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Re: Terse defence of Godel (is refuted)
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peteolcott
May 5, 2019, 11:20:21 AM5/5/19
to
On 5/5/2019 6:48 AM, xilog wrote:
> If I understand you correctly, your supposed refutation of Godel's incompleteness result rest upon redefining the notion of completeness.
>
> There are two problems here.
> The first is that if you re-define the terms and obtain a different result, you are not refuting the original, you are changing the subject. Even if you were correct, you would not have refuted Godel.
>
> The second is that, if I recall correctly, "incompleteness" is not a formal part of Godel's theorem, in fact what he proves is that in any formalisation of arithmetic (satisfying various conditions) there will always be formally undecidable sentences, i.e. ones which are neither provable nor refutable.
>
> Of course your method of redefinition could be applied to this, by defining "sentence" such that a formula is only a sentence if it is formally undecidable.
> But that would not refute model either, it would refute only a conjecture (as redefined by you) which Godel would never have considered proving.
>
Technically I am refuting the notion of formal proof that Gödel
used as insufficiently expressive to detect and reject semantic error.
Within the standard conventional sound deductive inference model
every conventionally undecidable sentence is excluded from the set
of deductively sound conclusions.
Deductively Sound Formal Proofs
https://www.researchgate.net/publication/332864362_Deductively_Sound_Formal_Proofs
--
Copyright 2019 Pete Olcott All rights reserved
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein
> If I understand you correctly, your supposed refutation of Godel's incompleteness result rest upon redefining the notion of completeness.
>
> There are two problems here.
> The first is that if you re-define the terms and obtain a different result, you are not refuting the original, you are changing the subject. Even if you were correct, you would not have refuted Godel.
>
> The second is that, if I recall correctly, "incompleteness" is not a formal part of Godel's theorem, in fact what he proves is that in any formalisation of arithmetic (satisfying various conditions) there will always be formally undecidable sentences, i.e. ones which are neither provable nor refutable.
>
> Of course your method of redefinition could be applied to this, by defining "sentence" such that a formula is only a sentence if it is formally undecidable.
> But that would not refute model either, it would refute only a conjecture (as redefined by you) which Godel would never have considered proving.
>
Technically I am refuting the notion of formal proof that Gödel
used as insufficiently expressive to detect and reject semantic error.
Within the standard conventional sound deductive inference model
every conventionally undecidable sentence is excluded from the set
of deductively sound conclusions.
Deductively Sound Formal Proofs
https://www.researchgate.net/publication/332864362_Deductively_Sound_Formal_Proofs
--
Copyright 2019 Pete Olcott All rights reserved
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein
peteolcott
May 7, 2019, 11:58:41 AM5/7/19
to
On 5/7/2019 5:54 AM, xilog wrote:
> You do yourself a disservice by claiming to refute Godel when in fact you are disagreeing with his terminology, which is well understood by logicians.
>
In the sound deductive inference model True(x) is defined as being provable
on the basis of true premises, thus it is impossible for Truth to diverge
from provability, therefore the third line of Tarski's proof is proven false:
(3) x ∉ Pr if and only if x ∈ Tr
Causing the rest of the whole proof to fail.
As far as the 1931 Incompleteness Theorem goes most people are like a
fish out-of-water when we take away Gödel numbers and diagonalization.
They have only learned these things by rote and never truly understood
the actual underlying mathematical relations involved.
Introduction to Mathematical logic Sixth edition Elliott Mendelson (2015)
1.4 An Axiom System for the Propositional Calculus page 28
A wf C is said to be a consequence in S of a set Γ of wfs if and only if there is a
sequence B1, …, Bk of wfs such that C is Bk and, for each i, either Bi is an axiom
or Bi is in Γ, or Bi is a direct consequence by some rule of inference of some of
the preceding wfs in the sequence. Such a sequence is called a proof (or deduction)
of C from Γ. The members of Γ are called the hypotheses or premisses of the proof.
We use Γ ⊢ C as an abbreviation for “C is a consequence of Γ”...
The first thing to know is that any "proof" that uses Gödel numbers and
diagonalization is not really a mathematical proof at all because it lacks:
[a sequence B1, …, Bk of wfs such that C is Bk] of inference steps. Instead
the Gödel numbers / diagonalization process leaps to the conclusion skipping
all of the intervening steps.
