a "Theorem" to only be a something provable inSo when we directly encode all semantics
the base system then it is merely a True Statement in the base system,
which the system can not be proven.
in the formal language such that
reCx ree F (Provable(F,x) rei True(F,x))
Then incompleteness ceases to exist
On 1/1/26 11:17 PM, olcott wrote:
On 1/1/2026 8:52 PM, Richard Damon wrote:
On 1/1/26 9:38 PM, olcott wrote:
On 1/1/2026 8:25 PM, Richard Damon wrote:
On 1/1/26 9:07 PM, olcott wrote:
On 1/1/2026 4:12 PM, Tristan Wibberley wrote:
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
F reo G_F rao -4Prov_F(riLG_FriY)
F proves that: G_F is equivalent to G_F is not provable in F
https://plato.stanford.edu/entries/goedel-incompleteness/
#FirIncTheCom
reaG ree WFF(F) (G rao (F re4 G))
There exists a G in F that is logically
equivalent to its own unprovability in F
reaG ree WFF(F) (G := (F re4 G))
There exists a G in F that asserts its own unprovability in F
The proof of G in F would seem to require a sequence
of inference steps in F that prove that they themselves
do not exist.
But that isn't what G is in the proof, so you are just using a bad
reference.
That you do not know exactly how semantics works in
linguistics (making sure to ignore all context) is
not my mistake. The reason that Ludwig Wittgenstein
was never understood is that none of his detractors
understood how language itself really works. Not
knowing how language really works results in
undetected muddled thinking.
No, YOU don't know how semantics work, or linqustics.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
Which is a statement in NATURAL LANGUAGE and you need to use Natural
Language "rules" to interpret it.
I have taken "interpretation" as a twisted lie since
I was 14. Semantics of linguistics agrees.
It has always been the exact meanings that are specified.
it has never been the way that people twist this in
their mind.
In other words, you just lie and are stupid.
The "interpreation" mentioned IS EXACTLY what is specified, but you are
just too stupid to understand,
a proposition which asserts its own unprovability
Does not mean a box of chocolates crushed on the floor.
It only means exactly one thing.
Right, but neither does it mean, in its context. what you try to make it.
And thus each word need to include its context.
Linguistic Semantics is required to exclude context
Nope, as context affect the semantics of a word.
Yes, sometimes "Semantics" is used to talk about giving the full list of possible meanings, but if you are using it that way, then you need to
list not just one meaning, but all the possible means in all possible contexts.
Context is only included in linguistic pragmatics.
Nope. Not unless you are meaning "Semantics" to give the list of
possible meaning and pragmatics to determine which one.
In which case, you can't use just "Semantics" as you base, as you thus
admit you don't actually know what the sentence means, just the wide assortment of possible meanings.
Your lack of knowledge never has been my mistake.
No, your stupidity is yours.
It seems you just don't know the actual meaning of what you are talking about as you start from an incomplete semantics and forget to apply pragmatics to it.
The proposition exists in both the base system and the meta system.
The assertion is just in the meta system, which understand the
"hidden" meaning of the relationship that the statement is based on.
The unprovabiliyt is just in the base system, which doesn't know this
meaning.
If you don't understand that you can't read a coded message without
the code book, you are just stupid.
G asserts its own unprovability.
Is what the above means semantically.
The proof of G does semantically entail a sequence
of inference steps that prove that they themselves
do not exist.
I two different systems.
I guuess to you cats are dog, Calulus is just 1st grade arithmatic.
Of course, it seems you can't understand either due to your stupidity.
I guess you are just showing that you think lying is correct logic.
It started as a statement, and then was proven true in the system (but
not by the system).
What else do you need to make it a Theorem?
I guess it depends on your definition of a "Theorem".
I am using the one that goes:
"A Theorem is a statement that has been proven."
note, no restriction that the proof was in the system the Theorem is
stated in, as long as the proof shows that it is actually True in that system.
Do you have a source that limits the proof to the system in question?
Perhaps this is just a diffence of schools of logic.
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the
mathematical
operations definable in the base system. What makes you think it
isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't think
the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a
statement which can be proven in a particular system.
True in the base system essentially means
a theorem of the base system.
On 1/1/26 11:22 PM, olcott wrote:
On 1/1/2026 9:45 PM, Richard Damon wrote:
On 1/1/26 10:33 PM, olcott wrote:
On 1/1/2026 8:45 PM, Andr|- G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the
mathematical
operations definable in the base system. What makes you think it >>>>>>>> isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't >>>>>>> think
the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a
statement which can be proven in a particular system.
This is the kind of clarity that we need.
True in the base system essentially means
a theorem of the base system.
Which s I explained, it is by at least the very normal definition.
It is a statement of fact in the base system.
And, that fact in the base system has been proven by a proof in some
system that knows of the base system.
Has always been irrelevant.
Nope. Got a reference?
Truth in the base system has always
actually been theorems of the base system.
But only if "Theorem" includes things proven to be true in the system
even if the proof is in another.
Truth DOES need to be based on the axioms of the base system, but allows
the truth to be established by an infinite chain of reasoning, unlike
proofs that need to be finite.
That is the way that
"true on the basis of meaning expressed in language"
has always worked. When math diverged math erred.
Nope. Not unless you mean by "meaning" to include the infinite chain for reasoning.
Note, "Formal Systems" don't work the way you want, as their "semanitcs"
are defined from the axioms and the operations of the system, possible continued for an infinite chain of operations.
Your problem is you just don't comprehend how infinity works, because
you mind is just to small.
If you want to limit a "Theorem" to only be a something provable inSo when we directly encode all semantics
the base system then it is merely a True Statement in the base
system, which the system can not be proven.
in the formal language such that
reCx ree F (Provable(F,x) rei True(F,x))
Then incompleteness ceases to exist
Nope, because you CAN'T do that unless you system can't support the
Natural Numbers.
Sorry, you just aren't allowed to ASSUME something like that.--
Your world is just exploded into a totally inconsistent mess.
Truth in the base system has always
actually been theorems of the base system.
But only if "Theorem" includes things proven to be true in the system
even if the proof is in another.
Truth DOES need to be based on the axioms of the base system, but allows
the truth to be established by an infinite chain of reasoning, unlike
proofs that need to be finite.
On 1/1/26 7:12 PM, Tristan Wibberley wrote:
On 01/01/2026 23:50, Richard Damon wrote:
On 1/1/26 6:17 PM, Tristan Wibberley wrote:
On 01/01/2026 22:42, Richard Damon wrote:
On 1/1/26 5:13 PM, Tristan Wibberley wrote:
On 01/01/2026 00:35, Richard Damon wrote:
THe statement G exist
Ah, I'm not so easily convinced
What did he do that might allow it not to exist?
He constructs it by the rules of F, and shows that for it to not be
true, F must be inconsistant.
You can't just complain that you don't think something exists, when it >>>>> was constructed by the system.
There's no symbol "G" in the system.
Sure there is, as system allow the creation of names for objects in
them.
Name a system that meets the basic requirements that doesn't allow the
creation of a "name" for a statement in the system.
Nope. The name is not a statement of the system, it's a statement of a
related system such as a meta-system or extension.
No, G is the statement created in the system, using the mathematical relationship defined in terms of operations in the system build in the
meta system.
G HAS to be in the system, so the PRR can refer to it.
OR, are you saying that in the system of arithmetic, we can't talk about
a variable "x" as it isn't defined in the system?
On 1/1/2026 8:38 PM, olcott wrote:
On 1/1/2026 8:25 PM, Richard Damon wrote:
On 1/1/26 9:07 PM, olcott wrote:
On 1/1/2026 4:12 PM, Tristan Wibberley wrote:
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
F reo G_F rao -4Prov_F(riLG_FriY)
F proves that: G_F is equivalent to G_F is not provable in F
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom >>>>
reaG ree WFF(F) (G rao (F re4 G))
There exists a G in F that is logically
equivalent to its own unprovability in F
reaG ree WFF(F) (G := (F re4 G))
There exists a G in F that asserts its own unprovability in F
The proof of G in F would seem to require a sequence
of inference steps in F that prove that they themselves
do not exist.
But that isn't what G is in the proof, so you are just using a bad
reference.
That you do not know exactly how semantics works in
linguistics (making sure to ignore all context) is
not my mistake. The reason that Ludwig Wittgenstein
was never understood is that none of his detractors
understood how language itself really works. Not
knowing how language really works results in
undetected muddled thinking.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G asserts its own unprovability.
Is what the above means semantically.
The proof of G does semantically entail a sequence
of inference steps that prove that they themselves
do not exist.
Ludwig Wittgenstein
8. I imagine someone asking my advice; he says:
"I have constructed a proposition (1 will use
'P' to designate it) in Russell's symbolism,
and by means of certain definitions and
transformations it can be so interpreted that
it says: 'P is not provable in Russell's system'.
On 02/01/2026 02:45, Andr|- G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the
mathematical
operations definable in the base system. What makes you think it
isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't think
the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a
statement which can be proven in a particular system.
We have to avoid "proven in a particular system" and choose "Derived in
a particular system" or "Derived of a particular system" or, since it's
well defined, "Theorem of a particular system".
The problem with "prove" is there are numerous episystems (HA being
popular) that provide for "proofs" of statements of systems they're
applied to. Technically, episystems may or may not prove the same set of statements that are theorem's of the system they're applied to.
On 02/01/2026 03:33, olcott wrote:
True in the base system essentially means
a theorem of the base system.
No, but a lot of people might say "true in the base system" when they
ought to say "a theorem of the base system" which means there is a
derivation from the axioms of the formal system using only the deduction rules of the formal system (which are restricted in what they can
possibly be).
"True" has such a variety of meanings that it should be avoided except
for when it describes the speaker's feelings about reality.
On 02/01/2026 04:45, Richard Damon wrote:
Truth in the base system has always
actually been theorems of the base system.
But only if "Theorem" includes things proven to be true in the system
even if the proof is in another.
If the statement is derived in another then it is a theorem of the other.
If it is merely "proved" by a proof episystem then it might not be a
theorem of either depending on the episystem and what is conventionally referred to as "proof" by that system. An intuitively safe episystem [my term, intended to carry some intuitive meaning] proves only its own
theorems and /labelled/ embeddings of just the theorems of the system
it's applied to, thus it provides alternative methods to find and
demonstrate theorems of the embedded system (and to reason about the theory-proper of the embedded system) while being clear about which
system(s) it reasons about.
I don't know of any that do the required labelling except that some
standard ones have such well established conventional symbols and are so small and intuitive (HA, HC, for example) that they are quite safe.
Haskell Curry tried in his 1950 Theory of Formal Deducibility to
establish some conventions around the use of the turnstile symbols but
it seems like they didn't take hold.
Truth DOES need to be based on the axioms of the base system, but allows
the truth to be established by an infinite chain of reasoning, unlike
proofs that need to be finite.
An infinite chain of reasoning is not completed at any time, least of
all this time. The limit of a chain of reasoning might be, episystems
could be useful for that, I wouldn't want to rule it out.
On 02/01/2026 00:23, Richard Damon wrote:
On 1/1/26 7:12 PM, Tristan Wibberley wrote:
On 01/01/2026 23:50, Richard Damon wrote:
On 1/1/26 6:17 PM, Tristan Wibberley wrote:
On 01/01/2026 22:42, Richard Damon wrote:
On 1/1/26 5:13 PM, Tristan Wibberley wrote:
On 01/01/2026 00:35, Richard Damon wrote:
THe statement G exist
Ah, I'm not so easily convinced
What did he do that might allow it not to exist?
He constructs it by the rules of F, and shows that for it to not be >>>>>> true, F must be inconsistant.
You can't just complain that you don't think something exists, when it >>>>>> was constructed by the system.
There's no symbol "G" in the system.
Sure there is, as system allow the creation of names for objects in
them.
Name a system that meets the basic requirements that doesn't allow the >>>> creation of a "name" for a statement in the system.
Nope. The name is not a statement of the system, it's a statement of a
related system such as a meta-system or extension.
No, G is the statement created in the system, using the mathematical
relationship defined in terms of operations in the system build in the
meta system.
G HAS to be in the system, so the PRR can refer to it.
OR, are you saying that in the system of arithmetic, we can't talk about
a variable "x" as it isn't defined in the system?
Godel's system P has variable objects, but no indeterminates. And it's namespace of Godel numbers is full up. You can do /some/ things like definitions using existential and universal quantification but the
character of the propositions is different than a definition of a new
symbol due to the Godel numbering; you have to be careful and not throw statements around like Goedel's introductory simile based on PM.
On 02/01/2026 03:26, olcott wrote:
On 1/1/2026 8:38 PM, olcott wrote:
On 1/1/2026 8:25 PM, Richard Damon wrote:
On 1/1/26 9:07 PM, olcott wrote:
On 1/1/2026 4:12 PM, Tristan Wibberley wrote:
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
F reo G_F rao -4Prov_F(riLG_FriY)
F proves that: G_F is equivalent to G_F is not provable in F
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom >>>>>
reaG ree WFF(F) (G rao (F re4 G))
There exists a G in F that is logically
equivalent to its own unprovability in F
reaG ree WFF(F) (G := (F re4 G))
There exists a G in F that asserts its own unprovability in F
The proof of G in F would seem to require a sequence
of inference steps in F that prove that they themselves
do not exist.
But that isn't what G is in the proof, so you are just using a bad
reference.
That you do not know exactly how semantics works in
linguistics (making sure to ignore all context) is
not my mistake. The reason that Ludwig Wittgenstein
was never understood is that none of his detractors
understood how language itself really works. Not
knowing how language really works results in
undetected muddled thinking.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G asserts its own unprovability.
Is what the above means semantically.
The proof of G does semantically entail a sequence
of inference steps that prove that they themselves
do not exist.
Ludwig Wittgenstein
8. I imagine someone asking my advice; he says:
"I have constructed a proposition (1 will use
'P' to designate it) in Russell's symbolism,
and by means of certain definitions and
transformations it can be so interpreted that
it says: 'P is not provable in Russell's system'.
False. He did not do that; he tried to do so then hallucinated that he succeeded. A contradiction follows from the negation of my
characterisation of his actions and so from the truth of the proposition
that he defined P so. That definitional proposition follows from the
axioms of inconsistent systems and not from those of useful consistent
ones. Typically it /is/ an axiom of inconsistent systems and not of consistent ones.
On 1/1/2026 10:45 PM, Richard Damon wrote:
On 1/1/26 11:22 PM, olcott wrote:
On 1/1/2026 9:45 PM, Richard Damon wrote:
On 1/1/26 10:33 PM, olcott wrote:
On 1/1/2026 8:45 PM, Andr|- G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the >>>>>>>>> mathematical
operations definable in the base system. What makes you think >>>>>>>>> it isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't >>>>>>>> think
the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a >>>>>> statement which can be proven in a particular system.
This is the kind of clarity that we need.
True in the base system essentially means
a theorem of the base system.
Which s I explained, it is by at least the very normal definition.
It is a statement of fact in the base system.
And, that fact in the base system has been proven by a proof in some
system that knows of the base system.
Has always been irrelevant.
Nope. Got a reference?
Truth in the base system has always
actually been theorems of the base system.
But only if "Theorem" includes things proven to be true in the system
even if the proof is in another.
Truth DOES need to be based on the axioms of the base system, but
allows the truth to be established by an infinite chain of reasoning,
unlike proofs that need to be finite.
That is the way that
"true on the basis of meaning expressed in language"
has always worked. When math diverged math erred.
Nope. Not unless you mean by "meaning" to include the infinite chain
for reasoning.
Note, "Formal Systems" don't work the way you want, as their
"semanitcs" are defined from the axioms and the operations of the
system, possible continued for an infinite chain of operations.
Your problem is you just don't comprehend how infinity works, because
you mind is just to small.
If you want to limit a "Theorem" to only be a something provable inSo when we directly encode all semantics
the base system then it is merely a True Statement in the base
system, which the system can not be proven.
in the formal language such that
reCx ree F (Provable(F,x) rei True(F,x))
Then incompleteness ceases to exist
Nope, because you CAN'T do that unless you system can't support the
Natural Numbers.
What do you think is missing from
"true on the basis of meaning expressed in language"
about natural numbers?
add/subtract/multiply/divide is all there
Sorry, you just aren't allowed to ASSUME something like that.
Your world is just exploded into a totally inconsistent mess.
On 1/2/2026 2:15 AM, Tristan Wibberley wrote:
On 02/01/2026 03:26, olcott wrote:
On 1/1/2026 8:38 PM, olcott wrote:
On 1/1/2026 8:25 PM, Richard Damon wrote:
On 1/1/26 9:07 PM, olcott wrote:
On 1/1/2026 4:12 PM, Tristan Wibberley wrote:
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
F reo G_F rao -4Prov_F(riLG_FriY)
F proves that: G_F is equivalent to G_F is not provable in F
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom >>>>>>
reaG ree WFF(F) (G rao (F re4 G))
There exists a G in F that is logically
equivalent to its own unprovability in F
reaG ree WFF(F) (G := (F re4 G))
There exists a G in F that asserts its own unprovability in F
The proof of G in F would seem to require a sequence
of inference steps in F that prove that they themselves
do not exist.
But that isn't what G is in the proof, so you are just using a bad
reference.
That you do not know exactly how semantics works in
linguistics (making sure to ignore all context) is
not my mistake. The reason that Ludwig Wittgenstein
was never understood is that none of his detractors
understood how language itself really works. Not
knowing how language really works results in
undetected muddled thinking.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G asserts its own unprovability.
Is what the above means semantically.
The proof of G does semantically entail a sequence
of inference steps that prove that they themselves
do not exist.
Ludwig Wittgenstein
8. I imagine someone asking my advice; he says:
"I have constructed a proposition (1 will use
'P' to designate it) in Russell's symbolism,
and by means of certain definitions and
transformations it can be so interpreted that
it says: 'P is not provable in Russell's system'.
False. He did not do that; he tried to do so then hallucinated that he
succeeded. A contradiction follows from the negation of my
characterisation of his actions and so from the truth of the proposition
that he defined P so. That definitional proposition follows from the
axioms of inconsistent systems and not from those of useful consistent
ones. Typically it /is/ an axiom of inconsistent systems and not of
consistent ones.
His paper is a convoluted mess hiding this simple fact
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
When we combine that with this:
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.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Foundations of Mathematical Logic 1977
Then G||del simply made a very convoluted analog
to the Liar Paradox.
On 02/01/2026 04:45, Richard Damon wrote:
Truth in the base system has always
actually been theorems of the base system.
But only if "Theorem" includes things proven to be true in the system
even if the proof is in another.
If the statement is derived in another then it is a theorem of the other.
If it is merely "proved" by a proof episystem then it might not be a
theorem of either depending on the episystem and what is conventionally referred to as "proof" by that system. An intuitively safe episystem [my term, intended to carry some intuitive meaning] proves only its own
theorems and /labelled/ embeddings of just the theorems of the system
it's applied to, thus it provides alternative methods to find and
demonstrate theorems of the embedded system (and to reason about the theory-proper of the embedded system) while being clear about which
system(s) it reasons about.
I don't know of any that do the required labelling except that some
standard ones have such well established conventional symbols and are so small and intuitive (HA, HC, for example) that they are quite safe.
Haskell Curry tried in his 1950 Theory of Formal Deducibility to
establish some conventions around the use of the turnstile symbols but
it seems like they didn't take hold.
Truth DOES need to be based on the axioms of the base system, but allows
the truth to be established by an infinite chain of reasoning, unlike
proofs that need to be finite.
An infinite chain of reasoning is not completed at any time, least of
all this time. The limit of a chain of reasoning might be, episystems
could be useful for that, I wouldn't want to rule it out.
On 02/01/2026 03:33, olcott wrote:
True in the base system essentially means
a theorem of the base system.
No, but a lot of people might say "true in the base system" when they
ought to say "a theorem of the base system" which means there is a
derivation from the axioms of the formal system using only the deduction rules of the formal system (which are restricted in what they can
possibly be).
"True" has such a variety of meanings that it should be avoided except
for when it describes the speaker's feelings about reality.
On 02/01/2026 02:45, Andr|- G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the
mathematical
operations definable in the base system. What makes you think it
isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't think
the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a
statement which can be proven in a particular system.
We have to avoid "proven in a particular system" and choose "Derived in
a particular system" or "Derived of a particular system" or, since it's
well defined, "Theorem of a particular system".
The problem with "prove" is there are numerous episystems (HA being
popular) that provide for "proofs" of statements of systems they're
applied to. Technically, episystems may or may not prove the same set of statements that are theorem's of the system they're applied to.
On 1/1/2026 11:54 PM, Tristan Wibberley wrote:
On 02/01/2026 02:45, Andr|- G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the
mathematical
operations definable in the base system. What makes you think it
isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't think >>>>> the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a
statement which can be proven in a particular system.
We have to avoid "proven in a particular system" and choose "Derived in
a particular system" or "Derived of a particular system" or, since it's
well defined, "Theorem of a particular system".
The problem with "prove" is there are numerous episystems (HA being
popular) that provide for "proofs" of statements of systems they're
applied to. Technically, episystems may or may not prove the same set of
statements that are theorem's of the system they're applied to.
The term "EpiSystems" most commonly refers to Epic Systems Corporation,
the dominant provider of Electronic Health Record (EHR) software in the United States. The reference to "HA being popular" likely means that
their software is very popular in Healthcare Academia, large hospital systems, and associated medical facilities.
