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devel / comp.theory / Godel EXTENDED Proof

Godel EXTENDED Proof

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Subject: Godel EXTENDED Proof
From: grahamco...@gmail.com (Graham Cooper)
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by: Graham Cooper - Tue, 7 Mar 2023 21:54 UTC

GODELS PROOF
The Incompleteness Theorem

g = NOT(EXIST(p) PROOF(p g))
ASSUME g<->FALSE ...[1]
EXIST(p) PROOF(p g)
g<->TRUE ...[2]
CONTRADICTION [1]<-/->[2]
g<->TRUE
NOT(EXIST(p) PROOF(p g))

NOT(EXIST(proof of TRUE statement))

CONCLUSION: There exists true mathematical statements that have no proof

---------------------------------------------------------------------

EXTENDED GODELS PROOF

(1) g = NOT(EXIST(p) PROOF(p g))

(2) ASSUME g<->FALSE
(3) EXIST(p) PROOF(p g) .....[2-->1]
(4) g<->TRUE .....[BY 3]
(5) CONTRADICTION .....[2]<-/->[4]

(6) ASSUME g<->TRUE
(7) EXIST(P) PROOF(G) .....[BY 6]
(8) NOT(EXIST(p) PROOF(p g)) .....[6-->1]
(9) CONTRADICTION .....[7]<-/->[8]

g IS NEITHER TRUE NOR FALSE
g HAS BOTH TRUTH VALUES
g IS SELF-CONTRADICTORY

(10) g<->FALSE .....[BY SPECIFICATION OF THEOREM PROVER]

---------------------------------------------------------------------

Re: Godel EXTENDED Proof

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Subject: Re: Godel EXTENDED Proof
From: jeffreyd...@gmail.com (Jeffrey Rubard)
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by: Jeffrey Rubard - Tue, 7 Mar 2023 23:46 UTC

On Tuesday, March 7, 2023 at 1:54:40 PM UTC-8, Graham Cooper wrote:
> GODELS PROOF
> The Incompleteness Theorem
>
> g = NOT(EXIST(p) PROOF(p g))
> ASSUME g<->FALSE ...[1]
> EXIST(p) PROOF(p g)
> g<->TRUE ...[2]
> CONTRADICTION [1]<-/->[2]
> g<->TRUE
> NOT(EXIST(p) PROOF(p g))
>
> NOT(EXIST(proof of TRUE statement))
>
> CONCLUSION: There exists true mathematical statements that have no proof
>
>
> ---------------------------------------------------------------------
>
> EXTENDED GODELS PROOF
>
> (1) g = NOT(EXIST(p) PROOF(p g))
>
> (2) ASSUME g<->FALSE
> (3) EXIST(p) PROOF(p g) .....[2-->1]
> (4) g<->TRUE .....[BY 3]
> (5) CONTRADICTION .....[2]<-/->[4]
>
> (6) ASSUME g<->TRUE
> (7) EXIST(P) PROOF(G) .....[BY 6]
> (8) NOT(EXIST(p) PROOF(p g)) .....[6-->1]
> (9) CONTRADICTION .....[7]<-/->[8]
>
> g IS NEITHER TRUE NOR FALSE
> g HAS BOTH TRUTH VALUES
> g IS SELF-CONTRADICTORY
>
> (10) g<->FALSE .....[BY SPECIFICATION OF THEOREM PROVER]
>
> ---------------------------------------------------------------------

"I think this is really not a good enough proof of the theorem."

Re: Godel EXTENDED Proof

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Subject: Re: Godel EXTENDED Proof
From: jeffreyd...@gmail.com (Jeffrey Rubard)
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by: Jeffrey Rubard - Thu, 9 Mar 2023 22:40 UTC

On Tuesday, March 7, 2023 at 3:46:59 PM UTC-8, Jeffrey Rubard wrote:
> On Tuesday, March 7, 2023 at 1:54:40 PM UTC-8, Graham Cooper wrote:
> > GODELS PROOF
> > The Incompleteness Theorem
> >
> > g = NOT(EXIST(p) PROOF(p g))
> > ASSUME g<->FALSE ...[1]
> > EXIST(p) PROOF(p g)
> > g<->TRUE ...[2]
> > CONTRADICTION [1]<-/->[2]
> > g<->TRUE
> > NOT(EXIST(p) PROOF(p g))
> >
> > NOT(EXIST(proof of TRUE statement))
> >
> > CONCLUSION: There exists true mathematical statements that have no proof
> >
> >
> > ---------------------------------------------------------------------
> >
> > EXTENDED GODELS PROOF
> >
> > (1) g = NOT(EXIST(p) PROOF(p g))
> >
> > (2) ASSUME g<->FALSE
> > (3) EXIST(p) PROOF(p g) .....[2-->1]
> > (4) g<->TRUE .....[BY 3]
> > (5) CONTRADICTION .....[2]<-/->[4]
> >
> > (6) ASSUME g<->TRUE
> > (7) EXIST(P) PROOF(G) .....[BY 6]
> > (8) NOT(EXIST(p) PROOF(p g)) .....[6-->1]
> > (9) CONTRADICTION .....[7]<-/->[8]
> >
> > g IS NEITHER TRUE NOR FALSE
> > g HAS BOTH TRUTH VALUES
> > g IS SELF-CONTRADICTORY
> >
> > (10) g<->FALSE .....[BY SPECIFICATION OF THEOREM PROVER]
> >
> > ---------------------------------------------------------------------
> "I think this is really not a good enough proof of the theorem."

"That's the point."
"The point is that you don't know how to do a mathematical proof?"

Subject	Author
Godel EXTENDED Proof	Graham Cooper
Godel EXTENDED Proof	Jeffrey Rubard
Godel EXTENDED Proof	Jeffrey Rubard