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From: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Mon, 2 May 2022 22:01:42 -0500

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View all headersFrom: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Mon, 2 May 2022 22:01:42 -0500

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On 5/2/2022 9:21 PM, Ben wrote:

It is OK that you are not a math guy.

If you were a math guy you would understand that if G is provable then that makes Gödel totally wrong. G is not Gödel's theorem, it is a key element of his theorem.

Incomplete T means that there exists a φ such that φ is not provable or refutable in formal system T.

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

olcott <polcott2@gmail.com> writes:

On 5/2/2022 6:43 PM, Ben wrote:

Aleksy Grabowski <hurufu@gmail.com> writes:

Thanks for confirmation, that's what exactly what I was trying to tell

to topic poster in one of my previous posts. Prolog in it's bare form

is a bad theorem solver. It wasn't designed a such.

If you want to deal with such problems maybe it is better to use Coq

theorem prover, I've never used it by myself, but it looks like one of

the best proving assistants out there.

And indeed there is a fully formalised proof of GIT in Coq (though I

think it's the slightly tighter Gödel-Rosser version).

It is true that G is not provable.

G is provable. Proofs abound. I was pointing out one in a proper proof

assistant, Coq.

It is OK that you are not a math guy.

If you were a math guy you would understand that if G is provable then that makes Gödel totally wrong. G is not Gödel's theorem, it is a key element of his theorem.

Incomplete T means that there exists a φ such that φ is not provable or refutable in formal system T.

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

Path: i2pn2.org!i2pn.org!aioe.org!news.uzoreto.com!news-out.netnews.com!news.alt.net!fdc2.netnews.com!peer02.ams1!peer.ams1.xlned.com!news.xlned.com!peer01.iad!feed-me.highwinds-media.com!news.highwinds-media.com!fx33.iad.POSTED!not-for-mail

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Subject: Re: Is this correct Prolog?

Content-Language: en-US

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

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From: Rich...@Damon-Family.org (Richard Damon)

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Subject: Re: Is this correct Prolog?

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On 5/2/22 11:01 PM, olcott wrote:

No, G IS provable, just not in the system F that G is described in, thus F is Incomplete by your definition above.

Part of the key of the Godel proof is that while G sort of refers to itself, it does it in a way that F can't handle, so in F, G doesn't refer to itself but just "some statement", but in a 'more advanced' version of F, say F', we can see that relationship, and show that G must be true, proving it in F', but not in F, thus F is incomplete.

We can then show that we can make a G' in F' with the same property, and thus show that there exists a system F'' where we can prove G'.

This is why you simplification doesn't work. In F, we can't convert G into the statement G says that G is unprovable, but we can in F', thus the statement in F' is that G says that G in unprovable in F, and that statement is provable in F'

You don't seem to be able to handle the concept of layers of logic systems.

On 5/2/2022 9:21 PM, Ben wrote:

olcott <polcott2@gmail.com> writes:

On 5/2/2022 6:43 PM, Ben wrote:

Aleksy Grabowski <hurufu@gmail.com> writes:

Thanks for confirmation, that's what exactly what I was trying to tell

to topic poster in one of my previous posts. Prolog in it's bare form

is a bad theorem solver. It wasn't designed a such.

If you want to deal with such problems maybe it is better to use Coq

theorem prover, I've never used it by myself, but it looks like one of

the best proving assistants out there.

And indeed there is a fully formalised proof of GIT in Coq (though I

think it's the slightly tighter Gödel-Rosser version).

It is true that G is not provable.

G is provable. Proofs abound. I was pointing out one in a proper proof

assistant, Coq.

It is OK that you are not a math guy.

If you were a math guy you would understand that if G is provable then that makes Gödel totally wrong. G is not Gödel's theorem, it is a key element of his theorem.

Incomplete T means that there exists a φ such that φ is not provable or refutable in formal system T.

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

No, G IS provable, just not in the system F that G is described in, thus F is Incomplete by your definition above.

Part of the key of the Godel proof is that while G sort of refers to itself, it does it in a way that F can't handle, so in F, G doesn't refer to itself but just "some statement", but in a 'more advanced' version of F, say F', we can see that relationship, and show that G must be true, proving it in F', but not in F, thus F is incomplete.

We can then show that we can make a G' in F' with the same property, and thus show that there exists a system F'' where we can prove G'.

This is why you simplification doesn't work. In F, we can't convert G into the statement G says that G is unprovable, but we can in F', thus the statement in F' is that G says that G in unprovable in F, and that statement is provable in F'

You don't seem to be able to handle the concept of layers of logic systems.

Path: i2pn2.org!i2pn.org!eternal-september.org!reader02.eternal-september.org!.POSTED!not-for-mail

From: ben.use...@bsb.me.uk (Ben)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Followup-To: comp.theory

Date: Tue, 03 May 2022 15:59:27 +0100

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View all headersFrom: ben.use...@bsb.me.uk (Ben)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

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Date: Tue, 03 May 2022 15:59:27 +0100

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olcott <polcott2@gmail.com> writes:

You are not a math guy. I am.

No. G is provable. Though I did make a mistake -- the link was to a

proof of G-RIT not G.

How are you getting on with E and specifying P? Have you given up?

--

Ben.

"le génie humain a des limites, quand la bêtise humaine n’en a pas"

Alexandre Dumas (fils)

On 5/2/2022 9:21 PM, Ben wrote:

olcott <polcott2@gmail.com> writes:

On 5/2/2022 6:43 PM, Ben wrote:

Aleksy Grabowski <hurufu@gmail.com> writes:

G is provable. Proofs abound. I was pointing out one in a proper proofThanks for confirmation, that's what exactly what I was trying to tell

to topic poster in one of my previous posts. Prolog in it's bare form

is a bad theorem solver. It wasn't designed a such.

If you want to deal with such problems maybe it is better to use Coq

theorem prover, I've never used it by myself, but it looks like one of

the best proving assistants out there.

And indeed there is a fully formalised proof of GIT in Coq (though I

think it's the slightly tighter Gödel-Rosser version).

It is true that G is not provable.

assistant, Coq.

It is OK that you are not a math guy.

You are not a math guy. I am.

If you were a math guy you would understand that if G is provable then

that makes Gödel totally wrong. G is not Gödel's theorem, it is a key

element of his theorem.

No. G is provable. Though I did make a mistake -- the link was to a

proof of G-RIT not G.

How are you getting on with E and specifying P? Have you given up?

--

Ben.

"le génie humain a des limites, quand la bêtise humaine n’en a pas"

Alexandre Dumas (fils)

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From: agis...@gm.invalid (André G. Isaak)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 09:18:35 -0600

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View all headersFrom: agis...@gm.invalid (André G. Isaak)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 09:18:35 -0600

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On 2022-05-02 18:57, olcott wrote:

No. The Liar can be used to construct an *identical* proof. Other antinomies could be used for similar proofs. He's already talking about The Liar.

You have some serious reading comprehension problems. I never denied the things Gödel wrote. I denied your conclusion because it does not follow.

Gödel starts by claiming there is a close relationship (*not* equivalence) between one particular antinomy, The Liar, and his G.

He then states that similar proofs could be constructed using any antinomy.

That entails that other antinomies could be used to construct similar proofs involving a similar close relation (again, *not* equivalence).

Gödel never claims *any* antinomy is equivalent to his G. Merely that a close relationship holds.

And all my comments concerned exactly what that relationship is.

André

--

To email remove 'invalid' & replace 'gm' with well known Google mail service.

On 5/2/2022 6:43 PM, Ben wrote:

Aleksy Grabowski <hurufu@gmail.com> writes:

IF you are defining that your logic system is limited to what Prolog can "Prove", that is fine. Just realize that you have just defined that your

logic system can't handle a lot of the real problems in the world, and in particular, it is very limited in the mathematics it can handle.

I am pretty sure that Prolog is NOT up to handling the logic needed to

handle the mathematics needed to express Godel's G, or the Halting Problem.

Thus, your "Proof" that these Theorems are "Wrong" is incorrect, you

have only proven that your limited logic system can't reach them in expressibility.

Thanks for confirmation, that's what exactly what I was trying to tell

to topic poster in one of my previous posts. Prolog in it's bare form

is a bad theorem solver. It wasn't designed a such.

If you want to deal with such problems maybe it is better to use Coq

theorem prover, I've never used it by myself, but it looks like one of

the best proving assistants out there.

And indeed there is a fully formalised proof of GIT in Coq (though I

think it's the slightly tighter Gödel-Rosser version).

It is true that G is not provable. G is not provable because it is semantically incorrect in the exactly same way that the Liar Paradox is semantically incorrect.

Gödel says:

14 Every epistemological antinomy can likewise be used for a similar undecidability proof

André denied this six times yesterday

The Liar Paradox is an epistemological antinomy, thus can likewise be used for a similar undecidability proof.

No. The Liar can be used to construct an *identical* proof. Other antinomies could be used for similar proofs. He's already talking about The Liar.

Which means that the Liar Paradox is sufficiently equivalent to Gödel's G. Which means if the basic mechanism of epistemological antinomy is shown to be semantically incorrect then Gödel's G is shown to be semantically incorrect.

You have some serious reading comprehension problems. I never denied the things Gödel wrote. I denied your conclusion because it does not follow.

Gödel starts by claiming there is a close relationship (*not* equivalence) between one particular antinomy, The Liar, and his G.

He then states that similar proofs could be constructed using any antinomy.

That entails that other antinomies could be used to construct similar proofs involving a similar close relation (again, *not* equivalence).

Gödel never claims *any* antinomy is equivalent to his G. Merely that a close relationship holds.

And all my comments concerned exactly what that relationship is.

André

--

To email remove 'invalid' & replace 'gm' with well known Google mail service.

Path: i2pn2.org!rocksolid2!i2pn.org!eternal-september.org!reader02.eternal-september.org!.POSTED!not-for-mail

From: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 11:08:53 -0500

Organization: A noiseless patient Spider

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View all headersFrom: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 11:08:53 -0500

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On 5/3/2022 10:18 AM, André G. Isaak wrote:

Really?

That you persisted (six times) on claiming that Gödel's statement about the Liar Paradox overrode and superseded his statement about the entire category that the Liar Paradox belongs to was despicably deceitful, unless you believe that "close relationship" is stronger than "similar undecidability proof". In that case you never lied about this. I really only want an honest dialogue so I am happy to admit my mistakes.

I take "similar undecidability proof" to mean isomorphic.

https://en.wikipedia.org/wiki/Isomorphism

Without carefully studying the philosophical underpinnings of the concept if incompleteness:

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

Incomplete T means that there exists a φ such that φ is not provable or refutable in formal system T.

The above is the precise measure of isomorphism. Anything meeting the above specification is isomorphic to Gödel's G.

One might not feel comfortable that isomorphic is what Gödel by "similar undecidability proof". When we examine the above definition of Incompleteness applied to the entire set of epistemological antinomies, then we realize that all of them are making the category mistake (thanks Flibble) of presuming that φ is a logic sentence / truth bearer.

Here is my first example of a category / type mismatch error that I wrote back in 2004: "What time is it (yes or no)?"

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

On 2022-05-02 18:57, olcott wrote:

On 5/2/2022 6:43 PM, Ben wrote:

Aleksy Grabowski <hurufu@gmail.com> writes:

IF you are defining that your logic system is limited to what Prolog can "Prove", that is fine. Just realize that you have just defined that your

logic system can't handle a lot of the real problems in the world, and in particular, it is very limited in the mathematics it can handle.

I am pretty sure that Prolog is NOT up to handling the logic needed to

handle the mathematics needed to express Godel's G, or the Halting Problem.

Thus, your "Proof" that these Theorems are "Wrong" is incorrect, you

have only proven that your limited logic system can't reach them in expressibility.

