Refuting Gödel:

[LeoMota]

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

[mickthinks]

You think you’ve refuted Gödel.

[Age]

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

Does the word 'himself' here refer to "godel" itself?

If yes, then saying and claiming that 'he' ["godel} took an 'x' not belonging to "himself/godel", is very, very different from saying and claiming that 'there is nothing that does not belong to itself'.

See, if "godel" took an 'x' not belonging to 'itself' [the 'x' itself] is very, very different from "godel" taking an 'x' not belonging to "himself" ["godel" itself].

So, which one, or which way, are you actually meaning here?

[Age]

You think you’ve refuted Gödel.

How do you 'know' that "leomota" only 'thinks' that it has refuted "godel"? After all "leomota" might actually 'know' that it has refuted "godel".

Just like the one that 'knew' that it had refuted the 'geocentric' view or belief. Some might have claimed that that one only 'thinks' it has refuted, or disproved, the 'geocentric' view, model, or belief. But, in Truth that one had actually refuted, or disproven, that 'old view', 'old model', or 'old belief'.

Do you 'know', or 'think', that "leomota" 'knows' or 'thinks' that it has refuted "godel"?

In other words do you 'know' or 'think' what you said and wrote here?

[alan1000]

I know - or at least I think I know - or at least I know that I think I know - that this thread may have originated with an intelligible idea.

[wtf]

I have written a book refuting Gödel

I read a book on Gödel once, but it was incomplete.

[Atla]

How can you refute someone who literally has "God" in his name?

[Harbal]

I have written a book refuting Gödel, so far no one has been able to counter my argument.

Is it really because they haven't been able to, or is it just that they haven't bothered to? There is a subtle difference.

[godelian]

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

The abstract of your book seems to suggest that Gödel's argument is akin to the liar paradox:

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

The focus of this work is to question whether the liar paradox represents something valid whose existence can be confirmed. Faced with a negative answer, we proceed to a critique contrary to the results obtained by Kurt Gödel, as he uses this paradox in the demonstrations of his famous theorems.

Liar paradox: "This statement is not true"

The liar paradox cannot be expressed in first-order arithmetic -- the system from which Gödel proves his theorems -- because of Alfred Tarski's undefinability of the truth. The true() predicate cannot be defined.

Gödel's sentence: "This statement is not provable"

Gödel goes to great length demonstrating how he defines the provable() predicate. Gödel creates an abstract database of all proofs indexed by logic sentence along with a lookup function that allows him to ascertain that a particular logic sentence has a proof. Hence, unlike the true() predicate, the provable() predicate can effectively be defined and implemented. In my opinion, Gödel does not make use of the liar paradox.

Jill Humphries concluded something similar:

https://projecteuclid.org/journals/notr ... 82658.full


Notre Dame Journal of Formal Logic
Volume XX, Number 3, July 1979
NDJFAM

GODEL'S PROOF AND THE LIAR PARADOX
JILL HUMPHRIES

Given this distinction between heterological and diagonal procedures, it can be shown that Gδdel's arguments are not related to the liar paradox.

Gödel's diagonal procedure is in fact just a database implemented using only arithmetic. That is why his database construction may appear confusingly hard. In fact, it is just a simple database.

Similarly, Gödel numbering is an encoding procedure -- using arithmetic operations only -- that we would not use in practice either, because it looks confusingly hard while practical modern encodings such as UTF-8 and MathJax are much simpler, much more efficient, and achieve exactly the same goal.

Gödel could only use arithmetic for implementing a system that would be trivially easy to implement if he had been allowed to use any imperative programming language. He was unfortunately not allowed to do that, because in that case, his theorems would not prove anything about first-order arithmetic.

[Eodnhoj7]

Question:

If all systems require something outside them that cannot be proven, doesn't this conceptualize an inherent spontaneity that grounds the system and as such is a proof that randomness guides a system thus paradoxically setting a foundation that is inherent within the system?