On 02/01/2026 00:23, Richard Damon wrote:
On 1/1/26 7:12 PM, Tristan Wibberley wrote:
On 01/01/2026 23:50, Richard Damon wrote:
On 1/1/26 6:17 PM, Tristan Wibberley wrote:
On 01/01/2026 22:42, Richard Damon wrote:
On 1/1/26 5:13 PM, Tristan Wibberley wrote:
On 01/01/2026 00:35, Richard Damon wrote:
THe statement G exist
Ah, I'm not so easily convinced
What did he do that might allow it not to exist?
He constructs it by the rules of F, and shows that for it to not be >>>>>> true, F must be inconsistant.
You can't just complain that you don't think something exists, when it >>>>>> was constructed by the system.
There's no symbol "G" in the system.
Sure there is, as system allow the creation of names for objects in
them.
Name a system that meets the basic requirements that doesn't allow the >>>> creation of a "name" for a statement in the system.
Nope. The name is not a statement of the system, it's a statement of a
related system such as a meta-system or extension.
No, G is the statement created in the system, using the mathematical
relationship defined in terms of operations in the system build in the
meta system.
G HAS to be in the system, so the PRR can refer to it.
OR, are you saying that in the system of arithmetic, we can't talk about
a variable "x" as it isn't defined in the system?
Godel's system P has variable objects, but no indeterminates. And it's namespace of Godel numbers is full up. You can do /some/ things like definitions using existential and universal quantification but the
character of the propositions is different than a definition of a new
symbol due to the Godel numbering; you have to be careful and not throw statements around like Goedel's introductory simile based on PM.
On 1/2/2026 12:20 AM, Tristan Wibberley wrote:
On 02/01/2026 00:23, Richard Damon wrote:
On 1/1/26 7:12 PM, Tristan Wibberley wrote:
On 01/01/2026 23:50, Richard Damon wrote:
On 1/1/26 6:17 PM, Tristan Wibberley wrote:
On 01/01/2026 22:42, Richard Damon wrote:
On 1/1/26 5:13 PM, Tristan Wibberley wrote:
On 01/01/2026 00:35, Richard Damon wrote:
THe statement G exist
Ah, I'm not so easily convinced
What did he do that might allow it not to exist?
He constructs it by the rules of F, and shows that for it to not be >>>>>>> true, F must be inconsistant.
You can't just complain that you don't think something exists,
when it
was constructed by the system.
There's no symbol "G" in the system.
Sure there is, as system allow the creation of names for objects in
them.
Name a system that meets the basic requirements that doesn't allow the >>>>> creation of a "name" for a statement in the system.
Nope. The name is not a statement of the system, it's a statement of a >>>> related system such as a meta-system or extension.
No, G is the statement created in the system, using the mathematical
relationship defined in terms of operations in the system build in the
meta system.
G HAS to be in the system, so the PRR can refer to it.
OR, are you saying that in the system of arithmetic, we can't talk about >>> a variable "x" as it isn't defined in the system?
Godel's system P has variable objects, but no indeterminates. And it's
namespace of Godel numbers is full up. You can do /some/ things like
definitions using existential and universal quantification but the
character of the propositions is different than a definition of a new
symbol due to the Godel numbering; you have to be careful and not throw
statements around like Goedel's introductory simile based on PM.
His paper is a convoluted mess hiding this simple fact
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
On 1/1/2026 10:32 PM, Richard Damon wrote:
On 1/1/26 11:17 PM, olcott wrote:
Context is only included in linguistic pragmatics.
Nope. Not unless you are meaning "Semantics" to give the list of
possible meaning and pragmatics to determine which one.
Compositionality is a concept in the philosophy of
language. A symbolic system is compositional if the
meaning of every complex expression E in that system
depends on, and depends only on, (i) ErCOs syntactic
structure and (ii) the meanings of ErCOs simple parts.
If a language is compositional, then the meaning of
a sentence S in that language cannot depend directly
on the context that sentence is used in or the intentions
of the speaker who uses it.
https://iep.utm.edu/compositionality-in-language/
Le 02/01/2026 |a 15:47, olcott a |-crit :
On 1/2/2026 2:15 AM, Tristan Wibberley wrote:
On 02/01/2026 03:26, olcott wrote:
On 1/1/2026 8:38 PM, olcott wrote:
On 1/1/2026 8:25 PM, Richard Damon wrote:
On 1/1/26 9:07 PM, olcott wrote:
On 1/1/2026 4:12 PM, Tristan Wibberley wrote:
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
F reo G_F rao -4Prov_F(riLG_FriY)
F proves that: G_F is equivalent to G_F is not provable in F
https://plato.stanford.edu/entries/goedel-incompleteness/
#FirIncTheCom
reaG ree WFF(F) (G rao (F re4 G))
There exists a G in F that is logically
equivalent to its own unprovability in F
reaG ree WFF(F) (G := (F re4 G))
There exists a G in F that asserts its own unprovability in F
The proof of G in F would seem to require a sequence
of inference steps in F that prove that they themselves
do not exist.
But that isn't what G is in the proof, so you are just using a bad >>>>>> reference.
That you do not know exactly how semantics works in
linguistics (making sure to ignore all context) is
not my mistake. The reason that Ludwig Wittgenstein
was never understood is that none of his detractors
understood how language itself really works. Not
knowing how language really works results in
undetected muddled thinking.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G asserts its own unprovability.
Is what the above means semantically.
The proof of G does semantically entail a sequence
of inference steps that prove that they themselves
do not exist.
Ludwig Wittgenstein
8. I imagine someone asking my advice; he says:
"I have constructed a proposition (1 will use
'P' to designate it) in Russell's symbolism,
and by means of certain definitions and
transformations it can be so interpreted that
it says: 'P is not provable in Russell's system'.
False. He did not do that; he tried to do so then hallucinated that he
succeeded. A contradiction follows from the negation of my
characterisation of his actions and so from the truth of the proposition >>> that he defined P so. That definitional proposition follows from the
axioms of inconsistent systems and not from those of useful consistent
ones. Typically it /is/ an axiom of inconsistent systems and not of
consistent ones.
His paper is a convoluted mess hiding this simple fact
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
When we combine that with this:
-a-a-a Let {T} be such a theory. Then the elementary
-a-a-a statements which belong to {T} we shall call the
-a-a-a elementary theorems of {T}; we also say that
-a-a-a these elementary statements are true for {T}.
-a-a-a Thus, given {T}, an elementary theorem is an
-a-a-a elementary statement which is true.
-a-a-a https://www.liarparadox.org/Haskell_Curry_45.pdf
Foundations of Mathematical Logic 1977
Then G||del simply made a very convoluted analog
to the Liar Paradox.
This is delusional wishful thinking on your part.
Your whole "work" is a defense of your ego you've forged from the fact
that you misunderstand G||del's articles (and many others).
The real mess is you, Peter.
On 02/01/2026 00:21, Richard Damon wrote:
It started as a statement, and then was proven true in the system (but
not by the system).
What else do you need to make it a Theorem?
You have to derive the statement from the axioms of the system using the deduction rules of the system. It's the actual definition of "Theorem of
a Formal System".
Deriving it or other demonstration in your choice of alternative system
does not make it a theorem of the base-system.
On 1/2/2026 9:11 AM, Python wrote:
Le 02/01/2026 |a 15:47, olcott a |-crit :
On 1/2/2026 2:15 AM, Tristan Wibberley wrote:
On 02/01/2026 03:26, olcott wrote:
On 1/1/2026 8:38 PM, olcott wrote:
On 1/1/2026 8:25 PM, Richard Damon wrote:
On 1/1/26 9:07 PM, olcott wrote:
On 1/1/2026 4:12 PM, Tristan Wibberley wrote:
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
F reo G_F rao -4Prov_F(riLG_FriY)
F proves that: G_F is equivalent to G_F is not provable in F
https://plato.stanford.edu/entries/goedel-incompleteness/
#FirIncTheCom
reaG ree WFF(F) (G rao (F re4 G))
There exists a G in F that is logically
equivalent to its own unprovability in F
reaG ree WFF(F) (G := (F re4 G))
There exists a G in F that asserts its own unprovability in F
The proof of G in F would seem to require a sequence
of inference steps in F that prove that they themselves
do not exist.
But that isn't what G is in the proof, so you are just using a bad >>>>>>> reference.
That you do not know exactly how semantics works in
linguistics (making sure to ignore all context) is
not my mistake. The reason that Ludwig Wittgenstein
was never understood is that none of his detractors
understood how language itself really works. Not
knowing how language really works results in
undetected muddled thinking.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G asserts its own unprovability.
Is what the above means semantically.
The proof of G does semantically entail a sequence
of inference steps that prove that they themselves
do not exist.
Ludwig Wittgenstein
8. I imagine someone asking my advice; he says:
"I have constructed a proposition (1 will use
'P' to designate it) in Russell's symbolism,
and by means of certain definitions and
transformations it can be so interpreted that
it says: 'P is not provable in Russell's system'.
False. He did not do that; he tried to do so then hallucinated that he >>>> succeeded. A contradiction follows from the negation of my
characterisation of his actions and so from the truth of the
proposition
that he defined P so. That definitional proposition follows from the
axioms of inconsistent systems and not from those of useful consistent >>>> ones. Typically it /is/ an axiom of inconsistent systems and not of
consistent ones.
His paper is a convoluted mess hiding this simple fact
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
When we combine that with this:
-a-a-a Let {T} be such a theory. Then the elementary
-a-a-a statements which belong to {T} we shall call the
-a-a-a elementary theorems of {T}; we also say that
-a-a-a these elementary statements are true for {T}.
-a-a-a Thus, given {T}, an elementary theorem is an
-a-a-a elementary statement which is true.
-a-a-a https://www.liarparadox.org/Haskell_Curry_45.pdf
Foundations of Mathematical Logic 1977
Then G||del simply made a very convoluted analog
to the Liar Paradox.
This is delusional wishful thinking on your part.
Your whole "work" is a defense of your ego you've forged from the fact
that you misunderstand G||del's articles (and many others).
The real mess is you, Peter.
LLM systems initially said this too.
They give me lots and lots of push-back.
When they finally understand my whole
system they always totally agree, 50 times now.
Two big advantages of LLM systems
(1) The have no egoic attachment to conventional wisdom
(2) They have deep knowledge across
(a) theory of computation
(b) foundations of mathematics
(c) foundations of logic
(d) Linguistic semantics
(e) Philosophy of all of the above.
With (1) and without (2)(d) and (2)(e) people
lack a sufficient basis to understand me.
This was the exact same issue for Ludwig Wittgenstein.
On 02/01/2026 03:09, Richard Damon wrote:
I guess it depends on your definition of a "Theorem".
I am using the one that goes:
"A Theorem is a statement that has been proven."
note, no restriction that the proof was in the system the Theorem is
stated in, as long as the proof shows that it is actually True in that
system.
Do you have a source that limits the proof to the system in question?
Perhaps this is just a diffence of schools of logic.
See Curry and Feys Combinatory Logic 1, Chapters 0-2. The term is well defined as it applies to formal systems. It's out of copyright and
available online.
Informal logic is another matter, where "theorem" /may/ be taken to mean
what you thought and where mistakes are commonplace, and pretty-much inevitable.
"Truth" is not part of formal systems except in AI where personal
intuitive philosophy is computationally modelled. "Truth" is personal as
any spiritualist can attest.
On 1/2/2026 9:11 AM, Python wrote:
Le 02/01/2026 |a 15:47, olcott a |-crit :
On 1/2/2026 2:15 AM, Tristan Wibberley wrote:
On 02/01/2026 03:26, olcott wrote:
On 1/1/2026 8:38 PM, olcott wrote:
On 1/1/2026 8:25 PM, Richard Damon wrote:
On 1/1/26 9:07 PM, olcott wrote:
On 1/1/2026 4:12 PM, Tristan Wibberley wrote:
On 31/12/2025 23:27, Richard Damon wrote:
So, how do you think you can prove it in F?
What does "F" refer to?
F reo G_F rao -4Prov_F(riLG_FriY)
F proves that: G_F is equivalent to G_F is not provable in F
https://plato.stanford.edu/entries/goedel-incompleteness/
#FirIncTheCom
reaG ree WFF(F) (G rao (F re4 G))
There exists a G in F that is logically
equivalent to its own unprovability in F
reaG ree WFF(F) (G := (F re4 G))
There exists a G in F that asserts its own unprovability in F
The proof of G in F would seem to require a sequence
of inference steps in F that prove that they themselves
do not exist.
But that isn't what G is in the proof, so you are just using a bad >>>>>>> reference.
That you do not know exactly how semantics works in
linguistics (making sure to ignore all context) is
not my mistake. The reason that Ludwig Wittgenstein
was never understood is that none of his detractors
understood how language itself really works. Not
knowing how language really works results in
undetected muddled thinking.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G asserts its own unprovability.
Is what the above means semantically.
The proof of G does semantically entail a sequence
of inference steps that prove that they themselves
do not exist.
Ludwig Wittgenstein
8. I imagine someone asking my advice; he says:
"I have constructed a proposition (1 will use
'P' to designate it) in Russell's symbolism,
and by means of certain definitions and
transformations it can be so interpreted that
it says: 'P is not provable in Russell's system'.
False. He did not do that; he tried to do so then hallucinated that he >>>> succeeded. A contradiction follows from the negation of my
characterisation of his actions and so from the truth of the proposition >>>> that he defined P so. That definitional proposition follows from the
axioms of inconsistent systems and not from those of useful consistent >>>> ones. Typically it /is/ an axiom of inconsistent systems and not of
consistent ones.
His paper is a convoluted mess hiding this simple fact
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
When we combine that with this:
-a-a-a Let {T} be such a theory. Then the elementary
-a-a-a statements which belong to {T} we shall call the
-a-a-a elementary theorems of {T}; we also say that
-a-a-a these elementary statements are true for {T}.
-a-a-a Thus, given {T}, an elementary theorem is an
-a-a-a elementary statement which is true.
-a-a-a https://www.liarparadox.org/Haskell_Curry_45.pdf
Foundations of Mathematical Logic 1977
Then G||del simply made a very convoluted analog
to the Liar Paradox.
This is delusional wishful thinking on your part.
Your whole "work" is a defense of your ego you've forged from the fact
that you misunderstand G||del's articles (and many others).
The real mess is you, Peter.
LLM systems initially said this too.
They give me lots and lots of push-back.
When they finally understand my whole
system they always totally agree, 50 times now.
Two big advantages of LLM systems
(1) The have no egoic attachment to conventional wisdom
(2) They have deep knowledge across
(a) theory of computation
(b) foundations of mathematics
(c) foundations of logic
(d) Linguistic semantics
(e) Philosophy of all of the above.
With (1) and without (2)(d) and (2)(e) people
lack a sufficient basis to understand me.
This was the exact same issue for Ludwig Wittgenstein.
On 1/2/26 1:14 AM, Tristan Wibberley wrote:
On 02/01/2026 04:45, Richard Damon wrote:
Truth in the base system has always
actually been theorems of the base system.
But only if "Theorem" includes things proven to be true in the system
even if the proof is in another.
If the statement is derived in another then it is a theorem of the other.
I will disagree with you here. Maybe it iw what you are trying to define "derived" as.
I can certainly use one system to guide me in building a statement in another. Or do you think that
is a task too hard?
I can certainly use one system that knows about another to show that a statement must be true in
that other.
If you want to reserve the lable "Theorem" for only things provable in taht system, I will let you,
but point out I think you are in the minority, and ask for your reference that specifies that.
On 02/01/2026 15:25, Richard Damon wrote:
On 1/2/26 1:14 AM, Tristan Wibberley wrote:
On 02/01/2026 04:45, Richard Damon wrote:
Truth in the base system has always
actually been theorems of the base system.
But only if "Theorem" includes things proven to be true in the system
even if the proof is in another.
If the statement is derived in another then it is a theorem of the
other.
I will disagree with you here. Maybe it iw what you are trying to
define "derived" as.
I can certainly use one system to guide me in building a statement in
another. Or do you think that is a task too hard?
I can certainly use one system that knows about another to show that a
statement must be true in that other.
If you want to reserve the lable "Theorem" for only things provable in
taht system, I will let you, but point out I think you are in the
minority, and ask for your reference that specifies that.
No, I'd say Tristan is spot on with how that's normally done.
While speaking informally, "theorem" can mean "a mathematical statement
that has a convincing argument for its truth" (e.g. Pythagoras'
theorem), in formal logic "Theorem" and "Theory" have a technical
meaning:-a "Theory" being the deductive closure of a set of axioms, and a Theorem being a sentence of the Theory.-a So every Theorem in the Theory
has a "derivation" from the theories axioms. -aIt is not directly to do
with "truth" in the formal system.-a [Of course, we want our system (including axioms) to be sound, so all Theorems will be true.]
-a <https://en.wikipedia.org/wiki/ Theory_(mathematical_logic)#Deductive_theories>
Of course, you could be learning from an author taking a different
approach, but I haven't personally come across one who would say that
the sentence represented by G was a "Theorem" of the underlying logical system!-a (That would (IMO) be grossly misleading...)
Similarly, the word "proof" can be informal (simply an argument that convinces people of the truth of a statement), or refer to the "proof calculus" of the formal system being discussed.-a Most authors I've come across seem to use "proof" more or less informally and for clarity
choose another word for whatever sequence of syntactic "proof steps" the formal system specifies.-a Often "derivation" is used, and that seems intuitive to me, so I try to always use that term here, and using
"proof" for more general mathematial arguments, e.g. proving that the G statement is "true" using some meta-theory.
Also just as an aside, I don't recall that Godel ever talked about
"truth" of his G statement.-a His proof was concerned with provability. (Neither the G sentence nor its negation is provable.)
Mike.
On 1/1/26 5:41 AM, Tristan Wibberley wrote:
On 31/12/2025 21:16, Pierre Asselin wrote:
In sci.logic Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
[ ... ]
Then he defines a new system "P" which he uses to get even more
muddled,
leaves out the crucial elements of his proof because it's too easy to
get wrong,
G||del, muddled? He was the most meticulous sonovabitch that ever
lived!
Have you heard about his musings on God?
and Stephen Meyer says he does get it wrong; he seems to be
the only person in the world that ever checked.
People have misunderstood G||del and proved it by their comments.
I don't know who Stephen Meyer is; my money is on G||del.
I misremembered, it was James Meyer. He has a website on it
http://www.jamesrmeyer.com . He's very angry about people telling him
he's wrong but who never checked like he did because they keep telling
him reasons it's right that he's certain are not reflected in the actual
work.
In other words, since he doesn't understand it, it must be wrong.
Since his page begins with a rejection of the axiom of Choice,What is the axiom of choice?
example he gives,
nature of infinite systems.
To expect that infinite systems behave just like we see finite systemssets.
work is a funamental error.
Yes, it seems to create paradoxes, but those paradoxes are only apparent
due to the lack of understanding about the actual nature of infinite
On 1/1/26 9:45 PM, Andr|- G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the
mathematical
operations definable in the base system. What makes you think it
isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't think
the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a
statement which can be proven in a particular system.
I guess it depends on your definition of a "Theorem".
I am using the one that goes:
"A Theorem is a statement that has been proven."
note, no restriction that the proof was in the system the Theorem is
stated in, as long as the proof shows that it is actually True in that system.
On 2026-01-01 20:09, Richard Damon wrote:
On 1/1/26 9:45 PM, Andr|- G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the
mathematical
operations definable in the base system. What makes you think it
isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't think >>>>> the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a
statement which can be proven in a particular system.
I guess it depends on your definition of a "Theorem".
I am using the one that goes:
"A Theorem is a statement that has been proven."
note, no restriction that the proof was in the system the Theorem is
stated in, as long as the proof shows that it is actually True in that
system.
A theorem is a statement that can be derived from the axioms of a
particular system. It may be true in other systems, but it is only a
theorem in systems in which it can be derived.
An obvious example to illustrate this would be the fact that there are
many theorems which can be derived in Euclidean geometry, but which are
not theorems of various non-Euclidean geometries. That is to say, not
only can they not be derived in those non-Euclidean geometries, but they
can be shown to be *false* in those non-Euclidean geometries.
Theoremhood is always tied to a particular formal system.
Andr|-
On 1/2/26 12:48 AM, Tristan Wibberley wrote:
On 02/01/2026 03:09, Richard Damon wrote:
I guess it depends on your definition of a "Theorem".
Perhaps this is just a diffence of schools of logic.
Perhaps I come from a somewhat later school of thought that has accepted
the concept of infinity and its implications.
On 1/2/26 9:43 PM, Andr|- G. Isaak wrote:
On 2026-01-01 20:09, Richard Damon wrote:
On 1/1/26 9:45 PM, Andr|- G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the
mathematical
operations definable in the base system. What makes you think it >>>>>>> isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't think >>>>>> the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a
statement which can be proven in a particular system.
I guess it depends on your definition of a "Theorem".
I am using the one that goes:
"A Theorem is a statement that has been proven."
note, no restriction that the proof was in the system the Theorem is
stated in, as long as the proof shows that it is actually True in
that system.
A theorem is a statement that can be derived from the axioms of a
particular system. It may be true in other systems, but it is only a
theorem in systems in which it can be derived.
Right, And the statement og Godel's G can be fully derived in the base system, as it is purely a mathematical relationship using the operations derivable in the system.
