Simply defining Gödel Incompleteness and Tarski Undefinability away V15

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

Simply defining Gödel Incompleteness and Tarski Undefinability away V15

Post by PeteOlcott »

Defining Gödel Incompleteness Away

We can simply define Gödel 1931 Incompleteness away by redefining the meaning of the standard
definition of Incompleteness: A theory T is incomplete if and only if there is some sentence φ such
that (T ⊬ φ) and (T ⊬ ¬φ).

This definition construes the existence of self-contradictory expressions in a formal system as proof that this formal system is incomplete because self-contradictory expressions are neither provable nor disprovable in this formal system.

Since self-contradictory expressions are neither provable nor disprovable only because they are self-contradictory we could define them as unsound instead of defining the formal system as incomplete.
According to Wittgenstein:
'True in Russell's system' means, as was said: proved in Russell's
system; and 'false in Russell's system' means: the opposite has
been proved in Russell's system. (Wittgenstein 1983,118-119)
Formalized by Olcott as:
∀F ∈ Formal_Systems ∀C ∈ WFF(F) (((F⊢C)) ↔ True(F, C))
∀F ∈ Formal_Systems ∀C ∈ WFF(F) (((F⊢¬C)) ↔ False(F, C))

We had to add that the proofs referred to by Wittgenstein must be to theorem consequences thus requiring the axioms of formal proofs to act as a proxy for the true premises of sound deduction.

We simply construe a formal proof to theorem consequences as isomorphic to deduction from a sound argument to a true conclusion. This requires the theorems of formal systems to be construed as the true premises of sound deduction.

[Simplified Gödel Sentence]
https://plato.stanford.edu/entries/goed ... rIncTheCom

(G) F ⊢ GF ↔ ¬ProvF(⌈GF⌉). // Original
(G) F ⊢ GF ↔ ¬ProvF(GF). // Remove arithmetization

// Adapt syntax and quantify:
∃F ∈ Formal_Systems ∃G ∈ WFF(F) (G ↔ (F ⊬ G))

When a WFF expresses that it is logically equivalent to its own unprovability: G ↔ (F ⊬ G) this expression is self-contradictory thus unsatisfiable. If we could prove that a sentence that asserts it is logically equivalent to its own unprovability is true this contradicts its assertion therefore we cannot prove that it is true.

Likewise with its negation: G ↔ (F ⊬ ¬G). If we could prove that a sentence that asserts it is logically equivalent to its own unprovability is false this contradicts its assertion therefore we cannot prove that it is false.

The conventional definition of incompleteness: A theory T is incomplete if and only if there is some sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ).

As we can see from the above neither (F ⊬ G) not (F ⊬ ¬G) can be satisfied only because they are both self contradictory. Because they are self-contradictory they meet the definition of Incompleteness. Within the sound deductive inference model unprovable expressions of language are simply construed as unsound arguments thus untrue.
Satisfiability
A formula is satisfiable if it is possible to find an interpretation
(model) that makes the formula true.
https://en.wikipedia.org/wiki/Satisfiability

Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a
formal language.
https://en.wikipedia.org/wiki/Interpretation_(logic)
Model theory
A model of a theory is a structure (e.g. an interpretation)
that satisfies the sentences of that theory.
https://en.wikipedia.org/wiki/Model_theory
Wittgenstein, Ludwig 1983. Remarks on the Foundations of Mathematics (Appendix III), 118-119. Cambridge, Massachusetts and London, England: The MIT Press http://www.liarparadox.org/Wittgenstein.pdf

Defining Gödel Incompleteness Away
https://www.researchgate.net/publicatio ... eness_Away

--
Copyright 2020 PL Olcott
Eodnhoj7
Posts: 6220
Joined: Mon Mar 13, 2017 3:18 am

Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V15

Post by Eodnhoj7 »

PeteOlcott wrote: Sat Jun 27, 2020 10:59 pm Defining Gödel Incompleteness Away

1. We can simply define Gödel 1931 Incompleteness away by redefining the meaning of the standard
definition of Incompleteness: A theory T is incomplete if and only if there is some sentence φ such
that (T ⊬ φ) and (T ⊬ ¬φ).



2. This definition construes the existence of self-contradictory expressions in a formal system as proof that this formal system is incomplete because self-contradictory expressions are neither provable nor disprovable in this formal system.

