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. The concept is named after John von Neumann, although it was first published by Ernst Zermelo in 1930.
The rank of a well-founded set is defined inductively as the smallest ordinal number greater than the ranks of all members of the set.[1] In particular, the rank of the empty set is zero, and every ordinal has a rank equal to itself. The sets in V are divided into the transfinite hierarchy Vα, called the cumulative hierarchy, based on their rank.
^Mirimanoff 1917; Moore 2013, pp. 261–262; Rubin 1967, p. 214.
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In set theory and related branches of mathematics, the vonNeumannuniverse, or vonNeumann hierarchy of sets, denoted by V, is the class of hereditary...
In modern set theory, it is common to restrict attention to the vonNeumannuniverse of pure sets, and many systems of axiomatic set theory are designed...
John vonNeumann (/vɒn ˈnɔɪmən/ von NOY-mən; Hungarian: Neumann János Lajos [ˈnɒjmɒn ˈjaːnoʃ ˈlɒjoʃ]; December 28, 1903 – February 8, 1957) was a Hungarian...
that Grothendieck universes exist. Constructible universeUniverse (mathematics) VonNeumannuniverse Streicher, Thomas (2006). "Universes in Toposes" (PDF)...
cumulative hierarchy V α {\displaystyle \mathrm {V} _{\alpha }} of the vonNeumannuniverse with V α + 1 = P ( W α ) {\displaystyle \mathrm {V} _{\alpha +1}={\mathcal...
}} , which denotes the ω {\displaystyle \omega } th stage of the vonNeumannuniverse. So here it is a countable set. In 1937, Wilhelm Ackermann introduced...
problematic if it is applied to class models of ZFC, such as the vonNeumannuniverse. The assertion "the real number x {\displaystyle x} is definable...
(ZFC) implies that the κ {\displaystyle \kappa } th level of the VonNeumannuniverse V κ {\displaystyle V_{\kappa }} is a model of ZFC whenever κ {\displaystyle...
nor its negation and is compatible with either. Indeed, using the vonNeumannuniverse, we can build a model of ZFC − Infinity + (¬Infinity). It is V ω...
VonNeumann cellular automata are the original expression of cellular automata, the development of which was prompted by suggestions made to John von...
downward infinite membership chains. The axiom is the contribution of vonNeumann (1925); it was adopted in a formulation closer to the one found in contemporary...
elementary embedding j : V → M {\displaystyle j:V\to M} from the VonNeumannuniverse V {\displaystyle V} into a transitive inner model M {\displaystyle...
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language change Sports: business chess, playoffs brackets Mathematics: VonNeumannuniverse Group theory: descendant trees There are many ways of visually representing...
Tree (descriptive set theory) Tree (set theory) Union (set theory) VonNeumannuniverse Zero sharp Analytical hierarchy Almost Ramsey cardinal Erdős cardinal...
formulations of set theory that are intended to be interpreted in the vonNeumannuniverse or to express the content of Zermelo–Fraenkel set theory, all sets...
Ordinarily these models are transitive subsets or subclasses of the vonNeumannuniverse V, or sometimes of a generic extension of V. Inner model theory studies...
supercompact. Large cardinals are understood in the context of the vonNeumannuniverse V, which is built up by transfinitely iterating the powerset operation...
mathematical objects. An example of such a class of sets could be the vonNeumannuniverse. But even when fixing the class of sets under consideration, it is...
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