Anyone already knowing the subject already knows that Gödel's G essentially
asserts that it is unprovable in F. Gödel's himself sums up this sentence as
"We are therefore confronted with a proposition which asserts its own unprovability." 15 (Gödel 1931:39-41)
The only last detail of this that most people are unaware of is that the only
reason why G is unprovable in F is that the proof of G is contradictory in F,
not that there is anything wrong with F as Gödel's proof concludes.
When we "unpack" G in the usual way all that we get is forty pages connected
math formulas prove that a certain positive integer is not derived by these
forty pages of formulas.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And Related Systems
I see yer point. Nice! I wonder if Gödel's theorems can be, you know, generalized to all of epistemology? No matter how good a "model" is, there'll always be at least one proposition that is undecidable in that model.
A is unprovable in M
A is a proposition, M is the model (corresponding to F)
When we define a formal system having expressions of language that are derived as the semantic consequence of other expressions of language that have been stipulated to be true, incompleteness and undecidability cannot exist within this system.
I was in mathematician mode back there, as best as I could manage that is. Generalize, generalize, generalize.
Anyway, what I feel is G can do a lot more than just prove Gödel's theorems.
I see yer point. Nice! I wonder if Gödel's theorems can be, you know, generalized to all of epistemology? No matter how good a "model" is, there'll always be at least one proposition that is undecidable in that model.
A is unprovable in M
A is a proposition, M is the model (corresponding to F)
When we define a formal system having expressions of language that are derived as the semantic consequence of other expressions of language that have been stipulated to be true, incompleteness and undecidability cannot exist within this system.
I was in mathematician mode back there, as best as I could manage that is. Generalize, generalize, generalize.
Anyway, what I feel is G can do a lot more than just prove Gödel's theorems.
https://www.liarparadox.org/G%C3%B6del_ ... (1931).pdf
The top of the page is Gödel's actual G and all that it does is
prove that a very specific integer number cannot be derived in
Peano Arithmetic. It needs very many connected formulas to
do this.
When we define a formal system having expressions of language that are derived as the semantic consequence of other expressions of language that have been stipulated to be true, incompleteness and undecidability cannot exist within this system.
I was in mathematician mode back there, as best as I could manage that is. Generalize, generalize, generalize.
Anyway, what I feel is G can do a lot more than just prove Gödel's theorems.
https://www.liarparadox.org/G%C3%B6del_ ... (1931).pdf
The top of the page is Gödel's actual G and all that it does is
prove that a very specific integer number cannot be derived in
Peano Arithmetic. It needs very many connected formulas to
do this.
Gracias. Above me pay grade friend. Neverthelss I'll download the pdf for reading later.
I'm merely relying on my intuition here so if this is a waste of your time do forgive me. Does the liar sentence fail in any way?
I was in mathematician mode back there, as best as I could manage that is. Generalize, generalize, generalize.
Anyway, what I feel is G can do a lot more than just prove Gödel's theorems.
https://www.liarparadox.org/G%C3%B6del_ ... (1931).pdf
The top of the page is Gödel's actual G and all that it does is
prove that a very specific integer number cannot be derived in
Peano Arithmetic. It needs very many connected formulas to
do this.
Gracias. Above me pay grade friend. Neverthelss I'll download the pdf for reading later.
I'm merely relying on my intuition here so if this is a waste of your time do forgive me. Does the liar sentence fail in any way?
It seem that our dialogue is much more effective than most because you are not burdened with many preconceived notions. Math guys seem to believe that the rote memorization of a bunch of complex steps is the exact same thing as fully understanding the problem.
An actual understanding of the problem comes from understanding the key essence of how the key elements of the problem fit together semantically. Math guys make sure to ignore this as irrelevant nonsense.
https://www.liarparadox.org/G%C3%B6del_ ... (1931).pdf
The top of the page is Gödel's actual G and all that it does is
prove that a very specific integer number cannot be derived in
Peano Arithmetic. It needs very many connected formulas to
do this.
Gracias. Above me pay grade friend. Neverthelss I'll download the pdf for reading later.
I'm merely relying on my intuition here so if this is a waste of your time do forgive me. Does the liar sentence fail in any way?
