Von Neumann universe

The von Neumann universe, commonly denoted by \(\textrm{WF}\), is a proper class defined by 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 \(\textrm{On}\) or ordinals. 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, although it is not actually itself a set.

Hierarchy
For a set \(X\), we denote by \(\mathcal{P}(X)\) the set of subsets 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 \(\bigcup_{\alpha \in \textrm{On}} V_{\alpha}\).

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.