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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.


See also

Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\zeta,\Sigma,\gamma\) (ordinals on infinite time Turing machine) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Buchholz's function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert's Psi function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal
Classes: \(V\) · \(L\) · \(\textrm{On}\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)

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