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The von Neumann universe, commonly denoted by $$\textrm{WF}$$, is a proper class defined as the union of a hierarchy $$(V_{\alpha})_{\alpha \in \textrm{On}}$$ of sets called von Nuemann hierarchy or cumulative hierarchy indexed by the proper class $$\textrm{On}$$ of ordinals.[1] We denote by $$V$$ the proper class of all sets. Since the equality $$V = \textrm{WF}$$ is provable under a sufficiently strong set theory such as ZFC set theory, we also naturally refer to $$V$$ as the von Neumann universe. In other words, the von Neumann universe could be thought of as the class of all sets. Note that $$V$$ itself is not a set, and hence it does not conflict the fact that there does not exist "the set of all sets".

## Hierarchy

For a set $$X$$, we denote by $$\mathcal{P}(X)$$ the set of subsets of $$X$$, also called the power set of $$X$$. We define the hierarchy $$(V_{\alpha})_{\alpha \in \textrm{On}}$$ of sets in the following transfinite inductive way: \begin{eqnarray*} V_{\alpha} = \bigcup_{\beta < \alpha} \mathcal{P}(V_{\beta}) = \bigcup \{\mathcal{P}(V_{\beta}) \mid \beta \in \alpha\} \end{eqnarray*} Then the von Neumann universe $$\textrm{WF}$$ is defined as its union $$\bigcup_{\alpha \in \textrm{On}} V_{\alpha}$$.

By the definition, for any $$x \in \textrm{WF}$$, there exists an ordinal $$\alpha$$ such that $$x \in V_{\alpha}$$, and the well-foundedness of $$(\textrm{On},\in)$$ implies the existence of the minimum $$\textrm{rank}(x)$$ of such an $$\alpha$$, which is called the rank of $$x$$. The assignment of the rank gives a map $$\textrm{WF} \to \textrm{On}$$. In $$\textrm{ZFC}$$ set theory, the equality $$V = \textrm{WF}$$ implies that the assignment is a map $$\textrm{rank} \colon V \to \textrm{On}$$.

The rank is useful because it allows us to convert the universal quantification $$\forall x : P(x)$$ for a predicate $$P(x)$$ on a set $$x$$ into the universal quantification $$\forall \alpha \in \textrm{On} : (\forall x : \textrm{rank}(x) = \alpha \rightarrow P(x))$$ for the predicate $$\forall x : \textrm{rank}(x) = \alpha \rightarrow P(x)$$ on an ordinal $$\alpha$$. Therefore the transfinite induction on $$(\textrm{On},\in)$$ implies the transfinite induction on $$(V,\in)$$.

## Examples

When $$\alpha = 0 = \emptyset$$, then the set $$\{\mathcal{P}(V_{\beta}) \mid \beta \in \alpha\}$$ is empty because there is no set $$\beta$$ satisfying $$\beta \in \alpha$$, and hence we have \begin{eqnarray*} V_0 = \bigcup \emptyset = \emptyset = \{\}. \end{eqnarray*} When $$\alpha = 1 = \{0\}$$, then the set $$\{\mathcal{P}(V_{\beta}) \mid \beta \in \alpha\}$$ coincides with the singleton $$\{\mathcal{P}(V_0)\}$$, and hence we have \begin{eqnarray*} V_1 = \bigcup \{\mathcal{P}(V_0)\} = \mathcal{P}(V_0) = \mathcal{P}(\{\}) = \{\{\}\} = \{0\} = 1. \end{eqnarray*} When $$\alpha = 2 = \{0,1\}$$, then the set $$\{\mathcal{P}(V_{\beta}) \mid \beta \in \alpha\}$$ coincides with the pair $$\{\mathcal{P}(V_0),\mathcal{P}(V_1)\}$$, and hence we have \begin{eqnarray*} & & V_2 = \bigcup \{\mathcal{P}(V_0),\mathcal{P}(V_1)\} = \mathcal{P}(V_0) \cup \mathcal{P}(V_1) = \mathcal{P}(\{\}) \cup \mathcal{P}(\{\{\}\}) \\ & = & \{\{\},\{\{\}\}\} = \{0,1\} = 2. \end{eqnarray*} Generally speaking, we have $$\mathcal{P}(V_{\beta}) \subsetneq \mathcal{P}(V_{\alpha})$$ for any ordinals $$\beta < \alpha$$, and hence $$V_{\alpha+1} = \mathcal{P}(V_{\alpha})$$ for any ordinal $$\alpha$$. In particular, we have \begin{eqnarray*} V_3 = \mathcal{P}(V_2) = \mathcal{P}(\{\{\},\{\{\}\}\}) = \{\{\},\{\{\}\},\{\{\{\}\}\},\{\{\},\{\{\}\}\}\} \supsetneq \{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\} = 3. \end{eqnarray*} From this level, the hierarchy contains many sets which are not ordinals.

## Application

The von Neuman hierarchy is frequently used to define models of set theories. For example, if $$\alpha$$ is an inaccessible cardinal, then $$V_{\alpha}$$ forms a model of $$\textrm{ZFC}$$ set theory. By the soundness of first order logic, it implies that $$\textrm{ZFC}$$ set theory augmented by the existence of an inaccessible cardinal proves the formalised consistency $$\textrm{Con}(\textrm{ZFC})$$ of $$\textrm{ZFC}$$ set theory itself. Therefore by Goedel's incompleteness theorem, the existence of an inaccessible cardinal is not provable under $$\textrm{ZFC}$$ set theory as long as it is consistent.

## References

1. K. Kunen, Set Theory: An Introduction to Independence Proofs, Studies in Logic and the Foundations of Mathematics, Volume 102, North Holland, 1983.