{"id":"banach-tarski-paradox","domain":"named-results","type":"Theorem","title":"Banach-Tarski Paradox","slug":"banach-tarski-paradox","url":"/mathematics/named-results/banach-tarski-paradox/","proved":"1924, Stefan Banach and Alfred Tarski","summary":"A solid ball can be decomposed into finitely many pieces (minimum 5) and reassembled into two balls identical to the original. A genuine theorem depending on the Axiom of Choice.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Banach, S. and Tarski, A. (1924). Sur la décomposition des ensembles de points. Fundamenta Mathematicae, 6, 244-277."},{"tier":1,"citation":"Wagon, S. (1985). The Banach-Tarski Paradox. Cambridge University Press."},{"tier":2,"citation":"Solovay, R.M. (1970). A model of set-theory in which every set of reals is Lebesgue measurable. Annals of Mathematics, 92(1), 1-56."},{"tier":3,"citation":"Su, F. (2000). The Banach-Tarski Paradox. Mudd Math Fun Facts."}],"relationships":[{"type":"consequence_of","target":"zfc-axioms","label":"ZFC Axioms — Axiom of Choice"},{"type":"uses","target":"group","label":"Group — free subgroup of rotations"},{"type":"related_to","target":"continuum-hypothesis","label":"Continuum Hypothesis — Cohen forcing, Solovay model"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"2.4"}