How Gödel's Proof Works

In 1931, Kurt Gödel stunned the mathematical world by proving that any axiomatic system for mathematics is either inconsistent or incomplete. His incompleteness theorems demonstrated that there will always be true mathematical statements that cannot be proven within the system, and no system can prove its own consistency. This article explains Gödel's ingenious method of mapping mathematical statements to numbers (Gödel numbering), how he constructed a self-referential statement that asserts its own unprovability, and why this shattered the dream of a complete, consistent foundation for mathematics.