## Simply defining Gödel Incompleteness away

What is the basis for reason? And mathematics?

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Skepdick
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### Re: Simply defining Gödel Incompleteness away

PeteOlcott wrote: Wed Jun 17, 2020 6:07 pm The humongous difference is that my formulation applies to the entire body of
analytical truth thus within its sound deductive inference model Gödel's 1931
Incompleteness theorem is simply incorrect.
No Pete. It doesn't apply to the "entire body of analytical truth".

The completeness theorem only holds for first order logic.. If you think human reasoning/knowledge can be represented in first order logic you are very very naive. Surely you understand the very notion of abstraction layers?

Your entire project is an attempt at stratifying English semantics. So then surely then a logic with stronger semantics is better for what it is that you are trying to achieve?

https://en.wikipedia.org/wiki/Higher-order_logic
In mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expressive, but their model-theoretic properties are less well-behaved than those of first-order logic.
And if you accept the idea that natural language (and the analytical knowledge we express in it) is a higher order logic then you need to take into account the curse of dimensionality.
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Simply defining Gödel Incompleteness away

Skepdick wrote: Wed Jun 17, 2020 6:15 pm
PeteOlcott wrote: Wed Jun 17, 2020 6:07 pm The humongous difference is that my formulation applies to the entire body of
analytical truth thus within its sound deductive inference model Gödel's 1931
Incompleteness theorem is simply incorrect.
No Pete. It doesn't apply to the "entire body of analytical truth".
My reformulation of the sound deductive inference model does apply to the whole body of analytical truth.

When we define sound deduction as:
(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.

My reformulation of the sound deductive inference model does show that Undecidable(F, X)
really does mean Invalid(F, X) and not Incomplete(F)

∀F ∈ Formal_System
∀X ∈ Language(F)
(((F ⊬ X) ∧ (F ⊬ ¬X)) ↔ Undecidable(F, X))
Skepdick
Posts: 5030
Joined: Fri Jun 14, 2019 11:16 am

### Re: Simply defining Gödel Incompleteness away

PeteOlcott wrote: Wed Jun 17, 2020 6:49 pm
Skepdick wrote: Wed Jun 17, 2020 6:15 pm
PeteOlcott wrote: Wed Jun 17, 2020 6:07 pm The humongous difference is that my formulation applies to the entire body of
analytical truth thus within its sound deductive inference model Gödel's 1931
Incompleteness theorem is simply incorrect.
No Pete. It doesn't apply to the "entire body of analytical truth".
My reformulation of the sound deductive inference model does apply to the whole body of analytical truth.

When we define sound deduction as:
(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.

My reformulation of the sound deductive inference model does show that Undecidable(F, X)
really does mean Invalid(F, X) and not Incomplete(F)

∀F ∈ Formal_System
∀X ∈ Language(F)
(((F ⊬ X) ∧ (F ⊬ ¬X)) ↔ Undecidable(F, X))
So you don't understand the difference between first order logic and higher order logic.

You think the WHOLE BODY OF ANALYTICAL TRUTH is representable as a first order model.

OK.

You are a fucking retard.
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Simply defining Gödel Incompleteness away

Skepdick wrote: Wed Jun 17, 2020 6:50 pm
PeteOlcott wrote: Wed Jun 17, 2020 6:49 pm
Skepdick wrote: Wed Jun 17, 2020 6:15 pm
No Pete. It doesn't apply to the "entire body of analytical truth".
My reformulation of the sound deductive inference model does apply to the whole body of analytical truth.

When we define sound deduction as:
(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.

My reformulation of the sound deductive inference model does show that Undecidable(F, X)
really does mean Invalid(F, X) and not Incomplete(F)

∀F ∈ Formal_System
∀X ∈ Language(F)
(((F ⊬ X) ∧ (F ⊬ ¬X)) ↔ Undecidable(F, X))
So you don't understand the difference between first order logic and higher order logic.

You think the WHOLE BODY OF ANALYTICAL TRUTH is representable as a first order model.

OK.

