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.
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In mathematics, the vonNeumannparadox, named after John vonNeumann, is the idea that one can break a planar figure such as the unit square into sets...
John vonNeumann (/vɒn ˈnɔɪmən/ von NOY-mən; Hungarian: Neumann János Lajos [ˈnɒjmɒn ˈjaːnoʃ ˈlɒjoʃ]; December 28, 1903 – February 8, 1957) was a Hungarian...
lottery with the highest expected utility. This result is called the vonNeumann–Morgenstern utility representation theorem. In other words, if an individual's...
In set theory and related branches of mathematics, the vonNeumann universe, or vonNeumann hierarchy of sets, denoted by V, is the class of hereditary...
balls, each of equal size to the first. The von Neumannparadox is a two-dimensional version. Paradoxical set: A set that can be partitioned into two sets...
vonNeumann entropy than ρ {\displaystyle \rho } .: 514 Just as the Schrödinger equation describes how pure states evolve in time, the vonNeumann equation...
the standard definition, suggested by John vonNeumann at the age of 19, now called definition of vonNeumann ordinals: "each ordinal is the well-ordered...
disproved in 1980. In 1929, during his work on the Banach–Tarski paradox, John vonNeumann defined the concept of amenable groups and showed that no amenable...
to have an Erdős number of 3. Stochastics ENIAC Colossus computer VonNeumannparadox ΒΑΡΒΟΓΛΗΣ, Χ (March 16, 2008). Ελληνική σφραγίδα στο πρώτο μηχανοργανωμένο...
above, one way to frame the information paradox is that Hawking's calculation appears to show that the vonNeumann entropy of Hawking radiation increases...
The Fermi paradox is the discrepancy between the lack of conclusive evidence of advanced extraterrestrial life and the apparently high likelihood of its...
paradox". The British Journal for the History of Science. 48 (1): 165–167. doi:10.1017/S0007087414000570. PMID 25833801. S2CID 206212526. vonNeumann...
by John vonNeumann in his 1925 axiom system for sets and classes. It formalizes the limitation of size principle, which avoids the paradoxes encountered...
After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems...
of Russell's paradox. In ZF it can be proven that the class ⋃ α V α {\displaystyle \bigcup _{\alpha }V_{\alpha }} , called the vonNeumann universe, is...
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