The implications of this statement can't be understood in the system,
but that isn't a requirment to be a Theorem.
An obvious example to illustrate this would be the fact that there are
many theorems which can be derived in Euclidean geometry, but which
are not theorems of various non-Euclidean geometries. That is to say,
not only can they not be derived in those non-Euclidean geometries,
but they can be shown to be *false* in those non-Euclidean geometries.
Right, but G isn't like this.
While speaking informally, "theorem" can mean "a mathematical statement
that has a convincing argument for its truth" (e.g. Pythagoras'
theorem), in formal logic "Theorem" and "Theory" have a technical
meaning:-a "Theory" being the deductive closure of a set of axioms, and a Theorem being a sentence of the Theory.-a
So every Theorem in the Theory
has a "derivation" from the theories axioms. It is not directly to do
with "truth" in the formal system. [Of course, we want our system
(including axioms) to be sound, so all Theorems will be true.]
Also just as an aside, I don't recall that Godel ever talked about
"truth" of his G statement.-a
The issue is that "derivation" doesn't actually imply a finiteness,
which is a necessity of "proof".
On 02/01/2026 17:54, Richard Damon wrote:
The issue is that "derivation" doesn't actually imply a finiteness,
which is a necessity of "proof".
Perhaps not. When you have written any other derivation down I would be interested to visit it someday.
On 1/2/26 9:43 PM, Andro G. Isaak wrote:
On 2026-01-01 20:09, Richard Damon wrote:
On 1/1/26 9:45 PM, Andro G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the mathematical
operations definable in the base system. What makes you think it isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't think >>>>>> the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a statement which can be proven
in a particular system.
I guess it depends on your definition of a "Theorem".
I am using the one that goes:
"A Theorem is a statement that has been proven."
note, no restriction that the proof was in the system the Theorem is stated in, as long as the
proof shows that it is actually True in that system.
A theorem is a statement that can be derived from the axioms of a particular system. It may be
true in other systems, but it is only a theorem in systems in which it can be derived.
Right, And the statement og Godel's G can be fully derived in the base system, as it is purely a
mathematical relationship using the operations derivable in the system.
The implications of this statement can't be understood in the system, but that isn't a requirment to
be a Theorem.
An obvious example to illustrate this would be the fact that there are many theorems which can be
derived in Euclidean geometry, but which are not theorems of various non-Euclidean geometries.
That is to say, not only can they not be derived in those non-Euclidean geometries, but they can
be shown to be *false* in those non-Euclidean geometries.
Right, but G isn't like this.
Theoremhood is always tied to a particular formal system.
Right, and the statement G needs nothing outside of the base system to be created.
What the meta-system provides is a "hidden" meaning to it.
Andro
It is sort of like given a binary of a program. The base computer still considers it a program, even
if the only way to figure out what this program does is to run it with all sorts of input. So, even
without a C compiler, that binary is a program.
WIth the C compiler and the C source code, we can understand much better what the program does, and
might not need to run it for a bunch of inputs.
The fact that program came out of the C compiler. doesn't make it not a program for that processor.
In the same way, G is a statement about using a specific set of operations defined in the base
system and whether any number given will statisfy it. WIth just the assembly code, that may be
impossible to determine, as there are an infinite number of possible inputs, so we can't test them all.
But, that sequence of operations that G uses, came out of a "compiler" in the meta-system, from
which we can see that this set of instructions are just a proof tester, to see if the number is a
representation of a give proof of the statement G in the base system, where every proof in the base
system produces a unique number, and every number produces a possible proof in the base system
(though many are just non-sense)
The fact we used a "compiler" to generate the statement doesn't make it not a program in the base
system, but does let us understand what it does.
On 03/01/2026 03:30, Richard Damon wrote:
On 1/2/26 9:43 PM, Andr|- G. Isaak wrote:
On 2026-01-01 20:09, Richard Damon wrote:
On 1/1/26 9:45 PM, Andr|- G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the
mathematical
operations definable in the base system. What makes you think it >>>>>>>> isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't >>>>>>> think
the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a
statement which can be proven in a particular system.
I guess it depends on your definition of a "Theorem".
I am using the one that goes:
"A Theorem is a statement that has been proven."
note, no restriction that the proof was in the system the Theorem is
stated in, as long as the proof shows that it is actually True in
that system.
A theorem is a statement that can be derived from the axioms of a
particular system. It may be true in other systems, but it is only a
theorem in systems in which it can be derived.
Right, And the statement og Godel's G can be fully derived in the base
system, as it is purely a mathematical relationship using the
operations derivable in the system.
Neither G nor -4G has a derivation (in your terms, a "formal prooof")
within the base system.-a That is what Godel proves, showing that the
base system is incomplete.
Mike.
The implications of this statement can't be understood in the system,
but that isn't a requirment to be a Theorem.
An obvious example to illustrate this would be the fact that there
are many theorems which can be derived in Euclidean geometry, but
which are not theorems of various non-Euclidean geometries. That is
to say, not only can they not be derived in those non-Euclidean
geometries, but they can be shown to be *false* in those non-
Euclidean geometries.
Right, but G isn't like this.
Theoremhood is always tied to a particular formal system.
Right, and the statement G needs nothing outside of the base system to
be created.
Sure.-a Just creating a statement doesn't mean the statement is a
Theorem.-a Theorems need a "formal proof" (aka, a derivation) in that
formal system.-a You have a basic misunderstanding somewhere!
Mike.
What the meta-system provides is a "hidden" meaning to it.
Andr|-
It is sort of like given a binary of a program. The base computer
still considers it a program, even if the only way to figure out what
this program does is to run it with all sorts of input. So, even
without a C compiler, that binary is a program.
WIth the C compiler and the C source code, we can understand much
better what the program does, and might not need to run it for a bunch
of inputs.
The fact that program came out of the C compiler. doesn't make it not
a program for that processor.
In the same way, G is a statement about using a specific set of
operations defined in the base system and whether any number given
will statisfy it. WIth just the assembly code, that may be impossible
to determine, as there are an infinite number of possible inputs, so
we can't test them all.
But, that sequence of operations that G uses, came out of a "compiler"
in the meta-system, from which we can see that this set of
instructions are just a proof tester, to see if the number is a
representation of a give proof of the statement G in the base system,
where every proof in the base system produces a unique number, and
every number produces a possible proof in the base system (though many
are just non-sense)
The fact we used a "compiler" to generate the statement doesn't make
it not a program in the base system, but does let us understand what
it does.
You have used lots of words to explain, in effect, that the G statement
is a statement of the base system.-a Well, most everybody here
understands that perfectly well.-a (Not PO though...)-a Your analogy with programs has nothing more than
-a program-a <---> statement in the language of the formal system
-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a (or what is often called a "sentence" in that language)
so program does not correlate with "Theorem" of the formal system.-a All
the stuff about compilers helping us to understand the purpose of
program is ok, but has nothing to do with whether the program
corresponds to a Theorem.
You seem to understand that Godel proves that the G statement does NOT
have a "derivation" (aka "formal proof using the specified proof rules
of the base system") in the base system.-a I.e. it is specifically NOT a "Theorem" of the base system, in the sense that people use that word /in
the realm of formal logic/.
Mike.
No, meta-system is NOT "full up" as there are an infinite number of
primes to use to define new objects.
Maybe you missed that part of it.
The base system is defined as it is, and thus can't change, but using a natural language lable to refer to a sentence in the system doesn't
change the system.
The meta-system CAN add the lable, which he uses in the building of the relationship, but the final results needs no reference to the statement itself, as that has been actually "encoded" into the relationship.
You seem to miss the fact that G is a label used to talk about the
sentence and not a "symbol" created in the base system.
When we combine that with this:
-a-a Let {T} be such a theory. Then the elementary
-a-a statements which belong to {T} we shall call the
-a-a elementary theorems of {T}; we also say that
-a-a these elementary statements are true for {T}.
-a-a Thus, given {T}, an elementary theorem is an
-a-a elementary statement which is true.
-a-a https://www.liarparadox.org/Haskell_Curry_45.pdf
Foundations of Mathematical Logic 1977
The real mess is you, Peter.
LLM systems initially said this too.
Then I should have used examples such as "Theorem of the theory proper
of a system".
You have to derive the statement from the axioms of the system using the
deduction rules of the system. It's the actual definition of "Theorem of
a Formal System".
And the statement G was.
On 02/01/2026 16:06, Richard Damon wrote:
You have to derive the statement from the axioms of the system using the >>> deduction rules of the system. It's the actual definition of "Theorem of >>> a Formal System".
And the statement G was.
Then a system can be complete.
On 02/01/2026 16:06, Richard Damon wrote:
You have to derive the statement from the axioms of the system using the >>> deduction rules of the system. It's the actual definition of "Theorem of >>> a Formal System".
And the statement G was.
Then a system can be complete.
On 1/3/2026 11:35 AM, Tristan Wibberley wrote:
On 02/01/2026 16:06, Richard Damon wrote:
You have to derive the statement from the axioms of the system using
the
deduction rules of the system. It's the actual definition of
"Theorem of
a Formal System".
And the statement G was.
Then a system can be complete.
If G is a theorem of F and G asserts its
own unprovability in F then a sequence
of inferences steps exist in F that
prove that they themselves do not exist.
An obvious example to illustrate this would be the fact that there are
many theorems which can be derived in Euclidean geometry, but which are
not theorems of various non-Euclidean geometries. That is to say, not
only can they not be derived in those non-Euclidean geometries, but they
can be shown to be *false* in those non-Euclidean geometries.
On 03/01/2026 03:30, Richard Damon wrote:
On 1/2/26 9:43 PM, Andr|- G. Isaak wrote:
On 2026-01-01 20:09, Richard Damon wrote:
On 1/1/26 9:45 PM, Andr|- G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the
mathematical
operations definable in the base system. What makes you think it >>>>>>>> isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't
think
the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a
statement which can be proven in a particular system.
I guess it depends on your definition of a "Theorem".
I am using the one that goes:
"A Theorem is a statement that has been proven."
note, no restriction that the proof was in the system the Theorem is
stated in, as long as the proof shows that it is actually True in
that system.
A theorem is a statement that can be derived from the axioms of a
particular system. It may be true in other systems, but it is only a
theorem in systems in which it can be derived.
Right, And the statement og Godel's G can be fully derived in the base
system, as it is purely a mathematical relationship using the
operations derivable in the system.
Neither G nor -4G has a derivation (in your terms, a "formal prooof")
within the base system.-a That is what Godel proves, showing that the
base system is incomplete.
On 03/01/2026 03:30, Richard Damon wrote:
On 1/2/26 9:43 PM, Andr|- G. Isaak wrote:
On 2026-01-01 20:09, Richard Damon wrote:
On 1/1/26 9:45 PM, Andr|- G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the
mathematical
operations definable in the base system. What makes you think it >>>>>>>> isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't >>>>>>> think
the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a
statement which can be proven in a particular system.
I guess it depends on your definition of a "Theorem".
I am using the one that goes:
"A Theorem is a statement that has been proven."
note, no restriction that the proof was in the system the Theorem is
stated in, as long as the proof shows that it is actually True in
that system.
A theorem is a statement that can be derived from the axioms of a
particular system. It may be true in other systems, but it is only a
theorem in systems in which it can be derived.
Right, And the statement og Godel's G can be fully derived in the base
system, as it is purely a mathematical relationship using the
operations derivable in the system.
Neither G nor -4G has a derivation (in your terms, a "formal prooof")
within the base system.-a That is what Godel proves, showing that the
base system is incomplete.
Mike.
The implications of this statement can't be understood in the system,
but that isn't a requirment to be a Theorem.
An obvious example to illustrate this would be the fact that there
are many theorems which can be derived in Euclidean geometry, but
which are not theorems of various non-Euclidean geometries. That is
to say, not only can they not be derived in those non-Euclidean
geometries, but they can be shown to be *false* in those non-
Euclidean geometries.
Right, but G isn't like this.
Theoremhood is always tied to a particular formal system.
Right, and the statement G needs nothing outside of the base system to
be created.
Sure.-a Just creating a statement doesn't mean the statement is a
Theorem.-a Theorems need a "formal proof" (aka, a derivation) in that
formal system.-a You have a basic misunderstanding somewhere!
Mike.
What the meta-system provides is a "hidden" meaning to it.
Andr|-
It is sort of like given a binary of a program. The base computer
still considers it a program, even if the only way to figure out what
this program does is to run it with all sorts of input. So, even
without a C compiler, that binary is a program.
WIth the C compiler and the C source code, we can understand much
better what the program does, and might not need to run it for a bunch
of inputs.
The fact that program came out of the C compiler. doesn't make it not
a program for that processor.
In the same way, G is a statement about using a specific set of
operations defined in the base system and whether any number given
will statisfy it. WIth just the assembly code, that may be impossible
to determine, as there are an infinite number of possible inputs, so
we can't test them all.
But, that sequence of operations that G uses, came out of a "compiler"
in the meta-system, from which we can see that this set of
instructions are just a proof tester, to see if the number is a
representation of a give proof of the statement G in the base system,
where every proof in the base system produces a unique number, and
every number produces a possible proof in the base system (though many
are just non-sense)
The fact we used a "compiler" to generate the statement doesn't make
it not a program in the base system, but does let us understand what
it does.
You have used lots of words to explain, in effect, that the G statement
is a statement of the base system.-a Well, most everybody here
understands that perfectly well.-a (Not PO though...)-a Your analogy with programs has nothing more than
-a program-a <---> statement in the language of the formal system
-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a (or what is often called a "sentence" in that language)
so program does not correlate with "Theorem" of the formal system.-a All
the stuff about compilers helping us to understand the purpose of
program is ok, but has nothing to do with whether the program
corresponds to a Theorem.
You seem to understand that Godel proves that the G statement does NOT
have a "derivation" (aka "formal proof using the specified proof rules
of the base system") in the base system.-a I.e. it is specifically NOT a "Theorem" of the base system, in the sense that people use that word /in
the realm of formal logic/.
Mike.
On 1/3/2026 10:32 AM, Mike Terry wrote:
On 03/01/2026 03:30, Richard Damon wrote:
On 1/2/26 9:43 PM, Andr|- G. Isaak wrote:
On 2026-01-01 20:09, Richard Damon wrote:
On 1/1/26 9:45 PM, Andr|- G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the >>>>>>>>> mathematical
operations definable in the base system. What makes you think >>>>>>>>> it isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't >>>>>>>> think
the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a >>>>>> statement which can be proven in a particular system.
I guess it depends on your definition of a "Theorem".
I am using the one that goes:
"A Theorem is a statement that has been proven."
note, no restriction that the proof was in the system the Theorem
is stated in, as long as the proof shows that it is actually True
in that system.
A theorem is a statement that can be derived from the axioms of a
particular system. It may be true in other systems, but it is only a
theorem in systems in which it can be derived.
Right, And the statement og Godel's G can be fully derived in the
base system, as it is purely a mathematical relationship using the
operations derivable in the system.
Neither G nor -4G has a derivation (in your terms, a "formal prooof")
within the base system.-a That is what Godel proves, showing that the
base system is incomplete.
Mike.
The implications of this statement can't be understood in the system,
but that isn't a requirment to be a Theorem.
An obvious example to illustrate this would be the fact that there
are many theorems which can be derived in Euclidean geometry, but
which are not theorems of various non-Euclidean geometries. That is
to say, not only can they not be derived in those non-Euclidean
geometries, but they can be shown to be *false* in those non-
Euclidean geometries.
Right, but G isn't like this.
Theoremhood is always tied to a particular formal system.
Right, and the statement G needs nothing outside of the base system
to be created.
Sure.-a Just creating a statement doesn't mean the statement is a
Theorem.-a Theorems need a "formal proof" (aka, a derivation) in that
formal system.-a You have a basic misunderstanding somewhere!
Mike.
What the meta-system provides is a "hidden" meaning to it.
Andr|-
It is sort of like given a binary of a program. The base computer
still considers it a program, even if the only way to figure out what
this program does is to run it with all sorts of input. So, even
without a C compiler, that binary is a program.
WIth the C compiler and the C source code, we can understand much
better what the program does, and might not need to run it for a
bunch of inputs.
The fact that program came out of the C compiler. doesn't make it not
a program for that processor.
In the same way, G is a statement about using a specific set of
operations defined in the base system and whether any number given
will statisfy it. WIth just the assembly code, that may be impossible
to determine, as there are an infinite number of possible inputs, so
we can't test them all.
But, that sequence of operations that G uses, came out of a
"compiler" in the meta-system, from which we can see that this set of
instructions are just a proof tester, to see if the number is a
representation of a give proof of the statement G in the base system,
where every proof in the base system produces a unique number, and
every number produces a possible proof in the base system (though
many are just non-sense)
The fact we used a "compiler" to generate the statement doesn't make
it not a program in the base system, but does let us understand what
it does.
You have used lots of words to explain, in effect, that the G
statement is a statement of the base system.-a Well, most everybody
here understands that perfectly well.-a (Not PO though...)-a Your
analogy with programs has nothing more than
-a-a program-a <---> statement in the language of the formal system
-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a (or what is often called a "sentence" in that language)
so program does not correlate with "Theorem" of the formal system.
All the stuff about compilers helping us to understand the purpose of
program is ok, but has nothing to do with whether the program
corresponds to a Theorem.
You seem to understand that Godel proves that the G statement does NOT
have a "derivation" (aka "formal proof using the specified proof rules
of the base system") in the base system.-a I.e. it is specifically NOT
a "Theorem" of the base system, in the sense that people use that
word /in the realm of formal logic/.
Mike.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
A proposition which asserts its own unprovability
literally means: G asserts its own unprovability.
This semantically entails that:
A proof of G requires a sequence of inference
steps that prove that they themselves do not exist.
...14 Every epistemological antinomy can likewise
be used for a similar undecidability proof...
(G||del 1931:40-41)
This literally means that the Liar Paradox can
likewise be used for a similar undecidability proof.
This semantically entails that:
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
Expands to not(true(not(true(not(true(not(true(...))))))))
Proves that the Liar Paradox is ungrounded, thus
neither G nor LP are truth bearers or propositions.
*Key difference between math and the philosophy of math*
The philosophy of math says maybe we have
been thinking about this stuff all wrong.
Math says of course we haven't been thinking
about this stuff all wrong everyone knows
that math is infallible.
*Hence the basis for disagreement over all these years*
On 03/01/2026 16:32, Mike Terry wrote:
On 03/01/2026 03:30, Richard Damon wrote:
On 1/2/26 9:43 PM, Andr|- G. Isaak wrote:
On 2026-01-01 20:09, Richard Damon wrote:
On 1/1/26 9:45 PM, Andr|- G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the
mathematical
operations definable in the base system. What makes you think it >>>>>>>>> isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't >>>>>>>> think
the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a >>>>>> statement which can be proven in a particular system.
I guess it depends on your definition of a "Theorem".
I am using the one that goes:
"A Theorem is a statement that has been proven."
note, no restriction that the proof was in the system the Theorem is >>>>> stated in, as long as the proof shows that it is actually True in
that system.
A theorem is a statement that can be derived from the axioms of a
particular system. It may be true in other systems, but it is only a
theorem in systems in which it can be derived.
Right, And the statement og Godel's G can be fully derived in the base
system, as it is purely a mathematical relationship using the
operations derivable in the system.
Neither G nor -4G has a derivation (in your terms, a "formal prooof")
within the base system.-a That is what Godel proves, showing that the
base system is incomplete.
That can't be what he meant can it? Lots of systems were known to have statements that had no derivation, all nonsense statements, for example.
Did he really mean that there's some level of completeness in which
there is meaninglessness (things that look like propositions but which
are not? Well, duh. But arithmetic isn't required for that, merely self-references such as non-ranked definitions and fixed-point
combinators (the meaning depends on a meaning that depends on a meaning that...).
Hang on, he had two incompleteness theorems and a completeness theorem.
Can we get some good terminology that distinguishes them because I think there's some referential ambiguity creeping in.
On 02/01/2026 15:35, Richard Damon wrote:
No, meta-system is NOT "full up" as there are an infinite number of
primes to use to define new objects.
I understood from reading it that those were occupied for variables.
Maybe you missed that part of it.
Perhaps.
The base system is defined as it is, and thus can't change, but using a
natural language lable to refer to a sentence in the system doesn't
change the system.
It didn't seem that you were doing that.
The meta-system CAN add the lable, which he uses in the building of the
relationship, but the final results needs no reference to the statement
itself, as that has been actually "encoded" into the relationship.
As long as you're doing that I'll hear more.
You seem to miss the fact that G is a label used to talk about the
sentence and not a "symbol" created in the base system.
Yes, I thought you were trying to do the latter informally.
On 1/3/26 11:47 AM, olcott wrote:
On 1/3/2026 10:32 AM, Mike Terry wrote:
On 03/01/2026 03:30, Richard Damon wrote:
On 1/2/26 9:43 PM, Andr|- G. Isaak wrote:
On 2026-01-01 20:09, Richard Damon wrote:
On 1/1/26 9:45 PM, Andr|- G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the >>>>>>>>>> mathematical
operations definable in the base system. What makes you think >>>>>>>>>> it isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't >>>>>>>>> think
the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is >>>>>>> a statement which can be proven in a particular system.
I guess it depends on your definition of a "Theorem".
