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An initial segment of the von Neumann universe. Ordinal multiplication is reversed from our usual convention; see Ordinal arithmetic.. The cumulative hierarchy is a collection of sets V α indexed by the class of ordinal numbers; in particular, V α is the set of all sets having ranks less than α.
A von Neumann architecture scheme. The von Neumann architecture—also known as the von Neumann model or Princeton architecture—is a computer architecture based on the First Draft of a Report on the EDVAC, [1] written by John von Neumann in 1945, describing designs discussed with John Mauchly and J. Presper Eckert at the University of Pennsylvania's Moore School of Electrical Engineering.
The structure N, 0, S is a model of the Peano axioms (Goldrei 1996). The existence of the set N is equivalent to the axiom of infinity in ZF set theory. The set N and its elements, when constructed this way, are an initial part of the von Neumann ordinals. Quine refer to these sets as "counter sets".
Sets in the von Neumann universe are organized into a cumulative hierarchy, based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (by transfinite recursion) an ordinal number, known as its rank.
Von Neumann published an introductory article on his axiom system in 1925. In 1928, he provided a detailed treatment of his system. [39] Von Neumann based his axiom system on two domains of primitive objects: functions and arguments. These domains overlap—objects that are in both domains are called argument-functions.
Von Neumann's mathematical analysis of the structure of self-replication preceded the discovery of the structure of DNA. [290] Ulam and von Neumann are also generally credited with creating the field of cellular automata , beginning in the 1940s, as a simplified mathematical model of biological systems.
The von Neumann universe is built from a cumulative hierarchy . The sets L α {\displaystyle \mathrm {L} _{\alpha }} of the constructible universe form a cumulative hierarchy. The Boolean-valued models constructed by forcing are built using a cumulative hierarchy.
John von Neumann. In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes. [1] It formalizes the limitation of size principle, which avoids the paradoxes encountered in earlier formulations of set theory by recognizing that some classes are too big to be sets.