{"id":"godels-incompleteness-theorems","domain":"named-results","also_in_domain":"foundations-logic","type":"Theorem","title":"Gödel's Incompleteness Theorems","slug":"godels-incompleteness-theorems","url":"/mathematics/named-results/godels-incompleteness-theorems/","proved":"1931, Kurt Gödel","summary":"Any consistent formal system powerful enough for arithmetic contains true unprovable statements (1st theorem) and cannot prove its own consistency (2nd theorem). Ended Hilbert's Programme.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38, 173-198."},{"tier":1,"citation":"Smullyan, R.M. (1992). Gödel's Incompleteness Theorems. Oxford University Press."},{"tier":2,"citation":"Nagel, E. and Newman, J.R. (2001). Gödel's Proof. Revised ed. NYU Press."},{"tier":3,"citation":"Hofstadter, D.R. (1979). Gödel, Escher, Bach. Basic Books."}],"relationships":[{"type":"related_to","target":null,"label":"Turing's Halting Problem"},{"type":"refutes","target":null,"label":"Hilbert's Programme"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.5"}