I am using the one that goes:
"A Theorem is a statement that has been proven."
note, no restriction that the proof was in the system the Theorem >>>>>> is stated in, as long as the proof shows that it is actually True >>>>>> in that system.
A theorem is a statement that can be derived from the axioms of a
particular system. It may be true in other systems, but it is only
a theorem in systems in which it can be derived.
Right, And the statement og Godel's G can be fully derived in the
base system, as it is purely a mathematical relationship using the
operations derivable in the system.
Neither G nor -4G has a derivation (in your terms, a "formal prooof")
within the base system.-a That is what Godel proves, showing that the
base system is incomplete.
Mike.
The implications of this statement can't be understood in the
system, but that isn't a requirment to be a Theorem.
An obvious example to illustrate this would be the fact that there
are many theorems which can be derived in Euclidean geometry, but
which are not theorems of various non-Euclidean geometries. That is >>>>> to say, not only can they not be derived in those non-Euclidean
geometries, but they can be shown to be *false* in those non-
Euclidean geometries.
Right, but G isn't like this.
Theoremhood is always tied to a particular formal system.
Right, and the statement G needs nothing outside of the base system
to be created.
Sure.-a Just creating a statement doesn't mean the statement is a
Theorem.-a Theorems need a "formal proof" (aka, a derivation) in that
formal system.-a You have a basic misunderstanding somewhere!
Mike.
What the meta-system provides is a "hidden" meaning to it.
Andr|-
It is sort of like given a binary of a program. The base computer
still considers it a program, even if the only way to figure out
what this program does is to run it with all sorts of input. So,
even without a C compiler, that binary is a program.
WIth the C compiler and the C source code, we can understand much
better what the program does, and might not need to run it for a
bunch of inputs.
The fact that program came out of the C compiler. doesn't make it
not a program for that processor.
In the same way, G is a statement about using a specific set of
operations defined in the base system and whether any number given
will statisfy it. WIth just the assembly code, that may be
impossible to determine, as there are an infinite number of possible
inputs, so we can't test them all.
But, that sequence of operations that G uses, came out of a
"compiler" in the meta-system, from which we can see that this set
of instructions are just a proof tester, to see if the number is a
representation of a give proof of the statement G in the base
system, where every proof in the base system produces a unique
number, and every number produces a possible proof in the base
system (though many are just non-sense)
The fact we used a "compiler" to generate the statement doesn't make
it not a program in the base system, but does let us understand what
it does.
You have used lots of words to explain, in effect, that the G
statement is a statement of the base system.-a Well, most everybody
here understands that perfectly well.-a (Not PO though...)-a Your
analogy with programs has nothing more than
-a-a program-a <---> statement in the language of the formal system
-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a (or what is often called a "sentence" in that
language)
so program does not correlate with "Theorem" of the formal system.
All the stuff about compilers helping us to understand the purpose of
program is ok, but has nothing to do with whether the program
corresponds to a Theorem.
You seem to understand that Godel proves that the G statement does
NOT have a "derivation" (aka "formal proof using the specified proof
rules of the base system") in the base system.-a I.e. it is
specifically NOT a "Theorem" of the base system, in the sense that
people use that word /in the realm of formal logic/.
Mike.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
WHich you don't seem to understand is the INTERPREATION of the statement made in the meta system about the statement.
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
A proposition which asserts its own unprovability
literally means: G asserts its own unprovability.
Nope. Just that it uses language too complicated for you to understand.
G, itself, asserts no such thing.
This semantically entails that:
A proof of G requires a sequence of inference
steps that prove that they themselves do not exist.
Right, so no such prove IN THE BASE SYSTEM can exist.
...14 Every epistemological antinomy can likewise
be used for a similar undecidability proof...
(G||del 1931:40-41)
Right, the core
This literally means that the Liar Paradox can
likewise be used for a similar undecidability proof.
And it was, but the FORM of the Liar, under a syntactic transform that actualy changes its meaning.
Of course, taht is beyound you understanding.
This semantically entails that:
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
Nope. You are just showing your stupidity.
Expands to not(true(not(true(not(true(not(true(...))))))))
Proves that the Liar Paradox is ungrounded, thus
neither G nor LP are truth bearers or propositions.
Right, because Prolog is unable to handle the needed Grammar, and since Prolog is
*Key difference between math and the philosophy of math*
The philosophy of math says maybe we have
been thinking about this stuff all wrong.
Nope.
Math says of course we haven't been thinking
about this stuff all wrong everyone knows
that math is infallible.
*Hence the basis for disagreement over all these years*
In other words, you are admitting you don't know what you are talking
about, and don't care.
Note, the "Philosophy of Math" isn't relevent here,
that would be used--
to help decide which rules of math we should use.
But the formal system has made that choice, and thus move past the philosophy.
Your "Philosophies" can't actually answer their questions, as they have
no actual foundations for what is true.
Formal Logic defines Truth within it, and thus can answer the question
in their systems.
On 1/3/2026 1:18 PM, Richard Damon wrote:
On 1/3/26 11:47 AM, olcott wrote:
On 1/3/2026 10:32 AM, Mike Terry wrote:
On 03/01/2026 03:30, Richard Damon wrote:
On 1/2/26 9:43 PM, Andr|- G. Isaak wrote:
On 2026-01-01 20:09, Richard Damon wrote:
On 1/1/26 9:45 PM, Andr|- G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the >>>>>>>>>>> mathematical
operations definable in the base system. What makes you think >>>>>>>>>>> it isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you
wouldn't think
the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is >>>>>>>> a statement which can be proven in a particular system.
I guess it depends on your definition of a "Theorem".
I am using the one that goes:
"A Theorem is a statement that has been proven."
note, no restriction that the proof was in the system the Theorem >>>>>>> is stated in, as long as the proof shows that it is actually True >>>>>>> in that system.
A theorem is a statement that can be derived from the axioms of a >>>>>> particular system. It may be true in other systems, but it is only >>>>>> a theorem in systems in which it can be derived.
Right, And the statement og Godel's G can be fully derived in the
base system, as it is purely a mathematical relationship using the
operations derivable in the system.
Neither G nor -4G has a derivation (in your terms, a "formal prooof") >>>> within the base system.-a That is what Godel proves, showing that the >>>> base system is incomplete.
Mike.
The implications of this statement can't be understood in the
system, but that isn't a requirment to be a Theorem.
An obvious example to illustrate this would be the fact that there >>>>>> are many theorems which can be derived in Euclidean geometry, but >>>>>> which are not theorems of various non-Euclidean geometries. That
is to say, not only can they not be derived in those non-Euclidean >>>>>> geometries, but they can be shown to be *false* in those non-
Euclidean geometries.
Right, but G isn't like this.
Theoremhood is always tied to a particular formal system.
Right, and the statement G needs nothing outside of the base system >>>>> to be created.
Sure.-a Just creating a statement doesn't mean the statement is a
Theorem.-a Theorems need a "formal proof" (aka, a derivation) in that >>>> formal system.-a You have a basic misunderstanding somewhere!
Mike.
What the meta-system provides is a "hidden" meaning to it.
Andr|-
It is sort of like given a binary of a program. The base computer
still considers it a program, even if the only way to figure out
what this program does is to run it with all sorts of input. So,
even without a C compiler, that binary is a program.
WIth the C compiler and the C source code, we can understand much
better what the program does, and might not need to run it for a
bunch of inputs.
The fact that program came out of the C compiler. doesn't make it
not a program for that processor.
In the same way, G is a statement about using a specific set of
operations defined in the base system and whether any number given
will statisfy it. WIth just the assembly code, that may be
impossible to determine, as there are an infinite number of
possible inputs, so we can't test them all.
But, that sequence of operations that G uses, came out of a
"compiler" in the meta-system, from which we can see that this set
of instructions are just a proof tester, to see if the number is a
representation of a give proof of the statement G in the base
system, where every proof in the base system produces a unique
number, and every number produces a possible proof in the base
system (though many are just non-sense)
The fact we used a "compiler" to generate the statement doesn't
make it not a program in the base system, but does let us
understand what it does.
You have used lots of words to explain, in effect, that the G
statement is a statement of the base system.-a Well, most everybody
here understands that perfectly well.-a (Not PO though...)-a Your
analogy with programs has nothing more than
-a-a program-a <---> statement in the language of the formal system
-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a (or what is often called a "sentence" in that
language)
so program does not correlate with "Theorem" of the formal system.
All the stuff about compilers helping us to understand the purpose
of program is ok, but has nothing to do with whether the program
corresponds to a Theorem.
You seem to understand that Godel proves that the G statement does
NOT have a "derivation" (aka "formal proof using the specified proof
rules of the base system") in the base system.-a I.e. it is
specifically NOT a "Theorem" of the base system, in the sense that
people use that word /in the realm of formal logic/.
Mike.
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
WHich you don't seem to understand is the INTERPREATION of the
statement made in the meta system about the statement.
That is not what the sentence literally says.
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
A proposition which asserts its own unprovability
literally means: G asserts its own unprovability.
Nope. Just that it uses language too complicated for you to understand.
G, itself, asserts no such thing.
This semantically entails that:
A proof of G requires a sequence of inference
steps that prove that they themselves do not exist.
Right, so no such prove IN THE BASE SYSTEM can exist.
...14 Every epistemological antinomy can likewise
be used for a similar undecidability proof...
(G||del 1931:40-41)
Right, the core
This literally means that the Liar Paradox can
likewise be used for a similar undecidability proof.
And it was, but the FORM of the Liar, under a syntactic transform that
actualy changes its meaning.
Not we don't need fifty pages of 85 formulas.
LP := ~True(LP)
Of course, taht is beyound you understanding.
This semantically entails that:
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
Nope. You are just showing your stupidity.
Expands to not(true(not(true(not(true(not(true(...))))))))
Proves that the Liar Paradox is ungrounded, thus
neither G nor LP are truth bearers or propositions.
Right, because Prolog is unable to handle the needed Grammar, and
since Prolog is
The Liar Paradox literally specifies infinite
recursion that never resolves to a truth value.
Your ignorance of Prolog is no rebuttal.
*Key difference between math and the philosophy of math*
The philosophy of math says maybe we have
been thinking about this stuff all wrong.
Nope.
Math says of course we haven't been thinking
about this stuff all wrong everyone knows
that math is infallible.
*Hence the basis for disagreement over all these years*
In other words, you are admitting you don't know what you are talking
about, and don't care.
Note, the "Philosophy of Math" isn't relevent here,
Sure it is. I am proposing the idea that the foundations
of math might be incorrect and you are essentially saying
that these foundations are impossibly incorrect because
they are inherently infallible.
My "error" is merely your own closed-mindedness.
that would be used to help decide which rules of math we should use.
But the formal system has made that choice, and thus move past the
philosophy.
Your "Philosophies" can't actually answer their questions, as they
have no actual foundations for what is true.
Formal Logic defines Truth within it, and thus can answer the question
in their systems.
His paper is a convoluted mess hiding this simple fact
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
When we combine that with this:
-a-a Let {T} be such a theory. Then the elementary
-a-a statements which belong to {T} we shall call the
-a-a elementary theorems of {T}; we also say that
-a-a these elementary statements are true for {T}.
-a-a Thus, given {T}, an elementary theorem is an
-a-a elementary statement which is true.
-a-a https://www.liarparadox.org/Haskell_Curry_45.pdf
Foundations of Mathematical Logic 1977
Then G||del simply made a very convoluted analog
to the Liar Paradox.
On 1/3/2026 10:58 AM, Tristan Wibberley wrote:
We begin by postulating a certain non void, definite
class {E} of statements, which we call elementary
statements...
We begin by postulating a certain non void, definite
class {E} of statements, which we call elementary
statements...
The statements of {E} are called elementary statements
to distinguish them from other statements which we may
form from them or about them in the U language...
A theory (over {E}) 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 {E} a certain
subclass of true statementsrCa
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}.
Curry, Haskell 1977. Foundations of Mathematical
Logic. New York: Dover Publications, 45 https://www.liarparadox.org/Haskell_Curry_45.pdf
In other words: reCx ree T ((True(T, x) rei (E reo x))
On 1/3/26 12:35 PM, Tristan Wibberley wrote:
On 02/01/2026 16:06, Richard Damon wrote:
You have to derive the statement from the axioms of the system using
the
deduction rules of the system. It's the actual definition of
"Theorem of
a Formal System".
And the statement G was.
Then a system can be complete.
No, because the derivation isn't finite, so isn't a proof.
It seems you are stuck in a finite world, in a logic that is infinite.
On 03/01/2026 16:32, Mike Terry wrote:
On 03/01/2026 03:30, Richard Damon wrote:
On 1/2/26 9:43 PM, Andr|- G. Isaak wrote:
On 2026-01-01 20:09, Richard Damon wrote:
On 1/1/26 9:45 PM, Andr|- G. Isaak wrote:
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the
mathematical
operations definable in the base system. What makes you think it >>>>>>>>> isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't >>>>>>>> think
the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a >>>>>> statement which can be proven in a particular system.
I guess it depends on your definition of a "Theorem".
I am using the one that goes:
"A Theorem is a statement that has been proven."
note, no restriction that the proof was in the system the Theorem is >>>>> stated in, as long as the proof shows that it is actually True in
that system.
A theorem is a statement that can be derived from the axioms of a
particular system. It may be true in other systems, but it is only a
theorem in systems in which it can be derived.
Right, And the statement og Godel's G can be fully derived in the base
system, as it is purely a mathematical relationship using the
operations derivable in the system.
Neither G nor -4G has a derivation (in your terms, a "formal prooof")
within the base system.-a That is what Godel proves, showing that the
base system is incomplete.
That can't be what he meant can it? Lots of systems were known to have statements that had no derivation, all nonsense statements, for example.
Did he really mean that there's some level of completeness in which
there is meaninglessness (things that look like propositions but which
are not? Well, duh. But arithmetic isn't required for that, merely self-references such as non-ranked definitions and fixed-point
combinators (the meaning depends on a meaning that depends on a meaning that...).
On 02/01/2026 14:47, olcott wrote:
His paper is a convoluted mess hiding this simple fact
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
When we combine that with this:
-a-a Let {T} be such a theory. Then the elementary
-a-a statements which belong to {T} we shall call the
-a-a elementary theorems of {T}; we also say that
-a-a these elementary statements are true for {T}.
-a-a Thus, given {T}, an elementary theorem is an
-a-a elementary statement which is true.
-a-a https://www.liarparadox.org/Haskell_Curry_45.pdf
Foundations of Mathematical Logic 1977
Then G||del simply made a very convoluted analog
to the Liar Paradox.
I also half suspect G||del's incompleteness theorem proof makes Richard's informal "G" shorthand be shorthand for an unbounded statement (not constructive).
I would like to see Richard's construction of the statement for which G
is shorthand. As it is mere shorthand then there is such a thing.
Obviously any finite individual can have an ordinary expression of the natural number it represents and I won't reject it (or some "n+xreU" with definite natural n for the non-determinate individuals like "15+xreU" or "3|u5+xreU" for "fffffffffffffffxreU", for example). Because otherwise I'd just be making it unreasonably difficult just for the typing. No "any
soln in x of x=sqrt(-1)" is not okay, nor is "lim{xraAreR}{x}". I won't accept an infinite sequence of 'f' because I haven't got time to wait
for the message to download.
I half suspect G||del's incompleteness proof was just using Hilbert's
methods socratically to prove Hilbert's formalism wrong so G||del's constructive approach would win out (correctly). (was it G||del's
doctrine, I think I recall Curry and Feys saying it was).
On 1/3/26 2:02 PM, Tristan Wibberley wrote:
On 03/01/2026 16:32, Mike Terry wrote:
On 03/01/2026 03:30, Richard Damon wrote:
On 1/2/26 9:43 PM, Andr|- G. Isaak wrote:
On 2026-01-01 20:09, Richard Damon wrote:
On 1/1/26 9:45 PM, Andr|- G. Isaak wrote:-a-a>
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the >>>>>>>>>> mathematical
operations definable in the base system. What makes you think it >>>>>>>>>> isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't >>>>>>>>> think
the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a >>>>>>> statement which can be proven in a particular system.
I guess it depends on your definition of a "Theorem".
I am using the one that goes:
"A Theorem is a statement that has been proven."
note, no restriction that the proof was in the system the Theorem is >>>>>> stated in, as long as the proof shows that it is actually True in
that system.
A theorem is a statement that can be derived from the axioms of a
particular system. It may be true in other systems, but it is only a >>>>> theorem in systems in which it can be derived.
Right, And the statement og Godel's G can be fully derived in the base >>>> system, as it is purely a mathematical relationship using the
operations derivable in the system.
Neither G nor -4G has a derivation (in your terms, a "formal prooof")
within the base system.-a That is what Godel proves, showing that the
base system is incomplete.
That can't be what he meant can it? Lots of systems were known to have
statements that had no derivation, all nonsense statements, for example.
Did he really mean that there's some level of completeness in which
there is meaninglessness (things that look like propositions but which
are not? Well, duh. But arithmetic isn't required for that, merely
self-references such as non-ranked definitions and fixed-point
combinators (the meaning depends on a meaning that depends on a meaning
that...).
Hang on, he had two incompleteness theorems and a completeness theorem.
Can we get some good terminology that distinguishes them because I think
there's some referential ambiguity creeping in.
The first incompleteness theorem was that all system of sufficent expressiveness (able to handle the Natural Numbers) have a statement
that IS true, but can not be proven.
On 1/3/26 11:54 AM, Tristan Wibberley wrote:
On 02/01/2026 15:35, Richard Damon wrote:
No, meta-system is NOT "full up" as there are an infinite number of
primes to use to define new objects.
I understood from reading it that those were occupied for variables.
or new terms, after all, we have an infinite number of them.
In fact, I thought he talked about assigning a number for proven
statements in the base system, so we could refer to them.
This is needed
to create the lable of "G" in the meta, so we can refer to it as we
build the proof checker, that at the end we finally know the "value" it represents.
On 02/01/2026 14:47, olcott wrote:
His paper is a convoluted mess hiding this simple fact
...We are therefore confronted with a proposition which
asserts its own unprovability. 15 rCa (G||del 1931:40-41)
G||del, Kurt 1931.
On Formally Undecidable Propositions of
Principia Mathematica And Related Systems
When we combine that with this:
-a-a Let {T} be such a theory. Then the elementary
-a-a statements which belong to {T} we shall call the
-a-a elementary theorems of {T}; we also say that
-a-a these elementary statements are true for {T}.
-a-a Thus, given {T}, an elementary theorem is an
-a-a elementary statement which is true.
-a-a https://www.liarparadox.org/Haskell_Curry_45.pdf
Foundations of Mathematical Logic 1977
Then G||del simply made a very convoluted analog
to the Liar Paradox.
I also half suspect G||del's incompleteness theorem proof makes Richard's informal "G" shorthand be shorthand for an unbounded statement (not constructive).
I would like to see Richard's construction of the statement for which G
is shorthand. As it is mere shorthand then there is such a thing.
Obviously any finite individual can have an ordinary expression of the natural number it represents and I won't reject it (or some "n+xreU" with definite natural n for the non-determinate individuals like "15+xreU" or "3|u5+xreU" for "fffffffffffffffxreU", for example). Because otherwise I'd just be making it unreasonably difficult just for the typing. No "any
soln in x of x=sqrt(-1)" is not okay, nor is "lim{xraAreR}{x}". I won't accept an infinite sequence of 'f' because I haven't got time to wait
for the message to download.
I half suspect G||del's incompleteness proof was just using Hilbert's
methods socratically to prove Hilbert's formalism wrong so G||del's constructive approach would win out (correctly). (was it G||del's
doctrine, I think I recall Curry and Feys saying it was).
On 03/01/2026 17:30, olcott wrote:
On 1/3/2026 10:58 AM, Tristan Wibberley wrote:
We begin by postulating a certain non void, definite
class {E} of statements, which we call elementary
statements...
I didn't write that.
On 03/01/2026 17:30, olcott wrote (quoting Curry):
We begin by postulating a certain non void, definite
class {E} of statements, which we call elementary
statements...
The statements of {E} are called elementary statements
to distinguish them from other statements which we may
form from them or about them in the U language...
Odd, in other places "elementary statement" also distinguishes them from "compound statements". Perhaps the 1977 work is restricted to his
simplified notion of formal system that eliminates binary predicates essentially leaving only logistic systems and maybe systems also with
nullary "primitive statements" as axioms not formed from formulas/terms.
A theory (over {E}) 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 {E} a certain
subclass of true statementsrCa
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}.
Curry, Haskell 1977. Foundations of Mathematical
Logic. New York: Dover Publications, 45
https://www.liarparadox.org/Haskell_Curry_45.pdf
In other words: reCx ree T ((True(T, x) rei (E reo x))
Curry would not approve of you formalising that without defining the
system in which you formalise it.
His notions of U-language and
A-language and progressive refinement of the U-language were carefully thought through leading to his incredible written lucidity, and the
immense benefit of reading his work carefully from the start.
On 03/01/2026 19:33, Richard Damon wrote:
On 1/3/26 11:54 AM, Tristan Wibberley wrote:
On 02/01/2026 15:35, Richard Damon wrote:
No, meta-system is NOT "full up" as there are an infinite number of
primes to use to define new objects.
I understood from reading it that those were occupied for variables.
or new terms, after all, we have an infinite number of them.
In fact, I thought he talked about assigning a number for proven
statements in the base system, so we could refer to them.
