# cumulative hierarchy

The 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 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.

 Title cumulative hierarchy Canonical name CumulativeHierarchy Date of creation 2013-03-22 16:18:43 Last modified on 2013-03-22 16:18:43 Owner yark (2760) Last modified by yark (2760) Numerical id 8 Author yark (2760) Entry type Definition Classification msc 03E99 Synonym iterative hierarchy Synonym Zermelo hierarchy Related topic CriterionForASetToBeTransitive Related topic ExampleOfUniverse Defines rank Defines rank of a set