In 1931, a 25-year-old Viennese logician named Kurt Gödel published a paper titled On Formally Undecidable Propositions of Principia Mathematica and Related Systems. It was nine pages long, technically immaculate, and destroyed the foundational programme that David Hilbert and Bertrand Russell had spent decades building. Mathematics had hoped to prove its own consistency by formal means. Gödel proved that no consistent formal system rich enough to express arithmetic could prove its own consistency. The dream of a complete, self-justifying mathematics was over.
Gödel's trick was self-reference made rigorous. He showed how to encode statements about a formal system within the system itself (Gödel numbering), then constructed a sentence that effectively says 'this sentence is not provable in this system.' If the system is consistent, the sentence cannot be proved (because then it would be false), and cannot be disproved (because then it would be both provable and not, by its own content). So true but unprovable. The first incompleteness theorem says any sufficiently expressive consistent system contains such sentences. The second says the consistency of the system is itself one of those unprovable statements. The result is not a defect of mathematics — it is a structural feature of formal systems that try to talk about themselves. Mathematicians went on doing mathematics; the foundationalist programme that wanted absolute certainty was permanently chastened. Hilbert lived another twelve years and never publicly accepted what had happened.
Incompleteness now appears in cognitive science, AI, and philosophy of mind arguments about whether mechanical systems can fully model themselves. Roger Penrose has argued (controversially) that human consciousness escapes Gödelian limits; most philosophers of mind disagree. The deeper cultural lesson — systems rich enough to be interesting are rich enough to have blind spots about themselves — has migrated well beyond mathematics, and is one of the most-cited results of the twentieth century even by people who do not know how the proof works.