To see what is truly going on with the Incompleteness Theorem requires us
to recreate it without Gödel numbers or diagonalization.
First incompleteness theorem
Any consistent formal system F within which a certain amount
of elementary arithmetic can be carried out is incomplete;
i.e., there are statements of the language of F which can
neither be proved nor disproved in F. (Raatikainen 2018)
https://plato.stanford.edu/entries/goedel-incompleteness/#Int
The above English paragraph does sum up the essence of the conclusion
of the 1931 Incompleteness Theorem. The following logic sentence
does formalize the above English paragraph without Gödel numbers:
∃F ∈ Formal_System ∃G ∈ Closed_WFF(F) (G ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))
Now we have the formalized essence of the conclusion of the 1931
Incompleteness Theorem with 100,000-fold of extraneous complexity
stripped away.
All that we have to do is see if there is any G of F that is materially
equivalent to its own unprovability and its own irrefutability.
If either one of these fail, then the 1931 Incompleteness theorem fails.
I will leave it here because I probably lost you when I said no Gödel
numbers. Once you understand (the mandatory prerequisites) up to this
point I will proceed.
> You do yourself a disservice by claiming to refute Godel when in fact you are disagreeing with his terminology, which is well understood by logicians.
>
In the sound deductive inference model True(x) is defined as being provable
on the basis of true premises, thus it is impossible for Truth to diverge
from provability, therefore the third line of Tarski's proof is proven false:
(3) x ∉ Pr if and only if x ∈ Tr
Causing the rest of the whole proof to fail.
As far as the 1931 Incompleteness Theorem goes most people are like a
fish out-of-water when we take away Gödel numbers and diagonalization.
They have only learned these things by rote and never truly understood
the actual underlying mathematical relations involved.
Introduction to Mathematical logic Sixth edition Elliott Mendelson (2015)
1.4 An Axiom System for the Propositional Calculus page 28
A wf C is said to be a consequence in S of a set Γ of wfs if and only if there is a
sequence B1, …, Bk of wfs such that C is Bk and, for each i, either Bi is an axiom
or Bi is in Γ, or Bi is a direct consequence by some rule of inference of some of
the preceding wfs in the sequence. Such a sequence is called a proof (or deduction)
of C from Γ. The members of Γ are called the hypotheses or premisses of the proof.
We use Γ ⊢ C as an abbreviation for “C is a consequence of Γ”...
The first thing to know is that any "proof" that uses Gödel numbers and
diagonalization is not really a mathematical proof at all because it lacks:
[a sequence B1, …, Bk of wfs such that C is Bk] of inference steps. Instead
the Gödel numbers / diagonalization process leaps to the conclusion skipping
all of the intervening steps.
To see what is truly going on with the Incompleteness Theorem requires us
to recreate it without Gödel numbers or diagonalization.
First incompleteness theorem
Any consistent formal system F within which a certain amount
of elementary arithmetic can be carried out is incomplete;
i.e., there are statements of the language of F which can
neither be proved nor disproved in F. (Raatikainen 2018)
https://plato.stanford.edu/entries/goedel-incompleteness/#Int
The above English paragraph does sum up the essence of the conclusion
of the 1931 Incompleteness Theorem. The following logic sentence
does formalize the above English paragraph without Gödel numbers:
∃F ∈ Formal_System ∃G ∈ Closed_WFF(F) (G ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))
Now we have the formalized essence of the conclusion of the 1931
Incompleteness Theorem with 100,000-fold of extraneous complexity
stripped away.
All that we have to do is see if there is any G of F that is materially
equivalent to its own unprovability and its own irrefutability.
If either one of these fail, then the 1931 Incompleteness theorem fails.
I will leave it here because I probably lost you when I said no Gödel
numbers. Once you understand (the mandatory prerequisites) up to this
point I will proceed.
peteolcott
May 7, 2019, 10:55:26 PM5/7/19
to
On 5/7/2019 5:38 PM, xilog wrote:
> I will pass over your comment on Tarski since I lack a specific reference to the theorem of which you speak.
> As to Godel:
>
> You must bear in mind the distinction between a strictly formal mathematical proof
> of the kind addressed by Mendelson and the normal standard of mathematical proof
> used in the development of mathematics and its publication in journals.
