How Gödel's Proof Works

In 1931, Kurt Gödel proved that any mathematical system based on a set of axioms will inevitably be incomplete—there will always be true statements about numbers that cannot be proved within that system. His incompleteness theorems destroyed the mathematical pursuit of a "theory of everything" and demonstrated that what mathematicians can prove depends on their starting assumptions rather than absolute truth. Since Gödel's discovery, mathematicians have identified numerous undecidable questions, such as the continuum hypothesis and the halting problem, confirming his theorems' predictions.