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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.
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC.
Von Neumann estimates the amount of memory required based on several classes of mathematical problems, including ordinary and partial differential equations, sorting and probability experiments. Of these, partial differential equations in two dimensions plus time will require the most memory, with three dimensions plus time being beyond what ...
Therefore, the axiom of limitation of size holds for the model V κ. The theorem stating that V κ has a well-ordering can be proved directly. Since κ is an ordinal of cardinality κ and |V κ | = κ, there is a one-to-one correspondence between κ and V κ. This correspondence produces a well-ordering of V κ. Von Neumann's proof is indirect.
defines V i for any ordinal number i. The union of all of the V i is the von Neumann universe V: :=. Every individual V i is a set, but their union V is a proper class. The axiom of foundation, which was added to ZF set theory at around the same time as the axiom of replacement, says that every set belongs to V.
Von Neumann's primitive operation is function application, denoted by [a, x] rather than a(x) where a is a function and x is an argument. This operation produces an argument. Von Neumann defined classes and sets using functions and argument-functions that take only two values, A and B. He defined x ∈ a if [a, x] ≠ A. [1]
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".
Von Neumann's model of an expanding economy considered the matrix pencil A − λB with nonnegative matrices A and B; von Neumann sought probability vectors p and q and a positive number λ that would solve the complementarity equation () = along with two inequality systems expressing economic efficiency.