Inside the system, it is neither provable nor disprovable. The system, then, is incomplete, because there is at least one true proposition about numbers (the one that says “I am not provable”) that cannot be proved within it. The conclusion—that no logical system can capture all the truths of mathematics—is known as the first incompleteness theorem. Gödel also proved that no logical system for mathematics could, by its own devices, be shown to be free from inconsistency, a result known as the second incompleteness theorem.