> Neither in mathematics in general nor in mathematical logic is it usual for mathematicians
> to prove their results formally, though it is held that a rigorous (but informal) proof,
> can be converted into a formal proof (at the cost of considerable additional labour).
> Mathematical logicians mainly reason _about_ formal systems rather than _in_ formal systems.
>
> Godel's incompleteness theorem is an informal proof about formal axiomatisations of
> arithmetic, which could be (and in fact has been) rendered fully formally.
> Such a formal rendering of Godel's incompleteness theorem would fully satisfy
> criteria similar to those given by Mendelson for the propositional calculus (though
> the system would be a predicate calculus, and somewhat more complicated).
> It would nevertheless still require the machinery of Godel numbering
> (or some equivalent method) in order to be able to talk about formal systems
> in a language which is strictly capable only of talking about numbers.
>
> The use of godel numbering is essential, and does not mark a departure from
> the notion of formal system described by Mendelson.
>
> Firstly, of course, there are undefined terms here, viz. "Formal_System" and "Closed_WFF".
> There is a difficulty in formalising the definition of these concepts in the language of
> arithmetic (which is the language in which Godel's theorem is to be understood), without
> using Godel numbers, because the only things which exist in that context are numbers.
Ah but we don't have to. Gödel's results apply to every formal system capable
of expressing arithmetic AND ABOVE. In other words they would apply equally
to a knowledge ontology of the sum total of all human knowledge.
In other words Gödel and Tarski "prove" that a complete and consistent
notion of truth itself is impossible.
All that we have to do to define a complete and consistent notion of
truth is to realize that truth itself is nothing more than a set of
mutually interlocking tautologies.
In logic, a tautology (from the Greek word ταυτολογία) is a formula or
assertion that is true in every possible interpretation.
How can tautologies possibly fail to hold?
If tautologies cannot possibly fail to hold, then Gödel and Tarski are WRONG !
This is the big picture grande scheme that unequivocally proves
that Tarski and Gödel are necessarily incorrect.
> I will pass over your comment on Tarski since I lack a specific reference to the theorem of which you speak.
> As to Godel:
>
> On 07/05/2019 16:58, peteolcott wrote:
>>
>> As far as the 1931 Incompleteness Theorem goes most people are like a
>> fish out-of-water when we take away Gödel numbers and diagonalization.
>> They have only learned these things by rote and never truly understood
>> the actual underlying mathematical relations involved.
>>
>> Introduction to Mathematical logic Sixth edition Elliott Mendelson (2015)
>> 1.4 An Axiom System for the Propositional Calculus page 28
>> A wf C is said to be a consequence in S of a set Γ of wfs if and only if there is a
>> sequence B1, …, Bk of wfs such that C is Bk and, for each i, either Bi is an axiom
>> or Bi is in Γ, or Bi is a direct consequence by some rule of inference of some of
>> the preceding wfs in the sequence. Such a sequence is called a proof (or deduction)
>> of C from Γ. The members of Γ are called the hypotheses or premisses of the proof.
>> We use Γ ⊢ C as an abbreviation for “C is a consequence of Γ”...
>>
>> The first thing to know is that any "proof" that uses Gödel numbers and
>> diagonalization is not really a mathematical proof at all because it lacks:
>> [a sequence B1, …, Bk of wfs such that C is Bk] of inference steps. Instead
>> the Gödel numbers / diagonalization process leaps to the conclusion skipping
>> all of the intervening steps.
>
> This is incorrect.
> On 07/05/2019 16:58, peteolcott wrote:
>>
>> As far as the 1931 Incompleteness Theorem goes most people are like a
>> fish out-of-water when we take away Gödel numbers and diagonalization.
>> They have only learned these things by rote and never truly understood
>> the actual underlying mathematical relations involved.
>>
>> Introduction to Mathematical logic Sixth edition Elliott Mendelson (2015)
>> 1.4 An Axiom System for the Propositional Calculus page 28
>> A wf C is said to be a consequence in S of a set Γ of wfs if and only if there is a
>> sequence B1, …, Bk of wfs such that C is Bk and, for each i, either Bi is an axiom
>> or Bi is in Γ, or Bi is a direct consequence by some rule of inference of some of
>> the preceding wfs in the sequence. Such a sequence is called a proof (or deduction)
>> of C from Γ. The members of Γ are called the hypotheses or premisses of the proof.
