cumulative hierarchy

The cumulative hierarchy of sets is defined by transfinite recursion as follows: we define $V_0=\mathrm{\varnothing }$ and for each ordinal $\alpha$ we define $V_{\alpha +1}=𝒫(V_{\alpha })$ 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