Measure theory | Theorems in the foundations of mathematics | Mathematical paradoxes | Group theory

Von Neumann paradox

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. (Wikipedia).

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Zermelo Fraenkel Introduction

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From playlist Zermelo Fraenkel axioms

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Newcomb's paradox | Famous Math Problems 7 | NJ Wildberger

Newcomb's paradox was first studied by American physicist William Newcomb, and popularized by articles by Robert Nozick and famously Martin Gardner in one of his 1974 Mathematical Games columns in Scientific American. The paradox involves notions of free will, determinism, choice, probabil

From playlist Famous Math Problems

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Even More Paradoxical: The Twin Paradox in Curved Spacetime

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From playlist Summer of Math Exposition Youtube Videos

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Relativity: how people get time dilation wrong

Einstein’s special theory of relativity is notorious for being easy to misuse, with the result that sometimes result in claims of paradoxes. When one digs more carefully into the theory, you find that no such paradoxes actually exist. In this video, Fermilab’s Dr. Don Lincoln describes a

From playlist Relativity

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Twins Paradox: The Complete Explanation

The twins paradox is easily the most famous paradoxes of all time. Using spacetime diagrams and the rules of relativity, we can show the paradox only happens because people are being lazy with special relativity. http://brilliant.org/ScienceAsylum ________________________________ VIDEO ANN

From playlist Einstein's Relativity

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Meet the Author - Ananyo Bhattacharya (Science Writer)

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From playlist Outreach

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The weirdest paradox in statistics (and machine learning)

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From playlist Novel topics (not in usual math curricula)

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From playlist Space Exploration

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Self-Replicating Robots and Galactic Domination

To check out any of the lectures available from Great Courses Plus go to http://ow.ly/dweH302dILJ We’ll soon be capable of building self-replicating robots. This will not only change humanity’s future but reshape the galaxy as we know it. Get your own Space Time t­shirt at http://bit.ly/

From playlist Futurism and Space Exploration

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Teach Astronomy - Von Neumann Machine

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From playlist 28. Interstellar Travel, SETI, and the Rarity of Life

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Russell's Paradox - A Ripple in the Foundations of Mathematics

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Creating and Using Entanglement (ICTS-IISc Joint Colloquium)

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From playlist Discussion Meeting: Entanglement from Gravity

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Turing Meets Paradoxes (History of Undecidability Part 3) - Computerphile

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From playlist The History of Undecidability

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From Analog to Digital and Back - George Dyson Keynote

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From playlist OSCON 2015

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How to Solve the Liar Paradox

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From playlist Summer of Math Exposition 2 videos

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Anthony Aguirre - Physics of Information

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From playlist Closer To Truth - Anthony Aguirre Interviews

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Statistical mechanical perspectives on cosmological puzzles by Christian Maes

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From playlist ICTS Colloquia

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Making Black Holes is HARDER than you think!

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From playlist Black Holes

Related pages

Hausdorff paradox | John von Neumann | Dense set | Subgroup | Unit square | Mathematics | Solvable group | Cardinality of the continuum | Generating set of a group | Banach–Tarski paradox | Amenable group | Free group | Banach measure | Group (mathematics) | SL2(R)