[wtf]

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.

You seem to have confused Gödel's incompleteness theorems with Russell's paradox. And then written a book based on your own confusion!

Nicely done.

Feel free to clarify.

[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).

[Eodnhoj7]

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?

[godelian]

Is this sarcasm or was it actually refuted?

Godel does not use the liar paradox. So, no, you cannot refute Godel on those grounds.

The sentence "This is not provable" is not the same as "This is not true" (liar paradox).

If you create the following table:

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

An example sentence would be "1+1=2" with the proof:

ChatGPT: Prove in Peano arithmetic that 1+1=2.

The proof that in Peano arithmetic is a classic exercise in formal mathematics. Here's the step-by-step derivation using Peano's axioms:

... bla bla ...

Conclusion

This proof demonstrates that using the axioms and recursive definitions in Peano arithmetic.

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.

This leads to Godel's first incompleteness theorem:

There exist false sentences that are provable from arithmetic or true sentences that are not provable.

Consequently, standard (Peano) arithmetic is inconsistent or incomplete.

The literature typically phrases it as:

If standard (Peano) arithmetic is consistent then it is incomplete.

That reflects the fact that we typically believe, without proof, that arithmetic is consistent.

By the way, if you encode the entire reasoning above while making solely use of arithmetic, then you get something rather complicated. The idea of creating a lookup table with all the proofs in arithmetic, however, is actually very simple.

Concerning the consistency of arithmetic, Godel's second incompleteness theorem proves that arithmetic cannot prove its own consistency because if it does, then it is provably inconsistent.

Nowhere in this reasoning, Godel ever makes use of the liar paradox. His canonical sentence "This sentence is not provable" is very similar to the liar paradox "This sentence is not true " but is not the liar paradox.

Truth and provability are not the same things. They are connected, however, through soundness theorem:

If a sentence is provable then it is true in all its interpretations.

The reverse, on the other hand, is not necessarily true.

It is not because a sentence is true that there is a proof for that.

This is why Godel's theorem is called the "incompleteness theorem".

[Eodnhoj7]

Is this sarcasm or was it actually refuted?

Godel does not use the liar paradox. So, no, you cannot refute Godel on those grounds.

The sentence "This is not provable" is not the same as "This is not true" (liar paradox).

If you create the following table:

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

An example sentence would be "1+1=2" with the proof:

ChatGPT: Prove in Peano arithmetic that 1+1=2.

The proof that in Peano arithmetic is a classic exercise in formal mathematics. Here's the step-by-step derivation using Peano's axioms:

... bla bla ...

Conclusion

This proof demonstrates that using the axioms and recursive definitions in Peano arithmetic.

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.

This leads to Godel's first incompleteness theorem:

There exist false sentences that are provable from arithmetic or true sentences that are not provable.

Consequently, standard (Peano) arithmetic is inconsistent or incomplete.

The literature typically phrases it as:

If standard (Peano) arithmetic is consistent then it is incomplete.

That reflects the fact that we typically believe, without proof, that arithmetic is consistent.

By the way, if you encode the entire reasoning above while making solely use of arithmetic, then you get something rather complicated. The idea of creating a lookup table with all the proofs in arithmetic, however, is actually very simple.

Concerning the consistency of arithmetic, Godel's second incompleteness theorem proves that arithmetic cannot prove its own consistency because if it does, then it is provably inconsistent.

Nowhere in this reasoning, Godel ever makes use of the liar paradox. His canonical sentence "This sentence is not provable" is very similar to the liar paradox "This sentence is not true " but is not the liar paradox.

Truth and provability are not the same things. They are connected, however, through soundness theorem:

If a sentence is provable then it is true in all its interpretations.

The reverse, on the other hand, is not necessarily true.

It is not because a sentence is true that there is a proof for that.

This is why Godel's theorem is called the "incompleteness theorem".

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