The Banach–Tarski Paradox

GPTKB entity

Predicate	Object
gptkbp:instanceOf	gptkb:paradox
gptkbp:appliesTo	gptkb:3-dimensional_Euclidean_space
gptkbp:category	counterintuitive result paradox in mathematics
gptkbp:compatibleWith	1-dimensional space 2-dimensional space
gptkbp:contradictsIntuition	volume preservation
gptkbp:dependsOn	axiom of choice
gptkbp:doesNotViolate	conservation of mass (in mathematics)
gptkbp:field	gptkb:geometry gptkb:set_theory
gptkbp:formedBy	gptkb:Alfred_Tarski gptkb:Stefan_Banach
https://www.w3.org/2000/01/rdf-schema#label	The Banach–Tarski Paradox
gptkbp:implies	non-measurable sets exist
gptkbp:influenced	gptkb:logic measure theory
gptkbp:involves	infinite sets rotations and translations isometries non-measurable sets
gptkbp:namedAfter	gptkb:Alfred_Tarski gptkb:Stefan_Banach
gptkbp:notPossibleWith	physical objects
gptkbp:publishedIn	gptkb:Fundamenta_Mathematicae
gptkbp:relatedTo	gptkb:Hausdorff_paradox axiom of choice controversy
gptkbp:state	A solid ball in 3‑dimensional space can be split into a finite number of pieces and reassembled into two identical copies of the original ball
gptkbp:uses	group theory paradoxical decomposition free group
gptkbp:yearProposed	1924
gptkbp:bfsParent	gptkb:Stan_Wagon
gptkbp:bfsLayer	7