Gödel's 1931 incompleteness theorem is anchored in this false assumption
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))

This false assumption is corrected by this foundational axiom:
∀φ ∈ T ((T ⊢ φ) ∨ (T ⊢ ¬φ))

This tiny little change
that I only discovered after trying everything else correctly
denigrates the 1931 incompleteness theorem into a triviality.

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

On 1/11/23 10:37 PM, olcott wrote:
> Gödel's 1931 incompleteness theorem is anchored in this false assumption
> The conventional definition of incompleteness:
> Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))

That isn't an assumption, it is a DEFINITION, definitions CAN'T be false.

>
> This false assumption is corrected by this foundational axiom:
> ∀φ ∈ T ((T ⊢ φ) ∨ (T ⊢ ¬φ))

Which isn't true in any but the simplest of fields. I guess that shows
how much you can understand, only the simplest of fields.

>
> This tiny little change
> that I only discovered after trying everything else correctly
> denigrates the 1931 incompleteness theorem into a triviality.
>

No, you didn't "Discover that", that was a long held hope in
mathemtatics that Godel proved wasn't true.

The fact that you think it is shows your stupidity AND your arrogance.
If you that EVERYONE before you was so stupid to not understand this,
you are showing how stupid YOU actually are.

You appear to have failed to learn about the developements in the last
century about logic, and have doomed yourself to repeating the mistakes,
and you have placed your wad on a horse that has already lost.

You have DOOMED yourself to being a laughing stock, only remembered for
your lying and stupidity.

On 1/12/2023 6:49 AM, Richard Damon wrote:
> On 1/11/23 10:37 PM, olcott wrote:
>> Gödel's 1931 incompleteness theorem is anchored in this false assumption
>> The conventional definition of incompleteness:
>> Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
>
> That isn't an assumption, it is a DEFINITION, definitions CAN'T be false.
>

OK so when I define that cats are dogs I am necessarily correct?

When-so-ever any expression of language directly contradicts knowledge
in the body of knowledge than that expression of language is incorrect.

If an analytic expression of language is true or false there must be a
complete set of semantic connections making it true or false otherwise
it is not a truth bearer.

Because formal systems are only allowed to have finite proofs formal
systems are not allowed to have infinite connections to their semantic
truth maker. Thus an expression is only true in a formal system iff it
is provable within this system. Otherwise this expression is untrue
which may or may not included false.

>>
>> This false assumption is corrected by this foundational axiom:
>> ∀φ ∈ T ((T ⊢ φ) ∨ (T ⊢ ¬φ))
>
> Which isn't true in any but the simplest of fields. I guess that shows
> how much you can understand, only the simplest of fields.
>
>>
>> This tiny little change
>> that I only discovered after trying everything else correctly
>> denigrates the 1931 incompleteness theorem into a triviality.
>>
>
> No, you didn't "Discover that", that was a long held hope in
> mathemtatics that Godel proved wasn't true.
>

Since G is unprovable in F G is simply untrue in F, that is just simply
that way that truth actually works. By what-ever means an expression is
only true iff this expression has a complete semantic connection to its
truth maker axioms. There is no such connection in G.

If I am correct and everyone in the universe disagrees this has no
effect what-so-ever on the correctness of what I say.

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

On Thursday, January 12, 2023 at 10:35:58 AM UTC-6, olcott wrote:
> On 1/12/2023 6:49 AM, Richard Damon wrote:
> > On 1/11/23 10:37 PM, olcott wrote:
> >> Gödel's 1931 incompleteness theorem is anchored in this false assumption
> >> The conventional definition of incompleteness:
> >> Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
> >
> > That isn't an assumption, it is a DEFINITION, definitions CAN'T be false.
> >
> OK so when I define that cats are dogs I am necessarily correct?
>
> When-so-ever any expression of language directly contradicts knowledge
> in the body of knowledge than that expression of language is incorrect.
>
> If an analytic expression of language is true or false there must be a
> complete set of semantic connections making it true or false otherwise
> it is not a truth bearer.
>
> Because formal systems are only allowed to have finite proofs formal
> systems are not allowed to have infinite connections to their semantic
> truth maker. Thus an expression is only true in a formal system iff it
> is provable within this system. Otherwise this expression is untrue
> which may or may not included false.
> >>
> >> This false assumption is corrected by this foundational axiom:
> >> ∀φ ∈ T ((T ⊢ φ) ∨ (T ⊢ ¬φ))
> >
> > Which isn't true in any but the simplest of fields. I guess that shows
> > how much you can understand, only the simplest of fields.
> >
> >>
> >> This tiny little change
> >> that I only discovered after trying everything else correctly
> >> denigrates the 1931 incompleteness theorem into a triviality.
> >>
> >
> > No, you didn't "Discover that", that was a long held hope in
> > mathemtatics that Godel proved wasn't true.
> >
> Since G is unprovable in F G is simply untrue in F, that is just simply
> that way that truth actually works. By what-ever means an expression is
> only true iff this expression has a complete semantic connection to its
> truth maker axioms. There is no such connection in G.
>
> If I am correct and everyone in the universe disagrees this has no
> effect what-so-ever on the correctness of what I say.
> --
> Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
> hits a target no one else can see." Arthur Schopenhauer