You are a fucking retard.
Do you think that this refers to first order logic?
By the theory of simple types I mean the doctrine which says that the objects of thought (or, in another interpretation, the symbolic expressions) are divided into types, namely: individuals, properties of individuals, relations between individuals, properties of such relations, etc. (with a similar hierarchy for extensions), and that sentences of the form: " a has the property φ ", " b bears the relation R to c ", etc. are meaningless, if a, b, c, R, φ are not of types fitting together.
https://en.wikipedia.org/wiki/History_o ... B6del_1944
Skepdick
Posts: 5030
Joined: Fri Jun 14, 2019 11:16 am

### Re: Simply defining Gödel Incompleteness away

PeteOlcott wrote: Wed Jun 17, 2020 7:30 pm
Skepdick wrote: Wed Jun 17, 2020 6:50 pm
PeteOlcott wrote: Wed Jun 17, 2020 6:49 pm

My reformulation of the sound deductive inference model does apply to the whole body of analytical truth.

When we define sound deduction as:
(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.

My reformulation of the sound deductive inference model does show that Undecidable(F, X)
really does mean Invalid(F, X) and not Incomplete(F)

∀F ∈ Formal_System
∀X ∈ Language(F)
(((F ⊬ X) ∧ (F ⊬ ¬X)) ↔ Undecidable(F, X))
So you don't understand the difference between first order logic and higher order logic.

You think the WHOLE BODY OF ANALYTICAL TRUTH is representable as a first order model.

OK.

You are a fucking retard.
Do you think that this refers to first order logic?
By the theory of simple types I mean the doctrine which says that the objects of thought (or, in another interpretation, the symbolic expressions) are divided into types, namely: individuals, properties of individuals, relations between individuals, properties of such relations, etc. (with a similar hierarchy for extensions), and that sentences of the form: " a has the property φ ", " b bears the relation R to c ", etc. are meaningless, if a, b, c, R, φ are not of types fitting together.
https://en.wikipedia.org/wiki/History_o ... B6del_1944
*sigh*

The property of a formal system where True ↔ Provable applies *ONLY* to first order logics.
If your system is a first order logic THEN True ↔ Provable.

https://en.wikipedia.org/wiki/Second-order_logic
In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic.[1] Second-order logic is in turn extended by higher-order logic and type theory.

First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, also quantifies over relations.
https://en.wikipedia.org/wiki/Higher-order_logic
The term "higher-order logic", abbreviated as HOL, is commonly used to mean higher-order simple predicate logic. Here "simple" indicates that the underlying type theory is the theory of simple types, also called the simple theory of types (see Type theory). Leon Chwistek and Frank P. Ramsey proposed this as a simplification of the complicated and clumsy ramified theory of types specified in the Principia Mathematica by Alfred North Whitehead and Bertrand Russell. Simple types is nowadays sometimes also meant to exclude polymorphic and dependent types
Here's the easiest way to test if your system is a HOL: can you do nested for-loops?

Can you iterate over existents, can you sub-iterate over the properties/qualities of those existents, can you iterate over the sub-qualities of those existents?

Starting with a "tree" how do you analytically arrive at its constituent quantum particles.
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Simply defining Gödel Incompleteness away

Skepdick wrote: Thu Jun 18, 2020 7:51 am
The property of a formal system where True ↔ Provable applies *ONLY* to first order logics.
If your system is a first order logic THEN True ↔ Provable.
When we eliminate all of the extraneous complexity that has been defined into the concept of formal systems and boil them down to their barest essence then the "cheap-trick" of Gödel's 1931 Incompleteness Theorem and the Tarski Undefinability Theorem are easily abolished:

(1) A VALID formal proof only applies truth preserving operations to expressions of language.

(2) A SOUND formal proof is a VALID formal proof that begins with a set of premises that are somehow** known to be true.

**This is addressed by a whole separate conversation regarding the philosophical foundationalism of analytical knowledge.

When we do these two things incompleteness and undefinability cannot possibly occur.

Truth always propagates from the premises to the conclusion or the argument is INVALID and the conclusion is always true or the argument is UNSOUND.

Eodnhoj7
Posts: 6220
Joined: Mon Mar 13, 2017 3:18 am

### Re: Simply defining Gödel Incompleteness away

PeteOlcott wrote: Thu Jun 18, 2020 4:07 pm
Skepdick wrote: Thu Jun 18, 2020 7:51 am
The property of a formal system where True ↔ Provable applies *ONLY* to first order logics.
If your system is a first order logic THEN True ↔ Provable.
When we eliminate all of the extraneous complexity that has been defined into the concept of formal systems and boil them down to their barest essence then the "cheap-trick" of Gödel's 1931 Incompleteness Theorem and the Tarski Undefinability Theorem are easily abolished:

(1) A VALID formal proof only applies truth preserving operations to expressions of language.
Truth preserving operations would require the operations as self referential where any truth statement fundamentally references itself otherwise the statement would depend upon an extraneous statement that is not proven. This circularity would require the liar's paradox to be true thus necessitating contradiction.

