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. 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.
Definition
The cumulative hierarchy is a collection of sets Vα indexed by the class of ordinal numbers; in particular, Vα is the set of all sets having ranks less than α. Thus there is one set Vα for each ordinal number α. Vα may be defined by transfinite recursion as follows:
Let V0 be the empty set: V 0 := ∅ . {\displaystyle V_{0}:=\varnothing .}
For any ordinal number β, let Vβ+1 be the power set of Vβ: V β + 1 := P ( V β ) . {\displaystyle V_{\beta +1}:={\mathcal {P}}(V_{\beta }).}
For any limit ordinal λ, let Vλ be the union of all the V-stages so far: V λ := ⋃ β < λ V β . {\displaystyle V_{\lambda }:=\bigcup _{\beta <\lambda }V_{\beta }.}
A crucial fact about this definition is that there is a single formula φ(α,x) in the language of ZFC that states "(α is an ordinal and) the set x is in Vα". The sets Vα are called stages or ranks. The class V is defined to be the union of all the V-stages:
V := ⋃ α V α . {\displaystyle V:=\bigcup _{\alpha }V_{\alpha }.}
Rank of a set The rank of a set S is the smallest α such that S ⊆ V α . {\displaystyle S\subseteq V_{\alpha }\,.} In other words, P ( V α ) {\displaystyle {\mathcal {P}}(V_{\alpha })} is the set of sets with rank ≤α. The stage Vα 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:
V α := ⋃ β < α P ( V β ) . {\displaystyle V_{\alpha }:=\bigcup _{\beta <\alpha }{\mathcal {P}}(V_{\beta }).}
This gives an equivalent definition of Vα by transfinite recursion. Substituting the above definition of Vα back into the definition of the rank of a set gives a self-contained recursive definition:
In other words,
rank ( S ) = ⋃ { rank ( z ) + 1 ∣ z ∈ S } . {\displaystyle \operatorname {rank} (S)=\bigcup \{\operatorname {rank} (z)+1\mid z\in S\}.}
Finite and low cardinality stages of the hierarchy The first five von Neumann stages V0 to V4 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.)
This sequence exhibits tetrational growth. The set V5 contains 216 = 65536 elements; the set V6 contains 265536 elements, which very substantially exceeds the number of atoms in the observable universe; and for any natural n {\displaystyle n} , the set Vn+1 contains 2 ↑↑ n {\displaystyle 2\uparrow \uparrow n} elements using Knuth's up-arrow notation. So the finite stages of the cumulative hierarchy cannot physically be written down explicitly after stage 5. The set Vω has the same cardinality as ω. The set Vω+1 has the same cardinality as the set of real numbers.
Applications and interpretations
Interpretation as the set-theoretical universe In the standard Zermelo–Fraenkel set theory, V is simply the universe, i.e., the class of all sets. It is a proper class, and thus not "the set of all sets", even though each individual stage Vα is a set, because the index α ranges over the class of all ordinals, a proper class. The universality of V also depends on the axiom of foundation (also known as the axiom of regularity). In non-well-founded set theories, the universe is larger than V since the former also contains non-well-founded sets. Often, V is defined as the universe, and then the formula V = ⋃αVα means "the universe of ZF sets is equal to the cumulative hierarchy"—not a definition, but a theorem equivalent to the axiom of regularity. Roitman states (without references) that the realization of this equivalence is due to von Neumann. By the modern definition, V also does not include urelements in the first stage, and thus only contains "pure sets". However, Zermelo's original construction of his transfinite recursive hierarchy in 1930 includes all urelements in V1 (P1 in his notation), with the empty set considered a special case of an urelement.
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