I missed that, it looked like they were taken for all naturals just for
an all the variables. I shall read that bit again.
This is needed
to create the lable of "G" in the meta, so we can refer to it as we
build the proof checker, that at the end we finally know the "value" it
represents.
And what number does it get? It should be expressible by the primitive symbols of the system P and the conventional shorthands that he defines.
If he merely defines the liar paradox then his system is inconsistent,
either P, or the meta-system.
On 1/3/2026 2:58 PM, Tristan Wibberley wrote:
On 03/01/2026 17:30, olcott wrote:
On 1/3/2026 10:58 AM, Tristan Wibberley wrote:
We begin by postulating a certain non void, definite
class {E} of statements, which we call elementary
statements...
I didn't write that.
That is part of how Curry defined True(x) rei Theorem(x) https://www.liarparadox.org/Haskell_Curry_45.pdf
On 1/3/26 5:23 PM, olcott wrote:
On 1/3/2026 2:58 PM, Tristan Wibberley wrote:
On 03/01/2026 17:30, olcott wrote:
On 1/3/2026 10:58 AM, Tristan Wibberley wrote:
We begin by postulating a certain non void, definite
class {E} of statements, which we call elementary
statements...
I didn't write that.
That is part of how Curry defined True(x) rei Theorem(x)
https://www.liarparadox.org/Haskell_Curry_45.pdf
But he doesn't define True(x) to be = Theorem(x)
On 03/01/2026 19:33, Richard Damon wrote:
On 1/3/26 11:54 AM, Tristan Wibberley wrote:
On 02/01/2026 15:35, Richard Damon wrote:
No, meta-system is NOT "full up" as there are an infinite number of
primes to use to define new objects.
I understood from reading it that those were occupied for variables.
or new terms, after all, we have an infinite number of them.
In fact, I thought he talked about assigning a number for proven
statements in the base system, so we could refer to them.
I missed that, it looked like they were taken for all naturals just for
an all the variables. I shall read that bit again.
This is needed
to create the lable of "G" in the meta, so we can refer to it as we
build the proof checker, that at the end we finally know the "value" it
represents.
And what number does it get? It should be expressible by the primitive symbols of the system P and the conventional shorthands that he defines.
If he merely defines the liar paradox then his system is inconsistent,
either P, or the meta-system.
On 1/3/2026 7:35 PM, Richard Damon wrote:
On 1/3/26 5:23 PM, olcott wrote:
On 1/3/2026 2:58 PM, Tristan Wibberley wrote:
On 03/01/2026 17:30, olcott wrote:
On 1/3/2026 10:58 AM, Tristan Wibberley wrote:
We begin by postulating a certain non void, definite
class {E} of statements, which we call elementary
statements...
I didn't write that.
That is part of how Curry defined True(x) rei Theorem(x)
https://www.liarparadox.org/Haskell_Curry_45.pdf
But he doesn't define True(x) to be = Theorem(x)
Thus, given {T}, an elementary theorem is an elementary statement
which is true.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Are you capable of ever paying complete attention?
I hyper-focus instead. This makes most everyone
else seem like they have severe attention deficit
by contrast.
On 1/3/26 8:45 PM, olcott wrote:
On 1/3/2026 7:35 PM, Richard Damon wrote:
On 1/3/26 5:23 PM, olcott wrote:
On 1/3/2026 2:58 PM, Tristan Wibberley wrote:
On 03/01/2026 17:30, olcott wrote:
On 1/3/2026 10:58 AM, Tristan Wibberley wrote:
We begin by postulating a certain non void, definite
class {E} of statements, which we call elementary
statements...
I didn't write that.
That is part of how Curry defined True(x) rei Theorem(x)
https://www.liarparadox.org/Haskell_Curry_45.pdf
But he doesn't define True(x) to be = Theorem(x)
Thus, given {T}, an elementary theorem is an elementary statement
which is true.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Are you capable of ever paying complete attention?
I hyper-focus instead. This makes most everyone
else seem like they have severe attention deficit
by contrast.
Which says that Theorems are true statement,
not that truth are proven statements.
On 1/3/2026 8:31 PM, Richard Damon wrote:
On 1/3/26 8:45 PM, olcott wrote:
On 1/3/2026 7:35 PM, Richard Damon wrote:
On 1/3/26 5:23 PM, olcott wrote:
On 1/3/2026 2:58 PM, Tristan Wibberley wrote:
On 03/01/2026 17:30, olcott wrote:
On 1/3/2026 10:58 AM, Tristan Wibberley wrote:
We begin by postulating a certain non void, definite
class {E} of statements, which we call elementary
statements...
I didn't write that.
That is part of how Curry defined True(x) rei Theorem(x)
https://www.liarparadox.org/Haskell_Curry_45.pdf
But he doesn't define True(x) to be = Theorem(x)
Thus, given {T}, an elementary theorem is an elementary statement
which is true.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Are you capable of ever paying complete attention?
I hyper-focus instead. This makes most everyone
else seem like they have severe attention deficit
by contrast.
Which says that Theorems are true statement, not that truth are proven
statements.
So you have no idea how true statements are derived
from other true statements ?
https://iep.utm.edu/val-snd/
On 1/3/26 9:49 PM, olcott wrote:
On 1/3/2026 8:31 PM, Richard Damon wrote:
On 1/3/26 8:45 PM, olcott wrote:
On 1/3/2026 7:35 PM, Richard Damon wrote:
On 1/3/26 5:23 PM, olcott wrote:
On 1/3/2026 2:58 PM, Tristan Wibberley wrote:
On 03/01/2026 17:30, olcott wrote:
On 1/3/2026 10:58 AM, Tristan Wibberley wrote:
We begin by postulating a certain non void, definite
class {E} of statements, which we call elementary
statements...
I didn't write that.
That is part of how Curry defined True(x) rei Theorem(x)
https://www.liarparadox.org/Haskell_Curry_45.pdf
But he doesn't define True(x) to be = Theorem(x)
Thus, given {T}, an elementary theorem is an elementary statement
which is true.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Are you capable of ever paying complete attention?
I hyper-focus instead. This makes most everyone
else seem like they have severe attention deficit
by contrast.
Which says that Theorems are true statement, not that truth are
proven statements.
So you have no idea how true statements are derived
from other true statements ?
https://iep.utm.edu/val-snd/
Right, but the chain can be infinite, and thus not a proof.
On 1/3/2026 9:07 PM, Richard Damon wrote:
On 1/3/26 9:49 PM, olcott wrote:
On 1/3/2026 8:31 PM, Richard Damon wrote:
On 1/3/26 8:45 PM, olcott wrote:
On 1/3/2026 7:35 PM, Richard Damon wrote:
On 1/3/26 5:23 PM, olcott wrote:
On 1/3/2026 2:58 PM, Tristan Wibberley wrote:
On 03/01/2026 17:30, olcott wrote:
On 1/3/2026 10:58 AM, Tristan Wibberley wrote:
We begin by postulating a certain non void, definite
class {E} of statements, which we call elementary
statements...
I didn't write that.
That is part of how Curry defined True(x) rei Theorem(x)
https://www.liarparadox.org/Haskell_Curry_45.pdf
But he doesn't define True(x) to be = Theorem(x)
Thus, given {T}, an elementary theorem is an elementary statement
which is true.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Are you capable of ever paying complete attention?
I hyper-focus instead. This makes most everyone
else seem like they have severe attention deficit
by contrast.
Which says that Theorems are true statement, not that truth are
proven statements.
So you have no idea how true statements are derived
from other true statements ?
https://iep.utm.edu/val-snd/
Right, but the chain can be infinite, and thus not a proof.
Right so we may never know if the Goldbach conjecture is true.
We do now that all paradoxes resolve to nonsense.
This means that True(L, x) can be defined for the
entire body of knowledge expressed in language.
"true on the basis of meaning expressed in language"
Eliminates a key issue that has plagued epistemology since 1951
https://www.theologie.uzh.ch/dam/jcr:ffffffff- fbd6-1538-0000-000070cf64bc/Quine51.pdf
On 1/3/26 10:36 PM, olcott wrote:
On 1/3/2026 9:07 PM, Richard Damon wrote:
On 1/3/26 9:49 PM, olcott wrote:
On 1/3/2026 8:31 PM, Richard Damon wrote:
On 1/3/26 8:45 PM, olcott wrote:
On 1/3/2026 7:35 PM, Richard Damon wrote:
On 1/3/26 5:23 PM, olcott wrote:
On 1/3/2026 2:58 PM, Tristan Wibberley wrote:
On 03/01/2026 17:30, olcott wrote:
On 1/3/2026 10:58 AM, Tristan Wibberley wrote:
We begin by postulating a certain non void, definite
class {E} of statements, which we call elementary
statements...
I didn't write that.
That is part of how Curry defined True(x) rei Theorem(x)
https://www.liarparadox.org/Haskell_Curry_45.pdf
But he doesn't define True(x) to be = Theorem(x)
Thus, given {T}, an elementary theorem is an elementary statement
which is true.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Are you capable of ever paying complete attention?
I hyper-focus instead. This makes most everyone
else seem like they have severe attention deficit
by contrast.
Which says that Theorems are true statement, not that truth are
proven statements.
So you have no idea how true statements are derived
from other true statements ?
https://iep.utm.edu/val-snd/
Right, but the chain can be infinite, and thus not a proof.
Right so we may never know if the Goldbach conjecture is true.
But it must be either True or False.
Your system can't handle that.
Unknown is a value of Knowledge, not Truth.
All you are doing is showing that you own system must be incomplete
becuase it can't even HANDLE some statements that we know must have a
truth value.
We do now that all paradoxes resolve to nonsense.
No, because the word "Paradox" just means an APPARENT contradiction.
For example, Zeno's paradox that seems to show that Achilies can't pass
the Tortoise is resolved by noting that while you went through an
infinite number of steps of logic, those only encompassed a finite
amount of time, and after that Achilies does pass the Tortoise.
The Liar's Paradox gets resolved by seeing that the statement just
doesn't have a Truth Value (Not all syntacticly valid statemente do) and thus isn't a Semantically valid statement, and the "Not" operator is
being given an invalid value (or Not(not-a-truth-value) is just not-a- truth-value).
This means that True(L, x) can be defined for the
*entire body of knowledge expressed in language*
No, because we can still express in that language statements that we can
not know if they are true, like the Goldbach conjecture.
Note, the True predicate has a domain of all syntactially valid
expressions, and returns false for any that are semantically invalid.
Thus True(L, "The Goldbach Conjecture") needs to resolve that actual
truth of that conjecture.
All you are showing is your inability to understand the rules of the
game you got in.
"true on the basis of meaning expressed in language"
Eliminates a key issue that has plagued epistemology since 1951
No, because it just admits its own limitation, and put forward a mis- defintion of Truth.
https://www.theologie.uzh.ch/dam/jcr:ffffffff-
fbd6-1538-0000-000070cf64bc/Quine51.pdf
Which is about Philosophy, not Logic, which is part of your problem, you don't understand the difference.
On 1/2/26 12:24 PM, Mike Terry wrote:
On 02/01/2026 15:25, Richard Damon wrote:
On 1/2/26 1:14 AM, Tristan Wibberley wrote:
On 02/01/2026 04:45, Richard Damon wrote:I will disagree with you here. Maybe it iw what you are trying to define "derived" as.
Truth in the base system has always
actually been theorems of the base system.
But only if "Theorem" includes things proven to be true in the system >>>>> even if the proof is in another.
If the statement is derived in another then it is a theorem of the other. >>>
I can certainly use one system to guide me in building a statement in another. Or do you think
that is a task too hard?
I can certainly use one system that knows about another to show that a statement must be true in
that other.
If you want to reserve the lable "Theorem" for only things provable in taht system, I will let
you, but point out I think you are in the minority, and ask for your reference that specifies that.
No, I'd say Tristan is spot on with how that's normally done.
While speaking informally, "theorem" can mean "a mathematical statement that has a convincing
argument for its truth" (e.g. Pythagoras' theorem), in formal logic "Theorem" and "Theory" have a
technical meaning:a "Theory" being the deductive closure of a set of axioms, and a Theorem being a
sentence of the Theory.a So every Theorem in the Theory has a "derivation" from the theories
axioms. aIt is not directly to do with "truth" in the formal system.a [Of course, we want our
system (including axioms) to be sound, so all Theorems will be true.]
aa <https://en.wikipedia.org/wiki/ Theory_(mathematical_logic)#Deductive_theories>
Note, the addition of the adjective DEDUCTIVE.
Of course, you could be learning from an author taking a different approach, but I haven't
personally come across one who would say that the sentence represented by G was a "Theorem" of the
underlying logical system!a (That would (IMO) be grossly misleading...)
I will somewhat agree here, because generally the term is reserved for statements about a more
general truth, as opposed to a statement about a specific fact. But the most generic definition is
just a statment that has been proven.
Similarly, the word "proof" can be informal (simply an argument that convinces people of the truth
of a statement), or refer to the "proof calculus" of the formal system being discussed.a Most
authors I've come across seem to use "proof" more or less informally and for clarity choose
another word for whatever sequence of syntactic "proof steps" the formal system specifies.a Often
"derivation" is used, and that seems intuitive to me, so I try to always use that term here, and
using "proof" for more general mathematial arguments, e.g. proving that the G statement is "true"
using some meta-theory.
The issue is that "derivation" doesn't actually imply a finiteness, which is a necessity of "proof".
The point is that the standard statement of "Incompleteness" talks about the provability of
statements in the system. Provability is inherently about the ability to create a proof in the system.
Yes, often an other will use a more confined word to establish the method of a proof.
Also just as an aside, I don't recall that Godel ever talked about "truth" of his G statement.
His proof was concerned with provability. (Neither the G sentence nor its negation is provable.)
Mike.
But there are many statements that they or their negation is provable, all you need is a statement
that isn't a truth bearer, for example, the liar paradox.
Incompleteness is about a statement that is true in the system and not being provable. That is the
ESSENCE of the concept of incompleteness.
It uses BOTH the concept of Truth, and Proof, so trying to say that these aren't terms used seems to
be a contradiction in your explaination.
On 1/4/2026 6:42 AM, Richard Damon wrote:
On 1/3/26 10:36 PM, olcott wrote:
On 1/3/2026 9:07 PM, Richard Damon wrote:
On 1/3/26 9:49 PM, olcott wrote:
On 1/3/2026 8:31 PM, Richard Damon wrote:
On 1/3/26 8:45 PM, olcott wrote:
On 1/3/2026 7:35 PM, Richard Damon wrote:
On 1/3/26 5:23 PM, olcott wrote:
On 1/3/2026 2:58 PM, Tristan Wibberley wrote:
On 03/01/2026 17:30, olcott wrote:
On 1/3/2026 10:58 AM, Tristan Wibberley wrote:
We begin by postulating a certain non void, definite
class {E} of statements, which we call elementary
statements...
I didn't write that.
That is part of how Curry defined True(x) rei Theorem(x)
https://www.liarparadox.org/Haskell_Curry_45.pdf
But he doesn't define True(x) to be = Theorem(x)
Thus, given {T}, an elementary theorem is an elementary statement >>>>>>> which is true.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Are you capable of ever paying complete attention?
I hyper-focus instead. This makes most everyone
else seem like they have severe attention deficit
by contrast.
Which says that Theorems are true statement, not that truth are
proven statements.
So you have no idea how true statements are derived
from other true statements ?
https://iep.utm.edu/val-snd/
Right, but the chain can be infinite, and thus not a proof.
Right so we may never know if the Goldbach conjecture is true.
But it must be either True or False.
Your system can't handle that.
Unknown is a value of Knowledge, not Truth.
All you are doing is showing that you own system must be incomplete
becuase it can't even HANDLE some statements that we know must have a
truth value.
It is not incomplete in the G||del sense.
We do now that all paradoxes resolve to nonsense.
No, because the word "Paradox" just means an APPARENT contradiction.
For example, Zeno's paradox that seems to show that Achilies can't
pass the Tortoise is resolved by noting that while you went through an
infinite number of steps of logic, those only encompassed a finite
amount of time, and after that Achilies does pass the Tortoise.
paradoxes resolve to nonsense.
The Liar's Paradox gets resolved by seeing that the statement just
doesn't have a Truth Value (Not all syntacticly valid statemente do)
and thus isn't a Semantically valid statement, and the "Not" operator
is being given an invalid value (or Not(not-a-truth-value) is just
not-a- truth-value).
Yes
This means that True(L, x) can be defined for the
*entire body of knowledge expressed in language*
No, because we can still express in that language statements that we
can not know if they are true, like the Goldbach conjecture.
Did you notice that those are not in the body of knowledge?
*entire body of knowledge expressed in language*
Note, the True predicate has a domain of all syntactially valid
expressions, and returns false for any that are semantically invalid.
If X is unknown or
semantically incoherent or
simply not encoded then True(X)==FALSE and True(~X)==FALSE
Thus True(L, "The Goldbach Conjecture") needs to resolve that actual
truth of that conjecture.
This is the domain
*entire body of knowledge expressed in language*
The Goldbach Conjecture's truth value is not in that domain
All you are showing is your inability to understand the rules of the
game you got in.
After 28 years I have finally got it.
"true on the basis of meaning expressed in language"
Eliminates a key issue that has plagued epistemology since 1951
No, because it just admits its own limitation, and put forward a mis-
defintion of Truth.
The analytic/synthetic distinction was broken by Quine
since 1951. I reframed it as the Analytic(Olcott) / Empirical
distinction.
https://www.theologie.uzh.ch/dam/jcr:ffffffff-
fbd6-1538-0000-000070cf64bc/Quine51.pdf
Which is about Philosophy, not Logic, which is part of your problem,
you don't understand the difference.
I defined the computable subset of knowledge.
On 02/01/2026 17:54, Richard Damon wrote:
On 1/2/26 12:24 PM, Mike Terry wrote:
On 02/01/2026 15:25, Richard Damon wrote:
On 1/2/26 1:14 AM, Tristan Wibberley wrote:
On 02/01/2026 04:45, Richard Damon wrote:
Truth in the base system has always
actually been theorems of the base system.
But only if "Theorem" includes things proven to be true in the system >>>>>> even if the proof is in another.
If the statement is derived in another then it is a theorem of the
other.
I will disagree with you here. Maybe it iw what you are trying to
define "derived" as.
I can certainly use one system to guide me in building a statement
in another. Or do you think that is a task too hard?
I can certainly use one system that knows about another to show that
a statement must be true in that other.
If you want to reserve the lable "Theorem" for only things provable
in taht system, I will let you, but point out I think you are in the
minority, and ask for your reference that specifies that.
No, I'd say Tristan is spot on with how that's normally done.
While speaking informally, "theorem" can mean "a mathematical
statement that has a convincing argument for its truth" (e.g.
Pythagoras' theorem), in formal logic "Theorem" and "Theory" have a
technical meaning:-a "Theory" being the deductive closure of a set of
axioms, and a Theorem being a sentence of the Theory.-a So every
Theorem in the Theory has a "derivation" from the theories axioms.
-aIt is not directly to do with "truth" in the formal system.-a [Of
course, we want our system (including axioms) to be sound, so all
Theorems will be true.]
-a-a <https://en.wikipedia.org/wiki/
Theory_(mathematical_logic)#Deductive_theories>
Note, the addition of the adjective DEDUCTIVE.
That's the type of theory most often encountered, I'd say.-a GIT is concerned with such a theory, where axioms are given etc. and we are
looking at what sentences can be proved from those axioms. Wikipedia
always tries to be as general as it can possibly be, making it an awful place for someone trying to /learn/ a maths topic.-a It's great if you're already a maths professor, then you can read some article and nod
wisely, and suggest edits to make it /even more/ general! :)
Of course, you could be learning from an author taking a different
approach, but I haven't personally come across one who would say that
the sentence represented by G was a "Theorem" of the underlying
logical system!-a (That would (IMO) be grossly misleading...)
I will somewhat agree here, because generally the term is reserved for
statements about a more general truth, as opposed to a statement about
a specific fact. But the most generic definition is just a statment
that has been proven.
Yes, but we are talking about formal systems, and "Theorem" in such a context implies a proof
/within the proof calculus of that formal system/.-a That is what GIT is concerned with.-a (I give an example of what I mean by "proof calculus" below, but I'm sure you understand the idea, just possibly not the
wording.)
Similarly, the word "proof" can be informal (simply an argument that
convinces people of the truth of a statement), or refer to the "proof
calculus" of the formal system being discussed.-a Most authors I've
come across seem to use "proof" more or less informally and for
clarity choose another word for whatever sequence of syntactic "proof
steps" the formal system specifies.-a Often "derivation" is used, and
that seems intuitive to me, so I try to always use that term here,
and using "proof" for more general mathematial arguments, e.g.
proving that the G statement is "true" using some meta-theory.
The issue is that "derivation" doesn't actually imply a finiteness,
which is a necessity of "proof".
Where do you get that idea?-a Are you thinking "derivation" is just an informal word?-a I'm using it in the technical sense previously explained.
Within a formal system there will be a set of rules which define what a valid "derivation" looks like.-a These would ensure that such derivations are finite.-a (I'm sure someone at some time has made a special study of "infinite proofs", but that is off the beaten track.)-a As explained in
my previous post, I'm using "derivation" as the technical term for
whatever passes as a "formal proof conforming to the requirements of the proof calculus of the system".-a This is so that the idea does not get muddled with your more general kind of proof = "convincing argument in
some meta-theory".
