cumulative hierarchy
The cumulative hierarchy of sets
is defined by transfinite recursion as follows:
we define
and for each ordinal we define
and for each limit ordinal
we define
.
Every set is a subset of for some ordinal , and the least such 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 is the least ordinal greater than the rank of every element of . For each ordinal , the set is the set of all sets of rank less than , and is the set of all sets of rank .
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
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 is a transitive set. Note that , and , but for the set is never an ordinal, because it has the element , 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 |