(2) A SOUND formal proof is a VALID formal proof that begins with a set of premises that are somehow** known to be true.

This "somehow" would require a system beyond it to prove itself as true as well as something beyond it as proven as true as well. This leads to an infinite regress. Analysis always results in the breaking down of its constituent parts into further parts thus any analysticla knowledge, through a regress of parts, requires an unprovability.

**This is addressed by a whole separate conversation regarding the philosophical foundationalism of analytical knowledge.

When we do these two things incompleteness and undefinability cannot possibly occur.

Truth always propagates from the premises to the conclusion or the argument is INVALID and the conclusion is always true or the argument is UNSOUND.

PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Simply defining Gödel Incompleteness away

Eodnhoj7 wrote: Thu Jun 18, 2020 6:03 pm
PeteOlcott wrote: Thu Jun 18, 2020 4:07 pm
Skepdick wrote: Thu Jun 18, 2020 7:51 am
The property of a formal system where True ↔ Provable applies *ONLY* to first order logics.
If your system is a first order logic THEN True ↔ Provable.
When we eliminate all of the extraneous complexity that has been defined into the concept of formal systems and boil them down to their barest essence then the "cheap-trick" of Gödel's 1931 Incompleteness Theorem and the Tarski Undefinability Theorem are easily abolished:

(1) A VALID formal proof only applies truth preserving operations to expressions of language.
Truth preserving operations would require the operations as self referential where any truth statement fundamentally references itself otherwise the statement would depend upon an extraneous statement that is not proven. This circularity would require the liar's paradox to be true thus necessitating contradiction.

(2) A SOUND formal proof is a VALID formal proof that begins with a set of premises that are somehow** known to be true.

This "somehow" would require a system beyond it to prove itself as true as well as something beyond it as proven as true as well. This leads to an infinite regress. Analysis always results in the breaking down of its constituent parts into further parts thus any analysticla knowledge, through a regress of parts, requires an unprovability.

**This is addressed by a whole separate conversation regarding the philosophical foundationalism of analytical knowledge.

When we do these two things incompleteness and undefinability cannot possibly occur.

Truth always propagates from the premises to the conclusion or the argument is INVALID and the conclusion is always true or the argument is UNSOUND.

No you merely have the philosophical foundationalism of analytical knowledge incorrectly.
Eodnhoj7
Posts: 6220
Joined: Mon Mar 13, 2017 3:18 am

### Re: Simply defining Gödel Incompleteness away

PeteOlcott wrote: Thu Jun 18, 2020 7:34 pm
Eodnhoj7 wrote: Thu Jun 18, 2020 6:03 pm
PeteOlcott wrote: Thu Jun 18, 2020 4:07 pm

When we eliminate all of the extraneous complexity that has been defined into the concept of formal systems and boil them down to their barest essence then the "cheap-trick" of Gödel's 1931 Incompleteness Theorem and the Tarski Undefinability Theorem are easily abolished:

(1) A VALID formal proof only applies truth preserving operations to expressions of language.
Truth preserving operations would require the operations as self referential where any truth statement fundamentally references itself otherwise the statement would depend upon an extraneous statement that is not proven. This circularity would require the liar's paradox to be true thus necessitating contradiction.

(2) A SOUND formal proof is a VALID formal proof that begins with a set of premises that are somehow** known to be true.

This "somehow" would require a system beyond it to prove itself as true as well as something beyond it as proven as true as well. This leads to an infinite regress. Analysis always results in the breaking down of its constituent parts into further parts thus any analysticla knowledge, through a regress of parts, requires an unprovability.

**This is addressed by a whole separate conversation regarding the philosophical foundationalism of analytical knowledge.

When we do these two things incompleteness and undefinability cannot possibly occur.

Truth always propagates from the premises to the conclusion or the argument is INVALID and the conclusion is always true or the argument is UNSOUND.

