Neumann

Universidad Simón Bolívar

Estructuras Discretas

John Von Neumann

Von Neumann integer A von Neumann integer is not an integer, but instead a construction of a natural number using some basic set notation. The von Neumann integers are defined inductively. The von Neumann integer zero is defined to be the empty set, ∅, and there are no smaller von Neumann integers. The von Neumann integer n is then the set of all von Neumann integers less than n. The set of von Neumann integers is the set of all finite von Neumann ordinals.

This form of construction from very basic notions of sets is applicable to various forms of set theory (for instance, Zermelo-Fraenkel set theory). While this construction suffices to define the set of natural numbers, a little more work must be done to define the set of all integers.

http://planetmath.org/vonneumanninteger Examples

0

=

∅

1

=

{0}={∅}

2

=

{0,1}={∅,{∅}}

3

=

{0,1,2}={∅,{∅},{{∅,{∅}}}}

⋮ n

=

{0,1,…,N−1}

John von Neumann was born on December 28, 1903. He was a Hungarian-American mathematician who made major contributions to a vast range of fields. The eldest of three brothers, von Neumann was born Neumann Janos Lajos. Von Neumann’s ancestors had originally immigrated to Hungary from Russia. John was a child prodigy who showed an aptitude for languages, memorization, and mathematics. By the age of six, he could exchange jokes in Classical Greek, memorize telephone directories, and displayed prodigious mental calculation abilities. He received his Ph.D. in mathematics from Pázmány Peter University in Budapest. That time he was 22 years of age. At the same time, he earned his diploma in chemical engineering from the ETH Zurich in Switzerland. John Neumann married twice. He married Mariette Kövesi in 1930, just before emigrating to the United States. They had one daughter. He then divorced her in 1937 and married Klari Dan in 1938.

In 1937, von Neumann became a naturalized citizen of the US. This was after migrating with his mother and brothers. In 1938, von Neumann was awarded the Bôcher Memorial Prize for his work in analysis.

Von Neumann also created the field of cellular automata without the aid of computers, constructing the first self-replicating automata with pencil and graph paper. Throughout his life von Neumann had a respect and admiration for business and government leaders; something which was often at variance with the inclinations of his scientific colleagues. Von Neumann’s interest in meteorological prediction led him to manipulating the environment by spreading colorants on the polar ice caps to enhance absorption of solar radiation, thereby raising global temperatures.

Von Neumann’s principal contribution to the atomic bomb itself was in the concept and design of the explosive lenses needed to compress the plutonium core of the Trinity test device.

Von Neumann’s hydrogen bomb work was also played out in the realm of computing, where he and Stanislaw Ulam developed simulations on von Neumann’s digital computers for the hydrodynamic computations. During this time he contributed to the development of the Monte Carlo method, which allowed complicated problems to be approximated using random numbers.

Von Neumann’s first significant contribution to economics was the minimax theorem of 1928. This theorem establishes that in certain zero sum games with perfect information, there exists a strategy for each player which allows both players to minimize their maximum losses.

An astoundingly creative mathematician, John von Neumann has played a rather important role in post-war economic theory. Death: John Neumann died in February 8, 1957 (aged 53) in Washington, D.C., United States. http://www.famousscientists.org/john-von-neumann/