It seem that our dialogue is much more effective than most because you are not burdened with many preconceived notions. Math guys seem to believe that the rote memorization of a bunch of complex steps is the exact same thing as fully understanding the problem.
An actual understanding of the problem comes from understanding the key essence of how the key elements of the problem fit together semantically. Math guys make sure to ignore this as irrelevant nonsense.
If I were working on this problem, I'd use the following format
Liar Sentence
1
2
3
.
.
.
.
Gödel Sentence
1
2
3
.
.
.
Propositions (in general with stress on inferential properties)
1
2
3
.
.
.
PeteOlcott wrote: ↑Thu Apr 20, 2023 11:17 pm
Thus it is dead obvious that the reason that G cannot be proved in F is that G is waaaay screwed up not that there is anything at all wrong with F so the conclusion of Gödel "incompleteness" theorem is dead wrong thus nullifying his whole proof.
Thus it is dead obvious that the reason that "no roads lead to Australia" cannot be proved on Earth is that Australia is waaay screwed up and not that there is anything at all wrong with Earth.
[Redacted]
You keep injecting moral terminology such as "right" and "wrong" when all we are talking about is the semantic properties of formal systems.
A system that is capable of derriving contradictions is not "wrong". It's just a system that is capable of derriving contradictions - an inconsistent system.
The part that this is "wrong" is just your cultural and socio-political bias.
I doubt that Gödel could be wrong. It's just that the issue/problem with the liar sentence doesn't carry over to the Gödel sentence. How did Gödel pull that off?
Agent Smith wrote: ↑Fri Apr 21, 2023 4:53 am
I doubt that Gödel could be wrong. It's just that the issue/problem with the liar sentence doesn't carry over to the Gödel sentence. How did Gödel pull that off?
It is trivial to prove that he is wrong after spending thousands of
hours over decades boiling it down to this simple English sentence.
G asserts its own unprovability in F
The reason that G cannot be proved in F is that this requires a
sequence of inference steps in F that proves no such sequence
of inference steps exists in F.
Do you understand that is a contradiction?
Do you understand why you cannot prove that you yourself never existed?
Agent Smith wrote: ↑Fri Apr 21, 2023 4:53 am
I doubt that Gödel could be wrong. It's just that the issue/problem with the liar sentence doesn't carry over to the Gödel sentence. How did Gödel pull that off?
It is trivial to prove that he is wrong after spending thousands of
hours over decades boiling it down to this simple English sentence.
G asserts its own unprovability in F
The reason that G cannot be proved in F is that this requires a
sequence of inference steps in F that proves no such sequence
of inference steps exists in F.
Do you understand that is a contradiction?
Do you understand why you cannot prove that you yourself never existed?
First off, your version - everybody has one, oui? - of G is, for me if not for others, the picture of clarity. Hats off to you. Second, in a very superficial sense, Gödel relies on self-contradiction; after all the liar paradox appears in his work per you.
My own take is that G is logically potent, unlike me , and I would like you to explore that aspect of G.
Agent Smith wrote: ↑Fri Apr 21, 2023 4:53 am
I doubt that Gödel could be wrong. It's just that the issue/problem with the liar sentence doesn't carry over to the Gödel sentence. How did Gödel pull that off?
It is trivial to prove that he is wrong after spending thousands of
hours over decades boiling it down to this simple English sentence.
G asserts its own unprovability in F
The reason that G cannot be proved in F is that this requires a
sequence of inference steps in F that proves no such sequence
of inference steps exists in F.
Do you understand that is a contradiction?
Do you understand why you cannot prove that you yourself never existed?
First off, your version - everybody has one, oui? - of G is, for me if not for others, the picture of clarity. Hats off to you. Second, in a very superficial sense, Gödel relies on self-contradiction; after all the liar paradox appears in his work per you.
My own take is that G is logically potent, unlike me , and I would like you to explore that aspect of G.
G is relatively stupid its only purpose is to prove that a certain specific
integer value cannot be derived from a certain very complex sequence of
formulas. Gödel's F is also pretty stupid it can only perform arithmetic.
I sum up the essence of his proof the same way that he does:
"We are therefore confronted with a proposition which asserts its own unprovability." 15 (Gödel 1931:39-41)
This summary more precisely matches the essence of the original proof: G asserts its own unprovability in F.