Thanks for confirmation, that's what exactly what I was trying to tell

to topic poster in one of my previous posts. Prolog in it's bare form

is a bad theorem solver. It wasn't designed a such.

If you want to deal with such problems maybe it is better to use Coq

theorem prover, I've never used it by myself, but it looks like one of

the best proving assistants out there.

And indeed there is a fully formalised proof of GIT in Coq (though I

think it's the slightly tighter Gödel-Rosser version).

It is true that G is not provable. G is not provable because it is semantically incorrect in the exactly same way that the Liar Paradox is semantically incorrect.

Gödel says:

14 Every epistemological antinomy can likewise be used for a similar undecidability proof

André denied this six times yesterday

The Liar Paradox is an epistemological antinomy, thus can likewise be used for a similar undecidability proof.

No. The Liar can be used to construct an *identical* proof.

Really?

Other antinomies could be used for similar proofs. He's already talking about The Liar.

Which means that the Liar Paradox is sufficiently equivalent to Gödel's G. Which means if the basic mechanism of epistemological antinomy is shown to be semantically incorrect then Gödel's G is shown to be semantically incorrect.

You have some serious reading comprehension problems. I never denied the things Gödel wrote. I denied your conclusion because it does not follow.

Gödel starts by claiming there is a close relationship (*not* equivalence) between one particular antinomy, The Liar, and his G.

He then states that similar proofs could be constructed using any antinomy.

That entails that other antinomies could be used to construct similar proofs involving a similar close relation (again, *not* equivalence).

That you persisted (six times) on claiming that Gödel's statement about the Liar Paradox overrode and superseded his statement about the entire category that the Liar Paradox belongs to was despicably deceitful, unless you believe that "close relationship" is stronger than "similar undecidability proof". In that case you never lied about this. I really only want an honest dialogue so I am happy to admit my mistakes.

Gödel never claims *any* antinomy is equivalent to his G. Merely that a close relationship holds.

I take "similar undecidability proof" to mean isomorphic.

https://en.wikipedia.org/wiki/Isomorphism

Without carefully studying the philosophical underpinnings of the concept if incompleteness:

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

Incomplete T means that there exists a φ such that φ is not provable or refutable in formal system T.

The above is the precise measure of isomorphism. Anything meeting the above specification is isomorphic to Gödel's G.

One might not feel comfortable that isomorphic is what Gödel by "similar undecidability proof". When we examine the above definition of Incompleteness applied to the entire set of epistemological antinomies, then we realize that all of them are making the category mistake (thanks Flibble) of presuming that φ is a logic sentence / truth bearer.

Here is my first example of a category / type mismatch error that I wrote back in 2004: "What time is it (yes or no)?"

And all my comments concerned exactly what that relationship is.

André

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

Path: i2pn2.org!i2pn.org!eternal-september.org!reader02.eternal-september.org!.POSTED!not-for-mail

From: agis...@gm.invalid (André G. Isaak)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 10:52:25 -0600

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View all headersFrom: agis...@gm.invalid (André G. Isaak)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 10:52:25 -0600

Organization: Christians and Atheists United Against Creeping Agnosticism

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On 2022-05-03 10:08, olcott wrote:

Of course really.

His original proof drew on The Liar for inspiration. So a proof which draws on The Liar would be the same proof.

He is saying that he could have used ANY antinomy.

IOW, he could have chosen a *different* antinomy from The Liar, call it Antinomy X, and constructed a *similar* proof around that. It would not be the same proof, but it would involve constructing some sentence G-Prime which held the same relation to Antinomy X as G hold to The Liar.

But there would not be an equivalence between G-Prime and X anymore than there is an equivalence between G and The Liar.

This is the point I keep trying to drive home. There is NO EQUIVALENCE between G and the The Liar. Only a close relationship.

Except I made no such claim. Not even once. You persisted (six times) in misreading my claim.

Fine. That would mean the proof involving the Liar and G would be isomorphic to the proof involving antinomy X and G-Prime.

That does *NOT* get you to the claim you were making which was that G in some sense equivalent to The Liar. It is not.

And your claim that G is not a truth bearer rests on your false claim that G and The Liar are somehow equivalent (though you refuse to say with respect to what). G is very clearly a truth-bearer. Go back and reread my original explanation.

Gödel's G is most definitely a truth bearer. It asserts that a specific polynomial equation has an integer solution. That claim must either be true or false.

André

--

To email remove 'invalid' & replace 'gm' with well known Google mail service.

On 5/3/2022 10:18 AM, André G. Isaak wrote:

On 2022-05-02 18:57, olcott wrote:

On 5/2/2022 6:43 PM, Ben wrote:

Aleksy Grabowski <hurufu@gmail.com> writes:

IF you are defining that your logic system is limited to what Prolog can "Prove", that is fine. Just realize that you have just defined that your

logic system can't handle a lot of the real problems in the world, and in particular, it is very limited in the mathematics it can handle.

I am pretty sure that Prolog is NOT up to handling the logic needed to

handle the mathematics needed to express Godel's G, or the Halting Problem.

Thus, your "Proof" that these Theorems are "Wrong" is incorrect, you

have only proven that your limited logic system can't reach them in expressibility.

Thanks for confirmation, that's what exactly what I was trying to tell

to topic poster in one of my previous posts. Prolog in it's bare form

is a bad theorem solver. It wasn't designed a such.

If you want to deal with such problems maybe it is better to use Coq

theorem prover, I've never used it by myself, but it looks like one of

the best proving assistants out there.

And indeed there is a fully formalised proof of GIT in Coq (though I

think it's the slightly tighter Gödel-Rosser version).

It is true that G is not provable. G is not provable because it is semantically incorrect in the exactly same way that the Liar Paradox is semantically incorrect.

Gödel says:

14 Every epistemological antinomy can likewise be used for a similar undecidability proof

André denied this six times yesterday

The Liar Paradox is an epistemological antinomy, thus can likewise be used for a similar undecidability proof.

No. The Liar can be used to construct an *identical* proof.

Really?

Of course really.

His original proof drew on The Liar for inspiration. So a proof which draws on The Liar would be the same proof.

He is saying that he could have used ANY antinomy.

IOW, he could have chosen a *different* antinomy from The Liar, call it Antinomy X, and constructed a *similar* proof around that. It would not be the same proof, but it would involve constructing some sentence G-Prime which held the same relation to Antinomy X as G hold to The Liar.

But there would not be an equivalence between G-Prime and X anymore than there is an equivalence between G and The Liar.

This is the point I keep trying to drive home. There is NO EQUIVALENCE between G and the The Liar. Only a close relationship.

Other antinomies could be used for similar proofs. He's already talking about The Liar.

Which means that the Liar Paradox is sufficiently equivalent to Gödel's G. Which means if the basic mechanism of epistemological antinomy is shown to be semantically incorrect then Gödel's G is shown to be semantically incorrect.

You have some serious reading comprehension problems. I never denied the things Gödel wrote. I denied your conclusion because it does not follow.

Gödel starts by claiming there is a close relationship (*not* equivalence) between one particular antinomy, The Liar, and his G.

He then states that similar proofs could be constructed using any antinomy.

That entails that other antinomies could be used to construct similar proofs involving a similar close relation (again, *not* equivalence).

That you persisted (six times) on claiming that Gödel's statement about the Liar Paradox overrode and superseded his statement about the entire category that the Liar Paradox belongs to was despicably deceitful,

Except I made no such claim. Not even once. You persisted (six times) in misreading my claim.

unless you believe that "close relationship" is stronger than "similar undecidability proof". In that case you never lied about this. I really only want an honest dialogue so I am happy to admit my mistakes.

Gödel never claims *any* antinomy is equivalent to his G. Merely that a close relationship holds.

I take "similar undecidability proof" to mean isomorphic.

Fine. That would mean the proof involving the Liar and G would be isomorphic to the proof involving antinomy X and G-Prime.

That does *NOT* get you to the claim you were making which was that G in some sense equivalent to The Liar. It is not.

And your claim that G is not a truth bearer rests on your false claim that G and The Liar are somehow equivalent (though you refuse to say with respect to what). G is very clearly a truth-bearer. Go back and reread my original explanation.

https://en.wikipedia.org/wiki/Isomorphism

Without carefully studying the philosophical underpinnings of the concept if incompleteness:

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

Incomplete T means that there exists a φ such that φ is not provable or refutable in formal system T.

The above is the precise measure of isomorphism. Anything meeting the above specification is isomorphic to Gödel's G.

One might not feel comfortable that isomorphic is what Gödel by "similar undecidability proof". When we examine the above definition of Incompleteness applied to the entire set of epistemological antinomies, then we realize that all of them are making the category mistake (thanks Flibble) of presuming that φ is a logic sentence / truth bearer.

Gödel's G is most definitely a truth bearer. It asserts that a specific polynomial equation has an integer solution. That claim must either be true or false.

André

--

To email remove 'invalid' & replace 'gm' with well known Google mail service.

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From: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 12:05:23 -0500

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View all headersFrom: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 12:05:23 -0500

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On 5/3/2022 11:52 AM, André G. Isaak wrote:

I have no idea what you mean by G-Prime.

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

It is the fact that the mathematical definition of Incompleteness simply assumes that φ is semantically correct that is the core mistake of the mathematical definition of Incompleteness.

When G is not provable in PA, how is it shown to be true?

If it is not shown to be true in PA then we have the strawman error.

When G is not provable in PA, how is it shown to be true?

If it is not shown to be true in PA then we have the strawman error.

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

On 2022-05-03 10:08, olcott wrote:

On 5/3/2022 10:18 AM, André G. Isaak wrote:

On 2022-05-02 18:57, olcott wrote:

logic system can't handle a lot of the real problems in the world, and in particular, it is very limited in the mathematics it can handle.

I am pretty sure that Prolog is NOT up to handling the logic needed to

handle the mathematics needed to express Godel's G, or the Halting Problem.

Thus, your "Proof" that these Theorems are "Wrong" is incorrect, you

have only proven that your limited logic system can't reach them in expressibility.

Thanks for confirmation, that's what exactly what I was trying to tell

to topic poster in one of my previous posts. Prolog in it's bare form

is a bad theorem solver. It wasn't designed a such.

If you want to deal with such problems maybe it is better to use Coq

theorem prover, I've never used it by myself, but it looks like one of

the best proving assistants out there.

And indeed there is a fully formalised proof of GIT in Coq (though I

think it's the slightly tighter Gödel-Rosser version).

It is true that G is not provable. G is not provable because it is semantically incorrect in the exactly same way that the Liar Paradox is semantically incorrect.

Gödel says:

14 Every epistemological antinomy can likewise be used for a similar undecidability proof

André denied this six times yesterday

The Liar Paradox is an epistemological antinomy, thus can likewise be used for a similar undecidability proof.

No. The Liar can be used to construct an *identical* proof.

Really?

Of course really.

His original proof drew on The Liar for inspiration. So a proof which draws on The Liar would be the same proof.

He is saying that he could have used ANY antinomy.

IOW, he could have chosen a *different* antinomy from The Liar, call it Antinomy X, and constructed a *similar* proof around that. It would not be the same proof, but it would involve constructing some sentence G-Prime which held the same relation to Antinomy X as G hold to The Liar.

But there would not be an equivalence between G-Prime and X anymore than there is an equivalence between G and The Liar.

This is the point I keep trying to drive home. There is NO EQUIVALENCE between G and the The Liar. Only a close relationship.

Other antinomies could be used for similar proofs. He's already talking about The Liar.

You have some serious reading comprehension problems. I never denied the things Gödel wrote. I denied your conclusion because it does not follow.

Gödel starts by claiming there is a close relationship (*not* equivalence) between one particular antinomy, The Liar, and his G.

He then states that similar proofs could be constructed using any antinomy.

That entails that other antinomies could be used to construct similar proofs involving a similar close relation (again, *not* equivalence).

