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

CONCLUSION: There exists true mathematical statements that have no proof

ABSTRACT

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

G states there does not exist a proof of G

The fact it was 1st encoded in archaic godel numbering nonsense
doesn't mean you cant represent G in a modern logic language

G is SELF REFERENTIAL

g is a statement about g
g is a statement about (a statement about g)

is where the incompleteness theorem ends

HOWEVER

g is a statement about (a statement about (a statement about g))
etc
ALSO give alternating TRUE FALSE values to g

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

SIMILAR TO

X = !X

X=TRUE
X=!X
X=FALSE
X=!X
X=TRUE
X=!X
X=FALSE
....

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

By standard analysis if

NOT(EXIST(p) PROOF(x))
THEN x IS FALSE

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

Godels Proof states

Since g is FALSE there is no proof of g
NOT(EXIST(p) PROOF(p g)) = TRUE

=============================

and that's where 21C LOGIC ENDS!

but it doesnt end there

What Godel actually proved

REMEMBER

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

so
NOT(EXIST(p) PROOF(p g))
==>

g
= NOT(EXIST(p) PROOF(p g)) ..........*FALSE*
= NOT(EXIST(p) PROOF(p NOT(EXIST(p) PROOF(p g)))) ..........*TRUE*

THAT IS THE INCOMPLETENESS THEOREM

====================================

BUT IT DOESNT STOP THERE

g
= NOT(EXIST(p) PROOF(p g)) ..........*FALSE*
= NOT(EXIST(p) PROOF(p NOT(EXIST(p) PROOF(p g)))) ..........*TRUE*
= NOT(EXIST(p) PROOF(p NOT(EXIST(p) PROOF(p NOT(EXIST(p) PROOF(p g)))))) ....*FALSE*

WHATS GOING ON HERE?

g = there is no proof of g
it = there is no proof of (there is no proof of g)
x = there is no proof of (there is no proof of (there is no proof of g))

THERE IS NO PROOF OF THE INCOMPLETENESS THEOREM !!!

HOW BIZARRE. Luckily X=FALSE

It cycles on and on.

There is no proof that there is no proof of the incompleteness theorem
is TRUE!

but they are all equal to g

ALL MODELS MUST GIVE [TRUE] TO ANY FORMULA
This makes the THEORY INCONSISTENT similarly to X=!X

> "This sentence cannot be proven"
> "This sentence is not true"
> are both self contradictory.
>
> *The whole metatheory nonsense is simply this*
> This sentence cannot be proven: "This sentence cannot be proven"
> and
> This sentence is not true: "This sentence is not true"
>
> The outer sentence escapes the self-contradiction
> allowing the outer sentence to be true.

So what is the SOLUTION?
g is an invalid formula just like X=!X so it is also "STRATIFIED" out of the THEORY

Can we state

ALL(logic) EXIST(p) PROOF(p logic) <-> TRUE(logic)

Not yet, as g is neither FALSE nor TRUE
Logicians argue that if g was FALSE then it has no proof which is what it states
Hence the term INCOMPLETNESS.

A LOGIC SOLVER with input

> g <-> !proof(g)
> g
[ENTER]

would return FALSE
but you would THINK *there is no proof of g*
g <-> !proof(g)
so g is RIGHT!

The only solution for a complete logic solver is TERNARY LOGIC

logicForumula(xy..) ==> {TF?}

g has both FALSE and TRUE values so the answer is ?

www.NEW-MATH.com

On 2/19/23 7:43 PM, 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
>
> CONCLUSION: There exists true mathematical statements that have no proof

Close, you are ignoring what is done in the theory, and what is done in
the meta-theory.

Note also, the statement in the meta-theory isn't that g is not
provable, but that g is not provable in the THEORY.

>
>
>
> ABSTRACT
>
> g = NOT(EXIST(p) PROOF(p g))

Which ISN'T the statement in the theory, only the "meaning" that can be
proved from g in the meta-theory, and the the statement is about the
lack of a proof of g in the THEORY.

>
> G states there does not exist a proof of G

Nope, from g, in the meta-theory, we can derive that if g is true there
is no proof of g in the theory, and if g is false, there IS a proof it
is true in the theory.

>
> The fact it was 1st encoded in archaic godel numbering nonsense
> doesn't mean you cant represent G in a modern logic language

WRONG. Because you aren't "translating" it correctly.

The correct interpretation (in the meta-theory) is that G can not be
proven IN THE THEORY.

>
> G is SELF REFERENTIAL

Nope, NOTHING in the statement of g actually refers to itself. It is
only additional knowledge in the meta-theory that allows us to DERIVE
information that G imputes about itself.

