Math's Fundamental Flaw

Caption notes

Veritasium uses Hilbert’s three questions to distinguish completeness, consistency, and decidability. The narrator uses Gödel numbering to explain how statements about formulas and proofs can become statements about arithmetic. For a consistent, effectively axiomatized formal system that can express enough arithmetic, incompleteness means some statements can neither be proved nor refuted within the system. The video then turns to Turing’s halting problem: no algorithm can always decide, for any program and input, whether it will halt.