That you persisted (six times) on claiming that Gödel's statement about the Liar Paradox overrode and superseded his statement about the entire category that the Liar Paradox belongs to was despicably deceitful,

Except I made no such claim. Not even once. You persisted (six times) in misreading my claim.

unless you believe that "close relationship" is stronger than "similar undecidability proof". In that case you never lied about this. I really only want an honest dialogue so I am happy to admit my mistakes.

Gödel never claims *any* antinomy is equivalent to his G. Merely that a close relationship holds.

I take "similar undecidability proof" to mean isomorphic.

Fine. That would mean the proof involving the Liar and G would be isomorphic to the proof involving antinomy X and G-Prime.

I have no idea what you mean by G-Prime.

That does *NOT* get you to the claim you were making which was that G in some sense equivalent to The Liar. It is not.

And your claim that G is not a truth bearer rests on your false claim that G and The Liar are somehow equivalent (though you refuse to say with respect to what).

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

It is the fact that the mathematical definition of Incompleteness simply assumes that φ is semantically correct that is the core mistake of the mathematical definition of Incompleteness.

G is very clearly a truth-bearer. Go back and reread my original explanation.

When G is not provable in PA, how is it shown to be true?

If it is not shown to be true in PA then we have the strawman error.

https://en.wikipedia.org/wiki/Isomorphism

Without carefully studying the philosophical underpinnings of the concept if incompleteness:

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

Incomplete T means that there exists a φ such that φ is not provable or refutable in formal system T.

The above is the precise measure of isomorphism. Anything meeting the above specification is isomorphic to Gödel's G.

One might not feel comfortable that isomorphic is what Gödel by "similar undecidability proof". When we examine the above definition of Incompleteness applied to the entire set of epistemological antinomies, then we realize that all of them are making the category mistake (thanks Flibble) of presuming that φ is a logic sentence / truth bearer.

Gödel's G is most definitely a truth bearer. It asserts that a specific polynomial equation has an integer solution. That claim must either be true or false.

André

When G is not provable in PA, how is it shown to be true?

If it is not shown to be true in PA then we have the strawman error.

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

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From: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re:_Is_this_correct_Prolog?_[_André_didn't_l

ie_after_all_]

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View all headersFrom: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re:_Is_this_correct_Prolog?_[_André_didn't_l

ie_after_all_]

Date: Tue, 3 May 2022 12:08:47 -0500

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On 5/3/2022 11:52 AM, André G. Isaak wrote:

OK, then I am very happy to say that my accusation that you lied was not justified, we were simply talking past each other.

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

On 2022-05-03 10:08, olcott wrote:

On 5/3/2022 10:18 AM, André G. Isaak wrote:

logic system can't handle a lot of the real problems in the world, and in particular, it is very limited in the mathematics it can handle.

I am pretty sure that Prolog is NOT up to handling the logic needed to

handle the mathematics needed to express Godel's G, or the Halting Problem.

Thus, your "Proof" that these Theorems are "Wrong" is incorrect, you

have only proven that your limited logic system can't reach them in expressibility.

Thanks for confirmation, that's what exactly what I was trying to tell

to topic poster in one of my previous posts. Prolog in it's bare form

is a bad theorem solver. It wasn't designed a such.

If you want to deal with such problems maybe it is better to use Coq

theorem prover, I've never used it by myself, but it looks like one of

the best proving assistants out there.

And indeed there is a fully formalised proof of GIT in Coq (though I

think it's the slightly tighter Gödel-Rosser version).

It is true that G is not provable. G is not provable because it is semantically incorrect in the exactly same way that the Liar Paradox is semantically incorrect.

Gödel says:

14 Every epistemological antinomy can likewise be used for a similar undecidability proof

André denied this six times yesterday

The Liar Paradox is an epistemological antinomy, thus can likewise be used for a similar undecidability proof.

No. The Liar can be used to construct an *identical* proof.

Really?

Of course really.

His original proof drew on The Liar for inspiration. So a proof which draws on The Liar would be the same proof.

OK, then I am very happy to say that my accusation that you lied was not justified, we were simply talking past each other.

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

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From: agis...@gm.invalid (André G. Isaak)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 11:17:19 -0600

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View all headersFrom: agis...@gm.invalid (André G. Isaak)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 11:17:19 -0600

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On 2022-05-03 11:05, olcott wrote:

It is defined directly above. Trying reading more carefully.

Here you're simply begging the question by assuming your own conclusion: that being true and being provable are the same. The whole point of Gödel's proof is that they cannot be the same (at least not for non-trivial systems).

The question is not whether it is true but whether it is a truth *bearer*.

You make the claim that The Liar is not a truth bearer (a plausible claim depending on one's definitions).

You then jump to the conclusion that G is not a truth bearer based on your assertion that it is "equivalent" to The Liar. But it is *NOT* equivalent. It merely bears a close relationship. But you refuse to actually consider what the nature of that relationship; there are both similiarites and differences.

Whereas the Liar has no content other than to assert its own falsity, Gödel's G has definite content. It does not assert its own unprovability, it asserts a very specific mathematical claim, one which must by its nature be either true or false. Therefore G *is* a truth bearer.

The formulation that G asserts its own unprovability is the Cliff-Notes version of the proof. It's not the substance of the actual proof.

André

--

To email remove 'invalid' & replace 'gm' with well known Google mail service.

On 5/3/2022 11:52 AM, André G. Isaak wrote:

On 2022-05-03 10:08, olcott wrote:

logic system can't handle a lot of the real problems in the world, and in particular, it is very limited in the mathematics it can handle.

I am pretty sure that Prolog is NOT up to handling the logic needed to

handle the mathematics needed to express Godel's G, or the Halting Problem.

Thus, your "Proof" that these Theorems are "Wrong" is incorrect, you

have only proven that your limited logic system can't reach them in expressibility.

Thanks for confirmation, that's what exactly what I was trying to tell

to topic poster in one of my previous posts. Prolog in it's bare form

is a bad theorem solver. It wasn't designed a such.

If you want to deal with such problems maybe it is better to use Coq

theorem prover, I've never used it by myself, but it looks like one of

the best proving assistants out there.

And indeed there is a fully formalised proof of GIT in Coq (though I

think it's the slightly tighter Gödel-Rosser version).

It is true that G is not provable. G is not provable because it is semantically incorrect in the exactly same way that the Liar Paradox is semantically incorrect.

Gödel says:

14 Every epistemological antinomy can likewise be used for a similar undecidability proof

André denied this six times yesterday

The Liar Paradox is an epistemological antinomy, thus can likewise be used for a similar undecidability proof.

No. The Liar can be used to construct an *identical* proof.

Really?

Of course really.

His original proof drew on The Liar for inspiration. So a proof which draws on The Liar would be the same proof.

He is saying that he could have used ANY antinomy.

IOW, he could have chosen a *different* antinomy from The Liar, call it Antinomy X, and constructed a *similar* proof around that. It would not be the same proof, but it would involve constructing some sentence G-Prime which held the same relation to Antinomy X as G hold to The Liar.

But there would not be an equivalence between G-Prime and X anymore than there is an equivalence between G and The Liar.

This is the point I keep trying to drive home. There is NO EQUIVALENCE between G and the The Liar. Only a close relationship.

You have some serious reading comprehension problems. I never denied the things Gödel wrote. I denied your conclusion because it does not follow.

Gödel starts by claiming there is a close relationship (*not* equivalence) between one particular antinomy, The Liar, and his G.

He then states that similar proofs could be constructed using any antinomy.

That entails that other antinomies could be used to construct similar proofs involving a similar close relation (again, *not* equivalence).

That you persisted (six times) on claiming that Gödel's statement about the Liar Paradox overrode and superseded his statement about the entire category that the Liar Paradox belongs to was despicably deceitful,

Except I made no such claim. Not even once. You persisted (six times) in misreading my claim.

unless you believe that "close relationship" is stronger than "similar undecidability proof". In that case you never lied about this. I really only want an honest dialogue so I am happy to admit my mistakes.

Gödel never claims *any* antinomy is equivalent to his G. Merely that a close relationship holds.

I take "similar undecidability proof" to mean isomorphic.

Fine. That would mean the proof involving the Liar and G would be isomorphic to the proof involving antinomy X and G-Prime.

I have no idea what you mean by G-Prime.

It is defined directly above. Trying reading more carefully.

That does *NOT* get you to the claim you were making which was that G in some sense equivalent to The Liar. It is not.

And your claim that G is not a truth bearer rests on your false claim that G and The Liar are somehow equivalent (though you refuse to say with respect to what).

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

It is the fact that the mathematical definition of Incompleteness simply assumes that φ is semantically correct that is the core mistake of the mathematical definition of Incompleteness.

G is very clearly a truth-bearer. Go back and reread my original explanation.

When G is not provable in PA, how is it shown to be true?

If it is not shown to be true in PA then we have the strawman error.

Here you're simply begging the question by assuming your own conclusion: that being true and being provable are the same. The whole point of Gödel's proof is that they cannot be the same (at least not for non-trivial systems).

The question is not whether it is true but whether it is a truth *bearer*.

You make the claim that The Liar is not a truth bearer (a plausible claim depending on one's definitions).

You then jump to the conclusion that G is not a truth bearer based on your assertion that it is "equivalent" to The Liar. But it is *NOT* equivalent. It merely bears a close relationship. But you refuse to actually consider what the nature of that relationship; there are both similiarites and differences.

Whereas the Liar has no content other than to assert its own falsity, Gödel's G has definite content. It does not assert its own unprovability, it asserts a very specific mathematical claim, one which must by its nature be either true or false. Therefore G *is* a truth bearer.

The formulation that G asserts its own unprovability is the Cliff-Notes version of the proof. It's not the substance of the actual proof.

André

--

To email remove 'invalid' & replace 'gm' with well known Google mail service.

Path: i2pn2.org!i2pn.org!eternal-september.org!reader02.eternal-september.org!.POSTED!not-for-mail

From: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 12:33:57 -0500

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View all headersFrom: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 12:33:57 -0500

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On 5/3/2022 12:17 PM, André G. Isaak wrote:

OK.

This is not any mere assumption.

The only way that any analytic expressions of language are correctly determined to be true is:

(a) They are defined to be true.

(b) They are derived from applying truth preserving operations to (a) or (b). Prolog knows this on the basis of its facts and rules. Facts are (a) and rules are (b). This is also known as sound deductive inference.

When G is not provable in PA, how is it shown to be true (wild guess)?

What is the precise basis for assessing that G is true? please provide ALL the steps.

Precise equivalence is defined by this:

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

The Liar Paradox can be neither proven nor refuted where T is the entire body of analytic knowledge and φ is LP.

It is isomorphic to the substance of the actual proof.

Gödel says:

since the undecidable proposition [R(q); q] states precisely that q belongs to K, i.e. according to (1), that [R(q); q] is not provable. We are therefore confronted with a proposition which asserts its own unprovability.

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

On 2022-05-03 11:05, olcott wrote:

On 5/3/2022 11:52 AM, André G. Isaak wrote:

On 2022-05-03 10:08, olcott wrote:

logic system can't handle a lot of the real problems in the world, and in particular, it is very limited in the mathematics it can handle.

I am pretty sure that Prolog is NOT up to handling the logic needed to

handle the mathematics needed to express Godel's G, or the Halting Problem.

Thus, your "Proof" that these Theorems are "Wrong" is incorrect, you

have only proven that your limited logic system can't reach them in expressibility.

Thanks for confirmation, that's what exactly what I was trying to tell

to topic poster in one of my previous posts. Prolog in it's bare form

is a bad theorem solver. It wasn't designed a such.

If you want to deal with such problems maybe it is better to use Coq

theorem prover, I've never used it by myself, but it looks like one of

the best proving assistants out there.

And indeed there is a fully formalised proof of GIT in Coq (though I

think it's the slightly tighter Gödel-Rosser version).