It is trivial to prove that he is wrong after spending thousands of
hours over decades boiling it down to this simple English sentence.
G asserts its own unprovability in F
The reason that G cannot be proved in F is that this requires a
sequence of inference steps in F that proves no such sequence
of inference steps exists in F.
Do you understand that is a contradiction?
Do you understand why you cannot prove that you yourself never existed?
First off, your version - everybody has one, oui? - of G is, for me if not for others, the picture of clarity. Hats off to you. Second, in a very superficial sense, Gödel relies on self-contradiction; after all the liar paradox appears in his work per you.
My own take is that G is logically potent, unlike me , and I would like you to explore that aspect of G.
G is relatively stupid its only purpose is to prove that a certain specific
integer value cannot be derived from a certain very complex sequence of
formulas. Gödel's F is also pretty stupid it can only perform arithmetic.
I sum up the essence of his proof the same way that he does:
"We are therefore confronted with a proposition which asserts its own unprovability." 15 (Gödel 1931:39-41)
This summary more precisely matches the essence of the original proof: G asserts its own unprovability in F.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And Related Systems
We must first agree on definitions of the pertinent, some more than others, words that appear in Gödel's proof. Second step is to exhaust all possible inferences.
I'm no fan of this method, but all philosophers ultimately end up using it, especially when they're cornered,
First off, your version - everybody has one, oui? - of G is, for me if not for others, the picture of clarity. Hats off to you. Second, in a very superficial sense, Gödel relies on self-contradiction; after all the liar paradox appears in his work per you.
My own take is that G is logically potent, unlike me , and I would like you to explore that aspect of G.
G is relatively stupid its only purpose is to prove that a certain specific
integer value cannot be derived from a certain very complex sequence of
formulas. Gödel's F is also pretty stupid it can only perform arithmetic.
I sum up the essence of his proof the same way that he does:
"We are therefore confronted with a proposition which asserts its own unprovability." 15 (Gödel 1931:39-41)
This summary more precisely matches the essence of the original proof: G asserts its own unprovability in F.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And Related Systems
We must first agree on definitions of the pertinent, some more than others, words that appear in Gödel's proof. Second step is to exhaust all possible inferences.
I'm no fan of this method, but all philosophers ultimately end up using it, especially when they're cornered,
I spent 20 years boiling the whole thing down to this:
G asserts its own unprovability in F.
When we even hypothesize that it is a correct basis it refutes Gödel's
proof (within this hypothesis) in a few more sentences .
PeteOlcott wrote: ↑Fri Apr 21, 2023 6:50 am
I spent 20 years boiling the whole thing down to this:
G asserts its own unprovability in F.
Whether G asserts its own unprovability in F.
Or whether you assert that G is unprovable in F.
It makes no difference if "G is unprovable in F" is a tautology e.g true in all models.
It is true in some models usually referred to as Meta_F.
The reason that G cannot be proved in F is that this requires
a sequence of inference steps in F that proves no such sequence
of inference steps exists in F.
G is relatively stupid its only purpose is to prove that a certain specific
integer value cannot be derived from a certain very complex sequence of
formulas. Gödel's F is also pretty stupid it can only perform arithmetic.
I sum up the essence of his proof the same way that he does:
"We are therefore confronted with a proposition which asserts its own unprovability." 15 (Gödel 1931:39-41)
This summary more precisely matches the essence of the original proof: G asserts its own unprovability in F.
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And Related Systems
We must first agree on definitions of the pertinent, some more than others, words that appear in Gödel's proof. Second step is to exhaust all possible inferences.
I'm no fan of this method, but all philosophers ultimately end up using it, especially when they're cornered,
I spent 20 years boiling the whole thing down to this:
G asserts its own unprovability in F.
When we even hypothesize that it is a correct basis it refutes Gödel's
proof (within this hypothesis) in a few more sentences .
I suggest that you keep Gödel and his incompleteness theorems on the backburner - for at least a week, say - and sink yer teeth into paradoxes in general. What can we learn from other antinomies? Go for the 10,000 feet view.
, and I would like you to explore that aspect of G.