A proof of G in F
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A proof of G in F
The scope of this post is ∃G ∈ F (G ↔ ¬(F ⊢ G))
There exists a G in F such that G is logically equivalent to its own
unprovability in F.
The idea here is to examine the philosophical foundation of the
mathematical notion of incompleteness making sure that it is coherent.
A proof of G in F that proves that G cannot be proved in F is simply
self-contradictory, thus no such G exists in F.
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
Thus proving that when the above G is neither provable nor refutable
in F it is because G is self-contradictory in F thus not because F is incomplete.
There exists a G in F such that G is logically equivalent to its own
unprovability in F.
The idea here is to examine the philosophical foundation of the
mathematical notion of incompleteness making sure that it is coherent.
A proof of G in F that proves that G cannot be proved in F is simply
self-contradictory, thus no such G exists in F.
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
Thus proving that when the above G is neither provable nor refutable
in F it is because G is self-contradictory in F thus not because F is incomplete.
Re: A proof of G in F
Did comp.theory get tired of your bullshit?PeteOlcott wrote: ↑Mon Mar 27, 2023 4:09 pm The scope of this post is ∃G ∈ F (G ↔ ¬(F ⊢ G))
There exists a G in F such that G is logically equivalent to its own
unprovability in F.
The idea here is to examine the philosophical foundation of the
mathematical notion of incompleteness making sure that it is coherent.
A proof of G in F that proves that G cannot be proved in F is simply
self-contradictory, thus no such G exists in F.
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
Thus proving that when the above G is neither provable nor refutable
in F it is because G is self-contradictory in F thus not because F is incomplete.
Here's the decision problem all over again. Only one of these expressions is true:
1. ∃G ∈ F (G ↔ ¬(F ⊢ G))
2. ¬∃G ∈ F (G ↔ ¬(F ⊢ G))
Give me the decision procedure for the above.
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Re: A proof of G in F
If there exists a proof of G in F that G is unprovable in F then this forms a contradiction and proves that there is no such G in F.
If a barber claims that he shaves all those that do not shave themselves then ZFC knows that this barber must be lying because he is saying that he both shaves himself and never shaves himself.
Re: A proof of G in F
Sure. So you've proven G ⊢ ¬GPeteOlcott wrote: ↑Mon Mar 27, 2023 5:32 pmIf there exists a proof of G in F that G is unprovable in F then this forms a contradiction and proves that there is no such G in F.
That's called proof of negation and it's a valid proof in Intuitionistic logic.
It's different to proof by contradiction which is ¬G ⊢G. Which is not generally valid in Intuitionistic logic.
Which logic are you using? Classical or intuitionistic?
https://math.andrej.com/2010/03/29/proo ... radiction/
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Re: A proof of G in F
I define my own system of correct reasoning:Skepdick wrote: ↑Mon Mar 27, 2023 6:22 pmSure. So you've proven G ⊢ ¬GPeteOlcott wrote: ↑Mon Mar 27, 2023 5:32 pmIf there exists a proof of G in F that G is unprovable in F then this forms a contradiction and proves that there is no such G in F.
That's called proof of negation and it's a valid proof in Intuitionistic logic.
It's different to proof by contradiction which is ¬G ⊢G. Which is not generally valid in Intuitionistic logic.
Which logic are you using? Classical or intuitionistic?
https://math.andrej.com/2010/03/29/proo ... radiction/
(by extending the notion of a syllogism)
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the semantic
property of Boolean true.
(b) Some expressions of language L are a semantically necessary consequence of others.
T is a subset of (a)
True(L,X) means X ∈ (a) or T ⊨□ X
False(L,X) means T ⊨□ ~X
This abolishes Tarski Undefinability and Gödel Incompleteness.
Re: A proof of G in F
What the fuck does this have to do with anything I said?PeteOlcott wrote: ↑Mon Mar 27, 2023 6:34 pmI define my own system of correct reasoning:Skepdick wrote: ↑Mon Mar 27, 2023 6:22 pmSure. So you've proven G ⊢ ¬GPeteOlcott wrote: ↑Mon Mar 27, 2023 5:32 pm
If there exists a proof of G in F that G is unprovable in F then this forms a contradiction and proves that there is no such G in F.
That's called proof of negation and it's a valid proof in Intuitionistic logic.
It's different to proof by contradiction which is ¬G ⊢G. Which is not generally valid in Intuitionistic logic.
Which logic are you using? Classical or intuitionistic?
https://math.andrej.com/2010/03/29/proo ... radiction/
(by extending the notion of a syllogism)
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the semantic
property of Boolean true.
(b) Some expressions of language L are a semantically necessary consequence of others.
T is a subset of (a)
True(L,X) means X ∈ (a) or T ⊨□ X
False(L,X) means T ⊨□ ~X
This abolishes Tarski Undefinability and Gödel Incompleteness.
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Re: A proof of G in F
"Which logic are you using? Classical or intuitionistic?"
Neither I am using my own system of correct reasoning.
Neither I am using my own system of correct reasoning.