>
> g is a statement about g

A LIE.

> g is a statement about (a statement about g)

A LIE

>
> is where the incompleteness theorem ends
>
> HOWEVER
>
> g is a statement about (a statement about (a statement about g))
> etc
> ALSO give alternating TRUE FALSE values to g

MORE LIES.

>
> ---------------------------------------------------------------------
>
>
> SIMILAR TO
>
> X = !X

NOPE.

And an interesting point, your arguement about tri-state logic falls
into that same issue, you are asserint that not ( e?) == e? so you end
up with the contradiction you are talking about, in actual fact.

>
> X=TRUE
> X=!X
> X=FALSE
> X=!X
> X=TRUE
> X=!X
> X=FALSE
> ...
>
>
> g = NOT(EXIST(p) PROOF(p g))

A LIE of a statement, since that is not G in the Theory.

>
> By standard analysis if
>
> NOT(EXIST(p) PROOF(x))
> THEN x IS FALSE

Which is a false analysis.

THe assumption that something not provable is false is not supportable
in anything that supports the math that Godel uses, as his proof shows.

>
> NOT(EXIST(p) PROOF(p g)) = FALSE
>
> Godels Proof states
>
> Since g is FALSE there is no proof of g

No, Godel proof show that g is TRUE as g is the statment that no such
number exist that meets the primative recursive relationship derived
from the "Proof Checker in the theory" algorithm for the statement g as
derieved in the meta-theory.

> NOT(EXIST(p) PROOF(p g)) = TRUE
>
> =============================
>
> and that's where 21C LOGIC ENDS!
>
> but it doesnt end there
>
> What Godel actually proved
>
> REMEMBER
>
> g = NOT(EXIST(p) PROOF(p g))
>
> so
> NOT(EXIST(p) PROOF(p g))
> ==>
>
> g
> = NOT(EXIST(p) PROOF(p g)) ..........*FALSE*
> = NOT(EXIST(p) PROOF(p NOT(EXIST(p) PROOF(p g)))) ..........*TRUE*
>
> THAT IS THE INCOMPLETENESS THEOREM
>
> ====================================
>
> BUT IT DOESNT STOP THERE

And you are missing that fact that the PROOF statement is about a proof
in the THEORY, but the logic you are doing is in the META-THEORY

Neglecting that is a source of a lot of your confusion.

>
> g
> = NOT(EXIST(p) PROOF(p g)) ..........*FALSE*
> = NOT(EXIST(p) PROOF(p NOT(EXIST(p) PROOF(p g)))) ..........*TRUE*
> = NOT(EXIST(p) PROOF(p NOT(EXIST(p) PROOF(p NOT(EXIST(p) PROOF(p g)))))) ....*FALSE*
>
>
> WHATS GOING ON HERE?
>
> g = there is no proof of g
> it = there is no proof of (there is no proof of g)
> x = there is no proof of (there is no proof of (there is no proof of g))
>
> THERE IS NO PROOF OF THE INCOMPLETENESS THEOREM !!!
>
> HOW BIZARRE. Luckily X=FALSE
>
> It cycles on and on.
>
> There is no proof that there is no proof of the incompleteness theorem
> is TRUE!
>
> but they are all equal to g
>
>
> ALL MODELS MUST GIVE [TRUE] TO ANY FORMULA
> This makes the THEORY INCONSISTENT similarly to X=!X
>
>
>> "This sentence cannot be proven"
>> "This sentence is not true"
>> are both self contradictory.
>>
>> *The whole metatheory nonsense is simply this*
>> This sentence cannot be proven: "This sentence cannot be proven"
>> and
>> This sentence is not true: "This sentence is not true"
>>
>> The outer sentence escapes the self-contradiction
>> allowing the outer sentence to be true.
>
> So what is the SOLUTION?
> g is an invalid formula just like X=!X so it is also "STRATIFIED" out of the THEORY
>
>
> Can we state
>
> ALL(logic) EXIST(p) PROOF(p logic) <-> TRUE(logic)
>
> Not yet, as g is neither FALSE nor TRUE
> Logicians argue that if g was FALSE then it has no proof which is what it states
> Hence the term INCOMPLETNESS.
>
> A LOGIC SOLVER with input
>
>> g <-> !proof(g)
>> g
> [ENTER]
>
> would return FALSE
> but you would THINK *there is no proof of g*
> g <-> !proof(g)
> so g is RIGHT!
>
> The only solution for a complete logic solver is TERNARY LOGIC
>
> logicForumula(xy..) ==> {TF?}
>
> g has both FALSE and TRUE values so the answer is ?
>
>
> www.NEW-MATH.com

