a Proof of G in F is contradictory

[olcott]

It turns out that the only reason that Gödel’s G is not provable in F is
that G is contradictory in F.

When Gödel’s G asserts that it is unprovable in F it is asserting that
there is no sequence of inference steps in F that derives G.

*A proof of G in F requires a sequence of inference steps in F that*
*proves there is no such sequence of inference steps in F, a*
*contradiction*

*This is like René Descartes saying*
“I think therefore thoughts do not exist”

The reason why G cannot be proved in F is that the proof of G in F is
contradictory in F, thus Gödel was wrong when he said the reason is that
F is incomplete. No formal system is ever supposed to be able to prove a
contradiction.

Now we can see both THAT G cannot be proved in F and perhaps for the
first time see WHY G cannot be proved in F. The “incompleteness”
conclusion has been refuted.

To be honest we would have to rename Gödel’s “incompleteness” theorem to
Olcott’s “can’t prove a contradiction” theorem.

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

[Richard Damon]

On 4/20/23 12:04 PM, olcott wrote:
> It turns out that the only reason that Gödel’s G is not provable in F is
> that G is contradictory in F.
>
> When Gödel’s G asserts that it is unprovable in F it is asserting that
> there is no sequence of inference steps in F that derives G.
>
> *A proof of G in F requires a sequence of inference steps in F that*
> *proves there is no such sequence of inference steps in F, a*
> *contradiction*
>
> *This is like René Descartes saying*
> “I think therefore thoughts do not exist”
>
> The reason why G cannot be proved in F is that the proof of G in F is
> contradictory in F, thus Gödel was wrong when he said the reason is that
> F is incomplete. No formal system is ever supposed to be able to prove a
> contradiction.
>
> Now we can see both THAT G cannot be proved in F and perhaps for the
> first time see WHY G cannot be proved in F. The “incompleteness”
> conclusion has been refuted.
>
> To be honest we would have to rename Gödel’s “incompleteness” theorem to
> Olcott’s “can’t prove a contradiction” theorem.
>

But you keep using the wrong statement for G, ikely because you just
don't understand the proof.

G does NOT assert that it is unprovable in F, that is just a conclusion
derived from G in Meta-F.

G is actually a statement that there does not exist a whole number that
satisfies a property expressed as a primitive recursive relationship.

Such a statement CAN'T be "contradictory" as either such a number exists
or it doesn't.

Of course, since you are too stupid to understand that statement, you
mix up which system you are talking in and what statement you are
working on.

[olcott]

On 4/21/2023 7:00 AM, Richard Damon wrote:
> On 4/20/23 12:04 PM, olcott wrote:
>> It turns out that the only reason that Gödel’s G is not provable in F is
>> that G is contradictory in F.
>>
>> When Gödel’s G asserts that it is unprovable in F it is asserting that
>> there is no sequence of inference steps in F that derives G.
>>
>> *A proof of G in F requires a sequence of inference steps in F that*
>> *proves there is no such sequence of inference steps in F, a*
>> *contradiction*
>>
>> *This is like René Descartes saying*
>> “I think therefore thoughts do not exist”
>>
>> The reason why G cannot be proved in F is that the proof of G in F is
>> contradictory in F, thus Gödel was wrong when he said the reason is that
>> F is incomplete. No formal system is ever supposed to be able to prove a
>> contradiction.
>>
>> Now we can see both THAT G cannot be proved in F and perhaps for the
>> first time see WHY G cannot be proved in F. The “incompleteness”
>> conclusion has been refuted.
>>
>> To be honest we would have to rename Gödel’s “incompleteness” theorem to
>> Olcott’s “can’t prove a contradiction” theorem.
>>
>
> But you keep using the wrong statement for G, ikely because you just
> don't understand the proof.
>
> G does NOT assert that it is unprovable in F, that is just a conclusion
> derived from G in Meta-F.
>

Gödel sums up his own G as simply:

"...a proposition which asserts its own unprovability." 15 (Gödel
1931:39-41)

Thus this summary is accurate.
*G asserts its own unprovability in F*

When you simply hypothesize that it is an accurate representation
of the essence of Gödel's G then it is easy to see that

*G asserts its own unprovability in F*
The reason that G cannot be proved in F is that this requires a
sequence of inference steps in F that proves no such sequence
of inference steps exists in F.

Since we already have the reason why G cannot be proved in F
(the proof of G is F is contradictory)
and Gödel said it was another different reason then Gödel was incorrect.

Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And
Related Systems

> G is actually a statement that there does not exist a whole  number that
> satisfies a property expressed as a primitive recursive relationship.
>
> Such a statement CAN'T be "contradictory" as either such a number exists
> or it doesn't.
>
> Of course, since you are too stupid to understand that statement, you
> mix up which system you are talking in and what statement you are
> working on.

[Richard Damon]

Right, which was shown in META-F, not F.

>
> Thus this summary is accurate.
> *G asserts its own unprovability in F*

Nope, the statement you are referencing is a statement that was DERIVED
from G, using knowledge available in Meta-F that isn't available in F.

>
> When you simply hypothesize that it is an accurate representation
> of the essence of Gödel's G then it is easy to see that

But you are mixing up the domains of the statements.

>
> *G asserts its own unprovability in F*
> The reason that G cannot be proved in F is that this requires a
> sequence of inference steps in F that proves no such sequence
> of inference steps exists in F.

Which isn't what G says, either in F or Meta-F. The statement is
something DERIVABLE from G in Meta-F.

>
> Since we already have the reason why G cannot be proved in F
> (the proof of G is F is contradictory)
> and Gödel said it was another different reason then Gödel was incorrect.

Nope. You are using incorrect logic, because you don't understand what
you are reading.

>
> Gödel, Kurt 1931.
> On Formally Undecidable Propositions of Principia Mathematica And
> Related Systems
>
>

Which you don't understand but are quoting statements out of context.

[olcott]

In exactly the same way that Tarski "proves" that the Liar Paradox is
true in his meta-theory.

This sentence is not true: "This sentence is not true" is true.

[Richard Damon]

You aren't reading what he is saying correctly if you are referencing
what I think you are refeencing.

I think the issue is you don't understand how a proof by contradiction
works.