Re: A proof of G in F
So lets call this system O for Olcott.PeteOlcott wrote: ↑Mon Mar 27, 2023 6:38 pm "Which logic are you using? Classical or intuitionistic?"
Neither I am using my own system of correct reasoning.
What's Correct(O) provable in?
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Re: A proof of G in F
Anything (such as the principle of explosion) that diverges from the foundation of correct reasoning that I have established is incorrect reasoning.Skepdick wrote: ↑Mon Mar 27, 2023 6:41 pmSo lets call this system O for Olcott.PeteOlcott wrote: ↑Mon Mar 27, 2023 6:38 pm "Which logic are you using? Classical or intuitionistic?"
Neither I am using my own system of correct reasoning.
What's Correct(O) provable in?
Foundation of correct reasoning
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the semantic property of Boolean true.
(b) Some expressions of language L are a semantically necessary consequence of others.
T is a subset of (a)
True(L,X) means X ∈ (a) or T ⊨□ X
False(L,X) means T ⊨□ ~X
The above Foundation of correct reasoning is simply how the body of analytic truth really works.
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Re: A proof of G in F
My god skep he's right. I checked his work.
I never thought I'd see it in my lifetime. But he's done it. He's bloody done it.
I never thought I'd see it in my lifetime. But he's done it. He's bloody done it.
Re: A proof of G in F
Is there any reason why you are avoiding my question?PeteOlcott wrote: ↑Mon Mar 27, 2023 6:57 pmAnything (such as the principle of explosion) that diverges from the foundation of correct reasoning that I have established is incorrect reasoning.Skepdick wrote: ↑Mon Mar 27, 2023 6:41 pmSo lets call this system O for Olcott.PeteOlcott wrote: ↑Mon Mar 27, 2023 6:38 pm "Which logic are you using? Classical or intuitionistic?"
Neither I am using my own system of correct reasoning.
What's Correct(O) provable in?
Foundation of correct reasoning
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the semantic property of Boolean true.
(b) Some expressions of language L are a semantically necessary consequence of others.
T is a subset of (a)
True(L,X) means X ∈ (a) or T ⊨□ X
False(L,X) means T ⊨□ ~X
The above Foundation of correct reasoning is simply how the body of analytic truth really works.
You said that you are using your system of "correct" reasoning. So to use your own operator: ⊨□ Correct(O)
In what system is the "Correctness" property of your system provable in?
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Re: A proof of G in F
"In what system is the "Correctness" property of your system provable in?"
The entire body of analytic truth specified in any formal or natural language that exists or ever will exist.
How do you know that baby kittens are a type of animal and not any type of ten story office building?
(a) Some expressions of language L are stipulated to have the semantic property of Boolean true.
(b) Some expressions of language L are a semantically necessary consequence of others.
The entire body of analytic truth specified in any formal or natural language that exists or ever will exist.
How do you know that baby kittens are a type of animal and not any type of ten story office building?
(a) Some expressions of language L are stipulated to have the semantic property of Boolean true.
(b) Some expressions of language L are a semantically necessary consequence of others.
Re: A proof of G in F
That's not a proof system.PeteOlcott wrote: ↑Mon Mar 27, 2023 8:55 pm "In what system is the "Correctness" property of your system provable in?"
The entire body of analytic truth specified in any formal or natural language that exists or ever will exist.
Idiot has mixed up kittens and Booleans.PeteOlcott wrote: ↑Mon Mar 27, 2023 8:55 pm How do you know that baby kittens are a type of animal and not any type of ten story office building?
(a) Some expressions of language L are stipulated to have the semantic property of Boolean true.
(b) Some expressions of language L are a semantically necessary consequence of others.
It's like he doesn't understand the difference between the abstract and the concrete.
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Re: A proof of G in F
"It's like he doesn't understand the difference between the abstract and the concrete."
Every expression of any language that can be completely verified as totally true entirely on the basis of its meaning is an element of the set of analytic truth.
Since the finite string "baby kittens" are a defined set of attributes and the finite string "ten story office buildings" are a defined set of attributes and these two sets are disjoint we know analytically that "baby kittens" are not any type of "ten story office building".
Every expression of any language that can be completely verified as totally true entirely on the basis of its meaning is an element of the set of analytic truth.
Since the finite string "baby kittens" are a defined set of attributes and the finite string "ten story office buildings" are a defined set of attributes and these two sets are disjoint we know analytically that "baby kittens" are not any type of "ten story office building".
Re: A proof of G in F
You are confusing the semantics of formal languages with what we call "meaning" in natural languages. They aren't even remotely related.PeteOlcott wrote: ↑Mon Mar 27, 2023 9:16 pm Every expression of any language that can be completely verified as totally true entirely on the basis of its meaning is an element of the set of analytic truth.
But don't take it from me: https://youtu.be/E4KhK3kktcM?t=2490
Here, I have a meme for you.Very likely that language doesn't have semantics in the technical sense. It has meaning, but meaning is just some broad loose thing - we don't know what that is.