KnowraBanach–Tarski paradoxLinked fromLinked fromThe 11 pages that link to Banach–Tarski paradox, each with the reason it gives.All 11Broader topic 6Related 3Compared with 2Axiom of choiceBroader topic: It exposes a counterintuitive consequence of allowing nonconstructive selections.Alfred TarskiCompared with: Tarski co-authored this striking result, distinct from his work on truth and model theory.Well-ordering theoremBroader topic: Its proof uses choice, whose equivalent well-ordering principle underwrites highly nonconstructive selections.Choice functionBroader topic: Its proof depends on choice principles that permit highly nonconstructive selections.Stefan BanachRelated: Banach co-authored this striking result with Alfred Tarski, extending his mathematical legacy beyond analysis.InfinityBroader topic: This counterintuitive result exposes consequences of infinite decompositions and the axiom of choice.Felix HausdorffRelated: Its proof draws on the kind of paradoxical decompositions Hausdorff first exhibited.Nonmeasurable setBroader topic: Its finite pieces are nonmeasurable, preventing ordinary volume from being preserved across the decomposition.Hausdorff paradoxBroader topic: It strengthens the sphere result by producing two balls from one using finitely many pieces.Solovay modelRelated: Its choice-dependent decompositions illustrate how weakening choice changes pathological consequences.Wallace–Bolyai–Gerwien theoremCompared with: Its counterintuitive set-theoretic decompositions contrast with finite polygonal dissections.