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- Von_Neumann_paradox abstract "In mathematics, the von Neumann paradox, named after John von Neumann, is the idea that one can break a planar figure such as the unit square into sets of points and subject each set to an area-preserving affine transformation such that the result is two planar figures of the same size as the original. This was proved in 1929 by John von Neumann, assuming the axiom of choice. It is based on the earlier Banach–Tarski paradox, which is in turn based on the Hausdorff paradox.Banach and Tarski had proved that, using isometric transformations, the result of taking apart and reassembling a two-dimensional figure would necessarily have the same area as the original. This would make creating two unit squares out of one impossible. But von Neumann realized that the trick of such so-called paradoxical decompositions was the use of a group of transformations that include as a subgroup a free group with two generators. The group of area-preserving transformations (whether the special linear group or the special affine group) contains such subgroups, and this opens the possibility of performing paradoxical decompositions using them.".
- Von_Neumann_paradox wikiPageID "20234262".
- Von_Neumann_paradox wikiPageLength "9142".
- Von_Neumann_paradox wikiPageOutDegree "31".
- Von_Neumann_paradox wikiPageRevisionID "668081755".
- Von_Neumann_paradox wikiPageWikiLink Amenable_group.
- Von_Neumann_paradox wikiPageWikiLink Axiom_of_choice.
- Von_Neumann_paradox wikiPageWikiLink Banach_measure.
- Von_Neumann_paradox wikiPageWikiLink Banach–Tarski_paradox.
- Von_Neumann_paradox wikiPageWikiLink Bijection.
- Von_Neumann_paradox wikiPageWikiLink Cantor-Bernstein-Schroeder_theorem.
- Von_Neumann_paradox wikiPageWikiLink Cardinality_of_the_continuum.
- Von_Neumann_paradox wikiPageWikiLink Category:Group_theory.
- Von_Neumann_paradox wikiPageWikiLink Category:Mathematics_paradoxes.
- Von_Neumann_paradox wikiPageWikiLink Category:Measure_theory.
- Von_Neumann_paradox wikiPageWikiLink Category:Theorems_in_the_foundations_of_mathematics.
- Von_Neumann_paradox wikiPageWikiLink Dense_set.
- Von_Neumann_paradox wikiPageWikiLink Free_group.
- Von_Neumann_paradox wikiPageWikiLink Generating_set_of_a_group.
- Von_Neumann_paradox wikiPageWikiLink Group_(mathematics).
- Von_Neumann_paradox wikiPageWikiLink Hausdorff_paradox.
- Von_Neumann_paradox wikiPageWikiLink Injection_(mathematics).
- Von_Neumann_paradox wikiPageWikiLink Injective_function.
- Von_Neumann_paradox wikiPageWikiLink Isometric_transformation.
- Von_Neumann_paradox wikiPageWikiLink Isometry.
- Von_Neumann_paradox wikiPageWikiLink John_von_Neumann.
- Von_Neumann_paradox wikiPageWikiLink Mathematics.
- Von_Neumann_paradox wikiPageWikiLink Miklós_Laczkovich.
- Von_Neumann_paradox wikiPageWikiLink One-to-one_correspondence.
- Von_Neumann_paradox wikiPageWikiLink Problem_of_measure.
- Von_Neumann_paradox wikiPageWikiLink SL2(R).
- Von_Neumann_paradox wikiPageWikiLink Schröder–Bernstein_theorem.
- Von_Neumann_paradox wikiPageWikiLink Solvable_group.
- Von_Neumann_paradox wikiPageWikiLink Special_affine_group.
- Von_Neumann_paradox wikiPageWikiLink Subgroup.
- Von_Neumann_paradox wikiPageWikiLink Unit_square.
- Von_Neumann_paradox wikiPageWikiLinkText "Von Neumann paradox".
- Von_Neumann_paradox wikiPageWikiLinkText "von Neumann paradox".
- Von_Neumann_paradox hasPhotoCollection Von_Neumann_paradox.
- Von_Neumann_paradox subject Category:Group_theory.
- Von_Neumann_paradox subject Category:Mathematics_paradoxes.
- Von_Neumann_paradox subject Category:Measure_theory.
- Von_Neumann_paradox subject Category:Theorems_in_the_foundations_of_mathematics.
- Von_Neumann_paradox hypernym Idea.
- Von_Neumann_paradox type Organisation.
- Von_Neumann_paradox type Theorem.
- Von_Neumann_paradox comment "In mathematics, the von Neumann paradox, named after John von Neumann, is the idea that one can break a planar figure such as the unit square into sets of points and subject each set to an area-preserving affine transformation such that the result is two planar figures of the same size as the original. This was proved in 1929 by John von Neumann, assuming the axiom of choice.".
- Von_Neumann_paradox label "Von Neumann paradox".
- Von_Neumann_paradox sameAs Paradoxe_de_von_Neumann.
- Von_Neumann_paradox sameAs m.04zvtwt.
- Von_Neumann_paradox sameAs Q3363330.
- Von_Neumann_paradox sameAs Q3363330.
- Von_Neumann_paradox wasDerivedFrom Von_Neumann_paradox?oldid=668081755.
- Von_Neumann_paradox isPrimaryTopicOf Von_Neumann_paradox.