cumulative hierarchy CumulativeHierarchy 2006-10-09 07:26:57 2008-03-09 07:55:53 Definition rank rank of a set \usepackage{amssymb} The \emph{cumulative hierarchy} of sets is defined by transfinite recursion as follows: we define $V_0=\varnothing$ and for each ordinal $\alpha$ we define $V_{\alpha+1}=\mathcal{P}(V_\alpha)$ and for each limit ordinal $\delta$ we define $V_\delta=\bigcup_{\alpha\in\delta}V_\alpha$. Every set is a subset of $V_\alpha$ for some ordinal $\alpha$, and the least such $\alpha$ is called the \emph{rank} of the set. It can be shown that the rank of an ordinal is itself, and in general the rank of a set $X$ is the least ordinal greater than the rank of every element of $X$. For each ordinal $\alpha$, the set $V_\alpha$ is the set of all sets of rank less than $\alpha$, and $V_{\alpha+1}\setminus V_\alpha$ is the set of all sets of rank $\alpha$. Note that the previous paragraph makes use of the Axiom of Foundation: if this axiom fails, then there are sets that are not subsets of any $V_\alpha$ and therefore have no rank. The previous paragraph also assumes that we are using a set theory such as ZF, in which elements of sets are themselves sets. Each $V_\alpha$ is a transitive set. Note that $V_0=0$, $V_1=1$ and $V_2=2$, but for $\alpha>2$ the set $V_\alpha$ is never an ordinal, because it has the element $\{1\}$, which is not an ordinal.