3. Since self-contradictory expressions are neither provable nor disprovable only because they are self-contradictory we could define them as unsound instead of defining the formal system as incomplete.
According to Wittgenstein:
'True in Russell's system' means, as was said: proved in Russell's
system; and 'false in Russell's system' means: the opposite has
been proved in Russell's system. (Wittgenstein 1983,118-119)
Formalized by Olcott as:
∀F ∈ Formal_Systems ∀C ∈ WFF(F) (((F⊢C)) ↔ True(F, C))
∀F ∈ Formal_Systems ∀C ∈ WFF(F) (((F⊢¬C)) ↔ False(F, C))

We had to add that the proofs referred to by Wittgenstein must be to theorem consequences thus requiring the axioms of formal proofs to act as a proxy for the true premises of sound deduction.

We simply construe a formal proof to theorem consequences as isomorphic to deduction from a sound argument to a true conclusion. This requires the theorems of formal systems to be construed as the true premises of sound deduction.

[Simplified Gödel Sentence]
https://plato.stanford.edu/entries/goed ... rIncTheCom

(G) F ⊢ GF ↔ ¬ProvF(⌈GF⌉). // Original
(G) F ⊢ GF ↔ ¬ProvF(GF). // Remove arithmetization

// Adapt syntax and quantify:
∃F ∈ Formal_Systems ∃G ∈ WFF(F) (G ↔ (F ⊬ G))

When a WFF expresses that it is logically equivalent to its own unprovability: G ↔ (F ⊬ G) this expression is self-contradictory thus unsatisfiable. If we could prove that a sentence that asserts it is logically equivalent to its own unprovability is true this contradicts its assertion therefore we cannot prove that it is true.

Likewise with its negation: G ↔ (F ⊬ ¬G). If we could prove that a sentence that asserts it is logically equivalent to its own unprovability is false this contradicts its assertion therefore we cannot prove that it is false.

The conventional definition of incompleteness: A theory T is incomplete if and only if there is some sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ).

As we can see from the above neither (F ⊬ G) not (F ⊬ ¬G) can be satisfied only because they are both self contradictory. Because they are self-contradictory they meet the definition of Incompleteness. Within the sound deductive inference model unprovable expressions of language are simply construed as unsound arguments thus untrue.
Satisfiability
A formula is satisfiable if it is possible to find an interpretation
(model) that makes the formula true.
https://en.wikipedia.org/wiki/Satisfiability

Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a
formal language.
https://en.wikipedia.org/wiki/Interpretation_(logic)
Model theory
A model of a theory is a structure (e.g. an interpretation)
that satisfies the sentences of that theory.
https://en.wikipedia.org/wiki/Model_theory
Wittgenstein, Ludwig 1983. Remarks on the Foundations of Mathematics (Appendix III), 118-119. Cambridge, Massachusetts and London, England: The MIT Press http://www.liarparadox.org/Wittgenstein.pdf

Defining Gödel Incompleteness Away
https://www.researchgate.net/publicatio ... eness_Away

--
Copyright 2020 PL Olcott
1. [/color]We can simply define Gödel 1931 Incompleteness away by redefining the meaning of the standard definition of Incompleteness: A theory T is incomplete if and only if there is some sentence φ such
that (T ⊬ φ) and (T ⊬ ¬φ).

The redefinition of incompleteness results in a system outside of the incompleteness theorem which in itself is undefined and assumed. The incompleteness theorem is in itself subject to it's own nature as incomplete.






color=#BF0000]2. [/color]This definition construes the existence of self-contradictory expressions in a formal system as proof that this formal system is incomplete because self-contradictory expressions are neither provable nor disprovable in this formal system.

[This requires a definition of "proof" which is neither provable or disprovable given any proof of "proof" is a circular reasoning thus leaving proof as strictly defined by variable beyond it which are neither provable or disprovable as well.


3. Since self-contradictory expressions are neither provable nor disprovable only because they are self-contradictory we could define them as unsound instead of defining the formal system as incomplete.

You would have to then defined soundness by an outside framework which is unsound. For soundness to describe proof would require proof to describe soundness and a circularity occurs. This circularity expands when you add in "incompleteness".

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

Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V15

Post by PeteOlcott »

Eodnhoj7 wrote: Sat Jun 27, 2020 11:35 pm You would have to then defined soundness by an outside framework which is unsound. For soundness to describe proof would require proof to describe soundness and a circularity occurs. This circularity expands when you add in "incompleteness".