It is true that G is not provable. G is not provable because it is semantically incorrect in the exactly same way that the Liar Paradox is semantically incorrect.

Gödel says:

14 Every epistemological antinomy can likewise be used for a similar undecidability proof

André denied this six times yesterday

The Liar Paradox is an epistemological antinomy, thus can likewise be used for a similar undecidability proof.

No. The Liar can be used to construct an *identical* proof.

Really?

Of course really.

His original proof drew on The Liar for inspiration. So a proof which draws on The Liar would be the same proof.

He is saying that he could have used ANY antinomy.

IOW, he could have chosen a *different* antinomy from The Liar, call it Antinomy X, and constructed a *similar* proof around that. It would not be the same proof, but it would involve constructing some sentence G-Prime which held the same relation to Antinomy X as G hold to The Liar.

But there would not be an equivalence between G-Prime and X anymore than there is an equivalence between G and The Liar.

This is the point I keep trying to drive home. There is NO EQUIVALENCE between G and the The Liar. Only a close relationship.

You have some serious reading comprehension problems. I never denied the things Gödel wrote. I denied your conclusion because it does not follow.

Gödel starts by claiming there is a close relationship (*not* equivalence) between one particular antinomy, The Liar, and his G.

He then states that similar proofs could be constructed using any antinomy.

That entails that other antinomies could be used to construct similar proofs involving a similar close relation (again, *not* equivalence).

That you persisted (six times) on claiming that Gödel's statement about the Liar Paradox overrode and superseded his statement about the entire category that the Liar Paradox belongs to was despicably deceitful,

Except I made no such claim. Not even once. You persisted (six times) in misreading my claim.

Gödel never claims *any* antinomy is equivalent to his G. Merely that a close relationship holds.

I take "similar undecidability proof" to mean isomorphic.

Fine. That would mean the proof involving the Liar and G would be isomorphic to the proof involving antinomy X and G-Prime.

I have no idea what you mean by G-Prime.

It is defined directly above. Trying reading more carefully.

OK.

That does *NOT* get you to the claim you were making which was that G in some sense equivalent to The Liar. It is not.

And your claim that G is not a truth bearer rests on your false claim that G and The Liar are somehow equivalent (though you refuse to say with respect to what).

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

It is the fact that the mathematical definition of Incompleteness simply assumes that φ is semantically correct that is the core mistake of the mathematical definition of Incompleteness.

G is very clearly a truth-bearer. Go back and reread my original explanation.

When G is not provable in PA, how is it shown to be true?

If it is not shown to be true in PA then we have the strawman error.

Here you're simply begging the question by assuming your own conclusion: that being true and being provable are the same.

This is not any mere assumption.

The only way that any analytic expressions of language are correctly determined to be true is:

(a) They are defined to be true.

(b) They are derived from applying truth preserving operations to (a) or (b). Prolog knows this on the basis of its facts and rules. Facts are (a) and rules are (b). This is also known as sound deductive inference.

The whole point of Gödel's proof is that they cannot be the same (at least not for non-trivial systems).

When G is not provable in PA, how is it shown to be true (wild guess)?

What is the precise basis for assessing that G is true? please provide ALL the steps.

The question is not whether it is true but whether it is a truth *bearer*.

You make the claim that The Liar is not a truth bearer (a plausible claim depending on one's definitions).

You then jump to the conclusion that G is not a truth bearer based on your assertion that it is "equivalent" to The Liar. But it is *NOT* equivalent. It merely bears a close relationship. But you refuse to actually consider what the nature of that relationship; there are both similiarites and differences.

Precise equivalence is defined by this:

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

The Liar Paradox can be neither proven nor refuted where T is the entire body of analytic knowledge and φ is LP.

Whereas the Liar has no content other than to assert its own falsity, Gödel's G has definite content. It does not assert its own unprovability, it asserts a very specific mathematical claim, one which must by its nature be either true or false. Therefore G *is* a truth bearer.

The formulation that G asserts its own unprovability is the Cliff-Notes version of the proof. It's not the substance of the actual proof.

André

It is isomorphic to the substance of the actual proof.

Gödel says:

since the undecidable proposition [R(q); q] states precisely that q belongs to K, i.e. according to (1), that [R(q); q] is not provable. We are therefore confronted with a proposition which asserts its own unprovability.

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

Path: i2pn2.org!i2pn.org!eternal-september.org!reader02.eternal-september.org!.POSTED!not-for-mail

From: agis...@gm.invalid (André G. Isaak)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 12:23:12 -0600

Organization: Christians and Atheists United Against Creeping Agnosticism

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View all headersFrom: agis...@gm.invalid (André G. Isaak)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 12:23:12 -0600

Organization: Christians and Atheists United Against Creeping Agnosticism

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On 2022-05-03 11:33, olcott wrote:

<snippage>

'True' and 'Correctly determined to be true' mean different things.

These are YOUR assumptions. They have not been demonstrated. And they are not consistent with the way in which the rest of the world talks about truth. You are talking about provability, not truth.

"True" and "Known to be true" are entirely different things.

Consider the equation Srt((2748 + 87)^3) = 150,948.776 (to 3 decimal places)

That equation is true but it is unlikely anyone knew this to be true until now since I very much doubt anyone had previously considered that specific equation. That's doesn't mean it wasn't true all along. Just that no one knew it was true.

There are all sorts of cases where we know one thing without knowing all things. I can know with certainty that (A ∨ B) is true meaning that I know that *at least* one of A or B must be true while still not knowing the truth value of either. These sorts of things occur all the time.

And not being provable in PA and not being provable are also two different things.

The Liar Paradox is a self-contradiction. The Gödel sentence is not.

No. It is not. G asserts a claim about mathematics. It does not assert anything about itself. However, within the metalanguage we can prove that G can only be true if it is not provable within the system under consideration.

We are confronted *in our analysis* with such a situation. But there is no proposition which directly asserts such a thing.

The problem is you refuse to look at the actual math and instead look only at the text commentary which is merely a guide to, not the actual substance of, the proof.

André

--

To email remove 'invalid' & replace 'gm' with well known Google mail service.

On 5/3/2022 12:17 PM, André G. Isaak wrote:

On 2022-05-03 11:05, olcott wrote:

On 5/3/2022 11:52 AM, André G. Isaak wrote:

<snippage>

G is very clearly a truth-bearer. Go back and reread my original explanation.

When G is not provable in PA, how is it shown to be true?

If it is not shown to be true in PA then we have the strawman error.

Here you're simply begging the question by assuming your own conclusion: that being true and being provable are the same.

This is not any mere assumption.

The only way that any analytic expressions of language are correctly determined to be true is:

'True' and 'Correctly determined to be true' mean different things.

(a) They are defined to be true.

(b) They are derived from applying truth preserving operations to (a) or (b). Prolog knows this on the basis of its facts and rules. Facts are (a) and rules are (b). This is also known as sound deductive inference.

These are YOUR assumptions. They have not been demonstrated. And they are not consistent with the way in which the rest of the world talks about truth. You are talking about provability, not truth.

The whole point of Gödel's proof is that they cannot be the same (at least not for non-trivial systems).

When G is not provable in PA, how is it shown to be true (wild guess)?

What is the precise basis for assessing that G is true? please provide ALL the steps.

"True" and "Known to be true" are entirely different things.

Consider the equation Srt((2748 + 87)^3) = 150,948.776 (to 3 decimal places)

That equation is true but it is unlikely anyone knew this to be true until now since I very much doubt anyone had previously considered that specific equation. That's doesn't mean it wasn't true all along. Just that no one knew it was true.

There are all sorts of cases where we know one thing without knowing all things. I can know with certainty that (A ∨ B) is true meaning that I know that *at least* one of A or B must be true while still not knowing the truth value of either. These sorts of things occur all the time.

And not being provable in PA and not being provable are also two different things.

The question is not whether it is true but whether it is a truth *bearer*.

You make the claim that The Liar is not a truth bearer (a plausible claim depending on one's definitions).

You then jump to the conclusion that G is not a truth bearer based on your assertion that it is "equivalent" to The Liar. But it is *NOT* equivalent. It merely bears a close relationship. But you refuse to actually consider what the nature of that relationship; there are both similiarites and differences.

Precise equivalence is defined by this:

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

The Liar Paradox can be neither proven nor refuted where T is the entire body of analytic knowledge and φ is LP.

The Liar Paradox is a self-contradiction. The Gödel sentence is not.

Whereas the Liar has no content other than to assert its own falsity, Gödel's G has definite content. It does not assert its own unprovability, it asserts a very specific mathematical claim, one which must by its nature be either true or false. Therefore G *is* a truth bearer.

The formulation that G asserts its own unprovability is the Cliff-Notes version of the proof. It's not the substance of the actual proof.

André

It is isomorphic to the substance of the actual proof.

No. It is not. G asserts a claim about mathematics. It does not assert anything about itself. However, within the metalanguage we can prove that G can only be true if it is not provable within the system under consideration.

Gödel says:

since the undecidable proposition [R(q); q] states precisely that q belongs to K, i.e. according to (1), that [R(q); q] is not provable. We are therefore confronted with a proposition which asserts its own unprovability.

We are confronted *in our analysis* with such a situation. But there is no proposition which directly asserts such a thing.

The problem is you refuse to look at the actual math and instead look only at the text commentary which is merely a guide to, not the actual substance of, the proof.

André

--

To email remove 'invalid' & replace 'gm' with well known Google mail service.

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From: jbb...@notatt.com (Jeff Barnett)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 12:33:19 -0600

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View all headersFrom: jbb...@notatt.com (Jeff Barnett)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 12:33:19 -0600

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On 5/3/2022 9:18 AM, André G. Isaak wrote:

IDIOT SNIPPED

times. I admire your tenacity but I must ask you a question: Why do you

think one more time will do either him or you any good? It's not like

this is an educational or soul-improving opportunity for either of you;

for you this must be frustrating no matter your intentions and for him

he is what he is and this is his only amusement. Whether troll or

buffoon, he's proud of his appearance as wasted space.

--

Jeff Barnett

On 2022-05-02 18:57, olcott wrote:

IDIOT SNIPPED

No. The Liar can be used to construct an *identical* proof. OtherI think that you have provided the troll a similar explanation over 100

antinomies could be used for similar proofs. He's already talking about

The Liar.

Gödel's G. Which means if the basic mechanism of epistemological

antinomy is shown to be semantically incorrect then Gödel's G is shown

to be semantically incorrect.

You have some serious reading comprehension problems. I never denied the

things Gödel wrote. I denied your conclusion because it does not follow.

Gödel starts by claiming there is a close relationship (*not*

equivalence) between one particular antinomy, The Liar, and his G.

He then states that similar proofs could be constructed using any antinomy.

That entails that other antinomies could be used to construct similar

proofs involving a similar close relation (again, *not* equivalence).

Gödel never claims *any* antinomy is equivalent to his G. Merely that a

close relationship holds.

And all my comments concerned exactly what that relationship is.

times. I admire your tenacity but I must ask you a question: Why do you

think one more time will do either him or you any good? It's not like

this is an educational or soul-improving opportunity for either of you;

for you this must be frustrating no matter your intentions and for him

he is what he is and this is his only amusement. Whether troll or

buffoon, he's proud of his appearance as wasted space.

--

Jeff Barnett

Path: i2pn2.org!i2pn.org!eternal-september.org!reader02.eternal-september.org!.POSTED!not-for-mail

From: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 13:59:06 -0500

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View all headersFrom: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 13:59:06 -0500

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On 5/3/2022 1:23 PM, André G. Isaak wrote:

Yes that seems to be correct.

On the other hand calling an expression of language true that has not be 'Correctly determined to be true' is an error.

If G is claimed to be true then this assertion must be supported by: 'Correctly determined to be true' otherwise the assessment of its truth is no more than a wild guess.

