# 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 ${V}_{\alpha +1}=\mathcal{P}({V}_{\alpha})$
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 |