structural:scholar_edition, structural:reformatting, quality:corruption, structural:summarization, information:highlighting
Proof sketch for the second theorem

The main difficulty in proving the second incompleteness theorem is to show that various facts about provability used in the proof of the first incompleteness theorem can be formalized within a system S using a formal predicate P for provability. Once this is done, the second incompleteness theorem follows by formalizing the entire proof of the first incompleteness theorem within the system S itself.

Let p stand for the undecidable sentence constructed above, and assume for purposes of obtaining a contradiction that the consistency of the system S can be proved from within the system S itself. This is equivalent to proving the statement "System S is consistent". Now consider the statement c, where c = "If the system S is consistent, then p is not provable". The proof of sentence c can be formalized within the system S, and therefore the statement c, " p is not provable", (or identically, "not P (p) ") can be proved in the system S.

Observe then, that if we can prove that the system S is consistent (ie. the statement in the hypothesis of c), then we have proved that p is not provable. But this is a contradiction since by the 1st Incompleteness Theorem, this sentence (ie. what is implied in the sentence c, "" p " is not provable") is what we construct to be unprovable. Notice that this is why we require formalizing the first Incompleteness Theorem in S: to prove the 2nd Incompleteness Theorem, we obtain a contradiction with the 1st Incompleteness Theorem which can do only by showing that the theorem holds in S. So we cannot prove that the system S is consistent. And the 2nd Incompleteness Theorem statement follows.