If G is claimed to be true then this assertion must be supported by: 'Correctly determined to be true' otherwise the assessment of its truth is no more than a wild guess.

Yes, however:

If G is claimed to be true then this assertion must be supported by: 'Correctly determined to be true' otherwise the assessment of its truth is no more than a wild guess.

(2748 + 87)^3 = 22,785,532,875

What are you sorting with Srt()?

That is not true. If A and B are syntactically correct expressions of a formal language yet neither one is semantically correct then we have the same case as the Liar Paradox not being a truth bearer, thus (A ∨ B) is neither true nor false.

I know that. Yet not being provable in PA also means that G cannot be derived in PA by applying truth preserving operations to the axioms Z of PA or other expressions Y derived from Z or Y.

If G is correctly determined to be true then there must be some process detailing the steps of how it is 'Correctly determined to be true'. Lacking these steps one cannot correctly assert that G is true, only that G might possibly be true.

Until you understand that G can only be correctly asserted to be true in X if G is provable in X. G may be true in X without a proof in X, yet it cannot be correctly asserted to be true in X without such a proof.

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

When both G and the LP exactly meet the official mathematical definition of incompleteness then G and LP are proven to be isomorphic.

You are letting weasel words leak semantic meaning.

My analysis pertaining to the official mathematical definition of incompleteness cuts through these weasel words.

This proves that G the LP and the following sentence are all isomorphic:

"This sentence is unprovable".

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

On 2022-05-03 11:33, olcott wrote:

On 5/3/2022 12:17 PM, André G. Isaak wrote:

On 2022-05-03 11:05, olcott wrote:

On 5/3/2022 11:52 AM, André G. Isaak wrote:

<snippage>

G is very clearly a truth-bearer. Go back and reread my original explanation.

When G is not provable in PA, how is it shown to be true?

If it is not shown to be true in PA then we have the strawman error.

Here you're simply begging the question by assuming your own conclusion: that being true and being provable are the same.

This is not any mere assumption.

The only way that any analytic expressions of language are correctly determined to be true is:

'True' and 'Correctly determined to be true' mean different things.

Yes that seems to be correct.

On the other hand calling an expression of language true that has not be 'Correctly determined to be true' is an error.

If G is claimed to be true then this assertion must be supported by: 'Correctly determined to be true' otherwise the assessment of its truth is no more than a wild guess.

(a) They are defined to be true.

(b) They are derived from applying truth preserving operations to (a) or (b). Prolog knows this on the basis of its facts and rules. Facts are (a) and rules are (b). This is also known as sound deductive inference.

These are YOUR assumptions. They have not been demonstrated. And they are not consistent with the way in which the rest of the world talks about truth. You are talking about provability, not truth.

If G is claimed to be true then this assertion must be supported by: 'Correctly determined to be true' otherwise the assessment of its truth is no more than a wild guess.

The whole point of Gödel's proof is that they cannot be the same (at least not for non-trivial systems).

When G is not provable in PA, how is it shown to be true (wild guess)?

What is the precise basis for assessing that G is true? please provide ALL the steps.

"True" and "Known to be true" are entirely different things.

Yes, however:

If G is claimed to be true then this assertion must be supported by: 'Correctly determined to be true' otherwise the assessment of its truth is no more than a wild guess.

Consider the equation Srt((2748 + 87)^3) = 150,948.776 (to 3 decimal places)

(2748 + 87)^3 = 22,785,532,875

What are you sorting with Srt()?

That equation is true but it is unlikely anyone knew this to be true until now since I very much doubt anyone had previously considered that specific equation. That's doesn't mean it wasn't true all along. Just that no one knew it was true.

There are all sorts of cases where we know one thing without knowing all things. I can know with certainty that (A ∨ B) is true meaning that I know that *at least* one of A or B must be true while still not knowing the truth value of either. These sorts of things occur all the time.

That is not true. If A and B are syntactically correct expressions of a formal language yet neither one is semantically correct then we have the same case as the Liar Paradox not being a truth bearer, thus (A ∨ B) is neither true nor false.

And not being provable in PA and not being provable are also two different things.

I know that. Yet not being provable in PA also means that G cannot be derived in PA by applying truth preserving operations to the axioms Z of PA or other expressions Y derived from Z or Y.

If G is correctly determined to be true then there must be some process detailing the steps of how it is 'Correctly determined to be true'. Lacking these steps one cannot correctly assert that G is true, only that G might possibly be true.

The question is not whether it is true but whether it is a truth *bearer*.

You make the claim that The Liar is not a truth bearer (a plausible claim depending on one's definitions).

You then jump to the conclusion that G is not a truth bearer based on your assertion that it is "equivalent" to The Liar. But it is *NOT* equivalent. It merely bears a close relationship. But you refuse to actually consider what the nature of that relationship; there are both similiarites and differences.

Precise equivalence is defined by this:

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

The Liar Paradox can be neither proven nor refuted where T is the entire body of analytic knowledge and φ is LP.

The Liar Paradox is a self-contradiction. The Gödel sentence is not.

Until you understand that G can only be correctly asserted to be true in X if G is provable in X. G may be true in X without a proof in X, yet it cannot be correctly asserted to be true in X without such a proof.

Whereas the Liar has no content other than to assert its own falsity, Gödel's G has definite content. It does not assert its own unprovability, it asserts a very specific mathematical claim, one which must by its nature be either true or false. Therefore G *is* a truth bearer.

The formulation that G asserts its own unprovability is the Cliff-Notes version of the proof. It's not the substance of the actual proof.

André

It is isomorphic to the substance of the actual proof.

No. It is not. G asserts a claim about mathematics. It does not assert anything about itself. However, within the metalanguage we can prove that G can only be true if it is not provable within the system under consideration.

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

When both G and the LP exactly meet the official mathematical definition of incompleteness then G and LP are proven to be isomorphic.

Gödel says:

since the undecidable proposition [R(q); q] states precisely that q belongs to K, i.e. according to (1), that [R(q); q] is not provable. We are therefore confronted with a proposition which asserts its own unprovability.

We are confronted *in our analysis* with such a situation. But there is no proposition which directly asserts such a thing.

You are letting weasel words leak semantic meaning.

My analysis pertaining to the official mathematical definition of incompleteness cuts through these weasel words.

This proves that G the LP and the following sentence are all isomorphic:

"This sentence is unprovable".

The problem is you refuse to look at the actual math and instead look only at the text commentary which is merely a guide to, not the actual substance of, the proof.

André

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

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From: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 14:12:03 -0500

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View all headersFrom: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 14:12:03 -0500

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On 5/3/2022 1:33 PM, Jeff Barnett wrote:

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

I used the above as the precise measure of isomorphism.

The whole error of the incompleteness theorem is entirely anchored in that the official mathematical definition of incompleteness incorrectly presumes that syntactically correct expressions of the language of formal system T are necessarily also semantically incorrect.

When T is the entire body of analytic knowledge and φ is the liar paradox, the official mathematical definition of incompleteness determines that the entire body of analytic knowledge is incomplete.

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

On 5/3/2022 9:18 AM, André G. Isaak wrote:

On 2022-05-02 18:57, olcott wrote:

IDIOT SNIPPED

No. The Liar can be used to construct an *identical* proof. Other antinomies could be used for similar proofs. He's already talking about The Liar.I think that you have provided the troll a similar explanation over 100 times. I admire your tenacity but I must ask you a question: Why do you think one more time will do either him or you any good? It's not like this is an educational or soul-improving opportunity for either of you; for you this must be frustrating no matter your intentions and for him he is what he is and this is his only amusement. Whether troll or buffoon, he's proud of his appearance as wasted space.

You have some serious reading comprehension problems. I never denied the things Gödel wrote. I denied your conclusion because it does not follow.

Gödel starts by claiming there is a close relationship (*not* equivalence) between one particular antinomy, The Liar, and his G.

He then states that similar proofs could be constructed using any antinomy.

That entails that other antinomies could be used to construct similar proofs involving a similar close relation (again, *not* equivalence).

Gödel never claims *any* antinomy is equivalent to his G. Merely that a close relationship holds.

And all my comments concerned exactly what that relationship is.

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

I used the above as the precise measure of isomorphism.

The whole error of the incompleteness theorem is entirely anchored in that the official mathematical definition of incompleteness incorrectly presumes that syntactically correct expressions of the language of formal system T are necessarily also semantically incorrect.

When T is the entire body of analytic knowledge and φ is the liar paradox, the official mathematical definition of incompleteness determines that the entire body of analytic knowledge is incomplete.

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

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From: agis...@gm.invalid (André G. Isaak)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 13:22:12 -0600

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View all headersFrom: agis...@gm.invalid (André G. Isaak)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 13:22:12 -0600

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On 2022-05-03 13:12, olcott wrote:

G is not a formalization of The Liar. There is no such formalization in the systems Gödel considers.

The Liar results in inconsistency whereas G results in incompleteness. This is a major difference between G and The Liar which is why you need to stop conflating the two. The Liar only exists in natural language, not in mathematics.

André

--

To email remove 'invalid' & replace 'gm' with well known Google mail service.

When T is the entire body of analytic knowledge and φ is the liar paradox, the official mathematical definition of incompleteness determines that the entire body of analytic knowledge is incomplete.

G is not a formalization of The Liar. There is no such formalization in the systems Gödel considers.

The Liar results in inconsistency whereas G results in incompleteness. This is a major difference between G and The Liar which is why you need to stop conflating the two. The Liar only exists in natural language, not in mathematics.

André

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From: agis...@gm.invalid (André G. Isaak)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 14:03:47 -0600

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View all headersFrom: agis...@gm.invalid (André G. Isaak)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 14:03:47 -0600

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On 2022-05-03 12:59, olcott wrote:

But calling it a "non-truth bearer" simply because it has not been determined to be true would equally be an error.

And it can be shown (i.e. correctly determined) that G is true, just not within the system for which it was constructed.

As I said, it can be shown in a higher order system.

But even if it could not be, if we can show that G can only be true in cases where it is not provable in T, then there are only two possible conclusions:

Either G is true and unprovable in T, in which case T is incomplete.

Or G is false and provable in T, in which case T is inconsistent.

Which is exactly what Gödel's theorem states: A system can be consistent or it can be complete but it cannot be both. An incomplete system is still useful. An inconsistent system is not. Ergo he phrases this as an consistent system must be incomplete [with the usual caveats about meeting some minimum threshold of expressive power].

That was obviously a type. Sqrt().

But your whole notion of 'semantically correct' is recognized by no one but you and is antithetical to the entire notion of a formal system. And what I describe above is a situation which *very* commonly arises in proofs. They're called proofs by dilemma and take the form:

A → C

B → C

(A ∨ B)

Therefore C

We draw a conclusion from A or B without knowing the truth value of either. This strategy, for example, was used by Euclid in his proof that no largest prime exists.

If you declare A and B to be 'non-truth bearers' simply because you don't know whether they are true or false, then this and many other proofs completely fall apart.

Just not in PA.

This is your confusion, not mine. We can assert that G is not provable and that ¬G is also not provable without asserting anything at all about whether G is true and still conclude that X is incomplete.

LP doesn't meet said definition. LP is a paradox of natural language. There is no 'isomorphism' between G and LP. And if you claim there is, you must state what that isomorphism actually is.

What does 'leak semantic meaning' even mean?

André

--

To email remove 'invalid' & replace 'gm' with well known Google mail service.

On 5/3/2022 1:23 PM, André G. Isaak wrote:

On 2022-05-03 11:33, olcott wrote:

On 5/3/2022 12:17 PM, André G. Isaak wrote:

On 2022-05-03 11:05, olcott wrote:

On 5/3/2022 11:52 AM, André G. Isaak wrote:

<snippage>

G is very clearly a truth-bearer. Go back and reread my original explanation.

When G is not provable in PA, how is it shown to be true?

