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Von Neumann universe
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== Definition == [[Image:Von Neumann Hierarchy.svg|thumb|right|upright=1.4|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''<sub>α</sub> indexed by the class of [[ordinal number]]s; in particular, ''V''<sub>α</sub> is the set of all sets having ranks less than α. Thus there is one set ''V''<sub>α</sub> for each ordinal number α. ''V''<sub>α</sub> may be defined by [[transfinite recursion]] as follows: * Let ''V''<sub>0</sub> be the [[empty set]]:<math display="block"> V_0 := \varnothing .</math> * For any [[ordinal number]] β, let ''V''<sub>β+1</sub> be the [[power set]] of ''V''<sub>β</sub>:<math display="block"> V_{\beta+1} := \mathcal{P} (V_\beta) .</math> * For any [[limit ordinal]] λ, let ''V''<sub>λ</sub> be the [[union (set theory)|union]] of all the ''V''-stages so far:<math display="block"> V_\lambda := \bigcup_{\beta < \lambda} V_\beta .</math> A crucial fact about this definition is that there is a single formula φ(α,''x'') in the language of ZFC that states "the set ''x'' is in ''V''<sub>α</sub>". The sets ''V''<sub>α</sub> are called '''stages''' or '''ranks'''. The class ''V'' is defined to be the union of all the ''V''-stages: <math display="block"> V := \bigcup_{\alpha} V_\alpha.</math> ===Rank of a set=== The '''rank''' of a set ''S'' is the smallest α such that <math>S \subseteq V_\alpha \,.</math> In other words, <math>\mathcal{P} (V_\alpha) </math> is the set of sets with rank ≤α. The stage ''V''<sub>α</sub> can also be characterized as the set of sets with rank strictly less than α, regardless of whether α is 0, a successor ordinal, or a limit ordinal: <math display="block">V_\alpha := \bigcup_{\beta < \alpha} \mathcal{P} (V_\beta).</math> This gives an equivalent definition of ''V''<sub>α</sub> by transfinite recursion. Substituting the above definition of ''V''<sub>α</sub> back into the definition of the rank of a set gives a self-contained recursive definition: {{block indent| The rank of a set is the smallest ordinal number strictly greater than the rank of all of its members.}} In other words, <math display="block">\operatorname{rank} (S) = \bigcup \{ \operatorname{rank} (z) + 1 \mid z \in S \}.</math> ===Finite and low cardinality stages of the hierarchy=== The first five von Neumann stages ''V''<sub>0</sub> to ''V''<sub>4</sub> may be visualized as follows. (An empty box represents the empty set. A box containing only an empty box represents the set containing only the empty set, and so forth.) [[File:Von Neumann universe 4.png|upright=3<!--[[WP:IAR]]: 1.8 would be too small-->|frameless|center|First 5 von Neumann stages]] This sequence exhibits [[tetrational]] growth. The set ''V''<sub>5</sub> contains 2<sup>16</sup> = 65536 elements; the set ''V''<sub>6</sub> contains 2<sup>65536</sup> elements, which very substantially exceeds the [[Observable universe#Matter content—number of atoms|number of atoms in the known universe]]; and for any natural ''n'', the set ''V''<sub>''n''+1</sub> contains 2 ⇈ ''n'' elements using [[Knuth's up-arrow notation]]. So the finite stages of the cumulative hierarchy cannot be written down explicitly after stage 5. The set ''V''<sub>ω</sub> has the same cardinality as ω. The set ''V''<sub>ω+1</sub> has the same cardinality as the set of real numbers.
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