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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.
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".
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 .
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.
However, it fails to account for one of the main differences between quantum mechanics and classical mechanics, that is, the effects of measurement. [13] The von Neumann description of quantum measurement of an observable A, when the system is prepared in a pure state ψ is the following (note, however, that von Neumann's description dates back ...
The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers. For a well-orderable set U , we define its cardinal number to be the smallest ordinal number equinumerous to U , using the von Neumann definition of an ordinal number.
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]
For example, in the spin 1/2 example discussed above, the system can be prepared in the state ψ by using measurement of σ 1 as a filter that retains only those particles such that σ 1 yields +1. By the von Neumann (so-called) postulates, immediately after the measurement the system is assuredly in the state ψ.