I want to ask if it is possible to deduce Godel's first incompleteness theorem from Chaitin's incompleteness theorem. I am reading the following AMS-Notice article. The authors claim that:
The main conceptual difficulty in G¨odel’s original proof is the self-reference of the statement “this statement has no proof.” A conceptually simpler proof of the first incompleteness theorem, based on Berry’s paradox, was given by Chaitin.....
But the statement they proved is the Chaitin's incompleteness theorem, which appeared different from the Godel's first incompleteness theorem. They also contendeded that:
A different proof for G¨odel’s first incompleteness theorem, also based on Berry’s paradox, was given by Boolos [Boolos89] (see also [Vopenka66, Kikuchi94]). Other proofs for the first incompleteness theorem are also known (for a recent survey, see [Kotlarski04]).
Does this mean that Chaitin's incompleteness theorem implies Godel's first incompleteness theorem? I ask at here as I feel kind of confused with the logical relationship.
The authors also made a dubious claim that:
G¨odel considered the related statement “this statement has no proof.” He showed that this statement can be expressed in any theory that is capable of expressing elementary arithmetic. If the statement has a proof, then it is false; but since in a consistent theory any statement that has a proof must be true, we conclude that if the theory is consistent, the statement has no proof. Since the statement has no proof, it is true (over N). Thus, if the theory is consistent, we have an example for a true statement (over N) that has no proof. The main conceptual difficulty in G¨odel’s original proof is the self-reference of the statement “this statement has no proof.”
which is very similar to Wittegenstein's remark on Godel's proof:
I imagine someone asking my advice; he says: “I have constructed a proposition (I will use ‘P’ to designate it) in Russell’s symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: ‘P is not provable in Russell’s system’. Must I not say that this proposition on the one hand is true, and on the other hand unprovable? For suppose it were false; then it is true that it is provable. And that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable.”
and it has been refuted by Godel himself:
"It is clear from the passages you cite that Wittgenstein did "not" understand [the first incompleteness theorem] (or pretended not to understand it). He interpreted it as a kind of logical paradox, while in fact is just the opposite, namely a mathematical theorem within an absolutely uncontroversial part of mathematics (finitary number theory or combinatorics)." (Wang 1996:197)
I think I should for someone else's opinion as I am quite confused - what is right, what is wrong at here?