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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.
In quantum mechanics, each physical system is associated with a Hilbert space, each element of which represents a possible state of the physical system.The approach codified by John von Neumann represents a measurement upon a physical system by a self-adjoint operator on that Hilbert space termed an "observable".
A major contribution von Neumann made to measure theory was the result of a paper written to answer a question of Haar regarding whether there existed an algebra of all bounded functions on the real number line such that they form "a complete system of representatives of the classes of almost everywhere-equal measurable bounded functions". [117]
In descriptive set theory the Jankov–von Neumann uniformization theorem is a result saying that every measurable relation on a pair of standard Borel spaces (with respect to the sigma algebra of analytic sets) admits a measurable section. It is named after V. A. Jankov and John von Neumann.
The primitive notions of his theory were function and argument. Using these notions, he defined class and set. [1] Paul Bernays reformulated von Neumann's theory by taking class and set as primitive notions. [2] Kurt Gödel simplified Bernays' theory for his relative consistency proof of the axiom of choice and the generalized continuum ...
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.
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.
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 .