# Why is it specified in the 2nd incompleteness theorem that a system of arithmetic cannot prove its own consistency?

I have seen in places I have read about Godel's incompleteness theorem that the second incompleteness theorem can be summarized as saying:

No axiomatic system with sufficiently strong arithmetic can prove its own consistency.

Can someone explain what that means? I would think it is trivially the case that no system could prove its own consistency, since consistency and completeness are "meta" notions, and aren't formalized in the object language.

• I think it follows from the first incompleteness theorem that there will be true statements that are not provable using a formal axiomatic system. To be consistent, a system should be able to prove or disprove any statement expressed in it's language. Jan 26, 2023 at 15:35
• @Vasili Oh okay that makes more sense. So the second theorem is more of a corollary to the first? Jan 26, 2023 at 16:07
• "consistency and completeness are "meta" notions, and aren't formalized in the object language" But they can be formalized in the object language. That's the whole trick behind the proof. Jan 26, 2023 at 16:09
• Minor nitpick but the "axiomatic system" needs to also be consistent and effective (recursively enumerable). Jan 26, 2023 at 16:33
• @Vasili You are conflating consistency with completeness. To be consistent, a system must NOT prove AND disprove $A$ for any sentence $A$ -- but it may fail to do both. A system which DOES prove or disprove each sentence $A$ is said to be complete.
– Ned
Jan 26, 2023 at 17:25