In 1931, the Austrian logician Kurt Gödel crushed mathematicians' dream of a complete, consistent foundation for math. He proved that any set of axioms will be incomplete: there will always be true facts about numbers that cannot be proved. He also showed that no set of axioms can prove its own consistency.\n\nGödel's main maneuver was to map statements about a system of axioms onto statements within the system using a unique number called a Gödel number.

For example, the formula 0 = 0 gets the Gödel number 243,000,000 (2⁶ × 3⁵ × 5⁶). This mapping allows a system to talk about itself.\n\nThe crux of the proof involves the self-referential statement G: “The formula with Gödel number sub(y, y, 17) cannot be proved.” By substituting the Gödel number of G into itself, G becomes a statement that says “I am not provable.” If G were provable, the system would be inconsistent; if unprovable, it is true but unprovable—demonstrating incompleteness.\n\nGödel's theorems foretold undecidable questions like the continuum hypothesis and the halting problem. They show that mathematics is not a closed system of absolute truth, but depends on starting assumptions.