If it is not shown to be true in PA then we have the strawman error.

Here you're simply begging the question by assuming your own conclusion: that being true and being provable are the same.

This is not any mere assumption.

The only way that any analytic expressions of language are correctly determined to be true is:

'True' and 'Correctly determined to be true' mean different things.

Yes that seems to be correct.

On the other hand calling an expression of language true that has not be 'Correctly determined to be true' is an error.

But calling it a "non-truth bearer" simply because it has not been determined to be true would equally be an error.

And it can be shown (i.e. correctly determined) that G is true, just not within the system for which it was constructed.

If G is claimed to be true then this assertion must be supported by: 'Correctly determined to be true' otherwise the assessment of its truth is no more than a wild guess.

(a) They are defined to be true.

(b) They are derived from applying truth preserving operations to (a) or (b). Prolog knows this on the basis of its facts and rules. Facts are (a) and rules are (b). This is also known as sound deductive inference.

These are YOUR assumptions. They have not been demonstrated. And they are not consistent with the way in which the rest of the world talks about truth. You are talking about provability, not truth.

If G is claimed to be true then this assertion must be supported by: 'Correctly determined to be true' otherwise the assessment of its truth is no more than a wild guess.

When G is not provable in PA, how is it shown to be true (wild guess)?

What is the precise basis for assessing that G is true? please provide ALL the steps.

"True" and "Known to be true" are entirely different things.

Yes, however:

If G is claimed to be true then this assertion must be supported by: 'Correctly determined to be true' otherwise the assessment of its truth is no more than a wild guess.

As I said, it can be shown in a higher order system.

But even if it could not be, if we can show that G can only be true in cases where it is not provable in T, then there are only two possible conclusions:

Either G is true and unprovable in T, in which case T is incomplete.

Or G is false and provable in T, in which case T is inconsistent.

Which is exactly what Gödel's theorem states: A system can be consistent or it can be complete but it cannot be both. An incomplete system is still useful. An inconsistent system is not. Ergo he phrases this as an consistent system must be incomplete [with the usual caveats about meeting some minimum threshold of expressive power].

Consider the equation Srt((2748 + 87)^3) = 150,948.776 (to 3 decimal places)

(2748 + 87)^3 = 22,785,532,875

What are you sorting with Srt()?

That was obviously a type. Sqrt().

That equation is true but it is unlikely anyone knew this to be true until now since I very much doubt anyone had previously considered that specific equation. That's doesn't mean it wasn't true all along. Just that no one knew it was true.

There are all sorts of cases where we know one thing without knowing all things. I can know with certainty that (A ∨ B) is true meaning that I know that *at least* one of A or B must be true while still not knowing the truth value of either. These sorts of things occur all the time.

That is not true. If A and B are syntactically correct expressions of a formal language yet neither one is semantically correct then we have the same case as the Liar Paradox not being a truth bearer, thus (A ∨ B) is neither true nor false.

But your whole notion of 'semantically correct' is recognized by no one but you and is antithetical to the entire notion of a formal system. And what I describe above is a situation which *very* commonly arises in proofs. They're called proofs by dilemma and take the form:

A → C

B → C

(A ∨ B)

Therefore C

We draw a conclusion from A or B without knowing the truth value of either. This strategy, for example, was used by Euclid in his proof that no largest prime exists.

If you declare A and B to be 'non-truth bearers' simply because you don't know whether they are true or false, then this and many other proofs completely fall apart.

And not being provable in PA and not being provable are also two different things.

I know that. Yet not being provable in PA also means that G cannot be derived in PA by applying truth preserving operations to the axioms Z of PA or other expressions Y derived from Z or Y.

If G is correctly determined to be true then there must be some process detailing the steps of how it is 'Correctly determined to be true'.

Just not in PA.

Lacking these steps one cannot correctly assert that G is true, only that G might possibly be true.

You make the claim that The Liar is not a truth bearer (a plausible claim depending on one's definitions).

You then jump to the conclusion that G is not a truth bearer based on your assertion that it is "equivalent" to The Liar. But it is *NOT* equivalent. It merely bears a close relationship. But you refuse to actually consider what the nature of that relationship; there are both similiarites and differences.

Precise equivalence is defined by this:

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

The Liar Paradox can be neither proven nor refuted where T is the entire body of analytic knowledge and φ is LP.

The Liar Paradox is a self-contradiction. The Gödel sentence is not.

Until you understand that G can only be correctly asserted to be true in X if G is provable in X. G may be true in X without a proof in X, yet it cannot be correctly asserted to be true in X without such a proof.

This is your confusion, not mine. We can assert that G is not provable and that ¬G is also not provable without asserting anything at all about whether G is true and still conclude that X is incomplete.

The formulation that G asserts its own unprovability is the Cliff-Notes version of the proof. It's not the substance of the actual proof.

André

It is isomorphic to the substance of the actual proof.

No. It is not. G asserts a claim about mathematics. It does not assert anything about itself. However, within the metalanguage we can prove that G can only be true if it is not provable within the system under consideration.

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

When both G and the LP exactly meet the official mathematical definition of incompleteness then G and LP are proven to be isomorphic.

LP doesn't meet said definition. LP is a paradox of natural language. There is no 'isomorphism' between G and LP. And if you claim there is, you must state what that isomorphism actually is.

Gödel says:

since the undecidable proposition [R(q); q] states precisely that q belongs to K, i.e. according to (1), that [R(q); q] is not provable. We are therefore confronted with a proposition which asserts its own unprovability.

We are confronted *in our analysis* with such a situation. But there is no proposition which directly asserts such a thing.

You are letting weasel words leak semantic meaning.

What does 'leak semantic meaning' even mean?

André

My analysis pertaining to the official mathematical definition of incompleteness cuts through these weasel words.

This proves that G the LP and the following sentence are all isomorphic:

"This sentence is unprovable".

The problem is you refuse to look at the actual math and instead look only at the text commentary which is merely a guide to, not the actual substance of, the proof.

André

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From: jbb...@notatt.com (Jeff Barnett)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 15:58:10 -0600

Organization: A noiseless patient Spider

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View all headersFrom: jbb...@notatt.com (Jeff Barnett)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 15:58:10 -0600

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On 5/3/2022 1:12 PM, olcott wrote:

--

Jeff Barnett

On 5/3/2022 1:33 PM, Jeff Barnett wrote:Polly want a cracker?

On 5/3/2022 9:18 AM, André G. Isaak wrote:

On 2022-05-02 18:57, olcott wrote:

IDIOT SNIPPED

No. The Liar can be used to construct an *identical* proof. OtherI think that you have provided the troll a similar explanation over

antinomies could be used for similar proofs. He's already talking

about The Liar.

Gödel's G. Which means if the basic mechanism of epistemological

antinomy is shown to be semantically incorrect then Gödel's G is

shown to be semantically incorrect.

You have some serious reading comprehension problems. I never denied

the things Gödel wrote. I denied your conclusion because it does not

follow.

Gödel starts by claiming there is a close relationship (*not*

equivalence) between one particular antinomy, The Liar, and his G.

He then states that similar proofs could be constructed using any

antinomy.

That entails that other antinomies could be used to construct similar

proofs involving a similar close relation (again, *not* equivalence).

Gödel never claims *any* antinomy is equivalent to his G. Merely that

a close relationship holds.

And all my comments concerned exactly what that relationship is.

100 times. I admire your tenacity but I must ask you a question: Why

do you think one more time will do either him or you any good? It's

not like this is an educational or soul-improving opportunity for

either of you; for you this must be frustrating no matter your

intentions and for him he is what he is and this is his only

amusement. Whether troll or buffoon, he's proud of his appearance as

wasted space.

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

I used the above as the precise measure of isomorphism.

The whole error of the incompleteness theorem is entirely anchored in

that the official mathematical definition of incompleteness incorrectly

presumes that syntactically correct expressions of the language of

formal system T are necessarily also semantically incorrect.

When T is the entire body of analytic knowledge and φ is the liar

paradox, the official mathematical definition of incompleteness

determines that the entire body of analytic knowledge is incomplete.

--

Jeff Barnett

Path: i2pn2.org!i2pn.org!eternal-september.org!reader02.eternal-september.org!.POSTED!not-for-mail

From: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 17:13:37 -0500

Organization: A noiseless patient Spider

Lines: 65

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View all headersFrom: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 17:13:37 -0500

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On 5/3/2022 4:58 PM, Jeff Barnett wrote:

In other words you have no rebuttal either because what I said is over your head or you want to make sure to never agree that I correctly proved my point. I vote for both.

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

On 5/3/2022 1:12 PM, olcott wrote:

On 5/3/2022 1:33 PM, Jeff Barnett wrote:Polly want a cracker?

On 5/3/2022 9:18 AM, André G. Isaak wrote:

On 2022-05-02 18:57, olcott wrote:

IDIOT SNIPPED

You have some serious reading comprehension problems. I never denied the things Gödel wrote. I denied your conclusion because it does not follow.

Gödel starts by claiming there is a close relationship (*not* equivalence) between one particular antinomy, The Liar, and his G.

He then states that similar proofs could be constructed using any antinomy.

That entails that other antinomies could be used to construct similar proofs involving a similar close relation (again, *not* equivalence).

Gödel never claims *any* antinomy is equivalent to his G. Merely that a close relationship holds.

And all my comments concerned exactly what that relationship is.

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

I used the above as the precise measure of isomorphism.

The whole error of the incompleteness theorem is entirely anchored in that the official mathematical definition of incompleteness incorrectly presumes that syntactically correct expressions of the language of formal system T are necessarily also semantically incorrect.

When T is the entire body of analytic knowledge and φ is the liar paradox, the official mathematical definition of incompleteness determines that the entire body of analytic knowledge is incomplete.

In other words you have no rebuttal either because what I said is over your head or you want to make sure to never agree that I correctly proved my point. I vote for both.

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

Path: i2pn2.org!i2pn.org!eternal-september.org!reader02.eternal-september.org!.POSTED!not-for-mail

From: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 21:53:01 -0500

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View all headersFrom: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 21:53:01 -0500

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On 5/3/2022 2:22 PM, André G. Isaak wrote:

Not according to the mathematical definition of incompleteness

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

Anything and everything that is neither provable nor refutable in some formal system proves that this formal system IS INCOMPLETE as long as φ is an expession of language of T.

That is not true, I think Tarski's way of formalizing the Liar Paradox may be the best. He describes exactly how it would be formalized and provides the exactly syntax for formalizing it. I will derive this and post it sometime soon. I will keep working on this until I am sure that I have it correctly.

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

On 2022-05-03 13:12, olcott wrote:

When T is the entire body of analytic knowledge and φ is the liar paradox, the official mathematical definition of incompleteness determines that the entire body of analytic knowledge is incomplete.

G is not a formalization of The Liar. There is no such formalization in the systems Gödel considers.

The Liar results in inconsistency whereas G results in incompleteness.

Not according to the mathematical definition of incompleteness

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

Anything and everything that is neither provable nor refutable in some formal system proves that this formal system IS INCOMPLETE as long as φ is an expession of language of T.

This is a major difference between G and The Liar which is why you need to stop conflating the two. The Liar only exists in natural language,

That is not true, I think Tarski's way of formalizing the Liar Paradox may be the best. He describes exactly how it would be formalized and provides the exactly syntax for formalizing it. I will derive this and post it sometime soon. I will keep working on this until I am sure that I have it correctly.

not in mathematics.

André

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

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Subject: Re: Is this correct Prolog?

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On 5/3/22 10:53 PM, olcott wrote:

Your missing that he is meaning that if the Liar is TRUE, it shows the logic system to be inconsistent.

While G being true just proves that the system is Incomplete.