Within the isomorphism between formal proofs and valid deduction
Formal-Proof.Unprovable(F, X) ≅ Valid-Deduction.Invalid-Argument(F, X).
Eodnhoj7
Posts: 6220
Joined: Mon Mar 13, 2017 3:18 am

Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V15

Post by Eodnhoj7 »

PeteOlcott wrote: Sat Jun 27, 2020 11:45 pm
Eodnhoj7 wrote: Sat Jun 27, 2020 11:35 pm You would have to then defined soundness by an outside framework which is unsound. For soundness to describe proof would require proof to describe soundness and a circularity occurs. This circularity expands when you add in "incompleteness".

Within the isomorphism between formal proofs and valid deduction
Formal-Proof.Unprovable(F, X) ≅ Valid-Deduction.Invalid-Argument(F, X).
You ignored my first two points.

The isomorphism is the inversion from one framework into a symmetrical opposite. This inversion requires both frameworks to be strictly assumed "as is" where outside frameworks must extend from each respective theory in order to define them. This continual extension of frameworks necessitates eventually one framework in being undefined. It does not escape the nature of assumption and infinite regress in justifying said frameworks, nor does it eliminate Godel's theorem.
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V15

Post by PeteOlcott »

Eodnhoj7 wrote: Sun Jun 28, 2020 12:11 am
PeteOlcott wrote: Sat Jun 27, 2020 11:45 pm
Eodnhoj7 wrote: Sat Jun 27, 2020 11:35 pm You would have to then defined soundness by an outside framework which is unsound. For soundness to describe proof would require proof to describe soundness and a circularity occurs. This circularity expands when you add in "incompleteness".

Within the isomorphism between formal proofs and valid deduction
Formal-Proof.Unprovable(F, X) ≅ Valid-Deduction.Invalid-Argument(F, X).
You ignored my first two points.

The isomorphism is the inversion from one framework into a symmetrical opposite. This inversion requires both frameworks to be strictly assumed "as is" where outside frameworks must extend from each respective theory in order to define them. This continual extension of frameworks necessitates eventually one framework in being undefined. It does not escape the nature of assumption and infinite regress in justifying said frameworks, nor does it eliminate Godel's theorem.
https://en.wikipedia.org/wiki/Isomorphi ... 22shape%22.

Sound and valid deductive inference map to formal proofs and formal proofs to theorem consequences respectively and the inverse mapping too.
Eodnhoj7
Posts: 6220
Joined: Mon Mar 13, 2017 3:18 am

Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V15

Post by Eodnhoj7 »

PeteOlcott wrote: Sun Jun 28, 2020 12:17 am
Eodnhoj7 wrote: Sun Jun 28, 2020 12:11 am
PeteOlcott wrote: Sat Jun 27, 2020 11:45 pm

Within the isomorphism between formal proofs and valid deduction
Formal-Proof.Unprovable(F, X) ≅ Valid-Deduction.Invalid-Argument(F, X).
You ignored my first two points.

The isomorphism is the inversion from one framework into a symmetrical opposite. This inversion requires both frameworks to be strictly assumed "as is" where outside frameworks must extend from each respective theory in order to define them. This continual extension of frameworks necessitates eventually one framework in being undefined. It does not escape the nature of assumption and infinite regress in justifying said frameworks, nor does it eliminate Godel's theorem.
https://en.wikipedia.org/wiki/Isomorphi ... 22shape%22.
Each respective framework needs to be defined by a framework beyond it for the isomorphism to occur. Dually each framework needs to be assumed. In each framework, P and Q, the Munchauseen trillema occurs.

P=P and Q=Q is not only circular but each assertion is assumed. Third, each framework both as composed of subframework and existing as subframeworks leads to an infinite regress. You cannot escape the trillema.
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V15

Post by PeteOlcott »

Eodnhoj7 wrote: Sun Jun 28, 2020 12:22 am
Each respective framework needs to be defined by a framework beyond it for the isomorphism to occur. Dually each framework needs to be assumed. In each framework, P and Q, the Munchauseen trillema occurs.

