Gödel's incompleteness theorems are what you're thinking of. https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems https://en.wikipedia.org/wiki/

Gödel's incompleteness theorems are what you're thinking of. https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_... That said, if you have a system X, it can't prove it's own consistency, but a stronger system Y can prove its consistency (and perhaps some other stronger system can prove Y's consistency. This gives us a chain of systems, each proving the consistency of some weaker system). That doesn't absolutely prove that the system is consistent--if Y was inconsistent, it could prove X is consistent (and also could prove that X is inconsistent). Nonetheless, it is still valuable. After all, part of our use of Y is the fact that we know of no inconsistency in it. And since formal systems often are subtly inconsistent, "consistent assuming some other system is consistent" is a lot better than "we have no proof whatsoever of consistency".