Derivations have G||del numbers.-a "Convincing arguments in the meta- theory" do not.-a The idea of a G||del number for an infinite derivation does not make sense.
The point is that the standard statement of "Incompleteness" talks
about the provability of statements in the system. Provability is
inherently about the ability to create a proof in the system.
"(FORMAL) Proof in the system" = "derivation".-a You want to use the word "proof" more generally, namely to cover arguments in a meta-theory which establish some meta-truth concerning the base system.-a That's ok [and
the reason I use "derivation" to distinguish from "informal proof"], but
not the type of "proof" that incompleteness refers to.
Yes, often an other will use a more confined word to establish the
method of a proof.
Right.-a The logical system will have some kind of "proof calculus" which says EXACTLY what constitutes a valid derivation for that system.-a E.g. perhaps something like:
-a A derivation is a finite sequence of sentences in the language of the formal system, such that
-a each sentence is either:
-a --a-a an axiom of the system,
-a --a-a or can be constructed from previous sentences by applying one of the
-a-a-a-a-a following "deduction rules":
-a-a-a-a-a --a-a [probably something equivalent to Modus Ponens]
-a-a-a-a-a --a-a [maybe other listed rules]
-a The last sentence of the derivation is the "result" of the derivation.
When G||del talks of provability in a system, he always means such a derivation, not some informal convincing argument.
From here on I'm just going to use the word derivation without
explaining it every time!
Also just as an aside, I don't recall that Godel ever talked about
"truth" of his G statement. His proof was concerned with provability.
(Neither the G sentence nor its negation is provable.)
Mike.
But there are many statements that they or their negation is provable,
all you need is a statement that isn't a truth bearer, for example,
the liar paradox.
We are talking about sentences in the language of some formal system.
You have been arguing for too long with PO!!
Incompleteness is about a statement that is true in the system and not
being provable. That is the ESSENCE of the concept of incompleteness.
There are different types of incompleteness, and different definitions. G||del's concerns there being a statement G such that neither G nor -4G
has a derivation in the system.-a There is no reference to "truth" in
that and I'd say his proof is essentially syntactical in nature.
All that aside, yes there are many statements of formal systems for
which neither they nor their negations are provable - those systems are incomplete, and its no big deal because largely speaking people did not expect them to be complete!-a The surprise with GIT is that it showed specific systems were incomplete, where we previously had no good reason
to think that might be the case.
In more modern terms e.g. we have FOL (first order logic) and PA
(Peano's Arithmetic - a FOL theory with its own symbols/axioms) and it
might well seem that all arithmetic truths could be proved within that system - after all, it has an induction axiom schema, and seems to be sufficient for our needs when we look at typical proofs we see in
school. What else could we need?-a Have we got "enough" axioms with PA? Mathematicians at the time of G||del would have liked to think so.-a But
GIT (more modern FOL version) shows that's not the case: PA is
incomplete. Also we see that extending PA adding a few more axioms isn't going to change this.
[It's strange - both you and Tristan expressed surprise that everyday theories might be incomplete. Incompleteness is not a "fatal flaw" in a theory in the way that inconsistency is - some theories are just
incomplete by their nature.)
It uses BOTH the concept of Truth, and Proof, so trying to say that
these aren't terms used seems to be a contradiction in your explaination.
You just SAY what you think incompleteness means, and then point out
that your definition used the word truth and mine doesn't!-a Very PO-
like :)-a-a A better way to go would be to look at G||del's paper and see how he defined incompleteness.
I'd say we have two notions:
a) A theory T is incomplete if there exists a sentence s such that
-a-a neither s nor -4s is a theorem of T
On 1/4/26 9:48 AM, olcott wrote:
On 1/4/2026 6:42 AM, Richard Damon wrote:
On 1/3/26 10:36 PM, olcott wrote:
On 1/3/2026 9:07 PM, Richard Damon wrote:
On 1/3/26 9:49 PM, olcott wrote:
On 1/3/2026 8:31 PM, Richard Damon wrote:
On 1/3/26 8:45 PM, olcott wrote:
On 1/3/2026 7:35 PM, Richard Damon wrote:
On 1/3/26 5:23 PM, olcott wrote:
On 1/3/2026 2:58 PM, Tristan Wibberley wrote:
On 03/01/2026 17:30, olcott wrote:
On 1/3/2026 10:58 AM, Tristan Wibberley wrote:
We begin by postulating a certain non void, definite
class {E} of statements, which we call elementary
statements...
I didn't write that.
That is part of how Curry defined True(x) rei Theorem(x)
https://www.liarparadox.org/Haskell_Curry_45.pdf
But he doesn't define True(x) to be = Theorem(x)
Thus, given {T}, an elementary theorem is an elementary statement >>>>>>>> which is true.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Are you capable of ever paying complete attention?
I hyper-focus instead. This makes most everyone
else seem like they have severe attention deficit
by contrast.
Which says that Theorems are true statement, not that truth are >>>>>>> proven statements.
So you have no idea how true statements are derived
from other true statements ?
https://iep.utm.edu/val-snd/
Right, but the chain can be infinite, and thus not a proof.
Right so we may never know if the Goldbach conjecture is true.
But it must be either True or False.
Your system can't handle that.
Unknown is a value of Knowledge, not Truth.
All you are doing is showing that you own system must be incomplete
becuase it can't even HANDLE some statements that we know must have a
truth value.
It is not incomplete in the G||del sense.
Then it is just inconsistant, and incomplete in the more general sense.
You CAN'T have you goal of "all general knowledge" and Truth is Provable
at the same time without having a broken system.
We do now that all paradoxes resolve to nonsense.
No, because the word "Paradox" just means an APPARENT contradiction.
For example, Zeno's paradox that seems to show that Achilies can't
pass the Tortoise is resolved by noting that while you went through
an infinite number of steps of logic, those only encompassed a finite
amount of time, and after that Achilies does pass the Tortoise.
paradoxes resolve to nonsense.
So, the sum of the number 1/2 + 1/4 + 1/8 + 1/16 ... is nonsense?
The Liar's Paradox gets resolved by seeing that the statement just
doesn't have a Truth Value (Not all syntacticly valid statemente do)
and thus isn't a Semantically valid statement, and the "Not" operator
is being given an invalid value (or Not(not-a-truth-value) is just
not-a- truth-value).
Yes
So (as the PREDICATE) True(LP) is false, and True(~LP) is also false.
But if X = ~True(X) can't use this excape, as the "True" preidcate is
ALWAYS a truth value, and thus ~True(X) is also ALWAYS a truth value.
This is the problem with a truth predicate, it looses the escape valve
of just using the not operator.
This means that True(L, x) can be defined for the
*entire body of knowledge expressed in language*
No, because we can still express in that language statements that we
can not know if they are true, like the Goldbach conjecture.
Did you notice that those are not in the body of knowledge?
*entire body of knowledge expressed in language*
So, are you saying you language can only express statements already know
to be true?
In other words, it isn't a "logic" that allows discovery?
The body of knowldege certainly understand the concepts of summing two numbers, of even numbers, and primes, so, if able to be inquisative, ask about the sums of primes and even numbers.
This shows that you system just can't do what you want it to do, and you view of "semantics" is just insufficent to do what you want it to,
Note, the True predicate has a domain of all syntactially valid
expressions, and returns false for any that are semantically invalid.
If X is unknown or
semantically incoherent or
simply not encoded then True(X)==FALSE and True(~X)==FALSE
Nope.
The fact we don't KNOW the truth of X doesn't affect the value returned
by True(x)
It seems you confuse Known with True, and not even go so far as Knowable.
That means truth values in you system CHANGE over time, which is unacceptable in an actual logic system.
Thus True(L, "The Goldbach Conjecture") needs to resolve that actual
truth of that conjecture.
This is the domain
*entire body of knowledge expressed in language*
The Goldbach Conjecture's truth value is not in that domain
But is expressible in that language.
I guess you are just asserting your system is just a repository of Knowledge, and WORTHLESS in dealing with statement not in its repository.
All you are showing is your inability to understand the rules of the
game you got in.
After 28 years I have finally got it.
Nop,e just showing you have lost it.
"true on the basis of meaning expressed in language"
Eliminates a key issue that has plagued epistemology since 1951
No, because it just admits its own limitation, and put forward a mis-
defintion of Truth.
The analytic/synthetic distinction was broken by Quine
since 1951. I reframed it as the Analytic(Olcott) / Empirical
distinction.
But that isn't part of Formal Logic, just general Philosophy.
It seems you don't even understand the scope of the field you are trying
to talk about.
https://www.theologie.uzh.ch/dam/jcr:ffffffff-
fbd6-1538-0000-000070cf64bc/Quine51.pdf
Which is about Philosophy, not Logic, which is part of your problem,
you don't understand the difference.
I defined the computable subset of knowledge.
No, you have failed to actually define anything.
You have a concept for a worthless system to record knowledge that you
can interograte to see if something was already known.
On 1/4/2026 1:21 PM, Richard Damon wrote:
On 1/4/26 9:48 AM, olcott wrote:
On 1/4/2026 6:42 AM, Richard Damon wrote:
On 1/3/26 10:36 PM, olcott wrote:
On 1/3/2026 9:07 PM, Richard Damon wrote:
On 1/3/26 9:49 PM, olcott wrote:
On 1/3/2026 8:31 PM, Richard Damon wrote:
On 1/3/26 8:45 PM, olcott wrote:
On 1/3/2026 7:35 PM, Richard Damon wrote:
On 1/3/26 5:23 PM, olcott wrote:
On 1/3/2026 2:58 PM, Tristan Wibberley wrote:
On 03/01/2026 17:30, olcott wrote:
On 1/3/2026 10:58 AM, Tristan Wibberley wrote:
We begin by postulating a certain non void, definite >>>>>>>>>>>>> class {E} of statements, which we call elementary
statements...
I didn't write that.
That is part of how Curry defined True(x) rei Theorem(x) >>>>>>>>>>> https://www.liarparadox.org/Haskell_Curry_45.pdf
But he doesn't define True(x) to be = Theorem(x)
Thus, given {T}, an elementary theorem is an elementary statement >>>>>>>>> which is true.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Are you capable of ever paying complete attention?
I hyper-focus instead. This makes most everyone
else seem like they have severe attention deficit
by contrast.
Which says that Theorems are true statement, not that truth are >>>>>>>> proven statements.
So you have no idea how true statements are derived
from other true statements ?
https://iep.utm.edu/val-snd/
Right, but the chain can be infinite, and thus not a proof.
Right so we may never know if the Goldbach conjecture is true.
But it must be either True or False.
Your system can't handle that.
Unknown is a value of Knowledge, not Truth.
All you are doing is showing that you own system must be incomplete
becuase it can't even HANDLE some statements that we know must have
a truth value.
It is not incomplete in the G||del sense.
Then it is just inconsistant, and incomplete in the more general sense.
You CAN'T have you goal of "all general knowledge" and Truth is
Provable at the same time without having a broken system.
It is categorically impossible to derive any element
of the body of knowledge that can be expressed in
language that is not entirely comprised of some relation
between finite strings.
We do now that all paradoxes resolve to nonsense.
No, because the word "Paradox" just means an APPARENT contradiction.
For example, Zeno's paradox that seems to show that Achilies can't
pass the Tortoise is resolved by noting that while you went through
an infinite number of steps of logic, those only encompassed a
finite amount of time, and after that Achilies does pass the Tortoise. >>>>
paradoxes resolve to nonsense.
So, the sum of the number 1/2 + 1/4 + 1/8 + 1/16 ... is nonsense?
The Liar's Paradox gets resolved by seeing that the statement just
doesn't have a Truth Value (Not all syntacticly valid statemente do)
and thus isn't a Semantically valid statement, and the "Not"
operator is being given an invalid value (or Not(not-a-truth-value)
is just not-a- truth-value).
Yes
So (as the PREDICATE) True(LP) is false, and True(~LP) is also false.
Indicating that LP is not a truth-bearer / proposition.
But if X = ~True(X) can't use this excape, as the "True" preidcate is
ALWAYS a truth value, and thus ~True(X) is also ALWAYS a truth value.
We just went over this:
LP is not a truth-bearer even when LP is called X.
This is the problem with a truth predicate, it looses the escape valve
of just using the not operator.
This means that True(L, x) can be defined for the
*entire body of knowledge expressed in language*
No, because we can still express in that language statements that we
can not know if they are true, like the Goldbach conjecture.
Did you notice that those are not in the body of knowledge?
*entire body of knowledge expressed in language*
So, are you saying you language can only express statements already
know to be true?
Language can express the the truth value of
the Goldbach conjecture is unknown.
In other words, it isn't a "logic" that allows discovery?
The body of knowldege certainly understand the concepts of summing two
numbers, of even numbers, and primes, so, if able to be inquisative,
ask about the sums of primes and even numbers.
No infinite proof completes in finite time.
Not even with the magic fairy dust of an
Oracle Machine.
This shows that you system just can't do what you want it to do, and
you view of "semantics" is just insufficent to do what you want it to,
I can't get my kitchen sink to bake me a birthday cake either.
Note, the True predicate has a domain of all syntactially valid
expressions, and returns false for any that are semantically invalid.
If X is unknown or
semantically incoherent or
simply not encoded then True(X)==FALSE and True(~X)==FALSE
Nope.
I stipulate that is an element of the architecture
that I am specifying. stipulated specifications
can only be incorrect when they are impossible
of incoheeent.
The fact we don't KNOW the truth of X doesn't affect the value
returned by True(x)
Unknowns are not in the domain of knowledge.
It seems you confuse Known with True, and not even go so far as Knowable.
domain of knowledge.
domain of knowledge.
domain of knowledge.
domain of knowledge.
domain of knowledge.
That means truth values in you system CHANGE over time, which is
unacceptable in an actual logic system.
Pluto being measured against updated criteria
is no longer a planet.
Thus True(L, "The Goldbach Conjecture") needs to resolve that actual
truth of that conjecture.
This is the domain
*entire body of knowledge expressed in language*
The Goldbach Conjecture's truth value is not in that domain
But is expressible in that language.
It is categorically impossible to derive any element
of the body of knowledge that can be expressed in
language that is not entirely comprised of some relation
between finite strings.
I guess you are just asserting your system is just a repository of
Knowledge, and WORTHLESS in dealing with statement not in its repository.
It is not an all knowing mind of God.
All you are showing is your inability to understand the rules of the
game you got in.
After 28 years I have finally got it.
Nop,e just showing you have lost it.
"true on the basis of meaning expressed in language"
Eliminates a key issue that has plagued epistemology since 1951
No, because it just admits its own limitation, and put forward a
mis- defintion of Truth.
The analytic/synthetic distinction was broken by Quine
since 1951. I reframed it as the Analytic(Olcott) / Empirical
distinction.
But that isn't part of Formal Logic, just general Philosophy.
My "true on the basis of meaning expressed in language"
within the body of knowledge specifies the precise subset
of knowledge that can be computed on the basis of relations
between finite strings. It also reframes the analytic/synthetic
distinction with an unequivocal line-of-demarcation between
Analytic(Olcott) and Empirical(Olcott).
It seems you don't even understand the scope of the field you are
trying to talk about.
https://www.theologie.uzh.ch/dam/jcr:ffffffff-
fbd6-1538-0000-000070cf64bc/Quine51.pdf
Which is about Philosophy, not Logic, which is part of your problem,
you don't understand the difference.
I defined the computable subset of knowledge.
No, you have failed to actually define anything.
It is categorically impossible to derive any element
of the body of knowledge that can be expressed in
language that is not entirely comprised of some relation
between finite strings.
You have a concept for a worthless system to record knowledge that you
can interograte to see if something was already known.
That you require an acceptable system to be the
omniscient mind of God is a category error.
On 1/4/26 3:21 PM, olcott wrote:
It is categorically impossible to derive any element
of the body of knowledge that can be expressed in
language that is not entirely comprised of some relation
between finite strings.
So?
The problem is we want to derive things that aren't yet in the body of knowledge.
And, the relationship between finite strings is often not what you
consider the "meaning of the words", as the strings often aren't just
words.
--
You have a concept for a worthless system to record knowledge that
you can interograte to see if something was already known.
That you require an acceptable system to be the
omniscient mind of God is a category error.
That you require the system to be impotent, and not able to talk about something unknow makes it worthless.
On 1/4/2026 2:23 PM, Richard Damon wrote:
On 1/4/26 3:21 PM, olcott wrote:
It is categorically impossible to derive any element
of the body of knowledge that can be expressed in
language that is not entirely comprised of some relation
between finite strings.
So?
That is the conclusive proof that I am correct.
The problem is we want to derive things that aren't yet in the body of
knowledge.
If you want to know the name of your wife's
mother and you have not met your wife yet
then the answer is not available by any means.\
Once the body of general knowledge is fully
populated an intelligent system can derive
brand new knowledge on the basis of semantic
entailment from this basis.
And, the relationship between finite strings is often not what you
consider the "meaning of the words", as the strings often aren't just
words.
I made sure to never limit it to words.
You have a concept for a worthless system to record knowledge that
you can interograte to see if something was already known.
That you require an acceptable system to be the
omniscient mind of God is a category error.
That you require the system to be impotent, and not able to talk about
something unknow makes it worthless.
On 02/01/2026 17:54, Richard Damon wrote:<SNIP>
On 1/2/26 12:24 PM, Mike Terry wrote:
On 02/01/2026 15:25, Richard Damon wrote:
On 1/2/26 1:14 AM, Tristan Wibberley wrote:
On 02/01/2026 04:45, Richard Damon wrote:
Similarly, the word "proof" can be informal (simply an argument that
convinces people of the truth of a statement), or refer to the "proof
calculus" of the formal system being discussed.-a Most authors I've
come across seem to use "proof" more or less informally and for
clarity choose another word for whatever sequence of syntactic "proof
steps" the formal system specifies.-a Often "derivation" is used, and
that seems intuitive to me, so I try to always use that term here,
and using "proof" for more general mathematial arguments, e.g.
proving that the G statement is "true" using some meta-theory.
The issue is that "derivation" doesn't actually imply a finiteness,
which is a necessity of "proof".
Where do you get that idea?-a Are you thinking "derivation" is just an informal word?-a I'm using it in the technical sense previously explained.
Within a formal system there will be a set of rules which define what a valid "derivation" looks like.-a These would ensure that such derivations are finite.-a (I'm sure someone at some time has made a special study of "infinite proofs", but that is off the beaten track.)-a As explained in
my previous post, I'm using "derivation" as the technical term for
whatever passes as a "formal proof conforming to the requirements of the proof calculus of the system".-a This is so that the idea does not get muddled with your more general kind of proof = "convincing argument in
some meta-theory". <SNIP>
On 1/4/2026 11:55 AM, Mike Terry wrote:
On 02/01/2026 17:54, Richard Damon wrote:-a-a <SNIP>
On 1/2/26 12:24 PM, Mike Terry wrote:
On 02/01/2026 15:25, Richard Damon wrote:
On 1/2/26 1:14 AM, Tristan Wibberley wrote:
On 02/01/2026 04:45, Richard Damon wrote:
Similarly, the word "proof" can be informal (simply an argument that
convinces people of the truth of a statement), or refer to the
"proof calculus" of the formal system being discussed.-a Most authors >>>> I've come across seem to use "proof" more or less informally and for
clarity choose another word for whatever sequence of syntactic
"proof steps" the formal system specifies.-a Often "derivation" is
used, and that seems intuitive to me, so I try to always use that
term here, and using "proof" for more general mathematial arguments,
e.g. proving that the G statement is "true" using some meta-theory.
The issue is that "derivation" doesn't actually imply a finiteness,
which is a necessity of "proof".
Where do you get that idea?-a Are you thinking "derivation" is just an
informal word?-a I'm using it in the technical sense previously explained. >>
Within a formal system there will be a set of rules which define what
a valid "derivation" looks like.-a These would ensure that such
derivations are finite.-a (I'm sure someone at some time has made a
special study of "infinite proofs", but that is off the beaten
track.)-a As explained in my previous post, I'm using "derivation" as
the technical term for whatever passes as a "formal proof conforming
to the requirements of the proof calculus of the system".-a This is so
that the idea does not get muddled with your more general kind of
proof = "convincing argument in some meta-theory".-a-a-a-a <SNIP>
There are examples of the following situations that I remember from discussions with logicians circa. 60 years ago. Assume we have a simple axiomatic system that allows us to express some facts about what we
believe to be natural numbers. Call the objects in a model of the system
N and we wish to prove, in our little system, that for all n in N p(n).
Now it turns out that for any n in N we can write a simple finite proof
of p(n) but are (provably in a larger or meta system) not able to prove
the universally quantified statement in the little system. Well
actually .... if you wrote a proof, in the little system, for each n in
N and joined them with and "&" operator you would prove the quantified statement albeit with an infinity proof. In a little larger system,
perhaps with an induction axiom, or a meta system it might be trivial to prove the whole thing with a finite effort.
