L is called consistent if no sentence is both provable and refutable in L, and inconsistent otherwise. Note, this definition does not refer to the set of true sentences. A sentence X is called decidable in L if it is either provable or refutable in L and undecidable otherwise. The system L is called complete if every sentence is decidable, and incomplete otherwise.
Theorem 1: If L is correct and if the set 
* is expressible in L, then L is incomplete.
Proof: Assume L is correct and that set * is expressible. From Lemma (D) it follows that there is a Gödel sentence for . This Gödel sentence is a sentence that says that it is true if and only if it is not provable. For L to be correct means every provable sentence is true and every refutable sentence is false. If our Gödel sentence were false, then it would be provable. But this contradicts the assumption that L is correct where every provable sentence is true. Therefore, our Gödel sentence must be true and not provable. This also means our Gödel sentence is not refutable because L is correct - only false sentences are refutable. So, our Gödel sentence is neither provable or refutable. By the definitions given above, our Gödel sentence is undecidable making system L incomplete. Q.E.D.1
1. Smullyan – GIT, Pgs. 10-11.
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L is called consistent if no sentence is both provable and refutable in L, and inconsistent otherwise. Note, this definition does not refer to the set of true sentences. A sentence X is called decidable in L if it is either provable or refutable in L and undecidable otherwise. The system L is called complete if every sentence is decidable, and incomplete otherwise.
