That seems like gibberish to me.Eodnhoj7 wrote: ↑Tue Jun 16, 2020 10:27 pmThere is no statement that is not self contradictory given that each statement is a localization of a continuum thus representing a part of the whole. As representative of a part, each statement always requires some statement, unrepresented, beyond it. It is this absence of a complete representation that necessitates a contradiction in terms given that something is always provable by what is not provable. With the increase in words comes an increase in contradictions. However with the increase in words, as an increase in contradictions, comes a necessary further increase in words to clarify the contradictions.PeteOlcott wrote: ↑Mon Jun 15, 2020 11:42 pmSelfcontradictory statements are rejected as incorrect.
"This sentence is not true" is indeed {not true} yet that does not make it
true because it has the selfcontradiction error.
Simply defining Gödel Incompleteness away

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Re: Simply defining Gödel Incompleteness away
Re: Simply defining Gödel Incompleteness away
1. What grammar is this?PeteOlcott wrote: ↑Wed Jun 17, 2020 4:04 pm ∀F ∈ Formal_System
∀X ∈ Language(F)
(((F ⊬ X) ∧ (F ⊬ ¬X)) ↔ Undecidable(F, X))
2. What do the symbols ∀, ⊬, ∈ and ↔ mean? e.g how are they evaluated?
3. I don't see a True() predicate anywhere.
4. Is Formal_System a predicate? If not  what is it?
Re: Simply defining Gödel Incompleteness away
There is always an assertion beyond the assertion which is unprovable given all assertions exist as continuums.PeteOlcott wrote: ↑Wed Jun 17, 2020 4:05 pmThat seems like gibberish to me.Eodnhoj7 wrote: ↑Tue Jun 16, 2020 10:27 pmThere is no statement that is not self contradictory given that each statement is a localization of a continuum thus representing a part of the whole. As representative of a part, each statement always requires some statement, unrepresented, beyond it. It is this absence of a complete representation that necessitates a contradiction in terms given that something is always provable by what is not provable. With the increase in words comes an increase in contradictions. However with the increase in words, as an increase in contradictions, comes a necessary further increase in words to clarify the contradictions.PeteOlcott wrote: ↑Mon Jun 15, 2020 11:42 pm
Selfcontradictory statements are rejected as incorrect.
"This sentence is not true" is indeed {not true} yet that does not make it
true because it has the selfcontradiction error.
Re: Simply defining Gödel Incompleteness away
What the hell does "premises known to be true" mean if notPeteOlcott wrote: ↑Wed Jun 17, 2020 4:04 pm From the sound deductive inference model we can see that this is incorrect. A sound deduction begins with premises that are known to be true
∀X ∈ Premises: X ↔ True(X)
How do you test/determine that your premises are "true" without a truthpredicate?!?
How do you solve the Garbage in  garbage out problem?

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Re: Simply defining Gödel Incompleteness away
Successor(Successor(0)) > Successor(0) is always true when evaluatedEodnhoj7 wrote: ↑Wed Jun 17, 2020 4:16 pmThere is always an assertion beyond the assertion which is unprovable given all assertions exist as continuums.PeteOlcott wrote: ↑Wed Jun 17, 2020 4:05 pmThat seems like gibberish to me.Eodnhoj7 wrote: ↑Tue Jun 16, 2020 10:27 pm
There is no statement that is not self contradictory given that each statement is a localization of a continuum thus representing a part of the whole. As representative of a part, each statement always requires some statement, unrepresented, beyond it. It is this absence of a complete representation that necessitates a contradiction in terms given that something is always provable by what is not provable. With the increase in words comes an increase in contradictions. However with the increase in words, as an increase in contradictions, comes a necessary further increase in words to clarify the contradictions.
on the basis of its pure semantics and can never be shown to be untrue or false.
Re: Simply defining Gödel Incompleteness away
Not "when evaluated", Pete. IF evaluated. IF!PeteOlcott wrote: ↑Wed Jun 17, 2020 4:34 pm Successor(Successor(0)) > Successor(0) is always true when evaluated
on the basis of its pure semantics and can never be shown to be untrue or false.
Some numbers are too large to evaluate in this universe's lifetime.
Re: Simply defining Gödel Incompleteness away
A > B > C > (x) with C being the final portion of the string and (x) being the unknown assertion beyond the string. The final variable of the string is always unproven.PeteOlcott wrote: ↑Wed Jun 17, 2020 4:34 pmSuccessor(Successor(0)) > Successor(0) is always true when evaluated
on the basis of its pure semantics and can never be shown to be untrue or false.
However if A> B > C > A the entire string is proven given through itself as a loop. However another outside string beyond it, required for proof, is accepted "as is" and is unproven.
Thus the statement as self referential is proven, however this self referential statement requires a self referential statement beyond it thus is simultaneously unproven.
All statements are both proven and unproven at the same time in different respects. Proof is dualistic.