>> We use Γ ⊢ C as an abbreviation for “C is a consequence of Γ”...
>>
>> The first thing to know is that any "proof" that uses Gödel numbers and
>> diagonalization is not really a mathematical proof at all because it lacks:
>> [a sequence B1, …, Bk of wfs such that C is Bk] of inference steps. Instead
>> the Gödel numbers / diagonalization process leaps to the conclusion skipping
>> all of the intervening steps.
>
>
> You must bear in mind the distinction between a strictly formal mathematical proof
> of the kind addressed by Mendelson and the normal standard of mathematical proof
> used in the development of mathematics and its publication in journals.
> Neither in mathematics in general nor in mathematical logic is it usual for mathematicians
> to prove their results formally, though it is held that a rigorous (but informal) proof,
> can be converted into a formal proof (at the cost of considerable additional labour).
> Mathematical logicians mainly reason _about_ formal systems rather than _in_ formal systems.
>
> Godel's incompleteness theorem is an informal proof about formal axiomatisations of
> arithmetic, which could be (and in fact has been) rendered fully formally.
> Such a formal rendering of Godel's incompleteness theorem would fully satisfy
> criteria similar to those given by Mendelson for the propositional calculus (though
> the system would be a predicate calculus, and somewhat more complicated).
> It would nevertheless still require the machinery of Godel numbering
> (or some equivalent method) in order to be able to talk about formal systems
> in a language which is strictly capable only of talking about numbers.
>
> The use of godel numbering is essential, and does not mark a departure from
> the notion of formal system described by Mendelson.
>
>> To see what is truly going on with the Incompleteness Theorem requires us
>> to recreate it without Gödel numbers or diagonalization.
>>
>> First incompleteness theorem
>> Any consistent formal system F within which a certain amount
>> of elementary arithmetic can be carried out is incomplete;
>> i.e., there are statements of the language of F which can
>> neither be proved nor disproved in F. (Raatikainen 2018)
>> https://plato.stanford.edu/entries/goedel-incompleteness/#Int
>>
>> The above English paragraph does sum up the essence of the conclusion
>> of the 1931 Incompleteness Theorem. The following logic sentence
>> does formalize the above English paragraph without Gödel numbers:
>> ∃F ∈ Formal_System ∃G ∈ Closed_WFF(F) (G ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))
>
> Not quite.
>> To see what is truly going on with the Incompleteness Theorem requires us
>> to recreate it without Gödel numbers or diagonalization.
>>
>> First incompleteness theorem
>> Any consistent formal system F within which a certain amount
>> of elementary arithmetic can be carried out is incomplete;
>> i.e., there are statements of the language of F which can
>> neither be proved nor disproved in F. (Raatikainen 2018)
>> https://plato.stanford.edu/entries/goedel-incompleteness/#Int
>>
>> The above English paragraph does sum up the essence of the conclusion
>> of the 1931 Incompleteness Theorem. The following logic sentence
>> does formalize the above English paragraph without Gödel numbers:
>> ∃F ∈ Formal_System ∃G ∈ Closed_WFF(F) (G ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))
>
>
> Firstly, of course, there are undefined terms here, viz. "Formal_System" and "Closed_WFF".
> There is a difficulty in formalising the definition of these concepts in the language of
> arithmetic (which is the language in which Godel's theorem is to be understood), without
> using Godel numbers, because the only things which exist in that context are numbers.
Ah but we don't have to. Gödel's results apply to every formal system capable
of expressing arithmetic AND ABOVE. In other words they would apply equally
to a knowledge ontology of the sum total of all human knowledge.
In other words Gödel and Tarski "prove" that a complete and consistent
notion of truth itself is impossible.
All that we have to do to define a complete and consistent notion of
truth is to realize that truth itself is nothing more than a set of
mutually interlocking tautologies.
In logic, a tautology (from the Greek word ταυτολογία) is a formula or
assertion that is true in every possible interpretation.
How can tautologies possibly fail to hold?
If tautologies cannot possibly fail to hold, then Gödel and Tarski are WRONG !
