Eodnhoj7 wrote: ↑Wed Jan 01, 2025 8:06 pm
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".