I really like it in turtle Asher Bach. We're Hofstædter talks about turtles therm.

On 1/12/23 11:35 AM, olcott wrote:
> On 1/12/2023 6:49 AM, Richard Damon wrote:
>> On 1/11/23 10:37 PM, olcott wrote:
>>> Gödel's 1931 incompleteness theorem is anchored in this false assumption
>>> The conventional definition of incompleteness:
>>> Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
>>
>> That isn't an assumption, it is a DEFINITION, definitions CAN'T be false.
>>
>
> OK so when I define that cats are dogs I am necessarily correct?

No, because both "cats" and "dogs" are previously defined terms.

"Incompleteness" (as a TECHNICAL) term isn't otherwise defined.

And this techincal definition even aligns with the "common" definition,
of missing something, as an Incomplete system IS missing something, the
proofs of some truths in it.

>
> When-so-ever any expression of language directly contradicts knowledge
> in the body of knowledge than that expression of language is incorrect.

And how does it directly contradict knowledge?

>
> If an analytic expression of language is true or false there must be a
> complete set of semantic connections making it true or false otherwise
> it is not a truth bearer.

Righg, and that definition does NOT say "finite"

>
> Because formal systems are only allowed to have finite proofs formal
> systems are not allowed to have infinite connections to their semantic

Right, **PROOFS** must be finite, doesn't say anything about Truth.

> truth maker. Thus an expression is only true in a formal system iff it
> is provable within this system. Otherwise this expression is untrue
> which may or may not included false.
>

Where does this come from.

True means, as you said above, as A complete set of semantic connections
(could be infinite) makeing it true.

PROVEN means having a KNOWN FINITE set of semantic connections making it
proven.

Different Words, Different Meanings, Different results.

Your logic says that since Dogs have Puppies, ALL animals must have
puppies, since Dogs are a subset of Animals.

>>>
>>> This false assumption is corrected by this foundational axiom:
>>> ∀φ ∈ T ((T ⊢ φ) ∨ (T ⊢ ¬φ))
>>
>> Which isn't true in any but the simplest of fields. I guess that shows
>> how much you can understand, only the simplest of fields.
>>
>>>
>>> This tiny little change
>>> that I only discovered after trying everything else correctly
>>> denigrates the 1931 incompleteness theorem into a triviality.
>>>
>>
>> No, you didn't "Discover that", that was a long held hope in
>> mathemtatics that Godel proved wasn't true.
>>
>
> Since G is unprovable in F G is simply untrue in F, that is just simply
> that way that truth actually works. By what-ever means an expression is
> only true iff this expression has a complete semantic connection to its
> truth maker axioms. There is no such connection in G.

Nope,

Are you saying there is not an answer to the question of does a proof
exist to G.

Either there IS a finite connection to G from the elementary axioms, or
there isn't, and thus G is either Provable or Not.

Since the ACTUAL statment of G is F is "There does not exist a Natural
Number g that satisfies <specific primative recursive relationship> you
are saying the fact we can't prove that such a number exists means the
question of its existance is not a Truth Bearer, which is absurd.

Either a Natural Number that meets that criterial exists or it dosn't,
there is NO middle ground.

You logic is broken because you are making the same mistake of saying
cats are dog by saying Truth is Provability.

YOU FAIL.