This is the big picture grande scheme that unequivocally proves
that Tarski and Gödel are necessarily incorrect.
peteolcott
May 10, 2019, 12:23:32 AM5/10/19
to
On 5/7/2019 5:38 PM, xilog wrote:
> I will pass over your comment on Tarski since I lack a specific reference to the theorem of which you speak.
> As to Godel:
>
> On 07/05/2019 16:58, peteolcott wrote:
>>
> I will pass over your comment on Tarski since I lack a specific reference to the theorem of which you speak.
> As to Godel:
>
> On 07/05/2019 16:58, peteolcott wrote:
>>
>> As far as the 1931 Incompleteness Theorem goes most people are like a
>> fish out-of-water when we take away Gödel numbers and diagonalization.
>> They have only learned these things by rote and never truly understood
>> the actual underlying mathematical relations involved.
>>
>> Introduction to Mathematical logic Sixth edition Elliott Mendelson (2015)
>> 1.4 An Axiom System for the Propositional Calculus page 28
>> A wf C is said to be a consequence in S of a set Γ of wfs if and only if there is a
>> sequence B1, …, Bk of wfs such that C is Bk and, for each i, either Bi is an axiom
>> or Bi is in Γ, or Bi is a direct consequence by some rule of inference of some of
>> the preceding wfs in the sequence. Such a sequence is called a proof (or deduction)
>> of C from Γ. The members of Γ are called the hypotheses or premisses of the proof.
>> We use Γ ⊢ C as an abbreviation for “C is a consequence of Γ”...
>>
>> The first thing to know is that any "proof" that uses Gödel numbers and
>> diagonalization is not really a mathematical proof at all because it lacks:
>> [a sequence B1, …, Bk of wfs such that C is Bk] of inference steps. Instead
>> the Gödel numbers / diagonalization process leaps to the conclusion skipping
>> all of the intervening steps.
>
>> fish out-of-water when we take away Gödel numbers and diagonalization.
>> They have only learned these things by rote and never truly understood
>> the actual underlying mathematical relations involved.
>>
>> Introduction to Mathematical logic Sixth edition Elliott Mendelson (2015)
>> 1.4 An Axiom System for the Propositional Calculus page 28
>> A wf C is said to be a consequence in S of a set Γ of wfs if and only if there is a
>> sequence B1, …, Bk of wfs such that C is Bk and, for each i, either Bi is an axiom
>> or Bi is in Γ, or Bi is a direct consequence by some rule of inference of some of
>> the preceding wfs in the sequence. Such a sequence is called a proof (or deduction)
>> of C from Γ. The members of Γ are called the hypotheses or premisses of the proof.
>> We use Γ ⊢ C as an abbreviation for “C is a consequence of Γ”...
>>
>> The first thing to know is that any "proof" that uses Gödel numbers and
>> diagonalization is not really a mathematical proof at all because it lacks:
>> [a sequence B1, …, Bk of wfs such that C is Bk] of inference steps. Instead
>> the Gödel numbers / diagonalization process leaps to the conclusion skipping
>> all of the intervening steps.
>
> This is incorrect.
>
> You must bear in mind the distinction between a strictly formal mathematical proof
> of the kind addressed by Mendelson and the normal standard of mathematical proof
> used in the development of mathematics and its publication in journals.
> Neither in mathematics in general nor in mathematical logic is it usual for mathematicians
> to prove their results formally, though it is held that a rigorous (but informal) proof,
> can be converted into a formal proof (at the cost of considerable additional labour).
> Mathematical logicians mainly reason _about_ formal systems rather than _in_ formal systems.
>
> Godel's incompleteness theorem is an informal proof about formal axiomatisations of
> arithmetic, which could be (and in fact has been) rendered fully formally.
> Such a formal rendering of Godel's incompleteness theorem would fully satisfy
> criteria similar to those given by Mendelson for the propositional calculus (though
> the system would be a predicate calculus, and somewhat more complicated).
> It would nevertheless still require the machinery of Godel numbering
> (or some equivalent method) in order to be able to talk about formal systems
> in a language which is strictly capable only of talking about numbers.
>
> The use of godel numbering is essential, and does not mark a departure from
> the notion of formal system described by Mendelson.