On 2/19/2023 6:43 PM, 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
>
> CONCLUSION: There exists true mathematical statements that have no proof
>
>
>
> ABSTRACT
>
> g = NOT(EXIST(p) PROOF(p g))
>
> G states there does not exist a proof of G
>
> The fact it was 1st encoded in archaic godel numbering nonsense
> doesn't mean you cant represent G in a modern logic language
>
> G is SELF REFERENTIAL
>
> g is a statement about g
> g is a statement about (a statement about g)
>
> is where the incompleteness theorem ends
>
> HOWEVER
>
> g is a statement about (a statement about (a statement about g))
> etc
> ALSO give alternating TRUE FALSE values to g
>
> ---------------------------------------------------------------------
>
>
> SIMILAR TO
>
> X = !X
>
> X=TRUE
> X=!X
> X=FALSE
> X=!X
> X=TRUE
> X=!X
> X=FALSE
> ...
>
>
> g = NOT(EXIST(p) PROOF(p g))
>
> By standard analysis if
>
> NOT(EXIST(p) PROOF(x))
> THEN x IS FALSE
>
> NOT(EXIST(p) PROOF(p g)) = FALSE
>
> Godels Proof states
>
> Since g is FALSE there is no proof of g
> NOT(EXIST(p) PROOF(p g)) = TRUE
>
> =============================
>
> and that's where 21C LOGIC ENDS!
>
> but it doesnt end there
>
> What Godel actually proved
>
> REMEMBER
>
> g = NOT(EXIST(p) PROOF(p g))
>
> so
> NOT(EXIST(p) PROOF(p g))
> ==>
>
> g
> = NOT(EXIST(p) PROOF(p g)) ..........*FALSE*
> = NOT(EXIST(p) PROOF(p NOT(EXIST(p) PROOF(p g)))) ..........*TRUE*
>
> THAT IS THE INCOMPLETENESS THEOREM
>
> ====================================
>
> BUT IT DOESNT STOP THERE
>
> g
> = NOT(EXIST(p) PROOF(p g)) ..........*FALSE*
> = NOT(EXIST(p) PROOF(p NOT(EXIST(p) PROOF(p g)))) ..........*TRUE*
> = NOT(EXIST(p) PROOF(p NOT(EXIST(p) PROOF(p NOT(EXIST(p) PROOF(p g)))))) ....*FALSE*
>
>
> WHATS GOING ON HERE?
>
> g = there is no proof of g
> it = there is no proof of (there is no proof of g)
> x = there is no proof of (there is no proof of (there is no proof of g))
>
> THERE IS NO PROOF OF THE INCOMPLETENESS THEOREM !!!
>
> HOW BIZARRE. Luckily X=FALSE
>
> It cycles on and on.
>
> There is no proof that there is no proof of the incompleteness theorem
> is TRUE!
>
> but they are all equal to g
>
>
> ALL MODELS MUST GIVE [TRUE] TO ANY FORMULA
> This makes the THEORY INCONSISTENT similarly to X=!X
>
>
>> "This sentence cannot be proven"
>> "This sentence is not true"
>> are both self contradictory.
>>
>> *The whole metatheory nonsense is simply this*
>> This sentence cannot be proven: "This sentence cannot be proven"
>> and
>> This sentence is not true: "This sentence is not true"
>>
>> The outer sentence escapes the self-contradiction
>> allowing the outer sentence to be true.
>
> So what is the SOLUTION?
> g is an invalid formula just like X=!X so it is also "STRATIFIED" out of the THEORY
>
>
> Can we state
>
> ALL(logic) EXIST(p) PROOF(p logic) <-> TRUE(logic)
>
> Not yet, as g is neither FALSE nor TRUE
> Logicians argue that if g was FALSE then it has no proof which is what it states
> Hence the term INCOMPLETNESS.
>
> A LOGIC SOLVER with input
>
>> g <-> !proof(g)
>> g
> [ENTER]
>
> would return FALSE
> but you would THINK *there is no proof of g*
> g <-> !proof(g)
> so g is RIGHT!
>
> The only solution for a complete logic solver is TERNARY LOGIC
>
> logicForumula(xy..) ==> {TF?}
>
> g has both FALSE and TRUE values so the answer is ?
>
>
> www.NEW-MATH.com

The actual correct TERNARY LOGIC is:
(a) True
(b) False
(c) not a truth bearer.

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 2/19/23 8:41 PM, olcott wrote:

>
> The actual correct TERNARY LOGIC is:
> (a) True
> (b) False
> (c) not a truth bearer.
>
>

That is ONE of them.