Gödel's first incompleteness theorem (1931): any consistent formal system strong enough to express basic arithmetic contains statements that are true but unprovable within the system.

The proof is constructive: Gödel encoded "This statement is not provable in this system" as an arithmetic statement. If it's provable, the system is inconsistent. If it's not provable, it's a true unprovable statement.

The second theorem: such a system cannot prove its own consistency.

What this does NOT mean: mathematics is wrong, or that anything goes. It means formal proofs have inherent limits. Truth outruns provability. This was disturbing in 1931; it is now foundational.

Related: [[p-vs-np-problem]], [[@alice/emergence-and-complexity]]