P=P and Q=Q is not only circular but each assertion is assumed. Third, each framework both as composed of subframework and existing as subframeworks leads to an infinite regress. You cannot escape the trillema.
A language L on Σ is said to be recursive if there exists a Turing
machine M that accepts L and halts on every w in Σ+. In other words, a
language is recursive if and only if there exists a membership algorithm
for it. (Linz 1990:288).
Linz, Peter 1990. An Introduction to Formal Languages and Automata.
Lexington/Toronto: D. C. Heath and Company, 288.

The axiomatic argument of the Munchauseen trillema is very easily avoided by defining recursive language operating on finite strings that has two membership algorithms: (a) Provable(F,X) and (b) Theorem(F,X) precisely corresponding to their conventional usage in symbolic logic. Isomorphism is proved in that the prior two membership algorithms are merely different names for Valid(F,X) and Sound(F,X), respectively.
Eodnhoj7
Posts: 6220
Joined: Mon Mar 13, 2017 3:18 am

Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V15

Post by Eodnhoj7 »

PeteOlcott wrote: Sun Jun 28, 2020 12:46 am
Eodnhoj7 wrote: Sun Jun 28, 2020 12:22 am
Each respective framework needs to be defined by a framework beyond it for the isomorphism to occur. Dually each framework needs to be assumed. In each framework, P and Q, the Munchauseen trillema occurs.

P=P and Q=Q is not only circular but each assertion is assumed. Third, each framework both as composed of subframework and existing as subframeworks leads to an infinite regress. You cannot escape the trillema.
A language L on Σ is said to be recursive if there exists a Turing
machine M that accepts L and halts on every w in Σ+. In other words, a
language is recursive if and only if there exists a membership algorithm
for it. (Linz 1990:288).
Linz, Peter 1990. An Introduction to Formal Languages and Automata.
Lexington/Toronto: D. C. Heath and Company, 288.

The axiomatic argument of the Munchauseen trillema is very easily avoided by defining recursive language operating on finite strings that has two membership algorithms: (a) Provable(F,X) and (b) Theorem(F,X) precisely corresponding to their conventional usage in symbolic logic. Isomorphism is proved in that the prior two membership algorithms are merely different names for Valid(F,X) and Sound(F,X), respectively.
Redefining recursive languages results in a regression where "recursion" is defined by some undefined phenomenon beyond it. You are merely assuming a definition of recursion and assuming other's know what you are talking about.

Building a platform on isomorphism alone does not negate the infinite regress, specifically your regress of terms as strings.

Corresponding to conventual usage demands a group agreed upon manner in how the symbols are used thus how they are assumed.

You cannot escape that necessity of regress in definition nor how all phenomenon are assumed.

You are still ignoring these two points:

The redefinition of incompleteness results in a system outside of the incompleteness theorem which in itself is undefined and assumed. The incompleteness theorem is in itself subject to its own nature as incomplete.



color=#BF0000]2. [/color]This definition construes the existence of self-contradictory expressions in a formal system as proof that this formal system is incomplete because self-contradictory expressions are neither provable nor disprovable in this formal system.

[This requires a definition of "proof" which is neither provable or disprovable given any proof of "proof" is a circular reasoning thus leaving proof as strictly defined by variable beyond it which are neither provable or disprovable as well.
Eodnhoj7
Posts: 6220
Joined: Mon Mar 13, 2017 3:18 am

Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V15

Post by Eodnhoj7 »

PeteOlcott wrote: Sun Jun 28, 2020 12:46 am
Eodnhoj7 wrote: Sun Jun 28, 2020 12:22 am
Each respective framework needs to be defined by a framework beyond it for the isomorphism to occur. Dually each framework needs to be assumed. In each framework, P and Q, the Munchauseen trillema occurs.

P=P and Q=Q is not only circular but each assertion is assumed. Third, each framework both as composed of subframework and existing as subframeworks leads to an infinite regress. You cannot escape the trillema.
A language L on Σ is said to be recursive if there exists a Turing
machine M that accepts L and halts on every w in Σ+. In other words, a
language is recursive if and only if there exists a membership algorithm
for it. (Linz 1990:288).
Linz, Peter 1990. An Introduction to Formal Languages and Automata.
Lexington/Toronto: D. C. Heath and Company, 288.

The axiomatic argument of the Munchauseen trillema is very easily avoided by defining recursive language operating on finite strings that has two membership algorithms: (a) Provable(F,X) and (b) Theorem(F,X) precisely corresponding to their conventional usage in symbolic logic. Isomorphism is proved in that the prior two membership algorithms are merely different names for Valid(F,X) and Sound(F,X), respectively.
1. Does your system requires P=P as a foundational axiom?

