Refuting Gödel:

[attofishpi]

I have written a book refuting Gödel, so far no one has been able to counter my argument.
Basically, he took an x not belonging to himself. However, there is nothing that does not belong to itself; he should have at least demonstrated the existence of such an x.

https://www.amazon.com/dp/B0CS8SX7KW

Salam

Nobody could refute Gödel until the advent of ASCII, compoooters and then the ASCII 256 character set that could incorporate his letter ö..

..nobody could do shit till then (wot a genius).

? Is this sarcasm or was it actually refuted?

Not sarcasm tis satire.

[godelian]

So to simplify things basically it comes down to a paradox of self reference relative to how proof occurs and it's meaning?

The self-reference is actually not essential. There are alternative incompleteness proofs that don't make use of it. So, the incompleteness theorem is not essentially a paradox of self-reference.

The incompleteness theorem is essentially about the following procedure:

Make the list of all proofs in arithmetic along with the target sentence that they prove, forming a table with two columns: proof and target sentence.

If none of these target sentences are outright false, then we know that there are true target sentences that do not appear in this table, simply, because there is no proof for it. That captures the notion of incompleteness.

The original proof uses self-reference but there are also so-called diagonal-free proofs that do not use it:

https://saeedsalehi.ir/pdf/conf/Tubingen-2021.pdf

It took until the 90ies to remove self-reference from the proof, but it can be done now.

[Eodnhoj7]

So to simplify things basically it comes down to a paradox of self reference relative to how proof occurs and it's meaning?

The self-reference is actually not essential. There are alternative incompleteness proofs that don't make use of it. So, the incompleteness theorem is not essentially a paradox of self-reference.

The incompleteness theorem is essentially about the following procedure:

Make the list of all proofs in arithmetic along with the target sentence that they prove, forming a table with two columns: proof and target sentence.

If none of these target sentences are outright false, then we know that there are true target sentences that do not appear in this table, simply, because there is no proof for it. That captures the notion of incompleteness.

The original proof uses self-reference but there are also so-called diagonal-free proofs that do not use it:

https://saeedsalehi.ir/pdf/conf/Tubingen-2021.pdf

It took until the 90ies to remove self-reference from the proof, but it can be done now.

Okay, so I had an old-school view.

[godelian]

Okay, so I had an old-school view.

I think that the old-school approach is still the most straightforward one.

The idea behind the old-school approach is very simple, on the condition that you leave out how exactly to encode it in arithmetic language because the details of doing that, are actually insanely convoluted. Godel's original paper is a horror story to read, exactly because the encoding details.

The worst part is the proof for the diagonal lemma.

The entire literature on the matter complains about how horribly unintuitive the proof is:

https://m.mathnet.ru/php/presentation.p ... n_lang=eng

Diagonal-free proofs of the Diagonal Lemma

Abstract: The Diagonal Lemma (of Gödel and Carnap) is one of the fundamental results in Mathematical Logic.

However, its proof (as presented in textbooks) is very unintuitive, and a kind of “pulling a rabbit out of a hat”.

The complaint is in fact about just a few lines:

https://proofwiki.org/wiki/Diagonal_Lemma

The proof for the diagonal lemma is very short, and I actually do understand it line by line, but every time I read it, it still leaves me completely baffled. I can certainly not remember it for more than a day at a time.

One problem is just that it proves "almost nothing".

If it wasn't the main stepping stone to prove Godel's incompleteness theorem and Tarski's undefinability theorem, nobody would be interested in this lemma. It really has a "who the hell cares?" vibe to it:

For every predicate about logic sentences, there exists a true sentence for which the predicate is true or a false sentence for which the predicate is false.

How uninteresting!

The diagonal lemma proves "almost nothing".

It does so by pointing out a few seemingly absurd things about it. In some ways, it is pure math at its worst.

It appears in the Godel's and Tarski's proof in its negative version:

For every predicate about logic sentences, there exists a true sentence for which the predicate is false or a false sentence for which the predicate is true.

Tarski's undefinability of the truth trivially follows from it too:

There exists a true sentence for which the truth predicate is false or a false sentence for which the truth predicate is true.

Hence, the truth cannot possibly be defined as a predicate.

[attofishpi]

Not sure whether U R pretending to be a Muslim or U R an actual Muslim, but crack on in this alias..

[godelian]

Not sure whether U R pretending to be a Muslim or U R an actual Muslim, but crack on in this alias..

