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INCOMPLETE INCOMPLETENESS THEOREM

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Subject: INCOMPLETE INCOMPLETENESS THEOREM
From: grahamco...@gmail.com (Graham Cooper)
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by: Graham Cooper - Sun, 26 Feb 2023 13:32 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]

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

!proof(g) ...................... BY INCOMPLETENESS THEOREM
proof(!proof(g)) ............... BY INCOMPLETENESS THEOREM
g<->!proof(g) .................. DEFINITION
g<->!proof( !proof(g) ) ........ SUBSTITION OF g

proof( !proof(g) )
^ !proof( !proof(g) )

Re: INCOMPLETE INCOMPLETENESS THEOREM

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Subject: Re: INCOMPLETE INCOMPLETENESS THEOREM
From: techfan5...@hotmail.com (Y A)
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by: Y A - Sun, 26 Feb 2023 20:23 UTC

Heyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy.....
Aitaksid mind rahaga ?

Telefoni number: groups.google.com/g/soc.culture.usa/c/QY0AWsnFpcQ

On Sunday, February 26, 2023 at 3:32:48 PM UTC+2, 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]
>
> ---------------------------------------------------------------------
>
> !proof(g) ...................... BY INCOMPLETENESS THEOREM
> proof(!proof(g)) ............... BY INCOMPLETENESS THEOREM
> g<->!proof(g) .................. DEFINITION
> g<->!proof( !proof(g) ) ........ SUBSTITION OF g
>
> proof( !proof(g) )
> ^
> !proof( !proof(g) )

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tech / sci.math / INCOMPLETE INCOMPLETENESS THEOREM

Subject	Author
INCOMPLETE INCOMPLETENESS THEOREM	Graham Cooper
Re: INCOMPLETE INCOMPLETENESS THEOREM	Y A