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Re: Simply defining Gödel Incompleteness away
Your reply is a dodge into an irrelevant tangent.Skepdick wrote: ↑Wed Jun 17, 2020 4:17 pmWhat the hell does "premises known to be true" mean if notPeteOlcott wrote: ↑Wed Jun 17, 2020 4:04 pm From the sound deductive inference model we can see that this is incorrect. A sound deduction begins with premises that are known to be true
∀X ∈ Premises: X ↔ True(X)
How do you test/determine that your premises are "true" without a truthpredicate?!?
How do you solve the Garbage in  garbage out problem?
I don't have to conclusively prove that Quine is incorrect in his TWO DOGMAS OF EMPIRICISM
or prove that the axiomatic argument of the Münchhausen trilemma is baseless to prove
that within this specified definition of sound deduction:
(a) Beginning with a set of premises that are known to be true.
(b) Applying truth preserving operations to these premises.
(c) Deriving a conclusion that is known to be true.
Undecidable(F, X) means Invalid(F,X) instead of Incomplete(F).
Providing the foundational basis of analytic knowledge is a whole
other issue for another day.
Re: Simply defining Gödel Incompleteness away
*sigh*PeteOlcott wrote: ↑Wed Jun 17, 2020 4:46 pmYour reply is a dodge into an irrelevant tangent.Skepdick wrote: ↑Wed Jun 17, 2020 4:17 pmWhat the hell does "premises known to be true" mean if notPeteOlcott wrote: ↑Wed Jun 17, 2020 4:04 pm From the sound deductive inference model we can see that this is incorrect. A sound deduction begins with premises that are known to be true
∀X ∈ Premises: X ↔ True(X)
How do you test/determine that your premises are "true" without a truthpredicate?!?
How do you solve the Garbage in  garbage out problem?
I don't have to conclusively prove that Quine is incorrect in his TWO DOGMAS OF EMPIRICISM
or prove that the axiomatic argument of the Münchhausen trilemma is baseless to prove
that within this specified definition of sound deduction:
(a) Beginning with a set of premises that are known to be true.
(b) Applying truth preserving operations to these premises.
(c) Deriving a conclusion that is known to be true.
Undecidable(F, X) means Invalid(F,X) instead of Incomplete(F).
Providing the foundational basis of analytic knowledge is a whole
other issue for another day.
I am ignoring points B and C because they DON'T FUCKING MATTER. So yes, you fucking have to address Quine.
You are being selective about the things you have to prove  I am not.
If True means Provable (which is what you claim) then I insist you PROVE that your premises are TRUE.
If you can't do that, you are playing some stupid equivocation dance.
You are confusing the a priori and a posteriori meaning of "decidable".
The way to DECIDE if your premises are true is to invoke the True() predicate on your premise.
If it halts  it's decidable.
UNTIL it halts  it's undecidable.

