Second Incompleteness Theorem

The Second Incompleteness Theorem states that any sufficiently powerful and consistent mathematical system cannot prove its own consistency. In other words, if the system is trustworthy and free of contradictions, it cannot demonstrate on its own that it won't produce errors. This means mathematicians cannot rely solely on the system’s internal proofs to confirm its correctness; they might need to step outside the system or accept it as an axiom. This theorem highlights fundamental limits in formal mathematics, showing that some truths about the system itself are inherently unprovable within the system.