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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.
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.
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.
Flow chart from von Neumann's "Planning and coding of problems for an electronic computing instrument", published in 1947. Von Neumann was the inventor, in 1945, of the merge sort algorithm, in which the first and second halves of an array are each sorted recursively and then merged. [281] [282]
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".
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.
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 α {\displaystyle \alpha } , known as its rank.
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.