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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 which 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 wikiPageRevisionID "584825230".
- 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 type Abstraction100002137.
- Von_Neumann_paradox type Communication100033020.
- Von_Neumann_paradox type Contradiction107206887.
- Von_Neumann_paradox type Falsehood106756407.
- Von_Neumann_paradox type MathematicsParadoxes.
- Von_Neumann_paradox type Message106598915.
- Von_Neumann_paradox type Paradox106724559.
- Von_Neumann_paradox type Proposition106750804.
- Von_Neumann_paradox type Statement106722453.
- Von_Neumann_paradox type Theorem106752293.
- Von_Neumann_paradox type TheoremsInTheFoundationsOfMathematics.
- 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 "Paradoxe de von Neumann".
- 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 sameAs Von_Neumann_paradox.
- Von_Neumann_paradox wasDerivedFrom Von_Neumann_paradox?oldid=584825230.
- Von_Neumann_paradox isPrimaryTopicOf Von_Neumann_paradox.