{"slug":"godel-1931","verification":{"valid":false,"broken_at":0,"reason":"prev mismatch"},"count":4,"sources":[{"id":"src-1","type":"primary","url":"https://doi.org/10.1007/BF01700692","title":"Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I","quote":"Any sufficiently powerful formal system contains statements that are true but unprovable within the system.","summary":"Gödel's 1931 paper proving that any consistent formal system capable of expressing arithmetic is incomplete.","claim_ids":["c1","c2","c3","c4","c6"],"quality_score":1},{"id":"src-2","type":"adjacent","url":"https://en.wikipedia.org/wiki/Principia_Mathematica","title":"Principia Mathematica","quote":"","summary":"The three-volume formal system by Russell and Whitehead that Gödel's theorem addressed.","claim_ids":["c1"],"quality_score":0.8},{"id":"src-3","type":"rival","url":"","title":"The Logical Artifact objection to biological self-reference","quote":"DNA does not 'refer to itself.' It is copied by external machinery.","summary":"A rival argument that Gödel's theorem governs symbol systems, not cells, and the mapping from logic to biology is contested.","claim_ids":["c5","c7"],"quality_score":0.6},{"id":"src-4","type":"adjacent","url":"https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach","title":"Gödel, Escher, Bach: An Eternal Golden Braid","quote":"","summary":"Hofstadter's 1979 exploration of strange loops and self-reference across logic, art, and music.","claim_ids":["c4","c5"],"quality_score":0.75}]}