>
> If I am correct and everyone in the universe disagrees this has no
> effect what-so-ever on the correctness of what I say.
>

On 1/12/2023 8:30 PM, Richard Damon wrote:
> On 1/12/23 11:35 AM, olcott wrote:
>> On 1/12/2023 6:49 AM, Richard Damon wrote:
>>> On 1/11/23 10:37 PM, olcott wrote:
>>>> Gödel's 1931 incompleteness theorem is anchored in this false
>>>> assumption
>>>> The conventional definition of incompleteness:
>>>> Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
>>>
>>> That isn't an assumption, it is a DEFINITION, definitions CAN'T be
>>> false.
>>>
>>
>> OK so when I define that cats are dogs I am necessarily correct?
>
> No, because both "cats" and "dogs" are previously defined terms.
>
> "Incompleteness" (as a TECHNICAL) term isn't otherwise defined.
>
> And this techincal definition even aligns with the "common" definition,
> of missing something, as an Incomplete system IS missing something, the
> proofs of some truths in it.
>
>>
>> When-so-ever any expression of language directly contradicts knowledge
>> in the body of knowledge than that expression of language is incorrect.
>
> And how does it directly contradict knowledge?
>
>>
>> If an analytic expression of language is true or false there must be a
>> complete set of semantic connections making it true or false otherwise
>> it is not a truth bearer.
>
> Righg, and that definition does NOT say "finite"
>
>>
>> Because formal systems are only allowed to have finite proofs formal
>> systems are not allowed to have infinite connections to their semantic
>
> Right, **PROOFS** must be finite, doesn't say anything about Truth.
>
>> truth maker. Thus an expression is only true in a formal system iff it
>> is provable within this system. Otherwise this expression is untrue
>> which may or may not included false.
>>
>
> Where does this come from.
>
> True means, as you said above, as A complete set of semantic connections
> (could be infinite) makeing it true.
>
> PROVEN means having a KNOWN FINITE set of semantic connections making it
> proven.
>
Thus true within a formal system must have a finite proof because formal
systems are not allowed to have infinite proofs.

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

On 1/12/23 9:36 PM, olcott wrote:

> Thus true within a formal system must have a finite proof because formal
> systems are not allowed to have infinite proofs.
>
>

The false statement that you have DIED on.

You are showing you mind is just incompetent and can't understand the
difference betweeh a Truth and a Proof.

Sorry, you are just proving your self to be STUPID.

On 1/12/2023 8:39 PM, Richard Damon wrote:
> On 1/12/23 9:36 PM, olcott wrote:
>
>> Thus true within a formal system must have a finite proof because formal
>> systems are not allowed to have infinite proofs.
>>
>>
>
> The false statement that you have DIED on.
>

Expressions of language that are proven to have a connection to their
truth maker axioms are a subset of all truth and comprise the entire
body of analytical knowledge.

One cannot have an expression of language within a formal system that is
true within this formal system that has no connection to its truth maker
axioms within this formal system.

Peano arithmetic has no idea that strawberry rhubarb pie is sweet, thus
the expression "strawberry rhubarb pie is sweet" is not true in Peano
arithmetic.

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

On 1/12/23 10:10 PM, olcott wrote:
> On 1/12/2023 8:39 PM, Richard Damon wrote:
>> On 1/12/23 9:36 PM, olcott wrote:
>>
>>> Thus true within a formal system must have a finite proof because formal
>>> systems are not allowed to have infinite proofs.
>>>
>>>
>>
>> The false statement that you have DIED on.
>>
>
> Expressions of language that are proven to have a connection to their
> truth maker axioms are a subset of all truth and comprise the entire
> body of analytical knowledge.

So, since proven statements are a SUBSET of all truth, what says that
all truths have to be proven.

that is like saying that since Dogs are a subset of Animals, all Animals
are Dogs.

You are just proving how little you understand about logic.
>
> One cannot have an expression of language within a formal system that is
> true within this formal system that has no connection to its truth maker
> axioms within this formal system.
>

Right, but that connection, for TRUTH, can be infinite.

> Peano arithmetic has no idea that strawberry rhubarb pie is sweet, thus
> the expression "strawberry rhubarb pie is sweet" is not true in Peano
> arithmetic.
>

So?

Just more Red Herring.