(G not being False in a system shows that the system is Inconsistent, as it says we can prove a false statement in the system)

On 5/3/2022 2:22 PM, André G. Isaak wrote:

On 2022-05-03 13:12, olcott wrote:

When T is the entire body of analytic knowledge and φ is the liar paradox, the official mathematical definition of incompleteness determines that the entire body of analytic knowledge is incomplete.

G is not a formalization of The Liar. There is no such formalization in the systems Gödel considers.

The Liar results in inconsistency whereas G results in incompleteness.

Not according to the mathematical definition of incompleteness

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

Anything and everything that is neither provable nor refutable in some formal system proves that this formal system IS INCOMPLETE as long as φ is an expession of language of T.

Your missing that he is meaning that if the Liar is TRUE, it shows the logic system to be inconsistent.

While G being true just proves that the system is Incomplete.

(G not being False in a system shows that the system is Inconsistent, as it says we can prove a false statement in the system)

This is a major difference between G and The Liar which is why you need to stop conflating the two. The Liar only exists in natural language,

That is not true, I think Tarski's way of formalizing the Liar Paradox may be the best. He describes exactly how it would be formalized and provides the exactly syntax for formalizing it. I will derive this and post it sometime soon. I will keep working on this until I am sure that I have it correctly.

not in mathematics.

André

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From: polco...@gmail.com (olcott)

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Subject: Re: Is this correct Prolog?

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View all headersFrom: polco...@gmail.com (olcott)

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Subject: Re: Is this correct Prolog?

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On 5/3/2022 3:03 PM, André G. Isaak wrote:

That if correct. If is is impossibly true or false then it is not a truth bearer.

unprovable in the system entails untrue in the system.

Sure, yet it is only true in this system which also makes it provable in that system.

It can't be true in T and unprovable in T.

If G is provable in U then it is true in U.

If G is unprovable in T then it is untrue in T.

G is unprovable in T because it is semantically incorrect.

It is not because they have unknown truth values.

OK great this is great progress!

Yes, but, only because the mathematical definition of Incomplete does not screen out semantically erroneous expression of the language of X:

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

It is a little iffy to say that the LP fits into the above formula making it isomorphic to Gödel's G unless we can at least specify the formal system that LP is a member of. Tarski seemed to have provided a very clean basis to formalize the Liar Paradox. I am working on deriving it.

You have to pay attention to this

since the undecidable proposition [R(q); q] states precisely that q

belongs to K, i.e. according to (1), that [R(q); q] is not provable.

It is the English language version of a formal logic sentence.

You were simply looking at the wrong lines of the Gödel quote.

On 2022-05-03 12:59, olcott wrote:

On 5/3/2022 1:23 PM, André G. Isaak wrote:

On 2022-05-03 11:33, olcott wrote:

On 2022-05-03 11:05, olcott wrote:

On 5/3/2022 11:52 AM, André G. Isaak wrote:

<snippage>

G is very clearly a truth-bearer. Go back and reread my original explanation.

When G is not provable in PA, how is it shown to be true?

If it is not shown to be true in PA then we have the strawman error.

Here you're simply begging the question by assuming your own conclusion: that being true and being provable are the same.

This is not any mere assumption.

The only way that any analytic expressions of language are correctly determined to be true is:

'True' and 'Correctly determined to be true' mean different things.

Yes that seems to be correct.

On the other hand calling an expression of language true that has not be 'Correctly determined to be true' is an error.

But calling it a "non-truth bearer" simply because it has not been determined to be true would equally be an error.

That if correct. If is is impossibly true or false then it is not a truth bearer.

And it can be shown (i.e. correctly determined) that G is true, just not within the system for which it was constructed.

unprovable in the system entails untrue in the system.

If G is claimed to be true then this assertion must be supported by: 'Correctly determined to be true' otherwise the assessment of its truth is no more than a wild guess.

(b) They are derived from applying truth preserving operations to (a) or (b). Prolog knows this on the basis of its facts and rules. Facts are (a) and rules are (b). This is also known as sound deductive inference.

These are YOUR assumptions. They have not been demonstrated. And they are not consistent with the way in which the rest of the world talks about truth. You are talking about provability, not truth.

If G is claimed to be true then this assertion must be supported by: 'Correctly determined to be true' otherwise the assessment of its truth is no more than a wild guess.

When G is not provable in PA, how is it shown to be true (wild guess)?

What is the precise basis for assessing that G is true? please provide ALL the steps.

"True" and "Known to be true" are entirely different things.

Yes, however:

If G is claimed to be true then this assertion must be supported by: 'Correctly determined to be true' otherwise the assessment of its truth is no more than a wild guess.

As I said, it can be shown in a higher order system.

Sure, yet it is only true in this system which also makes it provable in that system.

But even if it could not be, if we can show that G can only be true in cases where it is not provable in T, then there are only two possible conclusions:

It can't be true in T and unprovable in T.

Either G is true and unprovable in T, in which case T is incomplete.

Or G is false and provable in T, in which case T is inconsistent.

If G is provable in U then it is true in U.

If G is unprovable in T then it is untrue in T.

G is unprovable in T because it is semantically incorrect.

Which is exactly what Gödel's theorem states: A system can be consistent or it can be complete but it cannot be both. An incomplete system is still useful. An inconsistent system is not. Ergo he phrases this as an consistent system must be incomplete [with the usual caveats about meeting some minimum threshold of expressive power].

Consider the equation Srt((2748 + 87)^3) = 150,948.776 (to 3 decimal places)

(2748 + 87)^3 = 22,785,532,875

What are you sorting with Srt()?

That was obviously a type. Sqrt().

That equation is true but it is unlikely anyone knew this to be true until now since I very much doubt anyone had previously considered that specific equation. That's doesn't mean it wasn't true all along. Just that no one knew it was true.

There are all sorts of cases where we know one thing without knowing all things. I can know with certainty that (A ∨ B) is true meaning that I know that *at least* one of A or B must be true while still not knowing the truth value of either. These sorts of things occur all the time.

That is not true. If A and B are syntactically correct expressions of a formal language yet neither one is semantically correct then we have the same case as the Liar Paradox not being a truth bearer, thus (A ∨ B) is neither true nor false.

But your whole notion of 'semantically correct' is recognized by no one but you and is antithetical to the entire notion of a formal system. And what I describe above is a situation which *very* commonly arises in proofs. They're called proofs by dilemma and take the form:

A → C

B → C

(A ∨ B)

Therefore C

We draw a conclusion from A or B without knowing the truth value of either. This strategy, for example, was used by Euclid in his proof that no largest prime exists.

If you declare A and B to be 'non-truth bearers' simply because you don't know whether they are true or false, then this and many other proofs completely fall apart.

It is not because they have unknown truth values.

And not being provable in PA and not being provable are also two different things.

I know that. Yet not being provable in PA also means that G cannot be derived in PA by applying truth preserving operations to the axioms Z of PA or other expressions Y derived from Z or Y.

If G is correctly determined to be true then there must be some process detailing the steps of how it is 'Correctly determined to be true'.

Just not in PA.

OK great this is great progress!

Lacking these steps one cannot correctly assert that G is true, only that G might possibly be true.

You make the claim that The Liar is not a truth bearer (a plausible claim depending on one's definitions).

You then jump to the conclusion that G is not a truth bearer based on your assertion that it is "equivalent" to The Liar. But it is *NOT* equivalent. It merely bears a close relationship. But you refuse to actually consider what the nature of that relationship; there are both similiarites and differences.

Precise equivalence is defined by this:

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

The Liar Paradox can be neither proven nor refuted where T is the entire body of analytic knowledge and φ is LP.

The Liar Paradox is a self-contradiction. The Gödel sentence is not.

Until you understand that G can only be correctly asserted to be true in X if G is provable in X. G may be true in X without a proof in X, yet it cannot be correctly asserted to be true in X without such a proof.

This is your confusion, not mine. We can assert that G is not provable and that ¬G is also not provable without asserting anything at all about whether G is true and still conclude that X is incomplete.

Yes, but, only because the mathematical definition of Incomplete does not screen out semantically erroneous expression of the language of X:

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

The formulation that G asserts its own unprovability is the Cliff-Notes version of the proof. It's not the substance of the actual proof.

André

It is isomorphic to the substance of the actual proof.

No. It is not. G asserts a claim about mathematics. It does not assert anything about itself. However, within the metalanguage we can prove that G can only be true if it is not provable within the system under consideration.

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

When both G and the LP exactly meet the official mathematical definition of incompleteness then G and LP are proven to be isomorphic.

LP doesn't meet said definition. LP is a paradox of natural language. There is no 'isomorphism' between G and LP. And if you claim there is, you must state what that isomorphism actually is.

It is a little iffy to say that the LP fits into the above formula making it isomorphic to Gödel's G unless we can at least specify the formal system that LP is a member of. Tarski seemed to have provided a very clean basis to formalize the Liar Paradox. I am working on deriving it.

since the undecidable proposition [R(q); q] states precisely that q belongs to K, i.e. according to (1), that [R(q); q] is not provable. We are therefore confronted with a proposition which asserts its own unprovability.

We are confronted *in our analysis* with such a situation. But there is no proposition which directly asserts such a thing.

You have to pay attention to this

since the undecidable proposition [R(q); q] states precisely that q

belongs to K, i.e. according to (1), that [R(q); q] is not provable.

It is the English language version of a formal logic sentence.

You are letting weasel words leak semantic meaning.

What does 'leak semantic meaning' even mean?

You were simply looking at the wrong lines of the Gödel quote.

André

My analysis pertaining to the official mathematical definition of incompleteness cuts through these weasel words.

Click here to read the complete article

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From: polco...@gmail.com (olcott)

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Subject: Re: Is this correct Prolog?

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View all headersFrom: polco...@gmail.com (olcott)

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Subject: Re: Is this correct Prolog?

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On 5/3/2022 10:12 PM, Richard Damon wrote:

No the LP is semantically incorrect.

Someone here (I think its Andre) now seems to understand that if G is not provable in F then it is not true in F. If G is true it can only be true in an other different formal system than F. Tarski says this as theory and meta-theory.

G is semantically incorrect thus logically incoherent.

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

On 5/3/22 10:53 PM, olcott wrote:

On 5/3/2022 2:22 PM, André G. Isaak wrote:

On 2022-05-03 13:12, olcott wrote:

G is not a formalization of The Liar. There is no such formalization in the systems Gödel considers.

The Liar results in inconsistency whereas G results in incompleteness.

Not according to the mathematical definition of incompleteness

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

Anything and everything that is neither provable nor refutable in some formal system proves that this formal system IS INCOMPLETE as long as φ is an expession of language of T.

Your missing that he is meaning that if the Liar is TRUE, it shows the logic system to be inconsistent.

No the LP is semantically incorrect.

While G being true just proves that the system is Incomplete.

Someone here (I think its Andre) now seems to understand that if G is not provable in F then it is not true in F. If G is true it can only be true in an other different formal system than F. Tarski says this as theory and meta-theory.

(G not being False in a system shows that the system is Inconsistent, as it says we can prove a false statement in the system)

G is semantically incorrect thus logically incoherent.

This is a major difference between G and The Liar which is why you need to stop conflating the two. The Liar only exists in natural language,

That is not true, I think Tarski's way of formalizing the Liar Paradox may be the best. He describes exactly how it would be formalized and provides the exactly syntax for formalizing it. I will derive this and post it sometime soon. I will keep working on this until I am sure that I have it correctly.

not in mathematics.

André

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

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From: agis...@gm.invalid (André G. Isaak)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 21:54:53 -0600

Organization: Christians and Atheists United Against Creeping Agnosticism

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View all headersFrom: agis...@gm.invalid (André G. Isaak)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 21:54:53 -0600

Organization: Christians and Atheists United Against Creeping Agnosticism

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On 2022-05-03 21:24, olcott wrote:

Neither 'impossibly true' nor 'impossibly false' are meaningful English. I really wish you would stop using this adverb as if it somehow made sense. It doesn't.