No you merely have the philosophical foundationalism of analytical knowledge incorrectly.
The foundations are subject to further analysis, there are no foundations except the divergence of assertions. This is the only foundation of analysis. It is self referential as analysis is subject to analysis, the divergence of assertions can be observed under the further divergence of assertions.
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Simply defining Gödel Incompleteness away

Eodnhoj7 wrote: Thu Jun 18, 2020 9:05 pm
PeteOlcott wrote: Thu Jun 18, 2020 7:34 pm
Eodnhoj7 wrote: Thu Jun 18, 2020 6:03 pm
No you merely have the philosophical foundationalism of analytical knowledge incorrectly.
The foundations are subject to further analysis, there are no foundations except the divergence of assertions. This is the only foundation of analysis. It is self referential as analysis is subject to analysis, the divergence of assertions can be observed under the further divergence of assertions.
You failed to address any aspect of the actual issue of the foundational basis of analytical knowledge.

What is it that makes an expression of analytical knowledge true?
Eodnhoj7
Posts: 6220
Joined: Mon Mar 13, 2017 3:18 am

### Re: Simply defining Gödel Incompleteness away

PeteOlcott wrote: Thu Jun 18, 2020 9:11 pm
Eodnhoj7 wrote: Thu Jun 18, 2020 9:05 pm
PeteOlcott wrote: Thu Jun 18, 2020 7:34 pm

No you merely have the philosophical foundationalism of analytical knowledge incorrectly.
The foundations are subject to further analysis, there are no foundations except the divergence of assertions. This is the only foundation of analysis. It is self referential as analysis is subject to analysis, the divergence of assertions can be observed under the further divergence of assertions.
You failed to address any aspect of the actual issue of the foundational basis of analytical knowledge.

False, it is the divergence of assertions in continuums, no more no less.

What is it that makes an expression of analytical knowledge true?

All expressions are composed of grades of truth considering all assertions, baring those which contradict (yet the contradiction can be composed of true assertions thus even a contradiction has a truth value), are true through a recursion where the same thing is expressed in multiple ways.

For example 2+2=4 is true as 2+2 observes 4 as a variation of 2+2. However 2+2=5 is a contradiction for the very same reason, yet the assertions which 2+2=5 is composed of still express a recursive self referentiality under 1+1=2 and 1+1+1+1+1=5.

Skepdick
Posts: 5030
Joined: Fri Jun 14, 2019 11:16 am

### Re: Simply defining Gödel Incompleteness away

PeteOlcott wrote: Thu Jun 18, 2020 4:07 pm (1) A VALID formal proof only applies truth preserving operations to expressions of language.
You don't know what "truth" is in order to preserve it!
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Simply defining Gödel Incompleteness away

Skepdick wrote: Fri Jun 19, 2020 9:04 am
PeteOlcott wrote: Thu Jun 18, 2020 4:07 pm (1) A VALID formal proof only applies truth preserving operations to expressions of language.
You don't know what "truth" is in order to preserve it!
Sure I do. Analytical truth is membership in the body of analytical knowledge.
I also know how the axiomatic argument of the Münchhausen trilemma is overcome.
Eodnhoj7
Posts: 6220
Joined: Mon Mar 13, 2017 3:18 am

### Re: Simply defining Gödel Incompleteness away

PeteOlcott wrote: Fri Jun 19, 2020 3:15 pm
Skepdick wrote: Fri Jun 19, 2020 9:04 am
PeteOlcott wrote: Thu Jun 18, 2020 4:07 pm (1) A VALID formal proof only applies truth preserving operations to expressions of language.
You don't know what "truth" is in order to preserve it!
Sure I do. Analytical truth is membership in the body of analytical knowledge.
I also know how the axiomatic argument of the Münchhausen trilemma is overcome.
Then state it. The Munchauseen Trilemma is subject to itself, thus necessitating the trilemma to exist in in multiple forms where there is always a version of the trilemma that exists beyond the one addressed.
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Simply defining Gödel Incompleteness away

Eodnhoj7 wrote: Fri Jun 19, 2020 4:04 pm
PeteOlcott wrote: Fri Jun 19, 2020 3:15 pm
Skepdick wrote: Fri Jun 19, 2020 9:04 am
You don't know what "truth" is in order to preserve it!
Sure I do. Analytical truth is membership in the body of analytical knowledge.
I also know how the axiomatic argument of the Münchhausen trilemma is overcome.
Then state it. The Munchauseen Trilemma is subject to itself, thus necessitating the trilemma to exist in in multiple forms where there is always a version of the trilemma that exists beyond the one addressed.
Do you agree that membership in the body of analytical knowledge would indicate that an expression of language is true?