2. Define "proof".
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V15

Post by PeteOlcott »

Eodnhoj7 wrote: Sun Jun 28, 2020 2:39 am
PeteOlcott wrote: Sun Jun 28, 2020 12:46 am
Eodnhoj7 wrote: Sun Jun 28, 2020 12:22 am
Each respective framework needs to be defined by a framework beyond it for the isomorphism to occur. Dually each framework needs to be assumed. In each framework, P and Q, the Munchauseen trillema occurs.

P=P and Q=Q is not only circular but each assertion is assumed. Third, each framework both as composed of subframework and existing as subframeworks leads to an infinite regress. You cannot escape the trillema.
A language L on Σ is said to be recursive if there exists a Turing
machine M that accepts L and halts on every w in Σ+. In other words, a
language is recursive if and only if there exists a membership algorithm
for it. (Linz 1990:288).
Linz, Peter 1990. An Introduction to Formal Languages and Automata.
Lexington/Toronto: D. C. Heath and Company, 288.

The axiomatic argument of the Munchauseen trillema is very easily avoided by defining recursive language operating on finite strings that has two membership algorithms: (a) Provable(F,X) and (b) Theorem(F,X) precisely corresponding to their conventional usage in symbolic logic. Isomorphism is proved in that the prior two membership algorithms are merely different names for Valid(F,X) and Sound(F,X), respectively.
1. Does your system requires P=P as a foundational axiom?

2. Define "proof".
The most generic example of a formal system is finite string input produces Boolean output indicating that a decision has been made.
Eodnhoj7
Posts: 6220
Joined: Mon Mar 13, 2017 3:18 am

Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V15

Post by Eodnhoj7 »

PeteOlcott wrote: Sun Jun 28, 2020 7:37 am
Eodnhoj7 wrote: Sun Jun 28, 2020 2:39 am
PeteOlcott wrote: Sun Jun 28, 2020 12:46 am


Linz, Peter 1990. An Introduction to Formal Languages and Automata.
Lexington/Toronto: D. C. Heath and Company, 288.

The axiomatic argument of the Munchauseen trillema is very easily avoided by defining recursive language operating on finite strings that has two membership algorithms: (a) Provable(F,X) and (b) Theorem(F,X) precisely corresponding to their conventional usage in symbolic logic. Isomorphism is proved in that the prior two membership algorithms are merely different names for Valid(F,X) and Sound(F,X), respectively.
1. Does your system requires P=P as a foundational axiom?

2. Define "proof".
The most generic example of a formal system is finite string input produces Boolean output indicating that a decision has been made.
1. You ignored the first question.

2. I asked "what is proof?" not "what is a formal system?". Regardless, what is a finite string input and output other than a tautology where one thing is expressed under a new variation as a string of truth values?
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V15

Post by PeteOlcott »

Eodnhoj7 wrote: Sun Jun 28, 2020 4:41 pm
PeteOlcott wrote: Sun Jun 28, 2020 7:37 am
Eodnhoj7 wrote: Sun Jun 28, 2020 2:39 am

1. Does your system requires P=P as a foundational axiom?

2. Define "proof".
The most generic example of a formal system is finite string input produces Boolean output indicating that a decision has been made.
1. You ignored the first question.

2. I asked "what is proof?" not "what is a formal system?". Regardless, what is a finite string input and output other than a tautology where one thing is expressed under a new variation as a string of truth values?
The formal proof in such a formal system would simply be the sequential steps of the membership algorithm.

// Simplest possible formal system having a single axiom.
// (implemented as a membership algorithm)
void main(int argc, char *argv[])
{
if (argc != 2)
printf("Input a finite string such as: \"Cats are animals\"\n");
if (argc == 2)
{
std::string WFF = CompressSpaces(argv[1]);
std::transform(WFF.begin(), WFF.end(), WFF.begin(), ::toupper);
if (WFF == "CATS ARE ANIMALS")
printf("\"%s\": is True\n", argv[1]);
else
printf("\"%s\": is False\n", argv[1]);
}
}
Eodnhoj7
Posts: 6220
Joined: Mon Mar 13, 2017 3:18 am

Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V15

Post by Eodnhoj7 »

PeteOlcott wrote: Sun Jun 28, 2020 7:53 pm
Eodnhoj7 wrote: Sun Jun 28, 2020 4:41 pm
PeteOlcott wrote: Sun Jun 28, 2020 7:37 am

The most generic example of a formal system is finite string input produces Boolean output indicating that a decision has been made.
1. You ignored the first question.