The Islamic moral theory is actually a beautiful mathematical object. You can even test it with ChatGPT. The Christian one is not. It is not even closed under logical consequence. When I discovered Islamic moral theory, I quite quickly adopted it. Why use the Christian one, when you know that it doesn't even work?

[vamvam]

If you create the following table:

sentence1, proof1
sentence2, proof2
sentence3, proof3
...


In the table above, you end up with all the provable sentences in arithmetic along with their proofs.

So, the question, "Is sentence S provable?" amounts to trying to look up S in the table above. If you can find it, the answer is yes. Otherwise, the answer is no.

Imagine now that we are trying to look up S equal to "This sentence is not provable"?

There are two possible outcomes.

If you can find it in the table, then it is provable, and then the sentence "This sentence is not provable" is false. So, the system has managed to prove a false sentence. Hence, arithmetic is inconsistent.

If you cannot find it in the table, then the sentence is not provable, and then "This sentence is not provable" is true. So, the system has failed to prove a true sentence. Hence, arithmetic is incomplete.

Is the sentence "This number equals 2" true or false? Is it in the table with all the provable sentences?

"This" is a pronoun:

Pronoun
The part of speech that substitutes for nouns or noun phrases and designates persons or things asked for, previously specified, or understood from the context.

It is like a variable. It is a placeholder for some value. If you replace "this number" with "number 3" you get "Number 3 equals 2" and if you replace "this number" with "number 2" you get "Number 2 equals 2.".

So, "This number equals 2" is not in the above table with all the provable sentences not because it is not provable, true of false but because it is not a logical sentence.

Sentence
Mathematical logic

In mathematical logic, a sentence of a predicate logic is a Boolean-valued well-formed formula with no free variables.

There are two sentences A and B:
A: This sentence is not provable
B: This sentence is not provable

If we replace "this sentence" with "A" we get:
A: A is not provable
B: A is not provable

It looks like A and B are the same sentence. They are equal syntactically, but semantically they are not the same. This become more obvious when we write it down as:
A = A is not provable
B = A is not provable

Sentence A is self referencing and it claims to be not provable while B claims that A is not provable. Sentence A is not in the list of all the provable sentences while B is. So, conclusion is that we don't have one sentence that is both provable and not provable. There are two sentences that are syntactically equal, but semantically different. One is not provable and the other is.

p.s.
I don't actually know what "provable" means in this context so I try to use it the way you use it. Usually "provable" means "one can prove it is true" and it is the same as being true which means that "this sentence is not provable" is the same as "this sentence is false".

[godelian]

Is the sentence "This number equals 2" true or false?

"This sentence is not provable" is an interpretation of:

G ⟺ provable(⌜G⌝)

The general pattern is:

S ⟺ K(⌜S⌝)

Interpretation:

This sentence has property K.

A more literal interpretation is:

If S is true then K(⌜S⌝) is also true.
If K(⌜S⌝) is true then S is also true.

Godel's first incompleteness theorem is expressed as:

∃G (G ⟺ ¬provable(⌜G⌝))

Interpretation:

There exists a sentence G that is true but not provable or false and provable.

Alternatively:

If the theory at hand does not prove false sentences, then there exists a sentence that is true but not provable.

Concerning "this number is 2":

It is expressed in first-order logic as:

a=2

This is not a theorem, but it could be an axiom. The point is that there is nothing to prove about this sentence. If you axiomatize it to be true, then it is. Otherwise, the system has nothing to say about this sentence.

[vamvam]

Godel's first incompleteness theorem is expressed as:

∃G (G ⟺ ¬provable(⌜G⌝))

Interpretation:

There exists a sentence G that is true but not provable or false and provable.

Alternatively:

If the theory at hand does not prove false sentences, then there exists a sentence that is true but not provable.

If provable means one can find a truth value of a sentence then "This sentence is not provable" is provable and it is false.

[godelian]

Godel's first incompleteness theorem is expressed as:

∃G (G ⟺ ¬provable(⌜G⌝))

Interpretation:

There exists a sentence G that is true but not provable or false and provable.

Alternatively:

If the theory at hand does not prove false sentences, then there exists a sentence that is true but not provable.

If provable means one can find a truth value of a sentence then "This sentence is not provable" is provable and it is false.

We cannot find the truth value for G. In fact, we do not even need to know the truth value for G. We merely use the law of the excluded middle (LEM) in propositional calculus to assert that G is true or G is false. It could be either. If G is true then G is unprovable. If G is false then G is provable.

