The Incompleteness Theorems

The Incompleteness Theorems, proved by Kurt Gödel, show limits of mathematical systems. They state that in any consistent system powerful enough to describe basic arithmetic, there will be true statements that cannot be proven within that system. Additionally, such a system cannot prove its own consistency. This means no single set of rules can fully capture all mathematical truths or confirm its own reliability, highlighting inherent limitations in formal mathematical frameworks.