Godel's Second Incompleteness Theorem

Gödel's Second Incompleteness Theorem is a theorem in meta-mathematics which asserts that an axiomatic theory , which can prove anything that can, cannot prove its own consistency iff it is consistent. It is closely related to Godel's First Incompleteness Theorem, being in fact a stronger form, and easily derived (as shown below).

Proof

Firstly, if is not consistent, then can prove any statement, including its own consistency, because ex falso quodlibet.

Now suppose that is consistent. Then, by Godel's First Incompleteness Theorem, there is a statement , which is equivalent to the statement " cannot prove ", and cannot prove . But the proof of Gödel's First Incompleteness Theorem is itself a proof in , so that can prove the statement "If is consistent, then is equivalent to ' cannot prove '"; by modus tollens, can also prove "If is consistent, then is true." But if could prove its own consistency, then it could also prove , contradicting Gödel's First Incompleteness Theorem, and we are done.

See also

• Godel's First Incompleteness Theorem
• Proof
• Metamathematics
• Metalogic