Search results
Results from the WOW.Com Content Network
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.
This corresponds to the von Neumann architecture. SISD is one of the four main classifications as defined in Flynn's taxonomy . In this system, classifications are based upon the number of concurrent instructions and data streams present in the computer architecture.
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.
Von Neumann describes a detailed design of a "very high speed automatic digital computing system." He divides it into six major subdivisions: a central arithmetic part, CA; a central control part, CC; memory, M; input, I; output, O; and (slow) external memory, R, such as punched cards, Teletype tape, or magnetic wire or steel tape.
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.
Von Neumann's approach to limitation of size uses the axiom of limitation of size. As mentioned in § Implications of the axiom, von Neumann's axiom implies the axioms of separation, replacement, union, and choice. Like Fraenkel and Lévy, von Neumann had to add the axiom of infinity to his system since it cannot be proved from his other axioms.
The set N of natural numbers is defined in this system as the smallest set containing 0 and closed under the successor function S defined by S(n) = n ∪ {n}. The structure N , 0, S is a model of the Peano axioms ( Goldrei 1996 ).
The latter denotes the set in the von Neumann hierarchy indexed by the ordinal α 1. The class of all ordinal definable sets is denoted OD; it is not necessarily transitive, and need not be a model of ZFC because it might not satisfy the axiom of extensionality.