G is definitely a truth bearer. It states that a specific polynomial equation has at least one integer solution. Any statement of that form is either true or false. It is not possible for it to be both. It is not possible for it to be neither.

Maybe you should read the actual proof. We proof in some higher order system T' that G is unprovable in T and that G is true in T.

Again, try reading the actual proof. I mean the math part, not the text part.

Yes. It can. That is the entire point of the proof which you have not actually read. The problem is that you seem to have a personal definition of 'true' which essentially is the same as 'provable'. Yes, it can't be both provable in T and unprovable in T, but that's a truism. Not Gödel's theorem.

No one besides you recognizes this whole 'semantic incorrectness' notion. Until you can get people to accept that there is such a thing, there is no point bringing it up.

So then what is it? In the case of The Liar I am willing to accept your claim that it is not a truth-bearer on the grounds that The Liar effectively has no actual content apart from its own self-referential claim. That's about the only case I can see it being legitimate to refer to a declarative sentence as a non-truth bearer (under standard definitions, truth-bearer and declarative sentence are simply synonyms).

Gödel's G is not like that. It is not self-referential and it does have actual content.

There are no such semantically erroneous expressions. Claiming that they are 'erroneous' doesn't actually demonstrate an error. It's just an empty assertion on your part.

On 5/3/2022 3:03 PM, André G. Isaak wrote:

On 2022-05-03 12:59, olcott wrote:

On 5/3/2022 1:23 PM, André G. Isaak wrote:

On 2022-05-03 11:05, olcott wrote:

On 5/3/2022 11:52 AM, André G. Isaak wrote:

<snippage>

G is very clearly a truth-bearer. Go back and reread my original explanation.

When G is not provable in PA, how is it shown to be true?

If it is not shown to be true in PA then we have the strawman error.

Here you're simply begging the question by assuming your own conclusion: that being true and being provable are the same.

This is not any mere assumption.

The only way that any analytic expressions of language are correctly determined to be true is:

'True' and 'Correctly determined to be true' mean different things.

Yes that seems to be correct.

On the other hand calling an expression of language true that has not be 'Correctly determined to be true' is an error.

But calling it a "non-truth bearer" simply because it has not been determined to be true would equally be an error.

That if correct. If is is impossibly true or false then it is not a truth bearer.

Neither 'impossibly true' nor 'impossibly false' are meaningful English. I really wish you would stop using this adverb as if it somehow made sense. It doesn't.

G is definitely a truth bearer. It states that a specific polynomial equation has at least one integer solution. Any statement of that form is either true or false. It is not possible for it to be both. It is not possible for it to be neither.

And it can be shown (i.e. correctly determined) that G is true, just not within the system for which it was constructed.

unprovable in the system entails untrue in the system.

Maybe you should read the actual proof. We proof in some higher order system T' that G is unprovable in T and that G is true in T.

If G is claimed to be true then this assertion must be supported by: 'Correctly determined to be true' otherwise the assessment of its truth is no more than a wild guess.

(b) They are derived from applying truth preserving operations to (a) or (b). Prolog knows this on the basis of its facts and rules. Facts are (a) and rules are (b). This is also known as sound deductive inference.

These are YOUR assumptions. They have not been demonstrated. And they are not consistent with the way in which the rest of the world talks about truth. You are talking about provability, not truth.

If G is claimed to be true then this assertion must be supported by: 'Correctly determined to be true' otherwise the assessment of its truth is no more than a wild guess.

When G is not provable in PA, how is it shown to be true (wild guess)?

What is the precise basis for assessing that G is true? please provide ALL the steps.

"True" and "Known to be true" are entirely different things.

Yes, however:

If G is claimed to be true then this assertion must be supported by: 'Correctly determined to be true' otherwise the assessment of its truth is no more than a wild guess.

As I said, it can be shown in a higher order system.

Sure, yet it is only true in this system which also makes it provable in that system.

Again, try reading the actual proof. I mean the math part, not the text part.

But even if it could not be, if we can show that G can only be true in cases where it is not provable in T, then there are only two possible conclusions:

It can't be true in T and unprovable in T.

Yes. It can. That is the entire point of the proof which you have not actually read. The problem is that you seem to have a personal definition of 'true' which essentially is the same as 'provable'. Yes, it can't be both provable in T and unprovable in T, but that's a truism. Not Gödel's theorem.

Either G is true and unprovable in T, in which case T is incomplete.

Or G is false and provable in T, in which case T is inconsistent.

If G is provable in U then it is true in U.

If G is unprovable in T then it is untrue in T.

G is unprovable in T because it is semantically incorrect.

No one besides you recognizes this whole 'semantic incorrectness' notion. Until you can get people to accept that there is such a thing, there is no point bringing it up.

Which is exactly what Gödel's theorem states: A system can be consistent or it can be complete but it cannot be both. An incomplete system is still useful. An inconsistent system is not. Ergo he phrases this as an consistent system must be incomplete [with the usual caveats about meeting some minimum threshold of expressive power].

Consider the equation Srt((2748 + 87)^3) = 150,948.776 (to 3 decimal places)

(2748 + 87)^3 = 22,785,532,875

What are you sorting with Srt()?

That was obviously a type. Sqrt().

There are all sorts of cases where we know one thing without knowing all things. I can know with certainty that (A ∨ B) is true meaning that I know that *at least* one of A or B must be true while still not knowing the truth value of either. These sorts of things occur all the time.

That is not true. If A and B are syntactically correct expressions of a formal language yet neither one is semantically correct then we have the same case as the Liar Paradox not being a truth bearer, thus (A ∨ B) is neither true nor false.

But your whole notion of 'semantically correct' is recognized by no one but you and is antithetical to the entire notion of a formal system. And what I describe above is a situation which *very* commonly arises in proofs. They're called proofs by dilemma and take the form:

A → C

B → C

(A ∨ B)

Therefore C

We draw a conclusion from A or B without knowing the truth value of either. This strategy, for example, was used by Euclid in his proof that no largest prime exists.

If you declare A and B to be 'non-truth bearers' simply because you don't know whether they are true or false, then this and many other proofs completely fall apart.

It is not because they have unknown truth values.

So then what is it? In the case of The Liar I am willing to accept your claim that it is not a truth-bearer on the grounds that The Liar effectively has no actual content apart from its own self-referential claim. That's about the only case I can see it being legitimate to refer to a declarative sentence as a non-truth bearer (under standard definitions, truth-bearer and declarative sentence are simply synonyms).

Gödel's G is not like that. It is not self-referential and it does have actual content.

And not being provable in PA and not being provable are also two different things.

I know that. Yet not being provable in PA also means that G cannot be derived in PA by applying truth preserving operations to the axioms Z of PA or other expressions Y derived from Z or Y.

If G is correctly determined to be true then there must be some process detailing the steps of how it is 'Correctly determined to be true'.

Just not in PA.

OK great this is great progress!

Lacking these steps one cannot correctly assert that G is true, only that G might possibly be true.

You make the claim that The Liar is not a truth bearer (a plausible claim depending on one's definitions).

You then jump to the conclusion that G is not a truth bearer based on your assertion that it is "equivalent" to The Liar. But it is *NOT* equivalent. It merely bears a close relationship. But you refuse to actually consider what the nature of that relationship; there are both similiarites and differences.

Precise equivalence is defined by this:

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

The Liar Paradox can be neither proven nor refuted where T is the entire body of analytic knowledge and φ is LP.

The Liar Paradox is a self-contradiction. The Gödel sentence is not.

Until you understand that G can only be correctly asserted to be true in X if G is provable in X. G may be true in X without a proof in X, yet it cannot be correctly asserted to be true in X without such a proof.

This is your confusion, not mine. We can assert that G is not provable and that ¬G is also not provable without asserting anything at all about whether G is true and still conclude that X is incomplete.

Yes, but, only because the mathematical definition of Incomplete does not screen out semantically erroneous expression of the language of X:

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

There are no such semantically erroneous expressions. Claiming that they are 'erroneous' doesn't actually demonstrate an error. It's just an empty assertion on your part.

Whereas the Liar has no content other than to assert its own falsity, Gödel's G has definite content. It does not assert its own unprovability, it asserts a very specific mathematical claim, one which must by its nature be either true or false. Therefore G *is* a truth bearer.

The formulation that G asserts its own unprovability is the Cliff-Notes version of the proof. It's not the substance of the actual proof.

Click here to read the complete article

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From: agis...@gm.invalid (André G. Isaak)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 22:06:43 -0600

Organization: Christians and Atheists United Against Creeping Agnosticism

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View all headersFrom: agis...@gm.invalid (André G. Isaak)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Tue, 3 May 2022 22:06:43 -0600

Organization: Christians and Atheists United Against Creeping Agnosticism

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On 2022-05-03 21:12, Richard Damon wrote:

Actually, it doesn't matter whether it is true or false.

What's relevant here is that it is trivially easy to prove The Liar using a simple reductio ad absurdum. Unfortunately, it is equally trivially easy to prove ¬(The Liar) using a reductio of the same form.

That's very different from G, where no proof exists of either G or of ¬G.

The Liar leads to inconsistency.

The sentence 'This sentence is true' would be a natural language example which leads to incompleteness.

André

--

To email remove 'invalid' & replace 'gm' with well known Google mail service.

On 5/3/22 10:53 PM, olcott wrote:

On 5/3/2022 2:22 PM, André G. Isaak wrote:

G is not a formalization of The Liar. There is no such formalization in the systems Gödel considers.

The Liar results in inconsistency whereas G results in incompleteness.

Not according to the mathematical definition of incompleteness

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

Anything and everything that is neither provable nor refutable in some formal system proves that this formal system IS INCOMPLETE as long as φ is an expession of language of T.

Your missing that he is meaning that if the Liar is TRUE, it shows the logic system to be inconsistent.

Actually, it doesn't matter whether it is true or false.

What's relevant here is that it is trivially easy to prove The Liar using a simple reductio ad absurdum. Unfortunately, it is equally trivially easy to prove ¬(The Liar) using a reductio of the same form.

That's very different from G, where no proof exists of either G or of ¬G.

The Liar leads to inconsistency.

The sentence 'This sentence is true' would be a natural language example which leads to incompleteness.

André

--

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From: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Wed, 4 May 2022 01:17:03 -0500

Organization: A noiseless patient Spider

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View all headersFrom: polco...@gmail.com (olcott)

Newsgroups: comp.theory,comp.lang.prolog,comp.ai.philosophy

Subject: Re: Is this correct Prolog?

Date: Wed, 4 May 2022 01:17:03 -0500

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On 5/3/2022 11:06 PM, André G. Isaak wrote:

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

The Liar Paradox plugs right in to this

making it equivalent to Gödel's G.

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

On 2022-05-03 21:12, Richard Damon wrote:

On 5/3/22 10:53 PM, olcott wrote:

G is not a formalization of The Liar. There is no such formalization in the systems Gödel considers.

The Liar results in inconsistency whereas G results in incompleteness.

Not according to the mathematical definition of incompleteness

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

Anything and everything that is neither provable nor refutable in some formal system proves that this formal system IS INCOMPLETE as long as φ is an expession of language of T.

Your missing that he is meaning that if the Liar is TRUE, it shows the logic system to be inconsistent.

Actually, it doesn't matter whether it is true or false.

What's relevant here is that it is trivially easy to prove The Liar using a simple reductio ad absurdum. Unfortunately, it is equally trivially easy to prove ¬(The Liar) using a reductio of the same form.

Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)).

The Liar Paradox plugs right in to this

making it equivalent to Gödel's G.

That's very different from G, where no proof exists of either G or of ¬G.

The Liar leads to inconsistency.

The sentence 'This sentence is true' would be a natural language example which leads to incompleteness.

André

--

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;

Genius hits a target no one else can see." Arthur Schopenhauer

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