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Re: Simply defining Gödel Incompleteness away
My conclusion does logically follow from my premises thus if my premisesSkepdick wrote: ↑Wed Jun 17, 2020 4:52 pm*sigh*PeteOlcott wrote: ↑Wed Jun 17, 2020 4:46 pmYour reply is a dodge into an irrelevant tangent.Skepdick wrote: ↑Wed Jun 17, 2020 4:17 pm
What the hell does "premises known to be true" mean if not
∀X ∈ Premises: X ↔ True(X)
How do you test/determine that your premises are "true" without a truthpredicate?!?
How do you solve the Garbage in  garbage out problem?
I don't have to conclusively prove that Quine is incorrect in his TWO DOGMAS OF EMPIRICISM
or prove that the axiomatic argument of the Münchhausen trilemma is baseless to prove
that within this specified definition of sound deduction:
(a) Beginning with a set of premises that are known to be true.
(b) Applying truth preserving operations to these premises.
(c) Deriving a conclusion that is known to be true.
Undecidable(F, X) means Invalid(F,X) instead of Incomplete(F).
Providing the foundational basis of analytic knowledge is a whole
other issue for another day.
I am ignoring points B and C because they DON'T FUCKING MATTER. So yes, you fucking have to address Quine.
You are being selective about the things you have to prove  I am not.
are true then Gödel is wrong (within my assumptions) and you cannot possibly
show otherwise.
You can, however, digress into an infinite set of side issues making
closure forever impossible. This is incorrect software engineering.
Once one module is complete and correct then you move on to the next module.
Re: Simply defining Gödel Incompleteness away
You aren't fucking listening!!!PeteOlcott wrote: ↑Wed Jun 17, 2020 5:16 pm My conclusion does logically follow from my premises thus if my premises
are true then Gödel is wrong (within my assumptions) and you cannot possibly
show otherwise.
IF your premises are true. IF. IF.
But you said True means Provable. And you also said unprovable means untrue!
You can't prove your premises therefore your premises are untrue!
You own definitions disprove your own premises. I don't need to show it because you have shown it!
1000000000th time. You don't understand Godel.PeteOlcott wrote: ↑Wed Jun 17, 2020 5:16 pm You can, however, digress into an infinite set of side issues making
closure forever impossible. This is incorrect software engineering.
Once one module is complete and correct then you move on to the next module.
Godel's claims are about language which are (and hear me carefully here) POWERFUL ENOUGH to say things about THE NATURAL NUMBERS. In order for you to say anything about "THE NATURAL NUMBERS" you must be able to say things about infinite sets.
And that was precisely Godel's point.
You can't CLOSE an infinite set!
You can CLOSE a finite set!
Which is why computer scientists reject ABSOLUTE infinities.
But computer scientists also embrace transfinite numbers because they allow for transfinite recursion.
You know what regular English speakers call "transfinite recursion"? Induction! As in the problem of induction.
The only reason you think it's "incorrect software engineering" is because you don't know how to handle transfinite values in finite memory.
Re: Simply defining Gödel Incompleteness away
That was ALSO Godel's point!PeteOlcott wrote: ↑Wed Jun 17, 2020 5:16 pm You can, however, digress into an infinite set of side issues making
closure forever impossible.
Consistency OR completeness. CHOOSE one.
If you want closure CHOOSE completeness.
If you want closure GIVE UP consistency.
If you want CLOSURE then look no further than the Cartesian CLOSED categories
You seem uneasy with the consequences of this choice. Why?In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in that their internal language is the simply typed lambda calculus. They are generalized by closed monoidal categories, whose internal language, linear type systems, are suitable for both quantum and classical computation.

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Re: Simply defining Gödel Incompleteness away
I am asking: "Is my reasoning valid?"Skepdick wrote: ↑Wed Jun 17, 2020 5:19 pmYou aren't fucking listening!!!PeteOlcott wrote: ↑Wed Jun 17, 2020 5:16 pm My conclusion does logically follow from my premises thus if my premises
are true then Gödel is wrong (within my assumptions) and you cannot possibly
show otherwise.
IF your premises are true. IF. IF.
You keep saying that I have not proven that my reasoning is sound.
That is correct. I have not proven that my reasoning is sound.
I am not asking: "Is my reasoning sound?"
I am asking: "Is my reasoning valid?"
Re: Simply defining Gödel Incompleteness away
Like I have told you 100000 times before also.PeteOlcott wrote: ↑Wed Jun 17, 2020 5:55 pm I am asking: "Is my reasoning valid?"
You keep saying that I have not proven that my reasoning is sound.
That is correct. I have not proven that my reasoning is sound.
I am not asking: "Is my reasoning sound?"
I am asking: "Is my reasoning valid?"
In addition to Godel's incompleteness theorem. There is ALSO Godel's completeness theorem.
That theorem says EXACTLY what you have been saying.
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in firstorder logic.

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Re: Simply defining Gödel Incompleteness away
The humongous difference is that my formulation applies to the entire body ofSkepdick wrote: ↑Wed Jun 17, 2020 5:57 pmLike I have told you 100000 times before also.PeteOlcott wrote: ↑Wed Jun 17, 2020 5:55 pm I am asking: "Is my reasoning valid?"
You keep saying that I have not proven that my reasoning is sound.
That is correct. I have not proven that my reasoning is sound.
I am not asking: "Is my reasoning sound?"
I am asking: "Is my reasoning valid?"
In addition to Godel's incompleteness theorem. There is ALSO Godel's completeness theorem.
That theorem says EXACTLY what you have been saying.
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in firstorder logic.
analytical truth thus within its sound deductive inference model Gödel's 1931
Incompleteness theorem is simply incorrect.