<Li> If the system is ω - consistent, it can prove neither p nor its negation, and so p is undecidable . </Li> <Li> If the system is consistent, it may have the same situation, or it may prove the negation of p . In the later case, we have a statement ("not p") which is false but provable, and the system is not ω - consistent . </Li> <P> If one tries to "add the missing axioms" to avoid the incompleteness of the system, then one has to add either p or "not p" as axioms . But then the definition of "being a Gödel number of a proof" of a statement changes . which means that the formula Bew (x) is now different . Thus when we apply the diagonal lemma to this new Bew, we obtain a new statement p, different from the previous one, which will be undecidable in the new system if it is ω - consistent . </P> <P> George Boolos (1989) sketches an alternative proof of the first incompleteness theorem that uses Berry's paradox rather than the liar paradox to construct a true but unprovable formula . A similar proof method was independently discovered by Saul Kripke (Boolos 1998, p. 383). Boolos's proof proceeds by constructing, for any computably enumerable set S of true sentences of arithmetic, another sentence which is true but not contained in S . This gives the first incompleteness theorem as a corollary . According to Boolos, this proof is interesting because it provides a "different sort of reason" for the incompleteness of effective, consistent theories of arithmetic (Boolos 1998, p. 388). </P>

Who wrote there are essentially two parts to every speech a statement and its proof