On 1/4/26 5:13 PM, Jeff Barnett wrote:
On 1/4/2026 11:55 AM, Mike Terry wrote:
On 02/01/2026 17:54, Richard Damon wrote:-a-a-a <SNIP>
On 1/2/26 12:24 PM, Mike Terry wrote:
On 02/01/2026 15:25, Richard Damon wrote:
On 1/2/26 1:14 AM, Tristan Wibberley wrote:
On 02/01/2026 04:45, Richard Damon wrote:
Similarly, the word "proof" can be informal (simply an argument
that convinces people of the truth of a statement), or refer to the >>>>> "proof calculus" of the formal system being discussed.-a Most
authors I've come across seem to use "proof" more or less
informally and for clarity choose another word for whatever
sequence of syntactic "proof steps" the formal system specifies.
Often "derivation" is used, and that seems intuitive to me, so I
try to always use that term here, and using "proof" for more
general mathematial arguments, e.g. proving that the G statement is >>>>> "true" using some meta-theory.
The issue is that "derivation" doesn't actually imply a finiteness,
which is a necessity of "proof".
Where do you get that idea?-a Are you thinking "derivation" is just an
informal word?-a I'm using it in the technical sense previously
explained.
Within a formal system there will be a set of rules which define what
a valid "derivation" looks like.-a These would ensure that such
derivations are finite.-a (I'm sure someone at some time has made a
special study of "infinite proofs", but that is off the beaten
track.)-a As explained in my previous post, I'm using "derivation" as
the technical term for whatever passes as a "formal proof conforming
to the requirements of the proof calculus of the system".-a This is so
that the idea does not get muddled with your more general kind of
proof = "convincing argument in some meta-theory".-a-a-a-a <SNIP>
There are examples of the following situations that I remember from
discussions with logicians circa. 60 years ago. Assume we have a
simple axiomatic system that allows us to express some facts about
what we believe to be natural numbers. Call the objects in a model of
the system N and we wish to prove, in our little system, that for all
n in N p(n). Now it turns out that for any n in N we can write a
simple finite proof of p(n) but are (provably in a larger or meta
system) not able to prove the universally quantified statement in the
little system. Well actually .... if you wrote a proof, in the little
system, for each n in N and joined them with and "&" operator you
would prove the quantified statement albeit with an infinity proof. In
a little larger system, perhaps with an induction axiom, or a meta
system it might be trivial to prove the whole thing with a finite effort.
The issue how can you know or show that you can make such a proof for
all n within the system?
then you can built a finite proof based on those cases.
The problem is that the normal definition of "proof" requires it to be finite, as proofs are supposed to SHOW to a person that the statement is true, and we can't handle such an infinite series.
System that try to define "Proof" to include infinite logic, run into
the issue that infinte proofs can't be actually created or viewed, and
thus the system has decoupled "knoweledge" from proofs.
What you describe is sort of what Godel does, but points out that your infinite chain isn't a proof by the rules of standard logic, as the
proof in the meta system provides a reason why each of the numbers
applied to the relationship will fail. Thus, we DO have your concept of
an infinite "proof" in the system, which actually shows that the
statement must actually be true in the system (but this truth can't be
shown in the system except with infinite work).
On 1/4/2026 3:50 PM, Richard Damon wrote:
On 1/4/26 5:13 PM, Jeff Barnett wrote:
On 1/4/2026 11:55 AM, Mike Terry wrote:
On 02/01/2026 17:54, Richard Damon wrote:-a-a-a <SNIP>
On 1/2/26 12:24 PM, Mike Terry wrote:
On 02/01/2026 15:25, Richard Damon wrote:
On 1/2/26 1:14 AM, Tristan Wibberley wrote:
On 02/01/2026 04:45, Richard Damon wrote:
Similarly, the word "proof" can be informal (simply an argument
that convinces people of the truth of a statement), or refer to
the "proof calculus" of the formal system being discussed.-a Most >>>>>> authors I've come across seem to use "proof" more or less
informally and for clarity choose another word for whatever
sequence of syntactic "proof steps" the formal system specifies.
Often "derivation" is used, and that seems intuitive to me, so I
try to always use that term here, and using "proof" for more
general mathematial arguments, e.g. proving that the G statement
is "true" using some meta-theory.
The issue is that "derivation" doesn't actually imply a finiteness, >>>>> which is a necessity of "proof".
Where do you get that idea?-a Are you thinking "derivation" is just
an informal word?-a I'm using it in the technical sense previously
explained.
Within a formal system there will be a set of rules which define
what a valid "derivation" looks like.-a These would ensure that such
derivations are finite.-a (I'm sure someone at some time has made a
special study of "infinite proofs", but that is off the beaten
track.)-a As explained in my previous post, I'm using "derivation" as >>>> the technical term for whatever passes as a "formal proof conforming
to the requirements of the proof calculus of the system".-a This is
so that the idea does not get muddled with your more general kind of
proof = "convincing argument in some meta-theory".-a-a-a-a <SNIP>
There are examples of the following situations that I remember from
discussions with logicians circa. 60 years ago. Assume we have a
simple axiomatic system that allows us to express some facts about
what we believe to be natural numbers. Call the objects in a model of
the system N and we wish to prove, in our little system, that for all
n in N p(n). Now it turns out that for any n in N we can write a
simple finite proof of p(n) but are (provably in a larger or meta
system) not able to prove the universally quantified statement in the
little system. Well actually .... if you wrote a proof, in the little
system, for each n in N and joined them with and "&" operator you
would prove the quantified statement albeit with an infinity proof.
In a little larger system, perhaps with an induction axiom, or a meta
system it might be trivial to prove the whole thing with a finite
effort.
The issue how can you know or show that you can make such a proof for
all n within the system?
Some of the facts in the above are knowable in more powerful systems and
can only be appreciated from that viewpoint.
IF you can break the infinite number of n into a finite number ofcases,
then you can built a finite proof based on those cases.
If that were true in the examples I was shown, then none of the rest
would follow and I wouldn't have posted what I did.
The problem is that the normal definition of "proof" requires it to be
finite, as proofs are supposed to SHOW to a person that the statement
is true, and we can't handle such an infinite series.
That is simply not true. Period. If I recall correctly, the book
"Zermelo's Axiom of Choice -- Its Origins, Development and Influence" by Gregory H. Moore, talks about infinite proofs in a few places. (My copy
of the book is 40+ years ago and was unholy expensive. You can now get a paperback copy for $13.99 at US Amazon.con.) Where, other then in silly USENET, dose it say that all proofs in all formal systems must be
finite? Serious question. Maybe this is a point that you and Peter can
agree on, but don't implicate innocent bystanders.
System that try to define "Proof" to include infinite logic, run into
the issue that infinte proofs can't be actually created or viewed, and
thus the system has decoupled "knoweledge" from proofs.
Logicians qualified to discuss and develop theories of and using
infinitary proofs do not have such problems. (I'm using the term
"qualified logician" to distinguish an individual from myself and other want-a-bees participating in these idiotic discussions.) They know what
they are doing and often are exploring the interesting question of what
you can and cannot know in particular (styles of) formal systems. This
seems to be the main theme from all the want-a-bees but, for them, the discussion continues without a clue. By the way, do you have a single reference to a book published by one of the "technical" houses of repute
or a peer referenced journal around for at least 25 years that supports
a single thought expressed in your above paragraph that begins "System
(sic) that try to..."?
What you describe is sort of what Godel does, but points out that your
infinite chain isn't a proof by the rules of standard logic, as the
proof in the meta system provides a reason why each of the numbers
applied to the relationship will fail. Thus, we DO have your concept
of an infinite "proof" in the system, which actually shows that the
statement must actually be true in the system (but this truth can't be
shown in the system except with infinite work).
It's not what Godel does in any energetic sense. But it's a good "appeal
to authority", misguided but an appeal. As to standard logic -- there
are at least 20 want-a-bees in these discussions and no two of them
could write down the same precise definition! And the above is not my
idea; I'm really old and the idea was around before I was born. Unless
the system has appropriate rules for making and/or certifying the
infinity proof you can't prove it in the system. This, by the way, is
just another technique for showing that a system is incomplete, i.e.,
there is a fact that is true in all models and cannot be proven in the system. One last comment: Don't think of a proof as work. Its an entity
that exists by the definition of some formal system, e.g., a string that satisfies some mathematical predicate.
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(G||del 1931:40-41)
Thus the resolution of the Liar Paradox resolves
G||del incompleteness.
On 1/3/26 9:49 PM, olcott wrote:
So you have no idea how true statements are derived
from other true statements ?
https://iep.utm.edu/val-snd/
Right, but the chain can be infinite, and thus not a proof.
On 1/3/26 10:36 PM, olcott wrote:
We do now that all paradoxes resolve to nonsense.
No, because the word "Paradox" just means an APPARENT contradiction.
The Liar's Paradox gets resolved by seeing that the statement just
doesn't have a Truth Value (Not all syntacticly valid statemente do) and
thus isn't a Semantically valid statement, and the "Not" operator is
being given an invalid value (or Not(not-a-truth-value) is just not-a-truth-value).
On 03/01/2026 22:18, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(G||del 1931:40-41)
Thus the resolution of the Liar Paradox resolves
G||del incompleteness.
Misuse of "thus": ought to be "therefore".
Then you still have the
second statement of your syllogism missing. Perhaps you can construct it using a syllogism that derives the equivalence of all epistemological antinomies with the liar paradox.
So, the sum of the number 1/2 + 1/4 + 1/8 + 1/16 ... is nonsense?
On 1/4/26 8:48 PM, Jeff Barnett wrote:
On 1/4/2026 3:50 PM, Richard Damon wrote:
On 1/4/26 5:13 PM, Jeff Barnett wrote:
On 1/4/2026 11:55 AM, Mike Terry wrote:
On 02/01/2026 17:54, Richard Damon wrote:-a-a-a <SNIP>
On 1/2/26 12:24 PM, Mike Terry wrote:
On 02/01/2026 15:25, Richard Damon wrote:
On 1/2/26 1:14 AM, Tristan Wibberley wrote:
On 02/01/2026 04:45, Richard Damon wrote:
Similarly, the word "proof" can be informal (simply an argument >>>>>>> that convinces people of the truth of a statement), or refer to >>>>>>> the "proof calculus" of the formal system being discussed.-a Most >>>>>>> authors I've come across seem to use "proof" more or less
informally and for clarity choose another word for whatever
sequence of syntactic "proof steps" the formal system specifies. >>>>>>> Often "derivation" is used, and that seems intuitive to me, so I >>>>>>> try to always use that term here, and using "proof" for more
general mathematial arguments, e.g. proving that the G statement >>>>>>> is "true" using some meta-theory.
The issue is that "derivation" doesn't actually imply a
finiteness, which is a necessity of "proof".
Where do you get that idea?-a Are you thinking "derivation" is just >>>>> an informal word?-a I'm using it in the technical sense previously
explained.
Within a formal system there will be a set of rules which define
what a valid "derivation" looks like.-a These would ensure that such >>>>> derivations are finite.-a (I'm sure someone at some time has made a >>>>> special study of "infinite proofs", but that is off the beaten
track.)-a As explained in my previous post, I'm using "derivation"
as the technical term for whatever passes as a "formal proof
conforming to the requirements of the proof calculus of the
system".-a This is so that the idea does not get muddled with your
more general kind of proof = "convincing argument in some meta-
theory".-a-a-a-a <SNIP>
There are examples of the following situations that I remember from
discussions with logicians circa. 60 years ago. Assume we have a
simple axiomatic system that allows us to express some facts about
what we believe to be natural numbers. Call the objects in a model
of the system N and we wish to prove, in our little system, that for
all n in N p(n). Now it turns out that for any n in N we can write a
simple finite proof of p(n) but are (provably in a larger or meta
system) not able to prove the universally quantified statement in
the little system. Well actually .... if you wrote a proof, in the
little system, for each n in N and joined them with and "&" operator
you would prove the quantified statement albeit with an infinity
proof. In a little larger system, perhaps with an induction axiom,
or a meta system it might be trivial to prove the whole thing with a
finite effort.
The issue how can you know or show that you can make such a proof for
all n within the system?
Some of the facts in the above are knowable in more powerful systems
and can only be appreciated from that viewpoint.
-a-a> IF you can break the infinite number of n into a finite number of
cases,
Yes, As Tarski showed, SOME of the statements unprovable in a system can
be proven in a "higher order" meata-system, but not all. There will
ALWAYS be more statements, true in the original bases system, that can
not be proven in any system.
And the fact that some can be proven in a higher order system is why we
can assert them to be true, and thus (at least in the language I
learned) be called Theorems.
then you can built a finite proof based on those cases.
If that were true in the examples I was shown, then none of the rest
would follow and I wouldn't have posted what I did.
Yes, I was pointing out that we can't be talking about looking at
"infinite cases" that actual are just a finite number of generic cases.
The problem is that the normal definition of "proof" requires it to
be finite, as proofs are supposed to SHOW to a person that the
statement is true, and we can't handle such an infinite series.
That is simply not true. Period. If I recall correctly, the book
"Zermelo's Axiom of Choice -- Its Origins, Development and Influence"
by Gregory H. Moore, talks about infinite proofs in a few places. (My
copy of the book is 40+ years ago and was unholy expensive. You can
now get a paperback copy for $13.99 at US Amazon.con.) Where, other
then in silly USENET, dose it say that all proofs in all formal
systems must be finite? Serious question. Maybe this is a point that
you and Peter can agree on, but don't implicate innocent bystanders.
Note, Choice is an AXIOM because it can't actually be proven.
Those "proof" are just arguements on why the Axiom of Choice makes sense.
As for a reference, look at Wikipedia at https://en.wikipedia.org/wiki/ Formal_proof and the papers it references.
Note, Proofs are written down and presented. This can only be done if finite.
--System that try to define "Proof" to include infinite logic, run into
the issue that infinte proofs can't be actually created or viewed,
and thus the system has decoupled "knoweledge" from proofs.
Logicians qualified to discuss and develop theories of and using
infinitary proofs do not have such problems. (I'm using the term
"qualified logician" to distinguish an individual from myself and
other want-a-bees participating in these idiotic discussions.) They
know what they are doing and often are exploring the interesting
question of what you can and cannot know in particular (styles of)
formal systems. This seems to be the main theme from all the want-a-
bees but, for them, the discussion continues without a clue. By the
way, do you have a single reference to a book published by one of the
"technical" houses of repute or a peer referenced journal around for
at least 25 years that supports a single thought expressed in your
above paragraph that begins "System (sic) that try to..."?
What you describe is sort of what Godel does, but points out that
your infinite chain isn't a proof by the rules of standard logic, as
the proof in the meta system provides a reason why each of the
numbers applied to the relationship will fail. Thus, we DO have your
concept of an infinite "proof" in the system, which actually shows
that the statement must actually be true in the system (but this
truth can't be shown in the system except with infinite work).
It's not what Godel does in any energetic sense. But it's a good
"appeal to authority", misguided but an appeal. As to standard logic
-- there are at least 20 want-a-bees in these discussions and no two
of them could write down the same precise definition! And the above is
not my idea; I'm really old and the idea was around before I was born.
Unless the system has appropriate rules for making and/or certifying
the infinity proof you can't prove it in the system. This, by the way,
is just another technique for showing that a system is incomplete,
i.e., there is a fact that is true in all models and cannot be proven
in the system. One last comment: Don't think of a proof as work. Its
an entity that exists by the definition of some formal system, e.g., a
string that satisfies some mathematical predicate.
On 1/4/26 3:32 PM, olcott wrote:
On 1/4/2026 2:23 PM, Richard Damon wrote:
On 1/4/26 3:21 PM, olcott wrote:
It is categorically impossible to derive any element
of the body of knowledge that can be expressed in
language that is not entirely comprised of some relation
between finite strings.
So?
That is the conclusive proof that I am correct.
No it isn't. That doesn't make Truth computable, it makes everything computable true.
The problem is we want to derive things that aren't yet in the body
of knowledge.
If you want to know the name of your wife's
mother and you have not met your wife yet
then the answer is not available by any means.\
So? She still has a name.
On 1/3/2026 3:06 PM, Tristan Wibberley wrote:
On 03/01/2026 17:30, olcott wrote (quoting Curry):
In other words: reCx ree T ((True(T, x) rei (E reo x))
Curry would not approve of you formalising that without defining the
system in which you formalise it.
You have to read my quote of Curry to see that he
already defined {T} and {E}.
{E} is merely my own notion of atomic facts,
previously called base facts.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||-aHis notions of U-language and
A-language and progressive refinement of the U-language were carefully
thought through leading to his incredible written lucidity, and the
immense benefit of reading his work carefully from the start.
On 1/3/26 3:55 PM, Tristan Wibberley wrote:
I would like to see Richard's construction of the statement for which G
is shorthand. As it is mere shorthand then there is such a thing.
G is the statement
There does not exist a natural number (g) that satisfies the
relationship ...
Where the ... is the incredibly complicated formula that he derives
thorough most of the paper,
that can't be expressed in simple ascii.
On 03/01/2026 19:02, Tristan Wibberley wrote:
On 03/01/2026 16:32, Mike Terry wrote:
On 03/01/2026 03:30, Richard Damon wrote:
On 1/2/26 9:43 PM, Andr|- G. Isaak wrote:
On 2026-01-01 20:09, Richard Damon wrote:
On 1/1/26 9:45 PM, Andr|- G. Isaak wrote:-a-a>
On 2026-01-01 16:48, Richard Damon wrote:
On 1/1/26 6:13 PM, Tristan Wibberley wrote:
On 01/01/2026 22:40, Richard Damon wrote:
But it IS a theorem of the base system, as it uses ONLY the >>>>>>>>>> mathematical
operations definable in the base system. What makes you think it >>>>>>>>>> isn't a
Theorem in the base system.
It has no derivation in the base system, if it had you wouldn't >>>>>>>>> think
the base system were incomplete.
It has no PROOF in the base system.
Which means it is not a theorem of the base system. A theorem is a >>>>>>> statement which can be proven in a particular system.
I guess it depends on your definition of a "Theorem".
I am using the one that goes:
"A Theorem is a statement that has been proven."
note, no restriction that the proof was in the system the Theorem is >>>>>> stated in, as long as the proof shows that it is actually True in
that system.
A theorem is a statement that can be derived from the axioms of a
particular system. It may be true in other systems, but it is only a >>>>> theorem in systems in which it can be derived.
Right, And the statement og Godel's G can be fully derived in the base >>>> system, as it is purely a mathematical relationship using the
operations derivable in the system.
Neither G nor -4G has a derivation (in your terms, a "formal prooof")
within the base system.-a That is what Godel proves, showing that the
base system is incomplete.
That can't be what he meant can it? Lots of systems were known to have
statements that had no derivation, all nonsense statements, for example.
Yes it was what he meant!-a :/
His theorem was about formal systems of arithmetic.-a Such systems don't contain "nonsense statements".-a They have construction rules that define what constitues a well formed formula (WFF), and amongst those what so constitutes a "sentence".-a The semantics for the system define what
every sentence "means".-a It is not possible to create "nonsense" sentences.
G||del's concerns there being a statement G such that neither G nor -4G
has a derivation in the system.-a There is no reference to "truth" in
that and I'd say his proof is essentially syntactical in nature.
It's strange - both you and Tristan expressed surprise that everyday
theories might be incomplete
T is complete when for any sentence -a, either TrCareo -a or TrCareo -4-a. https://www.cairn.info/revue-philosophia-scientiae-2014-3-page-23.htm
According to the above when -a is self-contradictory this makes T
incomplete.
T is complete when for any sentence -a, either TrCareo -a or TrCareo -4-a. https://www.cairn.info/revue-philosophia-scientiae-2014-3-page-23.htm
According to the above when -a is self-contradictory this makes T
incomplete.
On 04/01/2026 19:21, Richard Damon wrote:
So, the sum of the number 1/2 + 1/4 + 1/8 + 1/16 ... is nonsense?
It's nondeterministic because "..." has more than one meaning for the
effect it has extending the series. It's not nonsense because it's not a statement.
+ureOreireU 2->rU+
/is/ deterministic, however... it's 1.
You might be mapping time nonlinearly whereby each imagined change
occurs in its imagined reality at a constant +oreL from the previous. It's
a common affliction among classical mediterranean philosophers.
On 04/01/2026 20:45, Richard Damon wrote:
On 1/4/26 3:32 PM, olcott wrote:
On 1/4/2026 2:23 PM, Richard Damon wrote:
On 1/4/26 3:21 PM, olcott wrote:
It is categorically impossible to derive any element
of the body of knowledge that can be expressed in
language that is not entirely comprised of some relation
between finite strings.
So?
That is the conclusive proof that I am correct.
No it isn't. That doesn't make Truth computable, it makes everything
computable true.
The problem is we want to derive things that aren't yet in the body
of knowledge.
If you want to know the name of your wife's
mother and you have not met your wife yet
then the answer is not available by any means.\
So? She still has a name.
Is this how the church banned divorce and also remarriage after a bereavement?
They'd made an AI knowledge-base and used the same axiom that you just did?
Henry VIII made a more sophisticated one and the rest is history. It was
a system that attested reality and was written in prolog: protestant.
On 04/01/2026 03:07, Richard Damon wrote:
On 1/3/26 9:49 PM, olcott wrote:
So you have no idea how true statements are derived
from other true statements ?
https://iep.utm.edu/val-snd/
Right, but the chain can be infinite, and thus not a proof.
It must have both a start and an end to be a derivation. I'm curious to
know how you came to think a derivation refers to a class of things that includes a variety that one would say was an infinite chain.
