Gödel's Incompleteness Theorem

A formal system (one that is consistent never yields a false statement) can not also be a complete system (containing all true statements)–there will always be statements that are unprovable yet true (i.e. G-statement).

The proof is derived from Principia Mathematica (PM) logic looking at number theory. You can build new theorems from axioms that are true infinitely but, because it is infinite, it can not contain every true statement.

That means there is always some truths that can’t be captured by a single system no matter how elaborate or seemingly robust. That’s why it’s important to have multiple systems and acknowledge you will always be limited in understanding the fullness of anything described by a formal system.

See also: