Two foundational results in logic demonstrating that any consistent, formal axiomatic system strong enough to describe arithmetic (like Peano arithmetic) is inherently incomplete.
- First theorem: Any consistent formal system powerful enough to express basic arithmetic contains true statements that cannot be proven within that system.
- Second theorem: Such a system cannot prove its own consistency.
Consider this statement: “this statement is not provable”
If the system proves it, the system is inconsistent. If it doesn’t, the statement is true but unprovable. Either way, the system is incomplete.