On 03/01/2026 22:27, olcott wrote:
On 1/3/2026 3:06 PM, Tristan Wibberley wrote:
On 03/01/2026 17:30, olcott wrote (quoting Curry):
In other words: reCx ree T ((True(T, x) rei (E reo x))
Curry would not approve of you formalising that without defining the
system in which you formalise it.
You have to read my quote of Curry to see that he
already defined {T} and {E}.
You forget the history of the posts of who you're talking to.
{E} is merely my own notion of atomic facts,
previously called base facts.
I think not. Elementary statements
are those made from predicates
adjoining terms but not adjoining other statements. Your atomic facts
are nullary predicates, or unary predicates adjoined to primitive terms,
or binary predicates adjoining terms to some world (perhaps if the world
is represented by a term - we reach the limits of my ready
understanding), etc...
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||-aHis notions of U-language and
A-language and progressive refinement of the U-language were carefully
thought through leading to his incredible written lucidity, and the
immense benefit of reading his work carefully from the start.
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
TAKE NOTE and also read his Theory of Formal Deducibility
On 04/01/2026 19:44, olcott wrote:
T is complete when for any sentence -a, either TrCareo -a or TrCareo -4-a. >> https://www.cairn.info/revue-philosophia-scientiae-2014-3-page-23.htm
According to the above when -a is self-contradictory this makes T
incomplete.
Is this now a totally conventional definition?
The unfortunate thing
about it is that it references negation even though there have been many notions of negation.
It uses "-4" implying the intuitionist negation but
should "incompleteness" be defined only for such systems?
There's an important stronger notion for positive intuitionist systems
that extending the system with an axiom extension either asserting -a or asserting -4-a is inconsistent. It's incompletable. Do you know the term
for that?
On 04/01/2026 19:44, olcott wrote:
T is complete when for any sentence -a, either TrCareo -a or TrCareo -4-a. >> https://www.cairn.info/revue-philosophia-scientiae-2014-3-page-23.htm
According to the above when -a is self-contradictory this makes T
incomplete.
I have to worry about that document:
"any sound and sufficiently strong and recursively enumerable theory is incomplete"--
That doesn't even mean there exists a description of the level of
strength over which a recursively enumerable theory is incomplete. I
suspect G||del showed there /does/ exist a description so they could have made a stronger statement. Then it /still/ wouldn't say that there
exists at least one system satisfying that description. I suspect G||del showed there /does/ exist at least one system that would satisfy such a description so they could have made an even stronger statement. As Mike
has said, it is /totally/ uninteresting that there are systems strong
enough to be incomplete (but ergh do we use "strong" like that? we fix inconsistency by weakening, even if that makes them incomplete - I think
they mean "sufficiently broad" or something like that).
Bodes ill for continued reading.
On 1/4/2026 11:55 AM, Mike Terry wrote:
On 02/01/2026 17:54, Richard Damon wrote:-a-a <SNIP>
On 1/2/26 12:24 PM, Mike Terry wrote:
On 02/01/2026 15:25, Richard Damon wrote:
On 1/2/26 1:14 AM, Tristan Wibberley wrote:
On 02/01/2026 04:45, Richard Damon wrote:
Similarly, the word "proof" can be informal (simply an argument that convinces people of the
truth of a statement), or refer to the "proof calculus" of the formal system being discussed.
Most authors I've come across seem to use "proof" more or less informally and for clarity choose
another word for whatever sequence of syntactic "proof steps" the formal system specifies.
Often "derivation" is used, and that seems intuitive to me, so I try to always use that term
here, and using "proof" for more general mathematial arguments, e.g. proving that the G
statement is "true" using some meta-theory.
The issue is that "derivation" doesn't actually imply a finiteness, which is a necessity of "proof".
Where do you get that idea?-a Are you thinking "derivation" is just an informal word?-a I'm using it
in the technical sense previously explained.
Within a formal system there will be a set of rules which define what a valid "derivation" looks
like.-a These would ensure that such derivations are finite.-a (I'm sure someone at some time has
made a special study of "infinite proofs", but that is off the beaten track.)-a As explained in my
previous post, I'm using "derivation" as the technical term for whatever passes as a "formal proof
conforming to the requirements of the proof calculus of the system".-a This is so that the idea
does not get muddled with your more general kind of proof = "convincing argument in some
meta-theory".-a-a-a-a <SNIP>
There are examples of the following situations that I remember from discussions with logicians
circa. 60 years ago. Assume we have a simple axiomatic system that allows us to express some facts
about what we believe to be natural numbers. Call the objects in a model of the system N and we wish
to prove, in our little system, that for all n in N p(n). Now it turns out that for any n in N we
can write a simple finite proof of p(n) but are (provably in a larger or meta system) not able to
prove the universally quantified statement in the little system.
Well actually .... if you wrote a
proof, in the little system, for each n in N and joined them with and "&" operator you would prove
the quantified statement albeit with an infinity proof.
In a little larger system, perhaps with an
induction axiom, or a meta system it might be trivial to prove the whole thing with a
finite effort.
On 04/01/2026 22:13, Jeff Barnett wrote:
On 1/4/2026 11:55 AM, Mike Terry wrote:
On 02/01/2026 17:54, Richard Damon wrote:-a-a-a <SNIP>
On 1/2/26 12:24 PM, Mike Terry wrote:
On 02/01/2026 15:25, Richard Damon wrote:
On 1/2/26 1:14 AM, Tristan Wibberley wrote:
On 02/01/2026 04:45, Richard Damon wrote:
Similarly, the word "proof" can be informal (simply an argument
that convinces people of the truth of a statement), or refer to the >>>>> "proof calculus" of the formal system being discussed. Most authors >>>>> I've come across seem to use "proof" more or less informally and
for clarity choose another word for whatever sequence of syntactic
"proof steps" the formal system specifies. Often "derivation" is
used, and that seems intuitive to me, so I try to always use that
term here, and using "proof" for more general mathematial
arguments, e.g. proving that the G statement is "true" using some
meta-theory.
The issue is that "derivation" doesn't actually imply a finiteness,
which is a necessity of "proof".
Where do you get that idea?-a Are you thinking "derivation" is just an
informal word?-a I'm using it in the technical sense previously
explained.
Within a formal system there will be a set of rules which define what
a valid "derivation" looks like.-a These would ensure that such
derivations are finite.-a (I'm sure someone at some time has made a
special study of "infinite proofs", but that is off the beaten
track.)-a As explained in my previous post, I'm using "derivation" as
the technical term for whatever passes as a "formal proof conforming
to the requirements of the proof calculus of the system".-a This is so
that the idea does not get muddled with your more general kind of
proof = "convincing argument in some meta-theory".-a-a-a-a <SNIP>
There are examples of the following situations that I remember from
discussions with logicians circa. 60 years ago. Assume we have a
simple axiomatic system that allows us to express some facts about
what we believe to be natural numbers. Call the objects in a model of
the system N and we wish to prove, in our little system, that for all
n in N p(n). Now it turns out that for any n in N we can write a
simple finite proof of p(n) but are (provably in a larger or meta
system) not able to prove the universally quantified statement in the
little system.
Yes, that can happen.-a It's known as -e-incompleteness.-a That's where we have
-a-a reo p(1)
-a-a reo p(2)
-a-a reo p(3)
-a-a ...
but not
-a-a reo reCn p(n)
There is a related phenomenon which is much more severe, which we have
-a-a reo p(1)
-a-a reo p(2)
-a-a reo p(3)
-a-a ...
and additionally
-a-a reo -4reCn p(n)
This is called -e-inconsistency.-a (Any inconsistent theory will also be -e-inconsistent, but it's possible for a consistent theory to be -e- inconsistent, so they are not equivalent concepts.)
-a <https://en.wikipedia.org/wiki/%CE%A9-consistent_theory>
So -e-incompleteness is saying our theory lacks some Theorems we would
like [regarding universal quantification for certain properties p],
while -e-inconsistency is much worse, actually saying that some n exists such that -4p(n) holds, despite p(1), p(2),... all holding.-a [Clearly
such an n can't be a (standard) natural number.]
I'm not really up on all this, but seem to (randomly!) recall:
1.-a -e-incompleteness is fairly common (not too worrying)
2.-a Godel's proof of GIT needed the assumption that his base system P was
-a-a-a /-e-consistent/, which is a stronger condition than just consistency. 3.-a A few years after GIT was published, someone published an improved proof,
-a-a-a along the same lines but "tweaked" so it required only that P is / consistent/,
-a-a-a rather than -e-consistent.-a (So nowadays when people discuss GIT they typically
-a-a-a don't even mention -e-consistency.)
Well actually .... if you wrote a"&" operator you would prove
proof, in the little system, for each n in N and joined them with and
the quantified statement albeit with an infinity proof.
Not sure if you're joining the proofs or the statements p(n) with an & operator.-a Well, you can't join proofs with '&', so you must mean
allowing infinitely long sentences like
-a p(0) & p(1) & p(2) &...
Or you could concatenating the proofs to make an infinitely long proof,
or both, I get where you're going.
Yeah, those sorts of ideas have all been studied, but it's not as simple
as you make out.
OK, so we string together an infinite number of proofs for p(1), p(2), p(3)... and/or we have an infinitely long conjunction as above, but how would that (of itself) actually prove reCn p(n) ?
Just from what's been said, it /wouldn't/, because we can build models
where all your infinitary stuff above happens, AND reCn p(n) is actually still false, and we wouldn't want to be proving false statements.
We're missing something /in our logical calculus/ that encapsulates the
idea that our domain is /only/ n=0,1,2,3... (i.e. the natural numbers)
So to actually prove reCn p(n), as well as /allowing/ infinitary proofs in some fashion, we'd also need to add /further/ infinitary logical
deduction rules somehow reflecting that our base domain is /only/ N.
(Just /saying/ "N is our preferred model" doesn't change things.)
In a little larger system, perhaps with anwhole thing with a
induction axiom, or a meta system it might be trivial to prove the
finite effort.
Yes that's common.-a For example there are common systems where for all (natural numbers) m,n we can prove
-a-a reo <m> + <n> = <n> + <m>
typically using induction over m,n (in the meta-system).-a [<n> means the numeral for n in our system, i.e. <3> is typically SSS0, and so on]. However it can fail to be the case for our sytem that
-a-a reo reCmreCn m + n = n + m
Mike.
On 04/01/2026 22:13, Jeff Barnett wrote:
On 1/4/2026 11:55 AM, Mike Terry wrote:
On 02/01/2026 17:54, Richard Damon wrote:-a-a-a <SNIP>
On 1/2/26 12:24 PM, Mike Terry wrote:
On 02/01/2026 15:25, Richard Damon wrote:
On 1/2/26 1:14 AM, Tristan Wibberley wrote:
On 02/01/2026 04:45, Richard Damon wrote:
Similarly, the word "proof" can be informal (simply an argument
that convinces people of the truth of a statement), or refer to the >>>>> "proof calculus" of the formal system being discussed. Most authors >>>>> I've come across seem to use "proof" more or less informally and
for clarity choose another word for whatever sequence of syntactic
"proof steps" the formal system specifies. Often "derivation" is
used, and that seems intuitive to me, so I try to always use that
term here, and using "proof" for more general mathematial
arguments, e.g. proving that the G statement is "true" using some
meta-theory.
The issue is that "derivation" doesn't actually imply a finiteness,
which is a necessity of "proof".
Where do you get that idea?-a Are you thinking "derivation" is just an
informal word?-a I'm using it in the technical sense previously
explained.
Within a formal system there will be a set of rules which define what
a valid "derivation" looks like.-a These would ensure that such
derivations are finite.-a (I'm sure someone at some time has made a
special study of "infinite proofs", but that is off the beaten
track.)-a As explained in my previous post, I'm using "derivation" as
the technical term for whatever passes as a "formal proof conforming
to the requirements of the proof calculus of the system".-a This is so
that the idea does not get muddled with your more general kind of
proof = "convincing argument in some meta-theory".-a-a-a-a <SNIP>
There are examples of the following situations that I remember from
discussions with logicians circa. 60 years ago. Assume we have a
simple axiomatic system that allows us to express some facts about
what we believe to be natural numbers. Call the objects in a model of
the system N and we wish to prove, in our little system, that for all
n in N p(n). Now it turns out that for any n in N we can write a
simple finite proof of p(n) but are (provably in a larger or meta
system) not able to prove the universally quantified statement in the
little system.
Yes, that can happen.-a It's known as -e-incompleteness.-a That's where we have
-a-a reo p(1)
-a-a reo p(2)
-a-a reo p(3)
-a-a ...
but not
-a-a reo reCn p(n)
There is a related phenomenon which is much more severe, which we have
-a-a reo p(1)
-a-a reo p(2)
-a-a reo p(3)
-a-a ...
and additionally
-a-a reo -4reCn p(n)
This is called -e-inconsistency.-a (Any inconsistent theory will also be -e-inconsistent, but it's possible for a consistent theory to be -e- inconsistent, so they are not equivalent concepts.)
-a <https://en.wikipedia.org/wiki/%CE%A9-consistent_theory>
So -e-incompleteness is saying our theory lacks some Theorems we would
like [regarding universal quantification for certain properties p],
while -e-inconsistency is much worse, actually saying that some n exists such that -4p(n) holds, despite p(1), p(2),... all holding.-a [Clearly
such an n can't be a (standard) natural number.]
I'm not really up on all this, but seem to (randomly!) recall:
1.-a -e-incompleteness is fairly common (not too worrying)
2.-a Godel's proof of GIT needed the assumption that his base system P was
-a-a-a /-e-consistent/, which is a stronger condition than just consistency. 3.-a A few years after GIT was published, someone published an improved proof,
-a-a-a along the same lines but "tweaked" so it required only that P is / consistent/,
-a-a-a rather than -e-consistent.-a (So nowadays when people discuss GIT they typically
-a-a-a don't even mention -e-consistency.)
Well actually .... if you wrote a"&" operator you would prove
proof, in the little system, for each n in N and joined them with and
the quantified statement albeit with an infinity proof.
Not sure if you're joining the proofs or the statements p(n) with an & operator.-a Well, you can't join proofs with '&', so you must mean
allowing infinitely long sentences like
-a p(0) & p(1) & p(2) &...
Or you could concatenating the proofs to make an infinitely long proof,
or both, I get where you're going.
Yeah, those sorts of ideas have all been studied, but it's not as simple
as you make out.
OK, so we string together an infinite number of proofs for p(1), p(2), p(3)... and/or we have an infinitely long conjunction as above, but how would that (of itself) actually prove reCn p(n) ?
Just from what's been said, it /wouldn't/, because we can build models
where all your infinitary stuff above happens, AND reCn p(n) is actually still false, and we wouldn't want to be proving false statements.
We're missing something /in our logical calculus/ that encapsulates the
idea that our domain is /only/ n=0,1,2,3... (i.e. the natural numbers)
So to actually prove reCn p(n), as well as /allowing/ infinitary proofs in some fashion, we'd also need to add /further/ infinitary logical
deduction rules somehow reflecting that our base domain is /only/ N.
(Just /saying/ "N is our preferred model" doesn't change things.)
Everything you've said above seems about right to me. As to joining the individual proofs with an "&", I probably should have specified ";"In a little larger system, perhaps with anwhole thing with a
induction axiom, or a meta system it might be trivial to prove the
finite effort.
Yes that's common.-a For example there are common systems where for all (natural numbers) m,n we can prove
-a-a reo <m> + <n> = <n> + <m>
typically using induction over m,n (in the meta-system).-a [<n> means the numeral for n in our system, i.e. <3> is typically SSS0, and so on]. However it can fail to be the case for our sytem that
-a-a reo reCmreCn m + n = n + m
On 04/01/2026 18:55, Mike Terry wrote:
G%del's concerns there being a statement G such that neither G nor 4G
has a derivation in the system.a There is no reference to "truth" in
that and I'd say his proof is essentially syntactical in nature.
From Curry and Feys very brief mention of the distinction I think
G%del's system P is a semantical system (it has numbers as objects
distinct from their presentation - which allows him to just make it all
the more complicated). Does that mean his proof must be semantical?
Also he relies on a meta-system which means embedding, does that force
the proof to be semantical even if derivations in P are syntactical?
I haven't got a handle on semantical vs syntactical.
On 1/4/2026 3:50 PM, Richard Damon wrote:
On 1/4/26 5:13 PM, Jeff Barnett wrote:
On 1/4/2026 11:55 AM, Mike Terry wrote:
On 02/01/2026 17:54, Richard Damon wrote:-a-a-a <SNIP>
On 1/2/26 12:24 PM, Mike Terry wrote:
On 02/01/2026 15:25, Richard Damon wrote:
On 1/2/26 1:14 AM, Tristan Wibberley wrote:
On 02/01/2026 04:45, Richard Damon wrote:
Similarly, the word "proof" can be informal (simply an argument
that convinces people of the truth of a statement), or refer to
the "proof calculus" of the formal system being discussed.-a Most
authors I've come across seem to use "proof" more or less
informally and for clarity choose another word for whatever
sequence of syntactic "proof steps" the formal system specifies.-a >>>>>> Often "derivation" is used, and that seems intuitive to me, so I
try to always use that term here, and using "proof" for more
general mathematial arguments, e.g. proving that the G statement
is "true" using some meta-theory.
The issue is that "derivation" doesn't actually imply a finiteness,
which is a necessity of "proof".
Where do you get that idea?-a Are you thinking "derivation" is just
an informal word?-a I'm using it in the technical sense previously
explained.
Within a formal system there will be a set of rules which define
what a valid "derivation" looks like.-a These would ensure that such
derivations are finite.-a (I'm sure someone at some time has made a
special study of "infinite proofs", but that is off the beaten
track.)-a As explained in my previous post, I'm using "derivation" as
the technical term for whatever passes as a "formal proof conforming
to the requirements of the proof calculus of the system".-a This is
so that the idea does not get muddled with your more general kind of
proof = "convincing argument in some meta-theory".-a-a-a-a <SNIP>
There are examples of the following situations that I remember from
discussions with logicians circa. 60 years ago. Assume we have a
simple axiomatic system that allows us to express some facts about
what we believe to be natural numbers. Call the objects in a model of
the system N and we wish to prove, in our little system, that for all
n in N p(n). Now it turns out that for any n in N we can write a
simple finite proof of p(n) but are (provably in a larger or meta
system) not able to prove the universally quantified statement in the
little system. Well actually .... if you wrote a proof, in the little
system, for each n in N and joined them with and "&" operator you
would prove the quantified statement albeit with an infinity proof.
In a little larger system, perhaps with an induction axiom, or a meta
system it might be trivial to prove the whole thing with a finite
effort.
The issue how can you know or show that you can make such a proof for
all n within the system?
Some of the facts in the above are knowable in more powerful systems and
can only be appreciated from that viewpoint.
IF you can break the infinite number of n into a finite number of cases,
then you can built a finite proof based on those cases.
If that were true in the examples I was shown, then none of the rest
would follow and I wouldn't have posted what I did.
The problem is that the normal definition of "proof" requires it to be
finite, as proofs are supposed to SHOW to a person that the statement
is true, and we can't handle such an infinite series.
That is simply not true. Period. If I recall correctly, the book
"Zermelo's Axiom of Choice -- Its Origins, Development and Influence" by Gregory H. Moore, talks about infinite proofs in a few places. (My copy
of the book is 40+ years ago and was unholy expensive. You can now get a paperback copy for $13.99 at US Amazon.con.) Where, other then in silly USENET, dose it say that all proofs in all formal systems must be
finite? Serious question. Maybe this is a point that you and Peter can
agree on, but don't implicate innocent bystanders.
Logicians qualified to discuss and develop theories of and using
infinitary proofs do not have such problems. (I'm using the term
"qualified logician" to distinguish an individual from myself and other want-a-bees participating in these idiotic discussions.)
there are at least 20 want-a-bees
... in these discussions and no two of them
could write down the same precise definition! And the above is not my
idea; I'm really old and the idea was around before I was born.
One last comment: Don't think of a proof as work. Its an entity
that exists by the definition of some formal system, e.g., a string that satisfies some mathematical predicate.
On 1/4/2026 7:23 PM, Richard Damon wrote:
Note, Proofs are written down and presented. This can only be done if
finite.
Perhaps you can find a legitimate reference that says that. Forget high school geometry books, and USENET newsgroups. I don't think you can! I
think you are making a lot of this up as you go along. Save that stuff
for debates with Peter, a past master at that. Your statement just above
is a proof description given to a high school class about how to get a
good grade. If you think you are speaking in the world of research in logic(s), it's nonsense.
On 04/01/2026 22:13, Jeff Barnett wrote:^^^
In a little larger system, perhaps with anwhole thing with a
induction axiom, or a meta system it might be trivial to prove the
finite effort.
Yes that's common.-a For example there are common systems where for all
(natural numbers) m,n we can prove^^^^^^^^^^^^^^^^^^^^^
-a-a reo <m> + <n> = <n> + <m>
typically using induction over m,n (in the meta-system).
There is a related phenomenon which is much more severe, which we have
-a-a reo p(1)
-a-a reo p(2)
-a-a reo p(3)
-a-a ...
and additionally
-a-a reo -4reCn p(n)
This is called -e-inconsistency.
Perhaps because things like a "derivative" in mathematics is a limit,
and thus goes to infinity.
It seems you belong to a tribe of reasoning that doesn't like the
infinite and thinks it needs to be excluded from common thinking.
... that is clearly "syntactic".
... syntactic or semantic concept?-a
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