This is an exclusively syntactic consequence of the expression G ⟺ ¬provable(⌜G⌝):

(G ⟺ ¬provable(⌜G⌝)) ⟺ ((G ∧¬provable(⌜G⌝)) ∨ (¬G ∧provable(⌜G⌝)))

It would be true of any propositional calculus expression of the type K ⟺ ¬M:

(K ⟺ ¬M) ⟺ ((K ∧ ¬M) ∨ (¬K ∧ M))

The real question is why G exists in the first place? That is an immediate consequence of the diagonal lemma:

∀φ(∃S (S ⟺ φ(⌜S⌝)))

For every property of logic sentences φ, there exists a true sentence S which has the property or a false sentence ¬S which does not have it.

Gödel's incompleteness theorem and its proof are exclusively about syntactic consequences in first-order logic. Furthermore, the only reason why arithmetic comes into play, is because calculations are needed to encode logic sentence S into natural number ⌜S⌝.

[vamvam]

G is truth value of the sentence and G is false.
provable(G) is predicate and it's truth value is true.
G <=> not provable(G) is true.

[godelian]

G <=> not provable(G) is true.

Yes, the expression is true because the diagonal lemma guarantees that it is true for at least one G.

However, we do not need to know for which G exactly this expression is true.

The expression is equivalent to:

(G and not provable(%G)) or (not G and provable(%G))

This means:

G is true and G is not provable, or
G is false and G is provable

This means that there exists:

A sentence that is true but not provable, or
A sentence that is false and provable.

If we assume that first-order arithmetic does not prove false sentences, it means that there exist a sentence that is true but not provable ("incompleteness theorem").

The hard part of the proof is to prove the diagonal lemma. This proof is quite short but generally considered incomprehensible:

https://proofwiki.org/wiki/Diagonal_Lemma

I personally never manage to remember this proof for longer than an hour.

Godel's incompleteness theorem follows trivially from the diagonal lemma (predicate at hand is "provable(n)"). Tarski's undefinability of the truth also follows trivially (predicate at hand is "true(n)").

G is truth value of the sentence and G is false.

No, G is an unknown sentence of which do not know whether it is true or false. We just know that G exists, and that the following is true about G:

G <=> not provable(%G)

What exactly G says, is unknown. What its truth value is, is unknown. But then again, it is irrelevant what exactly G says or what its truth value is. Godel's incompleteness theorem still necessarily follows from the simple fact that it exists.

The sentence "This is not provable" means "This unknown sentence is not provable".

Note that this is the classical version of the proof for Godel's incompleteness theorem. It is considered objectionable in terms of constructivism:

(1) It shamelessly makes use of the law of the excluded middle (LEM).
(2) It does not properly construct a witness, i.e. example, for its claim, because witness G is essentially unknown.

Note that G is obtained by diagonalization, which can somehow be considered to be constructive. However, there is no mention of any algorithm to actually locate such G. Therefore, it is not "properly" constructive.

ChatGPT: Is diagonalization a constructive algorithm?

Diagonalization, as a mathematical technique, is generally not considered a constructive algorithm in the sense of constructive mathematics or computability theory.

In the meanwhile, there are alternative versions available of the proof that are constructivistically acceptable, but they are also harder to explain than the proof in plain classical logic.

[vamvam]

If you replace provable(G) with G_is_provable you get the the formula

G <=> not G_is_provable

and this is just a simple propositional formula that does not and can not prove anything about provability in the first order logic. You are making a whole philosophy out of something trivial and dream that it proves something important.

[godelian]

If you replace provable(G) with G_is_provable you get the the formula

G <=> not G_is_provable

and this is just a simple propositional formula that does not and can not prove anything about provability in the first order logic. You are making a whole philosophy out of something trivial and dream that it proves something important.

provable(%S) needs to be a predicate operating on a natural number. Otherwise, the diagonal lemma does not apply. That is where Gödel numbering kicks in. You cannot replace provable(%S) by something that is not a predicate. Without the diagonal lemma, there is no guarantee that G actually exists.

[vamvam]

There is one thing I can't comprehend. If there is a sentence that is true and unprovable, how do you know that it is true? Did you prove it? If you dont prove it is true then it may be false and if you do prove it is true then it is not unprovable. Is there a rational solution of this dilemma?