{"id":"proof","domain":"foundations-logic","type":"Concept","title":"Proof","slug":"proof","url":"/mathematics/foundations-logic/proof/","summary":"A finite sequence of logical steps establishing a statement with certainty. Direct, contradiction, induction, construction, counterexample techniques. From Euclid to computer-verified proofs.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Velleman, D.J. (2019). How to Prove It. 3rd ed. Cambridge University Press."},{"tier":1,"citation":"Heath, T.L. (1956). Euclid: The Thirteen Books of the Elements. Dover."},{"tier":2,"citation":"Avigad, J. (2021). Mathematical logic and computation. Cambridge University Press."},{"tier":3,"citation":"Hersh, R. (1997). What Is Mathematics, Really? Oxford University Press."}],"relationships":[{"type":"built_on","target":"zfc-axioms","label":"ZFC Axioms"},{"type":"limited_by","target":"godels-incompleteness-theorems","label":"Gödel's Incompleteness Theorems"},{"type":"verification_tiers_tracked_in","target":"proof-status","label":"Proof Status Board"},{"type":"example_technique","target":"prime-number","label":"Prime Number — infinitude proof"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"2.3"}