Sure it does. It totally skips every single inference step and leaps
>
> You must bear in mind the distinction between a strictly formal mathematical proof
> of the kind addressed by Mendelson and the normal standard of mathematical proof
> used in the development of mathematics and its publication in journals.
> Neither in mathematics in general nor in mathematical logic is it usual for mathematicians
> to prove their results formally, though it is held that a rigorous (but informal) proof,
> can be converted into a formal proof (at the cost of considerable additional labour).
> Mathematical logicians mainly reason _about_ formal systems rather than _in_ formal systems.
>
> Godel's incompleteness theorem is an informal proof about formal axiomatisations of
> arithmetic, which could be (and in fact has been) rendered fully formally.
> Such a formal rendering of Godel's incompleteness theorem would fully satisfy
> criteria similar to those given by Mendelson for the propositional calculus (though
> the system would be a predicate calculus, and somewhat more complicated).
> It would nevertheless still require the machinery of Godel numbering
> (or some equivalent method) in order to be able to talk about formal systems
> in a language which is strictly capable only of talking about numbers.
>
> The use of godel numbering is essential, and does not mark a departure from
> the notion of formal system described by Mendelson.
directly to the conclusion. I don't think there are even any premises,
just a single leap to the conclusion.
>
>> To see what is truly going on with the Incompleteness Theorem requires us
>> to recreate it without Gödel numbers or diagonalization.
>>
>> First incompleteness theorem
>> Any consistent formal system F within which a certain amount
>> of elementary arithmetic can be carried out is incomplete;
>> i.e., there are statements of the language of F which can
>> neither be proved nor disproved in F. (Raatikainen 2018)
>> https://plato.stanford.edu/entries/goedel-incompleteness/#Int
>>
>> The above English paragraph does sum up the essence of the conclusion
>> of the 1931 Incompleteness Theorem. The following logic sentence
>> does formalize the above English paragraph without Gödel numbers:
>> ∃F ∈ Formal_System ∃G ∈ Closed_WFF(F) (G ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))
>
> Not quite.
>
> Firstly, of course, there are undefined terms here, viz. "Formal_System" and "Closed_WFF".
> There is a difficulty in formalising the definition of these concepts in the language of
> arithmetic (which is the language in which Godel's theorem is to be understood), without
> using Godel numbers, because the only things which exist in that context are numbers.
>
It is simply the conventional meaning of these terms.
>
> Firstly, of course, there are undefined terms here, viz. "Formal_System" and "Closed_WFF".
> There is a difficulty in formalising the definition of these concepts in the language of
> arithmetic (which is the language in which Godel's theorem is to be understood), without
> using Godel numbers, because the only things which exist in that context are numbers.
>
> Supposing that defect to be remedied the following is a better attempt:
>
> ∀F ∈ Formal_System. Condition(F) ⇒ ∃G ∈ Closed_WFF(F). ((F ⊬ G) ∧ (F ⊬ neg(G)))
>
> i.e. Godel showed something about EVERY formal system (satisfying certain conditions),
> viz. that there exists a sentence which is neither proven nor refuted by it.
In the sound deductive inference model this would be a Boolean
expression that is neither true nor false.
X > Y is true or false only when X and Y are tied to some
specific semantic meaning.
> It should be noted that in this the symbol "⊬" is the name of a binary relation over numbers,
As soon as we see that (within the sound deductive inference model)
(F ⊬ G) ∧ (F ⊬ neg(G)) specifies an expression that is neither true
nor false and (within the sound deductive inference model) there
can't be any Boolean expressions that are neither true nor false,
then it doesn't matter what the domain is.
> these being construed as sentences via some convention such as that established by a Godel numbering, and "neg" is an arithmetic function which maps the number for a formula to the number for the negation of the formula.
If you can't look at these as as predicate logic (within the sound deductive
inference model) then you won't understand that any sentence of predicate
logic must have exactly one Boolean value.
>
>> Now we have the formalized essence of the conclusion of the 1931
>> Incompleteness Theorem with 100,000-fold of extraneous complexity
>> stripped away.
>
>
We don't need Gödel numbers or diagonalization these add 100,000-fold
extraneous complexity. When we get rid of this and look a mathematical
logic from the POV of the (sound deductive inference model) it becomes glaringly
obvious that logic sentences that are neither true or false are WRONG !!!
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