Theorem 1°: If L is correct and if the set R* is expressible in L, then L is incomplete.
Proof: Assume L is correct and that R* is expressible. Let K be the predicate that expresses R*. The diagonalization of K is K(k), which by Lemma (D) is a Gödel sentence for R. What this means is that K(k) is true if and only if the Gödel number for K(k) is in R. Further, this says is that K(k) is true if and only if K(k) is refutable. This leaves us two possibilities: (1) K(k) is true and refutable or (2) K(k) is false and not refutable. Since L is correct, then (1) cannot be the case. Therefore, (2) holds. Since no false statements can be provable (because L is correct), then K(k) is neither provable or refutable. This means K(k) is undecidable, and L is incomplete. Q.E.D.1
Sentence K(k) can be thought of as saying “I am refutable” as opposed to our earlier H(h) which said, “I am not provable”.
1. Smullyan – GIT, Pg. 11.