2. I asked "what is proof?" not "what is a formal system?". Regardless, what is a finite string input and output other than a tautology where one thing is expressed under a new variation as a string of truth values?
The formal proof in such a formal system would simply be the sequential steps of the membership algorithm.

// Simplest possible formal system having a single axiom.
// (implemented as a membership algorithm)
void main(int argc, char *argv[])
{
if (argc != 2)
printf("Input a finite string such as: \"Cats are animals\"\n");
if (argc == 2)
{
std::string WFF = CompressSpaces(argv[1]);
std::transform(WFF.begin(), WFF.end(), WFF.begin(), ::toupper);
if (WFF == "CATS ARE ANIMALS")
printf("\"%s\": is True\n", argv[1]);
else
printf("\"%s\": is False\n", argv[1]);
}
}
This sequence leads to an infinite regress if the algorithm is to continue considering if would be both defined as a subset and composed of subsets. These subsets would have to continue. Dually the algorithm is assumed, and nothing outside of it considers itself as proof for the algorithm unless you accept a circularity where the algorithm is justified by proof and the proof is justified by the algorithm.

You still have not answered the question, is your logic grounded in the basic identity law of P=P (and P =/= -P and P v -P by default)?
PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V15

Post by PeteOlcott »

Eodnhoj7 wrote: Mon Jun 29, 2020 4:19 am
PeteOlcott wrote: Sun Jun 28, 2020 7:53 pm
Eodnhoj7 wrote: Sun Jun 28, 2020 4:41 pm

1. You ignored the first question.

2. I asked "what is proof?" not "what is a formal system?". Regardless, what is a finite string input and output other than a tautology where one thing is expressed under a new variation as a string of truth values?
The formal proof in such a formal system would simply be the sequential steps of the membership algorithm.

// Simplest possible formal system having a single axiom.
// (implemented as a membership algorithm)
void main(int argc, char *argv[])
{
if (argc != 2)
printf("Input a finite string such as: \"Cats are animals\"\n");
if (argc == 2)
{
std::string WFF = CompressSpaces(argv[1]);
std::transform(WFF.begin(), WFF.end(), WFF.begin(), ::toupper);
if (WFF == "CATS ARE ANIMALS")
printf("\"%s\": is True\n", argv[1]);
else
printf("\"%s\": is False\n", argv[1]);
}
}
This sequence leads to an infinite regress if the algorithm is to continue considering if would be both defined as a subset and composed of subsets. These subsets would have to continue.
That is ridiculous. There are not an infinite number of different categories of types of things in the world.
Eodnhoj7
Posts: 6220
Joined: Mon Mar 13, 2017 3:18 am

Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V15

Post by Eodnhoj7 »

PeteOlcott wrote: Mon Jun 29, 2020 4:30 pm
Eodnhoj7 wrote: Mon Jun 29, 2020 4:19 am
PeteOlcott wrote: Sun Jun 28, 2020 7:53 pm

The formal proof in such a formal system would simply be the sequential steps of the membership algorithm.

// Simplest possible formal system having a single axiom.
// (implemented as a membership algorithm)
void main(int argc, char *argv[])
{
if (argc != 2)
printf("Input a finite string such as: \"Cats are animals\"\n");
if (argc == 2)
{
std::string WFF = CompressSpaces(argv[1]);
std::transform(WFF.begin(), WFF.end(), WFF.begin(), ::toupper);
if (WFF == "CATS ARE ANIMALS")
printf("\"%s\": is True\n", argv[1]);
else
printf("\"%s\": is False\n", argv[1]);
}
}
This sequence leads to an infinite regress if the algorithm is to continue considering if would be both defined as a subset and composed of subsets. These subsets would have to continue.
That is ridiculous. There are not an infinite number of different categories of types of things in the world.
You still ignored my question of P=P.

The continual change, as the diverging of one category into another, necessitates a finite number approaching infinity. This finite number is always changing through a regress to infinity, ie infinite regress, considering the change itself is unending. Infinite, never ending change, results in an infinite regress.
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